(I’ve also significantly revised this paper about Extended TQFT since then, and it now includes some stuff which was the basis of my talk at Erlangen on cohomological twisting of the category .  I’ll get to that in the next post.  Also coming up, I’ll be describing some new things I’ve given some talks about recently which relate the Baez-Dolan groupoidification program to Khovanov-Lauda categorification of algebras – at least in one example, hopefully in a way which will generalize nicely.  There’s a draft of a paper which summarizes the basics here which Jamie Vicary and I are still polishing up – but more on that later.)

In the meantime, there were a few themes at the conference which bear on the Extended TQFT project in various ways, so in this post I’ll describe some of them.  (This isn’t an exhaustive description of all the talks: just of a selection of illustrative ones.)

Categories with Structures

A few talks were mainly about facts regarding the sorts of categories which get used in field theory contexts.  One important type, for instance, are fusion categories is a monoidal category which is enriched in vector spaces, generated by simple objects, and some other properties: essentially, monoidal 2-vector spaces.  The basic example would be categories of representations (of groups, quantum groups, algebras, etc.), but fusion categories are an abstraction of (some of) their properties.  Many of the standard properties are described and proved in this paper by Etingof, Nikshych, and Ostrik, which also poses one of the basic conjectures, the “ENO Conjecture”, which was referred to repeatedly in various talks.  This is the guess that every fusion category can be given a “pivotal” structure: an isomorphism from to .  It generalizes the theorem that there’s always such an isomorphism into .  More on this below.

Hendryk Pfeiffer talked about a combinatorial way to classify fusion categories in terms of certain graphs (see this paper here).  One way I understand this idea is to ask how much this sort of category really does generalize categories of representations, or actually comodules.  One starting point for this is the theorem that there’s a pair of functors between certain monoidal categories and weak Hopf algebras.  Specifically, the monoidal categories are , which consists of monoidal categories equipped with a forgetful functor into .  Then from this one can get (via a coend), a weak Hopf algebra over the base field (in the category ).  From a weak Hopf algebra , one can get back such a category by taking all the modules of .  These two processes form an adjunction: they’re not inverses, but we have maps between the two composites and the identity functors.

The new result Hendryk gave is that if we restrict our categories over to be abelian, and the functors between them to be linear, faithful, and exact (that is, roughly, that we’re talking about concrete monoidal 2-vector spaces), then this adjunction is actually an equivalence: so essentially, all such categories may as well be module categories for weak Hopf algebras.  Then he gave a characterization of these in terms of the “dimension graph” (in fact a quiver) for , where is one of the monoidal generators of .  The vertices of are labelled by the irreducible representations (i.e. set of generators of the category), and there’s a set of edges labelled by a basis of .  Then one can carry on and build a big graded algebra whose -graded part consists of length- paths in .  Then the point is that the weak Hopf algebra of which is (up to isomorphism) the module category will be a certain quotient of (after imposing some natural relations in a systematic way).

The point, then, is that the sort of categories mostly used in this area can be taken to be representation categories, but in general only of these weak Hopf algebras: groups and ordinary algebras are special cases, but they show up naturally for certain kinds of field theory.

Tensor Categories and Field Theories

There were several talks about the relationship between tensor categories of various sorts and particular field theories.  The idea is that local field theories can be broken down in terms of some kind of n-category: -dimensional regions get labelled by categories, -D boundaries between regions, or “defects”, are labelled by functors between the categories (with the idea that this shows how two different kinds of field can couple together at the defect), and so on (I think the highest-dimension that was discussed explicitly involved 3-categories, so one has junctions between defects, and junctions between junctions, which get assigned some higher-morphism data).  Alteratively, there’s the dual picture where categories are assigned to points, functors to 1-manifolds, and so on.  (This is just Poincaré duality in the case where the manifolds come with a decomposition into cells, which they often are if only for convenience).

Victor Ostrik gave a pair of talks giving an overview role tensor categories play in conformal field theory.  There’s too much material here to easily summarize, but the basics go like this: CFTs are field theories defined on cobordisms that have some conformal structure (i.e. notion of angles, but not distance), and on the algebraic side they are associated with vertex algebras (some useful discussion appears on mathoverflow, but in this context they can be understood as vector spaces equipped with exactly the algebraic operations needed to model cobordisms with some local holomorphic structure).

In particular, the irreducible representations of these VOA’s determine the “conformal blocks” of the theory, which tell us about possible correlations between observables (self-adjoint operators).  A VOA is “rational” if the category is semisimple (i.e. generated as finite direct sums of these conformal blocks).  For good VOA’s, will be a modular tensor category (MTC), which is a fusion category with a duality, braiding, and some other strucutre (see this for more).   So describing these gives us a lot of information about what CFT’s are possible.

The full data of a rational CFT are given by a vertex algebra, and a module category : that is, a fusion category is a sort of categorified ring, so it can act on as an ring acts on a module.  It turns out that choosing an is equivalent to finding a certain algebra (i.e. algebra object) , a “Lagrangian algebra” inside the centre of .  The Drinfel’d centre of a monoidal category is a sort of free way to turn a monoidal category into a braided one: but concretely in this case it just looks like .  Knowing the isomorphism class determines a “modular invariant”.  It gets “physics” meaning from how it’s equipped with an algebra structure (which can happen in more than one way), but in any case has an underlying vector space, which becomes the Hilbert space of states for the conformal field theory, which the VOA acts on in the natural way.

Now, that was all conformal field theory.  Christopher Douglas described some work with Chris Schommer-Pries and Noah Snyder about fusion categories and structured topological field theories.  These are functors out of cobordism categories, the most important of which are -categories, where the objects are points, morphisms are 1D cobordisms, and so on up to -morphisms which are -dimensional cobordisms.  To keep things under control, Chris Douglas talked about the case , which is where , and a “local” field theory is a 3-functor for some 3-category .  Now, the (Baez-Dolan) Cobordism Hypothesis, which was proved by Jacob Lurie, says that is, in a suitable sense, the free symmetric monoidal 3-category with duals.  What this amounts to is that a local field theory whose target 3-category is is “just” a dualizable object of .

The handy example which links this up to the above is when has objects which are tensor categories, morphisms which are bimodule categories (i.e. categories acted), 2-morphisms which are functors, and 3-morphisms which are natural transformations.  Then the issue is to classify what kind of tensor categories these objects can be.

The story is trickier if we’re talking about, not just topological cobordisms, but ones equipped with some kind of structure regulated by a structure group (for instance, orientation by , spin structure by its universal cover , and so on).  This means the cobordisms come equipped with a map into .  They take as the starting point, and then consider groups with a map to , and require that the map into is a lift of the map to .  Then one gets that a structured local field theory amounts to a dualizable objects of with a homotopy-fixed point for some -action – and this describes what gets assigned to the point by such a field theory.  What they then show is a correspondence between and classes of categories.  For instance, fusion categories are what one gets by imposing that the cobordisms be oriented.

Liang Kong talked about “Topological Orders and Tensor Categories”, which used the Levin-Wen models, from condensed matter phyiscs.  (Benjamin Balsam also gave a nice talk describing these models and showing how they’re equivalent to the Turaev-Viro and Kitaev models in appropriate cases.  Ingo Runkel gave a related talk about topological field theories with “domain walls”.).  Here, the idea of a “defect” (and topological order) can be understood very graphically: we imagine a 2-dimensional crystal lattice (of atoms, say), and the defect is a 1-dimensional place where the two lattices join together, with the internal symmetry of each breaking down at the boundary.  (For example, a square lattice glued where the edges on one side are offset and meet the squares on the other side in the middle of a face, as you typically see in a row of bricks – the slides linked above have some pictures).  The Levin-Wen models are built using a hexagonal lattice, starting with a tensor category with several properties: spherical (there are dualities satisfying some relations), fusion, and unitary: in fact, historically, these defining properties were rediscovered independently here as the requirement for there to be excitations on the boundary which satisfy physically-inspired consistency conditions.

These abstract the properties of a category of representations.  A generalization of this to “topological orders” in 3D or higher is an extended TFT in the sense mentioned just above: they have a target 3-category of tensor categories, bimodule categories, functors and natural transformations.  The tensor categories (say, , , etc.) get assigned to the bulk regions; to “domain walls” between different regions, namely defects between lattices, we assign bimodule categories (but, for instance, to a line within a region, we get understood as a -bimodule); then to codimension 2 and 3 defects we attach functors and natural transformations.  The algebra for how these combine expresses the ways these topological defects can go together.  On a lattice, this is an abstraction of a spin network model, where typically we have just one tensor category applied to the whole bulk, namely the representations of a Lie group (say, a unitary group).  Then we do calculations by breaking down into bases: on codimension-1 faces, these are simple objects of ; to vertices we assign a Hom space (and label by a basis for intertwiners in the special case); and so on.

Thomas Nickolaus spoke about the same kind of -equivariant Dijkgraaf-Witten models as at our workshop in Lisbon, so I’ll refer you back to my earlier post on that.  However, speaking of equivariance and group actions:

Michael Müger  spoke about “Orbifolds of Rational CFT’s and Braided Crossed -Categories” (see this paper for details).  This starts with that correspondence between rational CFT’s (strictly, rational chiral CFT’s) and modular categories .  (He takes to be the name of the CFT).  Then we consider what happens if some finite group acts on (if we understand as a functor, this is an action by natural transformations; if as an algebra, then ).  This produces an “orbifold theory” (just like a finite group action on a manifold produces an orbifold), which is the “-fixed subtheory” of , by taking -fixed points for every object, and is also a rational CFT.  But that means it corresponds to some other modular category , so one would like to know what category this is.

A natural guess might be that it’s , where is a “weak fixed-point” category that comes from a weak group action on a category .  Objects of are pairs where and is a specified isomorphism.  (This is a weak analog of , the set of fixed points for a group acting on a set).  But this guess is wrong – indeed, it turns out these categories have the wrong dimension (which is defined because the modular category has a trace, which we can sum over generating objects).  Instead, the right answer, denoted by , is the -fixed part of some other category.  It’s a braided crossed -category: one with a grading by , and a -action that gets along with it.  The identity-graded part of is just the original .

State Sum Models

This ties in somewhat with at least some of the models in the previous section.  Some of these were somewhat introductory, since many of the people at the conference were coming from a different background.  So, for instance, to begin the workshop, John Barrett gave a talk about categories and quantum gravity, which started by outlining the historical background, and the development of state-sum models.  He gave a second talk where he began to relate this to diagrams in Gray-categories (something he also talked about here in Lisbon in February, which I wrote about then).  He finished up with some discussion of spherical categories (and in particular the fact that there is a Gray-category of spherical categories, with a bunch of duals in the suitable sense).  This relates back to the kind of structures Chris Douglas spoke about (described above, but chronologically right after John).  Likewise, Winston Fairbairn gave a talk about state sum models in 3D quantum gravity – the Ponzano Regge model and Turaev-Viro model being the focal point, describing how these work and how they’re constructed.  Part of the point is that one would like to see that these fit into the sort of framework described in the section above, which for PR and TV models makes sense, but for the fancier state-sum models in higher dimensions, this becomes more complicated.

Higher Gauge Theory

There wasn’t as much on this topic as at our own workshop in Lisbon (though I have more remarks on higher gauge theory in one post about it), but there were a few entries.  Roger Picken talked about some work with Joao Martins about a cubical formalism for parallel transport based on crossed modules, which consist of a group and abelian group , with a map and an action of on satisfying some axioms.  They can represent categorical groups, namely group objects in (equivalently, categories internal to ), and are “higher” analogs of groups with a set of elements.  Roger’s talk was about how to understand holonomies and parallel transports in this context.  So, a “connection” lets on transport things with -symmetries along paths, and with -symmetries along surfaces.  It’s natural to describe this with squares whose edges are labelled by -elements, and faces labelled by -elements (which are the holonomies).  Then the “cubical approach” means that we can describe gauge transformations, and higher gauge transformations (which in one sense are the point of higher gauge theory) in just the same way: a gauge transformation which assigns -values to edges and -values to vertices can be drawn via the holonomies of a connection on a cube which extends the original square into 3D (so the edges become squares, and so get -values, and so on).  The higher gauge transformations work in a similar way.  This cubical picture gives a good way to understand the algebra of how gauge transformations etc. work: so for instance, gauge transformations look like “conjugation” of a square by four other squares – namely, relating the front and back faces of a cube by means of the remaining faces.  Higher gauge transformations can be described by means of a 4D hypercube in an analogous way, and their algebraic properties have to do with the 2D faces of the hypercube.

Derek Wise gave a short talk outlining his recent paper with John Baez in which they show that it’s possible to construct a higher gauge theory based on the Poincare 2-group which turns out to have fields, and dynamics, which are equivalent to teleparallel gravity, a slightly unusal theory which nevertheless looks in practice just like General Relativity.  I discussed this in a previous post.

So next time I’ll talk about the new additions to my paper on ETQFT which were the basis of my talk, which illustrates a few of the themes above.

The Ottawa meeting was organized by Alistair Savage, and Alex Hoffnung (like me, a former student of John Baez). Alistair gave a talk here at IST over the summer about a -deformation of Khovanov’s categorification of the Heisenberg Algebra I discussed in an earlier entry. A lot of the discussion at the workshop was based on the Khovanov-Lauda program, which began with categorifying quantum version of the classical Lie groups, and is now making lots of progress in the categorification of algebras, representation theory, and so on.

The point of this program is to describe “categorifications” of particular algebras. This means finding monoidal categories with the property that when you take the Grothendieck ring (the ring of isomorphism classes, with a multiplication given by the monoidal structure), you get back the integral form of some algebra. (And then recover the original by taking the tensor over with ). The key thing is how to represent the algebra by generators and relations. Since free monoidal categories with various sorts of structures can be presented as categories of string diagrams, it shouldn’t be surprising that the categories used tend to have objects that are sequences (i.e. monoidal products) of dots with various sorts of labelling data, and morphisms which are string diagrams that carry those labels on strands (actually, usually they’re linear combinations of such diagrams, so everything is enriched in vector spaces). Then one imposes relations on the “free” data given this way, by saying that the diagrams are considered the same morphism if they agree up to some local moves. The whole problem then is to find the right generators (labelling data) and relations (local moves). The result will be a categorification of a given presentation of the algebra you want.

So for instance, I was interested in Sabin Cautis and Anthony Licata‘s talks connected with this paper, “Heisenberg Categorification And Hilbert Schemes”. This is connected with a generalization of Khovanov’s categorification linked above, to include a variety of other algebras which are given a similar name. The point is that there’s such a “Heisenberg algebra” associated to different subgroups , which in turn are classified by Dynkin diagrams. The vertices of these Dynkin diagrams correspond to some generators of the Heisenberg algebra, and one can modify Khovanov’s categorification by having strands in the diagram calculus be labelled by these vertices. Rules for local moves involving strands with different labels will be governed by the edges of the Dynkin diagram. Their paper goes on to describe how to represent these categorifications on certain categories of Hilbert schemes.

Along the same lines, Aaron Lauda gave a talk on the categorification of the NilHecke algebra. This is defined as a subalgebra of endomorphisms of , generated by multiplications (by the ) and the divided difference operators . There are different from the usual derivative operators: in place of the differences between values of a single variable, they measure how a function behaves under the operation which switches variables and (that is, the reflection in the hyperplane where ). The point is that just like differentiation, this operator – together with multiplication – generates an algebra in . Aaron described how to categorify this presentation of the NilHecke algebra with a string-diagram calculus.

So anyway, there were a number of talks about the explosion of work within this general program – for instance, Marco Mackaay’s which I mentioned, as well as that of Pedro Vaz about the same project. One aspect of this program is that the relatively free “string diagram categories” are sometimes replaced with categories where the objects are bimodules and morphisms are bimodule homomorphisms. Making the relationship precise is then a matter of proving these satisfy exactly the relations on a “free” category which one wants, but sometimes they’re a good setting to prove one has a nice categorification. Thus, Ben Elias and Geordie Williamson gave two parts of one talk about “Soergel Bimodules and Kazhdan-Lusztig Theory” (see a blog post by Ben Webster which gives a brief intro to this notion, including pointing out that Soergel bimodules give a categorification of the Hecke algebra).

One of the reasons for doing this sort of thing is that one gets invariants for manifolds from algebras – in particular, things like the Jones polynomial, which is related to the Temperley-Lieb algebra. A categorification of it is Khovanov homology (which gives, for a manifold, a complex, with the property that the graded Euler characteristic of the complex is the Jones polynomial). The point here is that categorifying the algebra lets you raise the dimension of the kind of manifold your invariants are defined on.

So, for instance, Scott Morrison described “Invariants of 4-Manifolds from Khonanov Homology“.  This was based on a generalization of the relationship between TQFT’s and planar algebras.  The point is, planar algebras are described by the composition of diagrams of the following form: a big circle, containing some number of small circles.  The boundaries of each circle are labelled by some number of marked points, and the space between carries curves which connect these marked points in some way.  One composes these diagrams by gluing big circles into smaller circles (there’s some further discussion here including a picture, and much more in this book here).  Scott Morrison described these diagrams as “spaghetti and meatball” diagrams.  Planar algebras show up by associating a vector spaces to “the” circle with marked points, and linear maps to each way (up to isotopy) of filling in edges between such circles.  One can think of the circles and marked-disks as a marked-cobordism category, and so a functorial way of making these assignments is something like a TQFT.  It also gives lots of vector spaces and lots of linear maps that fit together in a particular way described by this category of marked cobordisms, which is what a “planar algebra” actually consists of.  Clearly, these planar algebras can be used to get some manifold invariants – namely the “TQFT” that corresponds to them.

Scott Morrison’s talk described how to get invariants of 4-dimensional manifolds in a similar way by boosting (almost) everything in this story by 2 dimensions.  You start with a 4-ball, whose boundary is a 3-sphere, and excise some number of 4-balls (with 3-sphere boundaries) from the interior.  Then let these 3D boundaries be “marked” with 1-D embedded links (think “knots” if you like).  These 3-spheres with embedded links are the objects in a category.  The morphisms are 4-balls which connect them, containing 2D knotted surfaces which happen to intersect the boundaries exactly at their embedded links.  By analogy with the image of “spaghetti and meatballs”, where the spaghetti is a collection of 1D marked curves, Morrison calls these 4-manifolds with embedded 2D surfaces “lasagna diagrams” (which generalizes to the less evocative case of “ pasta diagrams”, where we’ve just mentioned the and cases, with -dimensional “pasta” embedded in -dimensional balls).  Then the point is that one can compose these pasta diagrams by gluing the 4-balls along these marked boundaries.  One then gets manifold invariants from these sorts of diagrams by using Khovanov homology, which assigns to

Ben Webster talked about categorification of Lie algebra representations, in a talk called “Categorification, Lie Algebras and Topology“. This is also part of categorifying manifold invariants, since the Reshitikhin-Turaev Invariants are based on some monoidal category, which in this case is the category of representations of some algebra.  Categorifying this to a 2-category gives higher-dimensional equivalents of the RT invariants.  The idea (which you can check out in those slides) is that this comes down to describing the analog of the “highest-weight” representations for some Lie algebra you’ve already categorified.

The Lie theory point here, you might remember, is that representations of Lie algebras can be analyzed by decomposing them into “weight spaces” , associated to weights (where is the base field, which we can generally assume is ).  Weights turn Lie algebra elements into scalars, then.  So weight spaces generalize eigenspaces, in that acting by any element on a “weight vector” amounts to multiplying by .  (So that is an eigenvector for each , but the eigenvalue depends on , and is given by the weight.)  A weight can be the “highest” with respect to a natural order that can be put on weights ( if the difference is a nonnegative combination of simple weights).  Then a “highest weight representation” is one which is generated under the action of by a single weight vector , the “highest weight vector”.

The point of the categorification is to describe the representation in the same terms.  First, we introduce a special strand (which Ben Webster draws as a red strand) which represents the highest weight vector.  Then we say that the category that stands in for the highest weight representation is just what we get by starting with this red strand, and putting all the various string diagrams of the categorification of next to it.  One can then go on to talk about tensor products of these representations, where objects are found by amalgamating several such diagrams (with several red strands) together.  And so on.  These categorified representations are then supposed to be usable to give higher-dimensional manifold invariants.

Now, the flip side of higher categories that reproduce ordinary representation theory would be the representation theory of higher categories in their natural habitat, so to speak. Presumably there should be a fairly uniform picture where categorifications of normal representation theory will be special cases of this. Vlodymyr Mazorchuk gave an interesting talk called 2-representations of finitary 2-categories.  He gave an example of one of the 2-categories that shows up a lot in these Khovanov-Lauda categorifications, the 2-category of Soergel Bimodules mentioned above.  This has one object, which we can think of as a category of modules over the algebra (where I  is some ideal of homogeneous symmetric polynomials).  The morphisms are endofunctors of this category, which all amount to tensoring with certain bimodules – the irreducible ones being the Soergel bimodules.  The point of the talk was to explain the representations of 2-categories – that is, 2-functors from into some “classical” 2-category.  Examples would be 2-categories like “2-vector spaces”, or variants on it.  The examples he gave: (1) [small fully additive -linear categories], (2) the full subcategory of it with finitely many indecomposible elements, (3) [categories equivalent to module categories of finite dimensional associative -algebras].  All of these have some claim to be a 2-categorical analog of [vector spaces].  In general, Mazorchuk allowed representations of “FIAT” categories: Finitary (Two-)categories with Involutions and Adjunctions.

Part of the process involved getting a “multisemigroup” from such categories: a set with an operation which takes pairs of elements, and returns a subset of , satisfying some natural associativity condition.  (Semigroups are the case where the subset contains just one element – groups are the case where furthermore the operation is invertible).  The idea is that FIAT categories have some set of generators – indecomposable 1-morphisms – and that the multisemigroup describes which indecomposables show up in a composite.  (If we think of the 2-category as a monoidal category, this is like talking about a decomposition of a tensor product of objects).  So, for instance, for the 2-category that comes from the monoidal category of modules, we get the semigroup of nonnegative integers.  For the Soergel bimodule 2-category, we get the symmetric group.  This sort of thing helps characterize when two objects are equivalent, and in turn helps describe 2-representations up to some equivalence.  (You can find much more detail behind the link above.)

On the more classical representation-theoretic side of things, Joel Kamnitzer gave a talk called “Spiders and Buildings”, which was concerned with some geometric and combinatorial constructions in representation theory.  These involved certain trivalent planar graphs, called “webs”, whose edges carry labels between 1 and .  They’re embedded in a disk, and the outgoing edges, with labels determine a representation space for a group , say , namely the tensor product of a bunch of wedge products, , where acts on as usual.  Then a web determines an invariant vector in this space.  This comes about by having invariant vectors for each vertex (the basic case where ), and tensoring them together.  But the point is to interpret this construction geometrically.  This was a bit outside my grasp, since it involves the Langlands program and the geometric Satake correspondence, neither of which I know much of anything about, but which give geometric/topological ways of constructing representation categories.  One thing I did pick up is that it uses the “Langlands dual group” of to get a certain metric space called .  Then there’s a correspondence between the category of representations of and the category of (perverse, constructible) sheaves on this space.  This correspondence can be used to describe the vectors that come out of these webs.

Jim Dolan gave a couple of talks while I was there, which actually fit together as two parts of a bigger picture – one was during the workshop itself, and one at the logic seminar on the following Monday. It helped a lot to see both in order to appreciate the overall point, so I’ll mix them a bit indiscriminately. The first was called “Dimensional Analysis is Algebraic Geometry”, and the second “Toposes of Quasicoherent Sheaves on Toric Varieties”. For the purposes of the logic seminar, he gave the slogan of the second talk as “Algebraic Geometry is a branch of Categorical Logic”. Jim’s basic idea was inspired by Bill Lawvere’s concept of a “theory”, which is supposed to extend both “algebraic theories” (such as the “theory of groups”) and theories in the sense of physics.  Any given theory is some structured category, and “models” of the theory are functors into some other category to represent it – it thus has a functor category called its “moduli stack of models”.  A physical theory (essentially, models which depict some contents of the universe) has some parameters.  The “theory of elastic scattering”, for instance, has the masses, and initial and final momenta, of two objects which collide and “scatter” off each other.  The moduli space for this theory amounts to assignments of values to these parameters, which must satisfy some algebraic equations – conservation of energy and momentum (for example, , where ).  So the moduli space is some projective algebraic variety.  Jim explained how “dimensional analysis” in physics is the study of line bundles over such varieties (“dimensions” are just such line bundles, since a “dimension” is a 1-dimensional sort of thing, and “quantities” in those dimensions are sections of the line bundles).  Then there’s a category of such bundles, which are organized into a special sort of symmetric monoidal category – in fact, it’s contrained so much it’s just a graded commutative algebra.

In his second talk, he generalized this to talk about categories of sheaves on some varieties – and, since he was talking in the categorical logic seminar, he proposed a point of view for looking at algebraic geometry in the context of logic.  This view could be summarized as: Every (generalized) space studied by algebraic geometry “is” the moduli space of models for some theory in some doctrine.  The term “doctrine” is Bill Lawvere’s, and specifies what kind of structured category the theory and the target of its models are supposed to be (and of course what kind of functors are allowed as models).  Thus, for instance, toposes (as generalized spaces) are supposed to be thought of as “geometric theories”.  He explained that his “dimensional analysis doctrine” is a special case of this.  As usual when talking to Jim, I came away with the sense that there’s a very large program of ideas lurking behind everything he said, of which only the tip of the iceberg actually made it into the talks.

Next post, when I have time, will talk about the meeting at Erlangen…

As August is the month in which Portugal goes on vacation, and we had several family visitors toward the end of the summer, I haven’t posted in a while, but the term has now started up at IST, and seminars are underway, so there should be some interesting stuff coming up to talk about.

New Blog

First, I’ll point out that that Derek Wise has started a new blog, called simply “Simplicity“, which is (I imagine) what it aims to contain: things which seem complex explained so as to reveal their simplicity.  Unless I’m reading too much into the title.  As of this writing, he’s posted only one entry, but a lengthy one that gives a nice explanation of a program for categorified Klein geometries which he’s been thinking a bunch about.  Klein’s program for describing the geometry of homogeneous spaces (such as spherical, Euclidean, and hyperbolic spaces with constant curvature, for example) was developed at Erlangen, and goes by the name “The Erlangen Program”.  Since Derek is now doing a postdoc at Erlangen, and this is supposed to be a categorification of Klein’s approach, he’s referred to it the “2-Erlangen Program”.  There’s more discussion about it in a (somewhat) recent post by John Baez at the n-Category Cafe.  Both of them note the recent draft paper they did relating a higher gauge theory based on the Poincare 2-group to a theory known as teleparallel gravity.  I don’t know this theory so well, except that it’s some almost-equivalent way of formulating General Relativity

I’ll refer you to Derek’s own post for full details of what’s going on in this approach, but the basic motivation isn’t too hard to set out.  The Erlangen program takes the view that a homogeneous space is a space (let’s say we mean by this a topological space) which “looks the same everywhere”.  More precisely, there’s a group action by some , which we understand to be “symmetries” of the space, which is transitive.  Since every point is taken to every other point by some symmetry, the space is “homogeneous”.  Some symmetries leave certain points where they are – they form the stabilizer subgroup .  When the space is homogeneous, it is isomorphic to the coset space, .  So Klein’s idea is to say that any time you have a Lie group and a closed subgroup , this quotient will be called a “homogeneous space”.  A familiar example would be Euclidean space, , where is the Euclidean group and is the orthogonal group, but there are plenty of others.

This example indicates what Cartan geometry is all about, though – this is the next natural step after Klein geometry (Edit:  Derek’s blog now has a visual explanation of Cartan geometry, a.k.a. “generalized hamsterology”, new since I originally posted this).  We can say that Cartan is to Klein as Riemann is to Euclid.  (Or that Cartan is to Riemann as Klein is to Euclid – or if you want to get maybe too-precisely metaphorical, Cartan is the pushout of Klein and Riemann over Euclid).  The point is that Riemannian geometry studies manifolds – spaces which are not homogeneous, but look like Euclidean space locally.  Cartan geometry studies spaces which aren’t homogeneous, but can be locally modelled by Klein geometries.  Now, a Riemannian geometry is essentially a manifold with a metric, describing how it locally looks like Euclidean space.  An equivalent way to talk about it is a manifold with a bundle of Euclidean spaces (the tangent spaces) with a connection (the Levi-Civita connection associated to the metric).  A Cartan geometry can likewise be described as a -bundle with fibre with a connection

Then the point of the “2-Erlangen program” is to develop similar geometric machinery for 2-groups (a.k.a. categorical groups).  This is, as usual, a bit more complicated since actions of 2-groups are trickier than group-actions.  In their paper, though, the point is to look at spaces which are locally modelled by some sort of 2-Klein geometry which derives from the Poincare 2-group.  By analogy with Cartan geometry, one can talk about such Poincare 2-group connections on a space – that is, some kind of “higher gauge theory”.  This is the sort of framework where John and Derek’s draft paper formulates teleparallel gravity.  It turns out that the 2-group connection ends up looking like a regular connection with torsion, and this plays a role in that theory.  Their draft will give you a lot more detail.

Talk on Manifold Calculus

On a different note, one of the first talks I went to so far this semester was one by Pedro Brito about “Manifold Calculus and Operads” (though he ran out of time in the seminar before getting to talk about the connection to operads).  This was about motivating and introducing the Goodwillie Calculus for functors between categories of spaces.  (There are various references on this, but see for instance these notes by Hal Sadofsky). In some sense this is a generalization of calculus from functions to functors, and one of the main results Goodwillie introduced with this subject, is a functorial analog of Taylor’s theorem.  I’d seen some of this before, but this talk was a nice and accessible intro to the topic.

So the starting point for this “Manifold Calculus” is that we’d like to study functors from spaces to spaces (in fact this all applies to spectra, which are more general, but Pedro Brito’s talk was focused on spaces).  The sort of thing we’re talking about is a functor which, given a space , gives a moduli space of some sort of geometric structures we can put on , or of mappings from .  The main motivating example he gave was the functor

for some fixed manifold . Given a manifold , this gives the mapping space of all immersions of into .

(Recalling some terminology: immersions are maps of manifolds where the differential is nondegenerate – the induced map of tangent spaces is everywhere injective, meaning essentially that there are no points, cusps, or kinks in the image, but there might be self-intersections. Embeddings are, in addition, local homeomorphisms.)

Studying this functor means, among other things, looking at the various spaces of immersions of each into . We might first ask: can be immersed in at all – in other words, is nonempty?

So, for example, the Whitney Embedding Theorem says that if is at least , then there is an embedding of into (which is therefore also an immersion).

In more detail, we might want to know what is, which tells how many connected components of immersions there are: in other words, distinct classes of immersions which can’t be deformed into one another by a family of immersions. Or, indeed, we might ask about all the homotopy groups of , not just the zeroth: what’s the homotopy type of ? (Once we have a handle on this, we would then want to vary ).

It turns out this question is manageable, party due to a theorem of Smale and Hirsch, which is a generalization of Gromov’s h-principle – the original principle applies to solutions of certain kinds of PDE’s, saying that any solution can be deformed to a holomorphic one, so if you want to study the space of solutions up to homotopy, you may as well just study the holomorphic solutions.

The Smale-Hirsch theorem likewise gives a homotopy equivalence of two spaces, one of which is . The other is the space of “formal immersions”, called . It consists of all , where is smooth, and is a map of tangent spaces which restricts to , and is injective. These are “formally” like immersions, and indeed has an inclusion into , which happens to be a homotopy equivalence: it induces isomorphisms of all the homotopy groups. These come from homotopies taking each “formal immersion” to some actual immersion. So we’ve approximated , up to homotopy, by . (This “homotopy” of functors makes sense because we’re talking about an enriched functor – the source and target categories are enriched in spaces, where the concepts of homotopy theory are all available).

We still haven’t got to manifold calculus, but it will be all about approximating one functor by another – or rather, by a chain of functors which are supposed to be like the Taylor series for a function. The way to get this series has to do with sheafification, so first it’s handy to re-describe what the Smale-Hirsch theorem says in terms of sheaves. This means we want to talk about some category of spaces with a Grothendieck topology.

So lets let be the category whose objects are -dimensional manifolds and whose morphisms are embeddings (which, of course, are necessarily codimension 0). Now, the point here is that if is an embedding in , and has an immersion into , this induces an immersion of into . This amounst to saying is a contravariant functor:

That makes a presheaf. What the Smale-Hirsch theorem tells us is that this presheaf is a homotopy sheaf – but to understand that, we need a few things first.

First, what’s a homotopy sheaf? Well, the condition for a sheaf says that if we have an open cover of , then

So to say how is a homotopy sheaf, we have to give a topology, which means defining a “cover”, which we do in the obvious way – a cover is a collection of morphisms such that the union of all the images is just . The topology where this is the definition of a cover can be called , because it has the property that given any open cover and choice of 1 point in , that point will be in some of the cover.

This is part of a family of topologies, where only allows those covers with the property that given any choice of points in , some open set of the cover contains them all. These conditions, clearly, get increasingly restrictive, so we have a sequence of inclusions (a “filtration”):

Now, with respect to any given one of these topologies , we have the usual situation relating sheaves and presheaves.  Sheaves are defined relative to a given topology (i.e. a notion of cover).  A presheaf on  is just a contravariant functor from (in this case valued in spaces); a sheaf is one which satisfies a descent condition (I’ve discussed this before, for instance here, when I was running the Stacks Seminar at UWO).  The point of a descent condition, for a given topology is that if we can take the values of a functor “locally” – on the various objects of a cover for – and “glue” them to find the value for itself.  In particular, given a cover for , and a cover, there’s a diagram consisting of the inclusions of all the double-overlaps of sets in the cover into the original sets.  Then the descent condition for sheaves of spaces is that

The general fact is that there’s a reflective inclusion of sheaves into presheaves (see some discussion about reflective inclusions, also in an earlier post).  Any sheaf is a contravariant functor – this is the inclusion of into $latex PSh( \mathcal{E} )$.  The reflection has a left adjoint, sheafification, which takes any presheaf in to a sheaf which is the “best approximation” to it.  It’s the fact this is an adjoint which makes the inclusion “reflective”, and provides the sense in which the sheafification is an approximation to the original functor.

The way sheafification works can be worked out from the fact that it’s an adjoint to the inclusion, but it also has a fairly concrete description.  Given any one of the topologies ,  we have a whole collection of special diagrams, such as:

(using the usual notation where is the intersection of two sets in a cover, and the maps here are the inclusions of that intersection).  This and the various other diagrams involving these inclusions are special, given the topology .  The descent condition for a sheaf says that if we take the image of this diagram:

then we can “glue together” the objects and on the overlap to get one on the union.  That is, is a sheaf if is a colimit of the diagram above (intuitively, by “gluing on the overlap”).  In a presheaf, it would come equipped with some maps into the and : in a sheaf, this object and the maps satisfy some universal property.  Sheafification takes a presheaf to a sheaf which does this, essentially by taking all these colimits.  More accurately, since these sheaves are valued in spaces, what we really want are homotopy sheaves, where we can replace “colimit” with “homotopy colimit” in the above – which satisfies a universal property only up to homotopy, and which has a slightly weaker notion of “gluing”.   This (homotopy) sheaf is called because it depends on the topology which we were using to get the class of special diagrams.

One way to think about is that we take the restriction to manifolds which are made by pasting together at most open balls.  Then, knowing only this part of the functor , we extend it back to all manifolds by a Kan extension (this is the technical sense in which it’s a “best approximation”).

Now the point of all this is that we’re building a tower of functors that are “approximately” like , agreeing on ever-more-complicated manifolds, which in our motivating example is .  Whichever functor we use, we get a tower of functors connected by natural transformations:

This happens because we had that chain of inclusions of the topologies .  Now the idea is that if we start with a reasonably nice functor (like for example), then is just the limit of this diagram.  That is, it’s the universal thing which has a map into each commuting with all these connecting maps in the tower.  The tower of approximations – along with its limit (as a diagram in the category of functors) – is what Goodwillie called the “Taylor tower” for .  Then we say the functor is analytic if it’s just (up to homotopy!) the limit of this tower.

By analogy, think of an inclusion of a vector space with inner product into another such space which has higher dimension.  Then there’s an orthogonal projection onto the smaller space, which is an adjoint (as a map of inner product spaces) to the inclusion – so these are like our reflective inclusions.  So the smaller space can “reflect” the bigger one, while not being able to capture anything in the orthogonal complement.  Now suppose we have a tower of inclusions , where each space is of higher dimension, such that each of the is included into in a way that agrees with their maps to each other.  Then given a vector , we can take a sequence of approximations in the spaces.  If was “nice” to begin with, this series of approximations will eventually at least converge to it – but it may be that our tower of spaces doesn’t let us approximate every in this way.

That’s precisely what one does in calculus with Taylor series: we have a big vector space of smooth functions, and a tower of spaces we use to approximate.  These are polynomial functions of different degrees: first linear, then quadratic, and so forth.  The approximations to a function are orthogonal projections onto these smaller spaces.  The sequence of approximations, or rather its limit (as a sequence in the inner product space ), is just what we mean by a “Taylor series for “.  If is analytic in the first place, then this sequence will converge to it.

The same sort of phenomenon is happening with the Goodwillie calculus for functors: our tower of sheafifications of some functor are just “projections” onto smaller categories (of sheaves) inside the category of all contravariant functors.  (Actually, “reflections”, via the reflective inclusions of the sheaf categories for each of the topologies ).  The Taylor Tower for this functor is just like the Taylor series approximating a function.  Indeed, this analogy is fairly close, since the topologies will give approximations of which are in some sense based on points (so-called -excisive functors, which in our terminology here are sheaves in these topologies).  Likewise, a degree- polynomial approximation approximates a smooth function, in general in a way that can be made to agree at points.

Finally, I’ll point out that I mentioned that the Goodwillie calculus is actually more general than this, and applies not only to spaces but to spectra. The point is that the functor defines a kind of generalized cohomology theory – the cohomology groups for are the . So the point is, functors satisfying the axioms of a generalized cohomology theory are represented by spectra, whereas here is a special case that happens to be a space.

Lots of geometric problems can be thought of as classified by this sort of functor – if , the classifying space of a group, and we drop the requirement that the map be an immersion, then we’re looking at the functor that gives the moduli space of -connections on each .  The point is that the Goodwillie calculus gives a sense in which we can understand such functors by simpler approximations to them.

Often after attending a conference like that, I’d take the time to do a blog about all the talks – which I was tempted to do, but I’ve been busy with things I missed while I was away, and now it’s been quite a while.  I will probably come back at some point and think about the subject of conformal nets, because there were some interesting talks by Andre Henriques at one workshop I was at, and another by Roberto Longo at this one, which together got me interested in this subject.  But that’s not what I’m going to write about this time.

This time, I want to talk about a different kind of topic.  Talking  in Zurich with various people – John Barrett, John Baez, Laurent Freidel, Derek Wise, and some others, on and off – we kept coming back to kept coming back to various seemingly strange algebraic structures.  One such structure is a “loop“, also known (maybe less confusingly) as a “quasigroup” (in fact, a loop is a quasigroup with a unit).  This was especially confusing, because we were talking about these gadgets in the context of gauge theory, where you might want to think about assigning an element of one as the holonomy around a LOOP in spacetime.  Limitations of the written medium being what they are, I’ll just avoid the problem and say “quasigroup” henceforth, although actually I tend to use “loop” when I’m speaking.

The axioms for a quasigroup look just like the axioms for a group, except that the axiom of associativity is missing.  That is, it’s a set with a “multiplication” operation, and each element has a left and a right inverse, called and .  (I’m also assuming the quasigroup is unital from here on in).  Of course, in a group (which is a special kind of quasigroup where associativity holds), you can use associativity to prove that , but we don’t assume it’s true in a quasigroup.  Of course, you can consider the special case where it IS true: this is a “quasigroup with two-sided inverse”, which is a weaker assumption than associativity.

In fact, this is an example of a kind of question one often asks about quasigroups: what are some extra properties we can suppose which, if they hold for a quasigroup , make life easier?  Associativity is a strong condition to ask, and gives the special case of a group, which is a pretty well-understood area.  So mostly one looks for something weaker than associativity.  Probably the most well-known, among people who know about such things, is the Moufang axiom, named after Ruth Moufang, who did a lot of the pioneering work studying quasigroups.

There are several equivalent ways to state the Moufang axiom, but a nice one is:

Which you could derive from the associative law if you had it, but which doesn’t imply associativity.   With associators, one can go from a fully-right-bracketed to a fully-left-bracketed product of four things: .  There’s no associator here (a quasigroup is a set, not a category – though categorifying this stuff may be a nice thing to try), but the Moufang axiom says this is an equation when .  One might think of the stronger condition that says it’s true for all , but the Moufang axiom turns out to be the more handy one.

One way this is so is found in the division algebras.  A division algebra is a (say, real) vector space with a multiplication for which there’s an identity and a notion of division – that is, an inverse for nonzero elements.  We can generalize this enough that we allow different left and right inverses, but in any case, even if we relax this (and the assumption of associativity), it’s a well-known theorem that there are still only four finite dimensional ones.  Namely, they are , , , and : the real numbers, complex numbers, quaternions, and octonions, with real dimensions 1, 2, 4, and 8 respectively.

So the pattern goes like this.  The first two, and , are commutative and associative.  The quaternions are noncommutative, but still associative.  The octonions are neither commutative nor associative.  They also don’t satisfy that stronger axiom .  However, the octonions do satisfy the Moufang axiom.  In each case, you can get a quasigroup by taking the nonzero elements – or, using the fact that there’s a norm around in the usual way of presenting these algebras, the elements of unit norm.  The unit quaternions, in fact, form a group – specifically, the group .  The unit reals and complexes form abelian groups (respectively, , and ).  These groups all have familiar names.  The quasigroup of unit octonions doesn’t have any other more familiar name.  If you believe in the fundamental importance of this sequence of four division algebras, though, it does suggest that a natural sequence in which to weaken axioms for “multiplication” goes: commutative-and-associative, associative, Moufang.

The Moufang axiom does imply some other commonly suggested weakenings of associativity, as well.  For instance, a quasigroup that satisfies the Moufang axiom must also be alternative (a restricted form of associativity when two copies of one element are next to each other: i.e. the left alternative law , and right alternative law ).

Now, there are various ways one could go with this; the one I’ll pick is toward physics.  The first three entries in that sequence of four division algebras – and the corresponding groups – all show up all over the place in physics.  is the simplest nontrivial group, so this could hardly fail to be true, but at any rate, it appears as, for instance, the symmetry group of the set of orientations on a manifold, or the grading in supersymmetry (hence plays a role distinguishing bosons and fermions), and so on.  is, among any number of other things, the group in which action functionals take their values in Lagrangian quantum mechanics; in the Hamiltonian setup, it’s the group of phases that characterizes how wave functions evolve in time.  Then there’s , which is the (double cover of the) group of rotations of 3-space; as a consequence, its representation theory classifies the “spins”, or angular momenta, that a quantum particle can have.

What about the octonions – or indeed the quasigroup of unit octonions?  This is a little less clear, but I will mention this: John Baez has been interested in octonions for a long time, and in Zurich, gave a talk about what kind of role they might play in physics.  This is supposed to partially explain what’s going on with the “special dimensions” that appear in string theory – these occur where the dimension of a division algebra (and a Clifford algebra that’s associated to it) is the same as the codimension of a string worldsheet.  J.B.’s student, John Huerta, has also been interested in this stuff, and spoke about it here in Lisbon in February – it’s the subject of his thesis, and a couple of papers they’ve written.  The role of the octonions here is not nearly so well understood as elsewhere, and of course whether this stuff is actually physics, or just some interesting math that resembles it, is open to experiment – unlike those other examples, which are definitely physics if anything is!

So at this point, the foregoing sets us up to wonder about two questions.  First: are there any quasigroups that are actually of some intrinsic interest which don’t satisfy the Moufang axiom?  (This might be the next step in that sequence of successively weaker axioms).  Second: are there quasigroups that appear in genuine, experimentally tested physics?  (Supposing you don’t happen to like the example from string theory).

Well, the answer is yes on both counts, with one common example – a non-Moufang quasigroup which is of interest precisely because it has a direct physical interpretation.  This example is the composition of velocities in Special Relativity, and was pointed out to me by Derek Wise as a nice physics-based example of nonassociativity.  That it’s also non-Moufang is also true, and not too surprising once you start trying to check it by a direct calculation: in each case, the reason is that the interpretation of composition is very non-symmetric.  So how does this work?

Well, if we take units where the speed of light is 1, then Special Relativity tells us that relative velocities of two observers are vectors in the interior of .  That is, they’re 3-vectors with length less than 1, since the magnitude of the relative velocity must be less than the speed of light.  In any elementary course on Relativity, you’d learn how to compose these velocities, using the “gamma factor” that describes such things as time-dilation.  This can be derived from first principles, nor is it too complicated, but in any case the end result is a new “addition” for vectors:

where   is the reciprocal of the aforementioned “gamma” factor.  The vectors and are the components of the vector which are parallel to, and perpendicular to, , respectively.

The way this is interpreted is: if is the velocity of observer B as measured by observer A, and is the velocity of observer C as measured by observer B, then is the velocity of observer C as measured by observer A.

Clearly, there’s an asymmetry in how and are treated: the first vector, , is a velocity as seen by the same observer who sees the velocity in the final answer.  The second, , is a velocity as seen by an observer who’s vanished by the time we have in hand.  Just looking at the formular, you can see this is an asymmetric operation that distinguishes the left and right inputs.  So the fact (slightly onerous, but not conceptually hard, to check) that it’s noncommutative, and indeed nonassociative, and even non-Moufang, shouldn’t come as a big shock.

The fact that it makes into a quasigroup is a little less obvious, unless you’ve actually worked through the derivation – but from physical principles, is closed under this operation because the final relative velocity will again be less than the speed of light.  The fact that this has “division” (i.e. cancellation), is again obvious enough from physical principles: if we have , the relative velocity of A and C, and we have one of or – the relative velocity of B to either A or C – then the relative velocity of B to the other one of these two must exist, and be findable using this formula.  That’s the “division” here.

So in fact this non-Moufang quasigroup, exotic-sounding algebraic terminology aside, is one that any undergraduate physics student will have learned about and calculated with.

One point that Derek was making in pointing this example out to me was as a comment on a surprising claim someone (I don’t know who) had made, that mathematical abstractions like “nonassociativity” don’t really appear in physics.  I find the above a pretty convincing case that this isn’t true.

In fact, physics is full of Lie algebras, and the Lie bracket is a nonassociative multiplication (except in trivial cases).  But I guess there is an argument against this: namely, people often think of a Lie algebra as living inside its universal enveloping algebra.  Then the Lie bracket is defined as , using the underlying (associative!) multiplication.  So maybe one can claim that nonassociativity doesn’t “really” appear in physics because you can treat it as a derived concept.

An even simpler example of this sort of phenomenon: the integers with subtraction (rather than addition) are nonassociative, in that .  But this only suggests that subtraction is the wrong operation to use: it was derived from addition, which of course is commutative and associative.

In which case, the addition of velocities in relativity is also a derived concept.  Because, of course, really in SR there are no 3-space “velocities”: there are tangent vectors in Minkowski space, which is a 4-dimensional space.  Adding these vectors in is again, of course, commutative and associative.  The concept of “relative velocity” of two observers travelling along given vectors is a derived concept which gets its strange properties by treating the two arguments asymmetrically, just like like “commutator” and “subtraction” do: you first use one vector to decide on a way of slicing Minkowski spacetime into space and time, and then use this to measure the velocity of the other.

Even the octonions, seemingly the obvious “true” example of nonassociativity, could be brushed aside by someone who really didn’t want to accept any example: they’re constructed from the quaternions by the Cayley-Dickson construction, so you can think of them as pairs of quaternions (or 4-tuples of complex numbers).  Then the nonassociative operation is built from associative ones, via that construction.

So are there any “real” examples of “true” nonassociativity (let alone non-Moufangness) that can’t simply be dismissed as not a fundamental operation by someone sufficiently determined?  Maybe, but none I know of right now.  It may be quite possible to consistently hold that anything nonassociative can’t possibly be fundamental (for that matter, elements of noncommutative groups can be represented by matrices of commuting real numbers).  Maybe it’s just my attitude to fundamentals, but somehow this doesn’t move me much.  Even if there are no “fundamentals” examples, I think those given above suggest a different point: these derived operations have undeniable and genuine meaning – in some cases more immediate than the operations they’re derived from.  Whether or not subtraction, or the relative velocity measured by observers, or the bracket of (say) infinitesimal rotations, are “fundamental” ideas is less important than that they’re practical ones that come up all the time.

Motivation

The point of the category , initially, is that it extends the category of manifolds while having some nicer properties.  Thus, while all manifolds are smooth spaces, there are others, which allow to be closed under various operations.  These would include taking limits and colimits: for instance, any subset of a smooth space becomes a smooth space, and any quotient of a smooth space by an equivalence relation is a smooth space.  Then too, has exponentials (that is, if and are smooth spaces, so is ).

So, for instance, this is a good context for constructing loop spaces: a manifold is a smooth space, and so is its loop space , the space of all maps of the circle into .  This becomes important for talking about things like higher cohomology, gerbes, etc.  When starting with the category of manifolds, doing this requires you to go off and define infinite dimensional manifolds before can even be defined.  Likewise, the irrational torus is hard to talk about as a manifold: you take a torus, thought of as .  Then take a direction in with irrational slope, and identify any two points which are translates of each other in along the direction of this line.  The orbit of any point is then dense in the torus, so this is a very nasty space, certainly not a manifold.  But it’s a perfectly good smooth space.

Well, these examples motivate the kinds of things these nice categorical properties allow us to do, but wouldn’t deserve to be called a category of “smooth spaces” (Souriau’s original name for them) if they didn’t allow a notion of smooth maps, which is the basis for most of what we do with manifolds: smooth paths, derivatives of curves, vector fields, differential forms, smooth cohomology, smooth bundles, and the rest of the apparatus of differential geometry.  As with manifolds, this notion of smooth map ought to get along with the usual notion for in some sense.

Smooth Spaces

Thus, a smooth (i.e. diffeological) space consists of:

• A set (of “points”)
• A set (of “plots”) for every n and open such that:

1. All constant maps are plots
2. If is a plot, and is a smooth map, is a plot
3. If is an open cover of , and is a map, whose restrictions are all plots, so is

A smooth map between smooth spaces is one that gets along with all this structure (i.e. the composite with every plot is also a plot).

These conditions mean that smooth maps agree with the usual notion in , and we can glue together smooth spaces to produce new ones.  A manifold becomes a smooth space by taking all the usual smooth maps to be plots: it’s a full subcategory (we introduce new objects which aren’t manifolds, but no new morphisms between manifolds).  A choice of a set of plots for some space is a “diffeology”: there can, of course, be many different diffeologies on a given space.

So, in particular, diffeologies can encode a little more than the charts of a manifold.  Just for one example, a diffeology can have “stop signs”, as Dan put it – points with the property that any smooth map from which passes through them must stop at that point (have derivative zero – or higher derivatives, if you like).  Along the same lines, there’s a nonstandard diffeology on itself with the property that any smooth map from this into a manifold must have all derivatives zero at the endpoints.  This is a better object for defining smooth fundamental groups: you can concatenate these paths at will and they’re guaranteed to be smooth.

As a Quasitopos

An important fact about these smooth spaces is that they are concrete sheaves (i.e. sheaves with underlying sets) on the concrete site (i.e. a Grothendieck site where objects have underlying sets) whose objects are the .  This implies many nice things about the category .  One is that it’s a quasitopos.  This is almost the same as a topos (in particular, it has limits, colimits, etc. as described above), but where a topos has a “subobject classifier”, a quasitopos has a weak subobject classifier (which, perhaps confusingly, is “weak” because it only classifies the strong subobjects).

So remember that a subobject classifier is an object with a map from the terminal object, so that any monomorphism (subobject) is the pullback of along some map (the classifying map).  In the topos of sets, this is just the inclusion of a one-element set into a two-element set : the classifying map for a subset sends everything in (i.e. in the image of the inclusion map) to , and everything else to .  (That is, it’s the characteristic function.)  So pulling back

Any topos has one of these – in particular the topos of sheaves on the diffeological site has one.  But consists of the concrete sheaves, not all sheaves.  The subobject classifier of the topos won’t be concrete – but it does have a “concretification”, which turns out to be the weak subobject classifier.  The subobjects of a smooth space which it classifies (i.e. for which there’s a classifying map as above) are exactly the subsets equipped with the subspace diffeology.  (Which is defined in the obvious way: the plots are the plots of which land in ).

We’ll come back to this quasitopos shortly.  The main point is that Dan and his graduate student, Enxin Wu, have been trying to define a different kind of structure on .  We know it’s good for doing differential geometry.  The hope is that it’s also good for doing homotopy theory.

As a Model Category

The basic idea here is pretty well supported: naively, one can do a lot of the things done in homotopy theory in : to start with, one can define the “smooth homotopy groups” of a pointed space.  It’s a theorem by Dan and Enxin that several possible ways of doing this are equivalent.  But, for example, Iglesias-Zimmour defines them inductively, so that is the set of path-components of , and is defined recursively using loop spaces, mentioned above.  The point is that this all works in much as for topological spaces.

In particular, there are analogs for the for standard theorems like the long exact sequence of homotopy groups for a bundle.  Of course, you have to define “bundle” in – it’s a smooth surjective map , but saying a diffeological bundle is “locally trivial” doesn’t mean “over open neighborhoods”, but “under pullback along any plot”.  (Either of these converts a bundle over a whole space into a bundle over part of , where things are easy to define).

Less naively, the kind of category where homotopy theory works is a model category (see also here).  So the project Dan and Enxin have been working on is to give this sort of structure.  While there are technicalities behind those links, the essential point is that this means you have a closed category (i.e. with all limits and colimits, which does), on which you’ve defined three classes of morphisms: fibrations, cofibrations, and weak equivalences.  These are supposed to abstract the properties of maps in the homotopy theory of topological spaces – in that case weak equivalences being maps that induce isomorphisms of homotopy groups, the other two being defined by having some lifting properties (i.e. you can lift a homotopy, such as a path, along a fibration).

So to abstract the situation in , these classes have to satisfy some axioms (including an abstract form of the lifting properties).  There are slightly different formulations, but for instance, the “2 of 3″ axiom says that if two of , latex $g$ and are weak equivalences, so is the third.  Or, again, there should be a factorization for any morphism into a fibration and an acyclic cofibration (i.e. one which is also a weak equivalence), and also vice versa (that is, moving the adjective “acyclic” to the fibration).  Defining some classes of maps isn’t hard, but it tends to be that proving they satisfy all the axioms IS hard.

Supposing you could do it, though, you have things like the homotopy category (where you formally allow all weak equivalences to have inverses), derived functors(which come from a situation where homotopy theory is “modelled” by categories of chain complexes), and various other fairly powerful tools.  Doing this in would make it possible to use these things in a setting that supports differential geometry.  In particular, you’d have a lot of high-powered machinery that you could apply to prove things about manifolds, even though it doesn’t work in the category itself – only in the larger setting .

Dan and Enxin are still working on nailing down some of the proofs, but it appears to be working.  Their strategy is based on the principle that, for purposes of homotopy, topological spaces act like simplicial complexes.  So they define an affine “simplex”, .  These aren’t literally simplexes: they’re affine planes, which we understand as smooth spaces – with the subspace diffeology from .  But they behave like simplexes: there are face and degeneracy maps for them, and the like.  They form a “cosimplicial object”, which we can think of as a functor , where is the simplex category).

Then the point is one can look at, for a smooth space , the smooth singular simplicial set : it’s a simplicial set where the sets are sets of smooth maps from the affine simplex into .  Likewise, for a simplicial set , there’s a smooth space, the “geometric realization” .  These give two functors and , which are adjoints ( is the left adjoint).  And then, weak equivalences and fibrations being defined in simplicial sets (w.e. are homotopy equivalences of the realization in , and fibrations are “Kan fibrations”), you can just pull the definition back to : a smooth map is a w.e. if its image under is one.  The cofibrations get indirectly defined via the lifting properties they need to have relative to the other two classes.

So it’s still not completely settled that this definition actually gives a model category structure, but it’s pretty close.  Certainly, some things are known.  For instance, Enxin Wu showed that if you have a fibrant object (i.e. one where the unique map to the terminal object is a fibration – these are generally the “good” objects to define homotopy groups on), then the smooth homotopy groups agree with the simplicial ones for .  This implies that for these objects, the weak equivalences are exactly the smooth maps that give isomorphisms for homotopy groups.  And so forth.  But notice that even some fairly nice objects aren’t fibrant: two lines glued together at a point isn’t, for instance.

There are various further results.  One, a consquences of a result Enxin proved, is that all manifolds are fibrant objects, where these nice properties apply.  It’s interesting that this comes from the fact that, in , every (connected) manifold is a homogeneous space.  These are quotients of smooth groups, – the space is a space of cosets, and is understood to be the stabilizer of the point.  Usually one thinks of homogenous spaces as fairly rigid things: the Euclidean plane, say, where is the whole Euclidean group, and the rotations; or a sphere, where is all n-dimensional rotations, and the ones that fix some point on the sphere.  (Actually, this gives a projective plane, since opposite points on the sphere get identified.  But you get the idea).  But that’s for Lie groups.  The point is that , the space of diffeomorphisms from to itself, is a perfectly good smooth group.  Then the subgroup of diffeomorphisms that fix any point is a fine smooth subgroup, and is a homogeneous space in .  But that’s just , with acting transitively on it – any point can be taken anywhere on .

Cohesive Infinity-Toposes

One further thing I’d mention here is related to a related but more abstract approach to the question of how to incorporate homotopy-theoretic tools with a setting that supports differential geometry.  This is the notion of a cohesive topos, and more generally of a cohesive infinity-topos.  Urs Schreiber has advocated for this approach, for instance.  It doesn’t really conflict with the kind of thing Dan was talking about, but it gives a setting for it with lot of abstract machinery.  I won’t try to explain the details (which anyway I’m not familiar with), but just enough to suggest how the two seem to me to fit together, after discussing it a bit with Dan.

The idea of a cohesive topos seems to start with Bill Lawvere, and it’s supposed to characterize something about those categories which are really “categories of spaces” the way is.  Intuitively, spaces consist of “points”, which are held together in lumps we could call “pieces”.  Hence “cohesion”: the points of a typical space cohere together, rather than being a dust of separate elements.  When that happens, in a discrete space, we just say that each piece happens to have just one point in it – but a priori we distinguish the two ideas.  So we might normally say that has an “underlying set” functor , and its left adjoint, the “discrete space” functor (left adjoint since set maps from are the same as continuous maps from – it’s easy for maps out of to be continuous, since every subset is open).

In fact, any topos of sheaves on some site has a pair of functors like this (where becomes , the “set of global sections” functor), essentially because is the topos of sheaves on a single point, and there’s a terminal map from any site into the point.  So this adjoint pair is the “terminal geometric morphism” into .

But this omits there are a couple of other things that apply to : has a right adjoint, , where has only and as its open sets.  In , all the points are “stuck together” in one piece.  On the other hand, itself has a left adjoint, , which gives the set of connected components of a space.  is another kind of “underlying set” of a space.  So we call a topos “cohesive” when the terminal geometric morphism extends to a chain of four adjoint functors in just this way, which satisfy a few properties that characterize what’s happening here.  (We can talk about “cohesive sites”, where this happens.)

Now isn’t exactly a category of sheaves on a site: it’s the category of concrete sheaves on a (concrete) site.  There is a cohesive topos of all sheaves on the diffeological site.  (What’s more, it’s known to have a model category structure).  But now, it’s a fact that any cohesive topos has a subcategory of concrete objects (ones where the canonical unit map is mono: roughly, we can characterize the morphisms of by what they do to its points).  This category is always a quasitopos (and it’s a reflective subcategory of : see the previous post for some comments about reflective subcategories if interested…)  This is where fits in here.  Diffeologies define a “cohesion” just as topologies do: points are in the same “piece” if there’s some plot from a connected part of that lands on both.  Why is only a quasitopos?  Because in general, the subobject classifier in isn’t concrete – but it will have a “concretification”, which is the weak subobject classifier I mentioned above.

Where the “infinity” part of “infinity-topos” comes in is the connection to homotopy theory.  Here, we replace the topos with the infinity-topos of infinity-groupoids.  Then the “underlying” functor captures not just the set of points of a space , but its whole fundamental infinity-groupoid.  Its objects are points of , its morphisms are paths, 2-morphisms are homotopies of paths, and so on.  All the homotopy groups of live here.  So a cohesive inifinity-topos is defined much like above, but with playing the role of , and with that functor replaced by , something which, implicitly, gives all the homotopy groups of .  We might look for cohesive infinity-toposes to be given by the (infinity)-categories of simplicial sheaves on cohesive sites.

This raises a point Dan made in his talk over the diffeological site , we can talk about a cube of different structures that live over it, starting with presheaves: .  We can add different modifiers to this: the sheaf condition; the adjective “concrete”; the adjective “simplicial”.  Various combinations of these adjectives (e.g. simplicial presheaves) are known to have a model structure.  is the case where we have concrete sheaves on .  So far, it hasn’t been proved, but it looks like it shortly will be, that this has a model structure.  This is a particularly nice one, because these things really do seem a lot like spaces: they’re just sets with some easy-to-define and well-behaved (that’s what the sheaf condition does) structure on them, and they include all the examples a differential geometer requires, the manifolds.

This workshop was part of a semester-long program they’ve been having, sponsored by an EU network on noncommutative geometry.  After the workshop was done, Tim Porter and I stayed on for the rest of the week to give some informal seminars and talk to the various grad students who were visiting at the time.  The seminars started off being directed by questions, but ended up talking about TQFT’s and their relations to various kinds of algebras and higher categorical structures, via classifying spaces.  We also had some interesting discussions outside these, for example with Jennifer Maier, who’s been working with Thomas Nicklaus on equivariant Dijkgraaf-Witten theory; with Grace Kennedy, about planar algebras and their relationships to TQFT‘s. I’d also like to give some credit to Makoto Yamashita, who’s interested in noncommutative geometry (viz) and pointed out to me a paper of Alain Connes which gives an account of integration on groupoids, and what corresponds to measures in that setting, which thankfully agrees with what little of it I’d been able to work out on my own.

However, what I’d like to take the time to write up was from the earlier part of my trip, where I visited with Jamie Vicary at Oxford. While I was there, I gave a little lunch seminar about the bicategory (actually a tricategory), and some of the physics- and TQFT-related uses for it. That turned out to be very apropos, because they also had another visitor at the same time, namely Jean Benabou, the fellow who invented bicategories, and introduced the idea of bicategories of spans as one of the first examples.  He gave a talk while I was there which was about the relationship between spans and what he calls “distributors” (which are often called “profunctors“, but since he was anyway the one who introduced them and gave them that name in the first place, and since he has since decided that “profunctors” should refer to only a special class of these entities, I’ll follow his terminology).

(Edit: Thanks to Thomas Streicher for passing on a reference to lecture notes he prepared from lecture by Benabou on the same general topic.)

The question to answer is: what is the relation between spans of categories and distributors?

This is related to a slightly lower-grade question about the relationship between spans of sets, and relations, although the answer turns out to be more complicated.  So, remember that a span from a set to a set is just a diagram like this: .  They can be composed together – so that given a span from to , and from to , we can take fibre products over and get a span from to , consisting of pairs of elements from the sets which map down to the same .  We can do the same thing in any category with pullbacks, not just .

A span is a relation if the pair of arrows is “jointly monic”, which is to say that as a map , it is a monomorphism – which, since we’re talking about sets, essentially means “a subset”.  That is, up to isomorphism of spans, just picks out a bunch of pairs , which are the “related” pairs in this relation.  So there is an inclusion .  What’s more  the inclusion has a left adjoin, which turns a span into a corresponding relation.  It follows from the fact that has an “epi-mono factorization”: namely, the map that comes from the span (and the definition of product) will factor through the image.  That is, it is the composite , where the first part is surjective, and the second part is injective.  Then the inclusion is a relation.  So we say the inclusion of into is a reflection.  (This is a slightly misleading term: there’s an adjoint to the inclusion, but it’s not an adjoint equivalence.  “Reflecting” twice may not get you back where you started, or anywhere isomorphic to it.)

(Edit: Actually, this is a bit wrong.  See the comments below.  What’s true is that the hom-categories of have reflective inclusions into the hom-categories of .  Here, we think of as a 2-category because it’s naturally enriched in posets.  Then these reflective inclusions of hom-categories can be used to build  a lax functor from to – but not an actual functor.)

So a slightly more general question is: if is a monoidal category, and is a  “reflective subcategory“, can we make into a monoidal category just by defining (the product in ) to be the reflection of the original product?   This is the one-object version of a question about bicategories.  Namely, say that is a bicategory, and is a sub-bicategory such that every pair of objects gives a reflective subcategory: has a reflection.  Then can we “pull” the composition of morphisms in back to ?

The answer is no: this just doesn’t work in general.  For spans of sets, and relations, it works: composing spans essentially “counts paths” which relate elements and , whereas composing relations only keeps track of whether or not there is a path.  However, composing spans which come from relations, and then squashing them back down to relations again, agrees with the composite in (the squashing just tells whether the set of paths from to by a sequence of relations is empty or not).  But in the case of and some reflective subcategory – among other possible examples – associativity and unit axioms will break, unless the reflections are specially tuned.  This isn’t to say that we can’t make a monoidal category (or a bicategory).  It just means that pulling back or along the reflection won’t work.  But there is a theorem that says we can always promote such an inclusion into one where this works.

So what’s an instance of all this?  A distributor (again, often called “profunctor”) from a category to is actually a functor .  Then there’s a bicategory , where for each objects there’s a category .  Distributors represent, in some sense, a categorification of relations. (This observation follows the periodic table of category theory, in which a 1-category is a category, a 0-category is a set, and a (-1)-category is a truth value.  There’s a 1-category of relations, with hom-sets , and each one is a map from into truth values, specifying whether a pair is related.)

The most elementary example of a distributor is the “hom-set” construction, where , which is indeed covariant in and contravariant in .  A way to see the general case in that obviously determines a functor from into presheaves on : , where is the category .

In fact, given a functor , we can define two different distributors:

with

and

with

(Remember, these are functors from the product into : so they are just taking hom-sets here in in one direction or the other.)  This much is a tautology: putting a value in in leaves a free variable, but the point is that can be interpreted as a category of “big objects of “.  This is since the Yoneda embedding embeds by taking each object to the presentable presheaf which assigns each object the set of morphisms into , so has “extended” objects of .

So distributors like are “generalized functors” into – and the idea is that this is in roughly the same way that “distributions” are to be seen as “generalized functions”, hence the name.  (Benabou now prefers to use the name “profunctor” to refer only to those distributors which map to “pro-objects” in , which are just special presheaves, namely the “flat” ones.)

Now we have an idea that there is a bicategory , whose hom-categories consist of distributors (and natural transformations), and that the usual functors (which can be seen as distributors which only happen to land in the image of under the Yoneda embedding) form a sub-bicategory: that is, post-composition with turns a functor into a distributor.

But moreover, this operation has an adjoint: functors out of can be “lifted” to functors out of , just by taking the Kan extension of a functor along .  This will work (pointwise, even), as long as is cocomplete, so that we can basically “add up” contributions from the objects of by taking colimits.  In the special case where for some other category , then this tells us how to get composition of distributors .

Now, for a functor , there are straightforward unit and counit natural transformations which makes (the image of under the embedding of into ) a left adjoint for .  So we’ve embedded into in such a way that every functor has a right adjoint.  What about ?  In general, given a bicategory , we can construe as a tricategory, which contains , in such a way that every morphism of has an ambidextrous adjoint (both left and right adjoint).  (There’s work on this by Toby Kenney and Dorette Pronk, and Alex Hoffnung has also been looking at this recently.)  So how does relate to ?

One statement is that a distributor can be seen as a special kind of span, namely:

where consists of all the “elements of ” (in particular, pasting together all the images in of pairs and the set maps that come from morphisms between them in ).  (As an aside: Benabou also explained how a cospan, can be got from a distributor.  The objects of are just the disjoint union of those from and , and the hom-sets are just taken from either , or , or as the sets given by , depending on the situation.  Then the span we just described completes a pullback square opposite this cospan – it’s a comma category.)

These spans end up having some special properties that result from how they’re constructed.  In particular, will be an op-fibration and will be a fibration (this, for instance, is alifting property that let one lift morphisms – since the morphisms are found as the images of the original distributor, this makes sense).  Also, the fibres of are discrete (these are by definition the images of identity morphisms, so naturally they’re discrete categories).  Finally, these properties (fibration, op-fibration, and discrete fibres) are enough to guarantee that a given span is (isomorphic to) one that comes from a distributor.  So we have an embedding .

What’s more, it’s a reflective embedding, because we can always mangle any span to get a new one where these properties hold: it’s enough to force fibres to be discrete by taking their – the connected components.  The other properties will then follow.  But notice that this is a very nontrivial thing to do: in general, the fibres of could be any sort of category, and this operation turns them into sets (of isomorphism classes).  So there’s an adjunction between and , and is a reflective sub-bicategory of .  But the severity of ends up meaning that this doesn’t get along well with composition – the composition of distributors (described above) is not related to composition of spans (which works by weak pullback) via this reflection in a naive way.  However, the theorem mentioned above means that there will be SOME reflecction that makes the compositions get along.  It just may not be as nice as this one.

This is kind of surprising, and the ideal punchline to go here would be to say what that reflection is like, but I don’t know the answer to that question just now.  Anyone else know?

Thanks to Bob Coecke, here are some pictures of me, a few of the people from ComLab, and Jean Benabou at dinner at the Oxford University Club, with a variety of dopey expressions as Bob snapped the pictures unexpectedly.  Thanks Bob.

There is no abiding thing in what we know. We change from weaker to stronger lights, and each more powerful light pierces our hitherto opaque foundations and reveals fresh and different opacities below. We can never foretell which of our seemingly assured fundamentals the next change will not affect.

H.G. Wells, A Modern Utopia

So there’s a recent paper by some physicists, two of whom work just across the campus from me at IST, which purports to explain the Pioneer Anomaly, ultimately using a computer graphics technique, Phong shading. The point being that they use this to model more accurately than has been done before how much infrared radiation is radiating and reflecting off various parts of the Pioneer spacecraft. They claim that with the new, more accurate model, the net force from this radiation is just enough to explain the anomalous acceleration.

Well, plainly, any one paper needs to be rechecked before you can treat it as definitive, but this sort of result looks good for conventional General Relativity, when some people had suggested the anomaly was evidence some other theory was needed.  Other anomalies in the predictions of GR – the rotational profiles of galaxies, or redshift data, have also suggested alternative theories.  In order to preserve GR exactly on large scales, you have to introduce things like Dark Matter and Dark Energy, and suppose that something like 97% of the mass-energy of the universe is otherwise invisible.  Such Dark entities might exist, of course, but I worry it’s kind of circular to postulate them on the grounds that you need them to make GR explain observations, while also claiming this makes sense because GR is so well tested.

In any case, this refined calculation about Pioneer is a reminder that usually the more conservative extension of your model is better. It’s not so obvious to me whether a modified theory of gravity, or an unknown and invisible majority of the universe is more conservative.

And that’s the best segue I can think of into this next post, which is very different from recent ones.

Fundamentals

I was thinking recently about “fundamental” theories.  At the HGTQGR workshop we had several talks about the most popular physical ideas into which higher gauge theory and TQFT have been infiltrating themselves recently, namely string theory and (loop) quantum gravity.  These aren’t the only schools of thought about what a “quantum gravity” theory should look like – but they are two that have received a lot of attention and work.  Each has been described (occasionally) as a “fundamental” theory of physics, in the sense of one which explains everything else.  There has been a debate about this, since they are based on different principles.  The arguments against string theory are various, but a crucial one is that no existing form of string theory is “background independent” in the same way that General Relativity is. This might be because string theory came out of a community grounded in particle physics – it makes sense to perturb around some fixed background spacetime in that context, because no experiment with elementary particles is going to have a measurable effect on the universe at infinity. “M-theory” is supposed to correct this defect, but so far nobody can say just what it is.  String theorists criticize LQG on various grounds, but one of the more conceptually simple ones would be that it can’t be a unified theory of physics, since it doesn’t incorporate forces other than gravity.

There is, of course, some philosophical debate about whether either of these properties – background independence, or unification – is really crucial to a fundamental theory.   I don’t propose to answer that here (though for the record my hunch at he moment is that both of them are important and will hold up over time).  In fact, it’s “fundamental theory” itself that I’m thinking about here.

As I suggested in one of my first posts explaining the title of this blog, I expect that we’ll need lots of theories to get a grip on the world: a whole “atlas”, where each “map” is a theory, each dealing with a part of the whole picture, and overlapping somewhat with others. But theories are formal entities that involve symbols and our brain’s ability to manipulate symbols. Maybe such a construct could account for all the observable phenomena of the world – but a-priori it seems odd to assume that. The fact that they can provide various limits and approximations has made them useful, from an evolutionary point of view, and the tendency to confuse symbols and reality in some ways is a testament to that (it hasn’t hurt so much as to be selected out).

One little heuristic argument – not at all conclusive – against this idea involves Kolmogorov complexity: wanting to explain all the observed data about the universe is in some sense to “compress” the data.  If we can account for the observations – say, with a short description of some physical laws and a bunch of initial conditions, which is what a “fundamental theory” suggests – then we’ve found an upper bound on its Kolmogorov complexity.  If the universe actually contains such a description, then that must also be a lower bound on its complexity.  Thus, any complete description of the universe would have to be as big as the whole universe.

Well, as I said, this argument fails to be very convincing.  Partly because it assumes a certain form of the fundamental theory (in particular, a deterministic one), but mainly because it doesn’t rule out that there is indeed a very simple set of physical laws, but there are limits to the precision with which we could use them to simulate the whole world because we can’t encode the state of the universe perfectly.  We already knew that.  At most, that lack of precision puts some practical limits on our ability to confirm that a given set of physical laws we’ve written down is  empirically correct.  It doesn’t preclude there being one, or even our finding it (without necessarily being perfectly certain).  The way Einstein put it (in this address, by the way) was “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”  But a lack of certainty doesn’t mean they aren’t there.

However, this got me thinking about fundamental theories from the point of view of epistemology, and how we handle knowledge.

Reduction

First, there’s a practical matter. The idea of a fundamental theory is the logical limit of one version of reductionism. This is the idea that the behaviour of things should be explained in terms of smaller, simpler things. I have no problem with this notion, unless you then conclude that once you’ve found a “more fundamental” theory, the old one should be discarded.

For example: we have a “theory of chemistry”, which says that the constituents of matter are those found on the periodic table of elements.  This theory comes in various degrees of sophistication: for instance, you can start to learn the periodic table without knowing that there are often different isotopes of a given element, and only knowing the 91 naturally occurring elements (everything up to Uranium, except Technicium). This gives something like Mendeleev’s early version of the table. You could come across these later refinements by finding a gap in the theory (Technicium, say), or a disagreement with experiment (discovering isotopes by measuring atomic weights). But even a fairly naive version of the periodic table, along with some concepts about atomic bonds, gives a good explanation of a huge range of chemical reactions under normal conditions. It can’t explain, for example, how the Sun shines – but it explains a lot within its proper scope.

Where this theory fits in a fuller picture of the world has at least two directions: more fundamental, and less fundamental, theories.  What I mean by less “fundamental” is that some things are supposed to be explained by this theory of chemistry: the great abundance of proteins and other organic chemicals, say. The behaviour of the huge variety of carbon compounds predicted by basic chemistry is supposed to explain all these substances and account for how they behave.  The millions of organic compounds that show up in nature, and their complicated behaviour, is supposed to be explained in terms of just a few elements that they’re made of – mostly carbon, hydrogen, oxygen, nitrogen, sulfur, phosphorus, plus the odd trace element.

By “more fundamental”, I mean that the periodic table itself can start to seem fairly complicated, especially once you start to get more sophisticated, including transuranic elements, isotopes, radioactive decay rates, and the like. So it was explained in terms of a theory of the atom. Again, there are refinements, but the Bohr model of the atom ought to do the job: a nucleus made of protons and neutrons, and surrounded by shells of electrons.  We can add that these are governed by the Dirac equation, and then the possible states for electrons bound to a nucleus ought to explain the rows and columns of the periodic table. Better yet, they’re supposed to explain exactly the spectral lines of each element – the frequencies of light atoms absorb and emit – by the differences of energy levels between the shells.

Well, this is great, but in practice it has limits. Hardly anyone disputes that the Bohr model is approximately right, and should explain the periodic table etc. The problem is that it’s largely an intractable problem to actually solve the Schroedinger equation for the atom and use the results to predict the emission spectrum, chemical properties, melting point, etc. of, say, Vanadium…  On the other hand, it’s equally hard to use a theory of chemistry to adequately predict how proteins will fold. Protein conformation prediction is a hard problem, and while it’s chugging along and making progress, the point is a theory of chemistry alone isn’t enough: any successful method must rely on a whole extra body of knowledge.  This suggests our best bet at understanding all these phenomena is to have a whole toolbox of different theories, each one of which has its own body of relevant mathematics, its own domain-specific ontology, and some sense of how its concepts relate to those in other theories in the tookbox. (This suggests a view of how mathematics relates to the sciences which seems to me to reflect actual practice: it pervades all of them, in a different way than the way a “more fundamental” theory underlies a less fundamental one.  Which tends to spoil the otherwise funny XKCD comic on the subject…)

If one “explains” one theory in terms of another (or several others), then we may be able to put them into at least a partial order.  The mental image I have in mind is the “theoretical atlas” – a bunch of “charts” (the theories) which cover different parts of a globe (our experience, or the data we want to account for), and which overlap in places.  Some are subsets of others (are completely explained by them, in principle). Then we’d like to find a minimal (or is it maximal) element of this order: something which accounts for all the others, at least in principle.  In that mental image, it would be a map of the whole globe (or a dense subset of the surface, anyway).  Because, of course, the Bohr model, though in principle sufficient to account for chemistry, needs an explanation of its own: why are atoms made this way, instead of some other way? This ends up ramifying out into something like the Standard Model of particle physics.  Once we have that, we would still like to know why elementary particles work this way, instead of some other way…

An Explanatory Trilemma

There’s a problem here, which I think is unavoidable, and which rather ruins that nice mental image.  It has to do with a sort of explanatory version of Agrippa’s Trilemma, which is an observation in epistemology that goes back to Agrippa the Skeptic. It’s also sometimes called “Munchausen’s Trilemma”, and it was originally made about justifying beliefs.  I think a slightly different form of it can be applied to explanations, where instead of “how do I know X is true?”, the question you repeatedly ask is “why does it happen like X?”

So, the Agrippa Trilemma as classically expressed might lead to a sequence of questions about observation.  Q: How do we know chemical substances are made of elements? A: Because of some huge body of evidence. Q: How do we know this evidence is valid? A: Because it was confirmed by a bunch of experimental data. Q: How do we know that our experiments were done correctly? And so on. In mathematics, it might ask a series of questions about why a certain theorem is true, which we chase back through a series of lemmas, down to a bunch of basic axioms and rules of inference. We could be asked to justify these, but typically we just posit them. The Trilemma says that there are three ways this sequence of justifications can end up:

1. we arrive at an endpoint of premises that don’t require any justification
2. we continue indefinitely in a chain of justifications that never ends
3. we continue in a chain of justifications that eventually becomes circular

None of these seems to be satisfactory for an experimental science, which is partly why we say that there’s no certainty about empirical knowledge. In mathematics, the first option is regarded as OK: all statements in mathematics are “really” of the form if axioms A, B, C etc. are assumed, then conclusions X, Y, Z etc. eventually follow. We might eventually find that some axioms don’t apply to the things we’re interested in, and cease to care about those statements, but they’ll remain true. They won’t be explanations of anything very much, though.  If we’re looking at reality, it’s not enough to assume axioms A, B, C… We also want to check them, test them, see if they’re true – and we can’t be completely sure with only a finite amount of evidence.

The explanatory variation on Agrippa’s Trilemma, which I have in mind, deals with a slightly different problem.  Supposing the axioms seem to be true, and accepting provisionally that they are, we also have another question, which if anything is even more basic to science: we want to know WHY they’re true – we look for an explanation.

This is about looking for coherence, rather than confidence, in our knowledge (or at any rate, theories). But a similar problem appears. Suppose that elementary chemistry has explained organic chemistry; that atomic physics has explained why chemistry is how it is; and that the Standard model explains why atomic physics is how it is.  We still want to know why the Standard Model is the way it is, and so on. Each new explanation gives an account for one phenomenon in terms of different, more basic phenomenon. The Trilemma suggests the following options:

1. we arrive at an endpoint of premises that don’t require any explanation
2. we continue indefinitely in a chain of explanations that never ends
3. we continue in a chain of explanations that eventually becomes circular

Unless we accept option 1, we don’t have room for a “fundamental theory”.

Here’s the key point: this isn’t even a position about physics – it’s about epistemology, and what explanations are like, or maybe rather what our behaviour is like with regard to explanations. The standard version of Agrippa’s Trilemma is usually taken as an argument for something like fallibilism: that our knowledge is always uncertain. This variation isn’t talking about the justification of beliefs, but the sufficiency of explanation. It says that the way our mind works is such that there can’t be one final summation of the universe, one principle, which accounts for everything – because it would either be unaccounted for itself, or because it would have to account for itself by circular reasoning.

This might be a dangerous statement to make, or at least a theological one (theology isn’t as dangerous as it used to be): reasoning that things are the way they are “because God made it that way” is a traditional answer of the first type. True or not, I don’t think you can really call an “explanation”, since it would work equally well if things were some other way. In fact, it’s an anti-explanation: if you accept an uncaused-cause anywhere along the line, the whole motivation for asking after explanations unravels.  Maybe this sort of answer is a confession of humility and acceptance of limited understanding, where we draw the line and stop demanding further explanations. I don’t see that we all need to draw that line in the same place, though, so the problem hasn’t gone away.

What seems likely to me is that this problem can’t be made to go away.  That the situation we’ll actually be in is (2) on the list above.  That while there might not be any specific thing that scientific theories can’t explain, neither could there be a “fundamental theory” that will be satisfying to the curious forever.  Instead, we have an asymptotic approach to explanation, as each thing we want to explain gets picked up somewhere along the line: “We change from weaker to stronger lights, and each more powerful light pierces our hitherto opaque foundations and reveals fresh and different opacities below.”

Whereas Einstein’s gravity describes a regime where Newton’s gravitational constant is important but Planck’s constant is negligible, and (special-relativistic) quantum field theory assumes significant but not.  Both of these assume there is a special velocity scale, given by the speed of light , whereas classical mechanics assumes that all three can be neglected (i.e. and are zero, and is infinite).   The guiding assumption is that these are all approximations to some more fundamental theory, called “quantum gravity” just because it accepts that both and (as well as ) are significant in calculating physical effects.  So GR and QFT incorporate two of the three constants each, and classical mechanics incorporates neither.  The “principle of relative locality” arises when we consider a slightly different approximation to this underlying theory.

This approximation works with a regime where and are each negligible, but the ratio is not – this being related to the Planck mass .  The point is that this is an approximation with no special length scale (“Planck length”), but instead a special energy scale (“Planck mass”) which has to be preserved.   Since energy and momentum are different parts of a single 4-vector, this is also a momentum scale; we expect to see some kind of deformation of momentum space, at least for momenta that are bigger than this scale.  The existence of this scale turns out to mean that momenta don’t add linearly – at least, not unless they’re very small compared to the Planck scale.

So what is “Relative Locality”?  In the paper linked above, it’s stated like so:

Physics takes place in phase space and there is no invariant global projection that gives a description of processes in spacetime.  From their measurements local observers can construct descriptions of particles moving and interacting in a spacetime, but different observers construct different spacetimes, which are observer-dependent slices of phase space.

Motivation

This arises from taking the basic insight of general relativity – the requirement that physical principles should be invariant under coordinate transformations (i.e. diffeomorphisms) – and extend it so that instead of applying just to spacetime, it applies to the whole of phase space.  Phase space (which, in this limit where , replaces the Hilbert space of a truly quantum theory) is the space of position-momentum configurations (of things small enough to treat as point-like, in a given fixed approximation).  Having no means we don’t need to worry about any dynamical curvature of “spacetime” (which doesn’t exist), and having no Planck length means we can blithely treat phase space as a manifold with coordinates valued in the real line (which has no special scale).  Yet, having a special mass/momentum scale says we should see some purely combined “quantum gravity” effects show up.

The physical idea is that phase space is an accurate description of what we can see and measure locally.  Observers (whom we assume small enough to be considered point-like) can measure their own proper time (they “have a clock”) and can detect momenta (by letting things collide with them and measuring the energy transferred locally and its direction).  That is, we “see colors and angles” (i.e. photon energies and differences of direction).  Beyond this, one shouldn’t impose any particular theory of what momenta do: we can observe the momenta of separate objects and see what results when they interact and deduce rules from that.  As an extension of standard physics, this model is pretty conservative.  Now, conventionally, phase space would be the cotangent bundle of spacetime .  This model is based on the assumption that objects can be at any point, and wherever they are, their space of possible momenta is a vector space.  Being a bundle, with a global projection onto (taking to ), is exactly what this principle says doesn’t necessarily obtain.  We still assume that phase space will be some symplectic manifold.   But we don’t assume a priori that momentum coordinates give a projection whose fibres happen to be vector spaces, as in a cotangent bundle.

Now, a symplectic manifold  still looks locally like a cotangent bundle (Darboux’s theorem). So even if there is no universal “spacetime”, each observer can still locally construct a version of “spacetime”  by slicing up phase space into position and momentum coordinates.  One can, by brute force, extend the spacetime coordinates quite far, to distant points in phase space.  This is roughly analogous to how, in special relativity, each observer can put their own coordinates on spacetime and arrive at different notions of simultaneity.  In general relativity, there are issues with trying to extend this concept globally, but it can be done under some conditions, giving the idea of “space-like slices” of spacetime.  In the same way, we can construct “spacetime-like slices” of phase space.

Geometrizing Algebra

Now, if phase space is a cotangent bundle, momenta can be added (the fibres of the bundle are vector spaces).  Some more recent ideas about “quasi-Hamiltonian spaces” (initially introduced by Alekseev, Malkin and Meinrenken) conceive of momenta as “group-valued” – rather than taking values in the dual of some Lie algebra (the way, classically, momenta are dual to velocities, which live in the Lie algebra of infinitesimal translations).  For small momenta, these are hard to distinguish, so even group-valued momenta might look linear, but the premise is that we ought to discover this by experiment, not assumption.  We certainly can detect “zero momentum” and for physical reasons can say that given two things with two momenta , there’s a way of combining them into a combined momentum .  Think of doing this physically – transfer all momentum from one particle to another, as seen by a given observer.  Since the same momentum at the observer’s position can be either coming in or going out, this operation has a “negative” with .

We do have a space of momenta at any given observer’s location – the total of all momenta that can be observed there, and this space now has some algebraic structure.  But we have no reason to assume up front that is either commutative or associative (let alone that it makes momentum space at a given observer’s location into a vector space).  One can interpret this algebraic structure as giving some geometry.  The commutator for gives a metric on momentum space.  This is a bilinear form which is implicitly defined by the “norm” that assigns a kinetic energy to a particle with a given momentum. The associator given by , infinitesimally near where this makes sense, gives a connection.  This defines a “parallel transport” of a finite momentum in the direction of a momentum by saying infinitesimally what happens when adding to .

Various additional physical assumptions – like the momentum-space “duals” of the equivalence principle (that the combination of momenta works the same way for all kinds of matter regardless of charge), or the strong equivalence principle (that inertial mass and rest mass energy per the relation are the same) and so forth can narrow down the geometry of this metric and connection.  Typically we’ll find that it needs to be Lorentzian.  With strong enough symmetry assumptions, it must be flat, so that momentum space is a vector space after all – but even with fairly strong assumptions, as with general relativity, there’s still room for this “empty space” to have some intrinsic curvature, in the form of a momentum-space “dual cosmological constant”, which can be positive (so momentum space is closed like a sphere), zero (the vector space case we usually assume) or negative (so momentum space is hyperbolic).

This geometrization of what had been algebraic is somewhat analogous to what happened with velocities (i.e. vectors in spacetime)) when the theory of special relativity came along.  Insisting that the “invariant” scale be the same in every reference system meant that the addition of velocities ceased to be linear.  At least, it did if you assume that adding velocities has an interpretation along the lines of: “first, from rest, add velocity v to your motion; then, from that reference frame, add velocity w”.  While adding spacetime vectors still worked the same way, one had to rephrase this rule if we think of adding velocities as observed within a given reference frame – this became (scaling so and assuming the velocities are in the same direction).  When velocities are small relative to , this looks roughly like linear addition.  Geometrizing the algebra of momentum space is thought of a little differently, but similar things can be said: we think operationally in terms of combining momenta by some process.  First transfer (group-valued) momentum to a particle, then momentum – the connection on momentum space tells us how to translate these momenta into the “reference frame” of a new observer with momentum shifted relative to the starting point.  Here again, the special momentum scale (which is also a mass scale since a momentum has a corresponding kinetic energy) is a “deformation” parameter – for momenta that are small compared to this scale, things seem to work linearly as usual.

There’s some discussion in the paper which relates this to DSR (either “doubly” or “deformed” special relativity), which is another postulated limit of quantum gravity, a variation of SR with both a special velocity and a special mass/momentum scale, to consider “what SR looks like near the Planck scale”, which treats spacetime as a noncommutative space, and generalizes the Lorentz group to a Hopf algebra which is a deformation of it.  In DSR, the noncommutativity of “position space” is directly related to curvature of momentum space.  In the “relative locality” view, we accept a classical phase space, but not a classical spacetime within it.

Physical Implications

We should understand this scale as telling us where “quantum gravity effects” should start to become visible in particle interactions.  This is a fairly large scale for subatomic particles.  The Planck mass as usually given is about 21 micrograms: small for normal purposes, about the size of a small sand grain, but very large for subatomic particles.  Converting to momentum units with , this is about 6 kg m/s: on the order of the momentum of a kicked soccer ball or so.  For a subatomic particle this is a lot.

This scale does raise a question for many people who first hear this argument, though – that quantum gravity effects should become apparent around the Planck mass/momentum scale, since macro-objects like the aforementioned soccer ball still seem to have linearly-additive momenta.  Laurent explained the problem with this intuition.  For interactions of big, extended, but composite objects like soccer balls, one has to calculate not just one interaction, but all the various interactions of their parts, so the “effective” mass scale where the deformation would be seen becomes where is the number of particles in the soccer ball.  Roughly, the point is that a soccer ball is not a large “thing” for these purposes, but a large conglomeration of small “things”, whose interactions are “fundamental”.  The “effective” mass scale tells us how we would have to alter the physical constants to be able to treat it as a “thing”.  (This is somewhat related to the question of “effective actions” and renormalization, though these are a bit more complicated.)

There are a number of possible experiments suggested in the paper, which Laurent mentioned in the talk.  One involves a kind of “twin paradox” taking place in momentum space.  In “spacetime”, a spaceship travelling a large loop at high velocity will arrive where it started having experienced less time than an observer who remained there (because of the Lorentzian metric) – and a dual phenomenon in momentum space says that particles travelling through loops (also in momentum space) should arrive displaced in space because of the relativity of localization.  This could be observed in particle accelerators where particles make several transits of a loop, since the effect is cumulative.  Another effect could be seen in astronomical observations: if an observer is observing some distant object via photons of different wavelengths (hence momenta), she might “localize” the object differently – that is, the two photons travel at “the same speed” the whole way, but arrive at different times because the observer will interpret the object as being at two different distances for the two photons.

This last one is rather weird, and I had to ask how one would distinguish this effect from a variable speed of light (predicted by certain other ideas about quantum gravity).  How to distinguish such effects seems to be not quite worked out yet, but at least this is an indication that there are new, experimentally detectible, effects predicted by this “relative locality” principle.  As Laurent emphasized, once we’ve noticed that not accepting this principle means making an a priori assumption about the geometry of momentum space (even if only in some particular approximation, or limit, of a true theory of quantum gravity), we’re pretty much obliged to stop making that assumption and do the experiments.  Finding our assumptions were right would simply be revealing which momentum space geometry actually obtains in the approximation we’re studying.

A final note about the physical interpretation: this “relative locality” principle can be discovered by looking (in the relevant limit) at a Lagrangian for free particles, with interactions described in terms of momenta.  It so happens that one can describe this without referencing a “real” spacetime: the part of the action that allows particles to interact when “close” only needs coordinate functions, which can certainly exist here, but are an observer-dependent construct.  The conservation of (non-linear) momenta is specified via a Lagrange multiplier.  The whole Lagrangian formalism for the mechanics of colliding particles works without reference to spacetime.  Now, even though all the interactions (specified by the conservation of momentum terms) happen “at one location”, in that there will be an observer who sees them happening in the momentum space of her own location.  But an observer at a different point may disagree about whether the interaction was local – i.e. happened at a single point in spacetime.  Thus “relativity of localization”.

Again, this is no more bizarre (mathematically) than the fact that distant, relatively moving, observers in special relativity might disagree about simultaneity, whether two events happened at the same time.  They have their own coordinates on spacetime, and transferring between them mixes space coordinates and time coordinates, so they’ll disagree whether the time-coordinate values of two events are the same.  Similarly, in this phase-space picture, two different observers each have a coordinate system for splitting phase space into “spacetime” and “energy-momentum” coordinates, but switching between them may mix these two pieces.  Thus, the two observers will disagree about whether the spacetime-coordinate values for the different interacting particles are the same.  And so, one observer says the interaction is “local in spacetime”, and the other says it’s not.  The point is that it’s local for the particles themselves (thinking of them as observers).  All that’s going on here is the not-very-astonishing fact that in the conventional picture, we have no problem with interactions being nonlocal in momentum space (particles with very different momenta can interact as long as they collide with each other)… combined with the inability to globally and invariantly distinguish position and momentum coordinates.

What this means, philosophically, can be debated, but it does offer some plausibility to the claim that space and time are auxiliary, conceptual additions to what we actually experience, which just account for the relations between bits of matter.  These concepts can be dispensed with even where we have a classical-looking phase space rather than Hilbert space (where, presumably, this is even more true).

…

Edit: On a totally unrelated note, I just noticed this post by Alex Hoffnung over at the n-Category Cafe which gives a lot of detail on issues relating to spans in bicategories that I had begun to think more about recently in relation to developing a higher-gauge-theoretic version of the construction I described for ETQFT. In particular, I’d been thinking about how the 2-group analog of restriction and induction for representations realizes the various kinds of duality properties, where we have adjunctions, biadjunctions, and so forth, in which units and counits of the various adjunctions have further duality. This observation seems to be due to Jim Dolan, as far as I can see from a brief note in HDA II. In that case, it’s really talking about the star-structure of the span (tri)category, but looking at the discussion Alex gives suggests to me that this theme shows up throughout this subject. I’ll have to take a closer look at the draft paper he linked to and see if there’s more to say…

Now that we’re back, I’ll attempt to wrap up with the summaries of things discussed at the workshop on Higher Gauge Theory, TQFT, and Quantum Gravity.  In the previous post I described talks which I roughly gathered under TQFT and Higher Gauge Theory, but the latter really ramifies out in a few different ways.  As began to be clear before, higher bundles are classified by higher cohomology of manifolds, and so are gerbes – so in fact these are two slightly different ways of talking about the same thing.  I also remarked, in the summary of Konrad Waldorf’s talk, the idea that the theory of gerbes on a manifold is equivalent to ordinary gauge theory on its loop space – which is one way to make explicit the idea that categorification “raises dimension”, in this case from parallel transport of points to that of 1-dimensional loops.  Next we’ll expand on that theme, and then finally reach the “Quantum Gravity” part, and draw the connection between this and higher gauge theory toward the end.

Gerbes and Cohomology

The very first workshop speaker, in fact, was Paolo Aschieri, who has done a lot of work relating noncommutative geometry and gravity.  In this case, though, he was talking about noncommutative gerbes, and specifically referred to this work with some of the other speakers.  To be clear, this isn’t about gerbes with noncommutative group , but about gerbes on noncommutative spaces.  To begin with, it’s useful to express gerbes in the usual sense in the right language.  In particular, he explain what a gerbe on a manifold is in concrete terms, giving Hitchin’s definition (viz).  A gerbe can be described as “a cohomology class” but it’s more concrete to present it as:

• a collection of line bundles associated with double overlaps .  Note this gets an algebraic structure (multiplication of bundles is pointwise , with an inverse given by the dual, , so we can require…
• , which helps define…
• transition functions on triple overlaps , which are sections of .  If this product is trivial, there’d be a 1-cocycle condition here, but we only insist on the 2-cocycle condition…
• 

This is a -gerbe on a commutative space.  The point is that one can make a similar definition for a noncommutative space.  If the space is associated with the algebra of smooth functions, then a line bundle is a module for , so if is noncommutative (thought of as a “space” ), a “bundle over is just defined to be an -module.  One also has to define an appropriate “covariant derivative” operator on this module, and the -product must be defined as well, and will be noncommutative (we can think of it as a deformation of the above).  The transition functions are sections: that is, elements of the modules in question.  his means we can describe a gerbe in terms of a big stack of modules, with a chosen algebraic structure, together with some elements.  The idea then is that gerbes can give an interpretation of cohomology of noncommutative spaces as well as commutative ones.

Mauro Spera spoke about a point of view of gerbes based on “transgressions”.  The essential point is that an -gerbe on a space can be seen as the obstruction to patching together a family of  -gerbes.  Thus, for instance, a 0-gerbe is a -bundle, which is to say a complex line bundle.  As described above, a 1-gerbe can be understood as describing the obstacle to patching together a bunch of line bundles, and the obstacle is the ability to find a cocycle satisfying the requisite conditions.  This obstacle is measured by the cohomology of the space.  Saying we want to patch together -gerbes on the fibre.  He went on to discuss how this manifests in terms of obstructions to string structures on manifolds (already discussed at some length in the post on Hisham Sati’s school talk, so I won’t duplicate here).

A talk by Igor Bakovic, “Stacks, Gerbes and Etale Groupoids”, gave a way of looking at gerbes via stacks (see this for instance).  The organizing principle is the classification of bundles by the space maps into a classifying space – or, to get the category of principal -bundles on, the category , where is the category of sheaves on and is the classifying topos of -sets.  (So we have geometric morphisms between the toposes as the objects.)  Now, to get further into this, we use that is equivalent to the category of Étale spaces over – this is a refinement of the equivalence between bundles and presheaves.  Taking stalks of a presheaf gives a bundle, and taking sections of a bundle gives a presheaf – and these operations are adjoint.

The issue at hand is how to categorify this framework to talk about 2-bundles, and the answer is there’s a 2-adjunction between the 2-category of such things, and , the 2-category of fibred categories over .  (That is, instead of looking at “sheaves of sets”, we look at “sheaves of categories” here.)  The adjunction, again, involves talking stalks one way, and taking sections the other way.  One hard part of this is getting a nice definition of “stalk” for stacks (i.e. for the “sheaves of categories”), and a good part of the talk focused on explaining how to get a nice tractable definition which is (fibre-wise) equivalent to the more natural one.

Bakovic did a bunch of this work with Branislav Jurco, who was also there, and spoke about “Nonabelian Bundle 2-Gerbes“.  The paper behind that link has more details, which I’ve yet to entirely absorb, but the essential point appears to be to extend the description of “bundle gerbes” associated to crossed modules up to 2-crossed modules.  Bundles, with a structure-group , are classified by the cohomology with coefficients in ; and whereas “bundle-gerbes” with a structure-crossed-module can likewise be described by cohomology .  Notice this is a bit different from the description in terms of higher cohomology for a -gerbe, which can be understood as a bundle-gerbe using the shifted crossed module (when is abelian.  The goal here is to generalize this part to nonabelian groups, and also pass up to “bundle 2-gerbes” based on a 2-crossed module, or crossed complex of length 2, as I described previously for Joao Martins’ talk.  This would be classified in terms of cohomology valued in the 2-crossed module.  The point is that one can describe such a thing as a bundle over a fibre product, which (I think – I’m not so clear on this part) deals with the same structure of overlaps as the higher cohomology in the other way of describing things.

Finally,  a talk that’s a little harder to classify than most, but which I’ve put here with things somewhat related to string theory, was Alexander Kahle‘s on “T-Duality and Differential K-Theory”, based on work with Alessandro Valentino.  This uses the idea of the differential refinement of cohomology theories – in this case, K-theory, which is a generalized cohomology theory, which is to say that K-theory satisfies the Eilenberg-Steenrod axioms (with the dimension axiom relaxed, hence “generalized”).  Cohomology theories, including generalized ones, can have differential refinements, which pass from giving topological to geometrical information about a space.  So, while K-theory assigns to a space the Grothendieck ring of the category of vector bundles over it, the differential refinement of K-theory does the same with the category of vector bundles with connection.  This captures both local and global structures, which turns out to be necessary to describe fields in string theory – specifically, Ramond-Ramond fields.  The point of this talk was to describe what happens to these fields under T-duality.  This is a kind of duality in string theory between a theory with large strings and small strings.  The talk describes how this works, where we have a manifold with fibres at each point with fibres strings of radius and with radius .  There’s a correspondence space , which has projection maps down into the two situations.  Fields, being forms on such a fibration, can be “transferred” through this correspondence space by a “pull-back and push-forward” (with, in the middle, a wedge with a form that mixes the two directions, ).  But to be physically the right kind of field, these “forms” actually need to be representing cohomology classes in the differential refinement of K-theory.

Quantum Gravity etc.

Now, part of the point of this workshop was to try to build, or anyway maintain, some bridges between the kind of work in geometry and topology which I’ve been describing and the world of physics.  There are some particular versions of physical theories where these ideas have come up.  I’ve already touched on string theory along the way (there weren’t many talks about it from a physicist’s point of view), so this will mostly be about a different sort of approach.

Benjamin Bahr gave a talk outlining this approach for our mathematician-heavy audience, with his talk on “Spin Foam Operators” (see also for instance this paper).  The point is that one approach to quantum gravity has a theory whose “kinematics” (the description of the state of a system at a given time) is described by “spin networks” (based on gauge theory), as described back in the pre-school post.  These span a Hilbert space, so the “dynamical” issue of such models is how to get operators between Hilbert spaces from “foams” that interpolate between such networks – that is, what kind of extra data they might need, and how to assign amplitudes to faces and edges etc. to define an operator, which (assuming a “local” theory where distant parts of the foam affect the result independently) will be of the form:

where is a particular complex (foam), is a way of assigning irreps to faces of the foam, and is the assignment of intertwiners to edges.  Later on, one can take a discrete version of a path integral by summing over all these .  Here we have a product over faces and one over vertices, with an amplitude assigned (somehow – this is the issue) to faces.  The trace is over all the representation spaces assigned to the edges that are incident to a vertex (this is essentially the only consistent way to assign an amplitude to a vertex).  If we also consider spacetimes with boundary, we need some amplitudes at the boundary edges, as well.  A big part of the work with such models is finding such amplitudes that meet some nice conditions.

Some of these conditions are inherently necessary – to ensure the theory is invariant under gauge transformations, or (formally) changing orientations of faces.  Others are considered optional, though to me “functoriality” (that the way of deriving operators respects the gluing-together of foams) seems unavoidable – it imposes that the boundary amplitudes have to be found from the in one specific way.  Some other nice conditions might be: that depends only on the topology of (which demands that the operators be projections); that is invariant under subdivision of the foam (which implies the amplitudes have to be ).

Assuming all these means the only choice is exactly which sub-projection is of the projection onto the gauge-invariant part of the representation space for the faces attached to edge .  The rest of the talk discussed this, including some examples (models for BF-theory, the Barrett-Crane model and the more recent EPRL/FK model), and finished up by discussing issues about getting a nice continuum limit by way of “coarse graining”.

On a related subject, Bianca Dittrich spoke about “Dynamics and Diffeomorphism Symmetry in Discrete Quantum Gravity”, which explained the nature of some of the hard problems with this sort of discrete model of quantum gravity.  She began by asking what sort of models (i.e. which choices of amplitudes) in such discrete models would actually produce a nice continuum theory – since gravity, classically, is described in terms of spacetimes which are continua, and the quantum theory must look like this in some approximation.  The point is to think of these as “coarse-graining” of a very fine (perfect, in the limit) approximation to the continuum by a triangulation with a very short length-scale for the edges.  Coarse graining means discarding some of the edges to get a coarser approximation (perhaps repeatedly).  If the happens to be triangulation-independent, then coarse graining makes no difference to the result, nor does the converse process of refining the triangulation.  So one question is:  if we expect the continuum limit to be diffeomorphism invariant (as is General Relativity), what does this say at the discrete level?  The relation between diffeomorphism invariance and triangulation invariance has been described by Hendryk Pfeiffer, and in the reverse direction by Dittrich et al.

Actually constructing the dynamics for a system like this in a nice way (“canonical dynamics with anomaly-free constraints”) is still a big problem, which Bianca suggested might be approached by this coarse-graining idea.  Now, if a theory is topological (here we get the link to TQFT), such as electromagnetism in 2D, or (linearized) gravity in 3D, coarse graining doesn’t change much.  But otherwise, changing the length scale means changing the action for the continuum limit of the theory.  This is related to renormalization: one starts with a “naive” guess at a theory, then refines it (in this case, by the coarse-graining process), which changes the action for the theory, until arriving at (or approximating to) a fixed point.  Bianca showed an example, which produces a really huge, horrible action full of very complicated terms, which seems rather dissatisfying.  What’s more, she pointed out that, unless the theory is topological, this always produces an action which is non-local – unlike the “naive” discrete theory.  That is, the action can’t be described in terms of a bunch of non-interacting contributions from the field at individual points – instead, it’s some function which couples the field values at distant points (albeit in a way that falls off exponentially as the points get further apart).

In a more specific talk, Aleksandr Mikovic discussed “Finiteness and Semiclassical Limit of EPRL-FK Spin Foam Models”, looking at a particular example of such models which is the (relatively) new-and-improved candidate for quantum gravity mentioned above.  This was a somewhat technical talk, which I didn’t entirely follow, but  roughly, the way he went at this was through the techniques of perturbative QFT.  That is, by looking at the theory in terms of an “effective action”, instead of some path integral over histories with action – which looks like .  Starting with some classical history – a stationary point of the action – the effective action is an integral over small fluctuations around it of .

He commented more on the distinction between the question of triangulation independence (which is crucial for using spin foams to give invariants of manifolds) and the question of whether the theory gives a good quantum theory of gravity – that’s the “semiclassical limit” part.  (In light of the above, this seems to amount to asking if “diffeomorphism invariance” really extends through to the full theory, or is only approximately true, in the limiting case).  Then the “finiteness” part has to do with the question of getting decent asymptotic behaviour for some of those weights mentioned above so as to give a nice effective action (if not necessarily triangulation independence).  So, for instance, in the Ponzano-Regge model (which gives a nice invariant for manifolds), the vertex amplitudes are found by the 6j-symbols of representations.  The asymptotics of the 6j symbols then becomes an issue – Alekandr noted that to get a theory with a nice effective action, those 6j-symbols need to be scaled by a certain factor.  This breaks triangulation independence (hence means we don’t have a good manifold invariant), but gives a physically nicer theory.  In the case of 3D gravity, this is not what we want, but as he said, there isn’t a good a-priori reason to think it can’t give a good theory of 4D gravity.

Now, making a connection between these sorts of models and higher gauge theory, Aristide Baratin spoke about “2-Group Representations for State Sum Models”.  This is a project Baez, Freidel, and Wise, building on work by Crane and Sheppard (see my previous post, where Derek described the geometry of the representation theory for some 2-groups).  The idea is to construct state-sum models where, at the kinematical level, edges are labelled by 2-group representations, faces by intertwiners, and tetrahedra by 2-intertwiners.  (This assumes the foam is a triangulation – there’s a certain amount of back-and-forth in this area between this, and the Poincaré dual picture where we have 4-valent vertices).  He discussed this in a couple of related cases – the Euclidean and Poincaré 2-groups, which are described by crossed modules with base groups or respectively, acting on the abelian group (of automorphisms of the identity) in the obvious way.  Then the analogy of the 6j symbols above, which are assigned to tetrahedra (or dually, vertices in a foam interpolating two kinematical states), are now 10j symbols assigned to 4-simplexes (or dually, vertices in the foam).

One nice thing about this setup is that there’s a good geometric interpretation of the kinematics – irreducible representations of these 2-groups pick out orbits of the action of the relevant on .  These are “mass shells” – radii of spheres in the Euclidean case, or proper length/time values that pick out hyperboloids in the Lorentzian case of .  Assigning these to edges has an obvious geometric meaning (as a proper length of the edge), which thus has a continuous spectrum.  The areas and volumes interpreting the intertwiners and 2-intertwiners start to exhibit more of the discreteness you see in the usual formulation with representations of the groups themselves.  Finally, Aristide pointed out that this model originally arose not from an attempt to make a quantum gravity model, but from looking at Feynman diagrams in flat space (a sort of “quantum flat space” model), which is suggestively interesting, if not really conclusively proving anything.

Finally, Laurent Freidel gave a talk, “Classical Geometry of Spin Network States” which was a way of challenging the idea that these states are exclusively about “quantum geometries”, and tried to give an account of how to interpret them as discrete, but classical.  That is, the quantization of the classical phase space (the cotangent bundle of connections-mod-gauge) involves first a discretization to a spin-network phase space , and then a quantization to get a Hilbert space , and the hard part is the first step.  The point is to see what the classical phase space is, and he describes it as a (symplectic) quotient , which starts by assigning $T^*(SU(2))$ to each edge, then reduced by gauge transformations.  The puzzle is to interpret the states as geometries with some discrete aspect.

The answer is that one thinks of edges as describing (dual) faces, and vertices as describing some polytopes.  For each , there’s a -dimensional “shape space” of convex polytopes with -faces and a given fixed area .  This has a canonical symplectic structure, where lengths and interior angles at an edge are the canonically conjugate variables.  Then the whole phase space describes ways of building geometries by gluing these things (associated to vertices) together at the corresponding faces whenever the two vertices are joined by an edge.  Notice this is a bit strange, since there’s no particular reason the faces being glued will have the same shape: just the same area.  An area-1 pentagon and an area-1 square associated to the same edge could be glued just fine.  Then the classical geometry for one of these configurations is build of a bunch of flat polyhedra (i.e. with a flat metric and connection on them).  Measuring distance across a face in this geometry is a little strange.  Given two points inside adjacent cells, you measure orthogonal distance to the matched faces, and add in the distance between the points you arrive at (orthogonally) – assuming you glued the faces at the centre.  This is a rather ugly-seeming geometry, but it’s symplectically isomorphic to the phase space of spin network states – so it’s these classical geometries that spin-foam QG is a quantization of.  Maybe the ugliness should count against this model of quantum gravity – or maybe my aesthetic sense just needs work.

(Laurent also gave another talk, which was originally scheduled as one of the school talks, but ended up being a very interesting exposition of the principle of “Relativity of Localization”, which is hard to shoehorn into the themes I’ve used here, and was anyway interesting enough that I’ll devote a separate post to it.)

TQFT and Variations

I start here for no better reason than the personal one that it lets me put my talk first, so I’m on familiar ground to start with, for which reason also I’ll probably give more details here than later on.  So: a TQFT is a linear representation of the category of cobordisms – that is, a (symmetric monoidal) functor , in the notation I mentioned in the first school post.  An Extended TQFT is a higher functor , representing a category of cobordisms with corners into a higher category of k-Vector spaces (for some definition of same).  The essential point of my talk is that there’s a universal construction that can be used to build one of these at , which relies on some way of representing into , whose objects are groupoids, and whose morphisms in are pairs of groupoid homomorphisms .  The 2-morphisms have an analogous structure.  The point is that there’s a 2-functor which is takes representations of groupoids, at the level of objects; for morphisms, there is a “pull-push” operation that just uses the restricted and induced representation functors to move a representation across a span; the non-trivial (but still universal) bit is the 2-morphism map, which uses the fact that the restriction and induction functors are bi-ajdoint, so there are units and counits to use.  A construction using gauge theory gives groupoids of connections and gauge transformations for each manifold or cobordism.  This recovers a form of the Dijkgraaf-Witten model.  In principle, though, any way of getting a groupoid (really, a stack) associated to a space functorially will give an ETQFT this way.  I finished up by suggesting what would need to be done to extend this up to higher codimension.  To go to codimension 3, one would assign an object (codimension-3 manifold) a 3-vector space which is a representation 2-category of 2-groupoids of connections valued in 2-groups, and so on.  There are some theorems about representations of n-groupoids which would need to be proved to make this work.

The fact that different constructions can give groupoids for spaces was used by the next speaker, Thomas Nicklaus, whose talk described another construction that uses the I mentioned above.  This one produces “Equivariant Dijkgraaf-Witten Theory”.  The point is that one gets groupoids for spaces in a new way.  Before, we had, for a space a groupoid whose objects are -connections (or, put another way, bundles-with-connection) and whose morphisms are gauge transformations.  Now we suppose that there’s some group which acts weakly (i.e. an action defined up to isomorphism) on .  We think of this as describing “twisted bundles” over .  This is described by a quotient stack (which, as a groupoid, gets some extra isomorphisms showing where two objects are related by the -action).  So this gives a new map , and applying gives a TQFT.  The generating objects for the resulting 2-vector space are “twisted sectors” of the equivariant DW model.  There was some more to the talk, including a description of how the DW model can be further mutated using a cocycle in the group cohomology of , but I’ll let you look at the slides for that.

Next up was Jamie Vicary, who was talking about “(1,2,3)-TQFT”, which is another term for what I called “Extended” TQFT above, but specifying that the objects are 1-manifolds, the morphisms 2-manifolds, and the 2-morphisms are 3-manifolds.  He was talking about a theorem that identifies oriented TQFT’s of this sort with “anomaly-free modular tensor categories” – which is widely believed, but in fact harder than commonly thought.  It’s easy enough that such a TQFT corresponds to a MTC – it’s the category assigned to the circle.  What’s harder is showing that the TQFT’s are equivalent functors iff the categories are equivalent.  This boils down, historically, to the difficulty of showing the category is rigid.  Jamie was talking about a project with Bruce Bartlett and Chris Schommer-Pries, whose presentation of the cobordism category (described in the school post) was the basis of their proof.

Part of it amounts to giving a description of the TQFT in terms of certain string diagrams.  Jamie kindly credited me with describing this point of view to him: that the codimension-2 manifolds in a TQFT can be thought of as “boundaries in space” – codimension-1 manifolds are either time-evolving boundaries, or else slices of space in which the boundaries live; top-dimension cobordisms are then time-evolving slices of space-with-boundary.  (This should be only a heuristic way of thinking – certainly a generic TQFT has no literal notion of “time-evolution”, though in that (2+1) quantum gravity can be seen as a TQFT, there’s at least one case where this picture could be taken literally.)  Then part of their proof involves showing that the cobordisms can be characterized by taking vector spaces on the source and target manifolds spanned by the generating objects, and finding the functors assigned to cobordisms in terms of sums over all “string diagrams” (particle worldlines, if you like) bounded by the evolving boundaries.  Jamie described this as a “topological path integral”.  Then one has to describe the string diagram calculus – ridigidy follows from the “yanking” rule, for instance, and this follows from Morse theory as in Chris’ presentation of the cobordism category.

There was a little more discussion about what the various properties (proved in a similar way) imply.  One is “cloaking” – the fact that a 2-morphism which “creates a handle” is invisible to the string diagrams in the sense that it introduces a sum over all diagrams with a string “looped” around the new handle, but this sum gives a result that’s equal to the original map (in any “pivotal” tensor category, as here).

Chronologically before all these, one of the first talks on such a topic was by Rafael Diaz, on Homological Quantum Field Theory, or HLQFT for short, which is a rather different sort of construction.  Remember that Homotopy QFT, as described in my summary of Tim Porter’s school sessions, is about linear representations of what I’ll for now call , whose morphisms are -dimensional cobordisms equipped with maps into a space up to homotopy.  HLQFT instead considers cobordisms equipped with maps taken up to homology.

Specifically, there’s some space , say a manifold, with some distinguished submanifolds (possibly boundary components; possibly just embedded submanifolds; possibly even all of for a degenerate case).  Then we define to have objects which are -manifolds equipped with maps into which land on the distinguished submanifolds (to make composition work nicely, we in fact assume they map to a single point).  Morphisms in are trickier, and look like : a cobordism in this category is likewise equipped with a map from its boundary into which recovers the maps on its objects.  That is a homology class of maps from to , which agrees with .  This forms a monoidal category as with standard cobordisms.  Then HLQFT is about representations of this category.  One simple case Rafael described is the dimension-1 case, where objects are (ordered sets of) points equipped with maps that pick out chosen submanifolds of , and morphisms are just braids equipped with homology classes of “paths” joining up the source and target submanifolds.  Then a representation might, e.g., describe how to evolve a homology class on the starting manifold to one on the target by transporting along such a path-up-to-homology.  In higher dimensions, the evolution is naturally more complicated.

A slightly looser fit to this section is the talk by Thomas Krajewski, “Quasi-Quantum Groups from Strings” (see this) – he was talking about how certain algebraic structures arise from “string worldsheets”, which are another way to describe cobordisms.  This does somewhat resemble the way an algebraic structure (Frobenius algebra) is related to a 2D TQFT, but here the string worldsheets are interacting with 3-form field, (the curvature of that 2-form field of string theory) and things needn’t be topological, so the result is somewhat different.

Part of the point is that quantizing such a thing gives a higher version of what happens for quantizing a moving particle in a gauge field.  In the particle case, one comes up with a line bundle (of which sections form the Hilbert space) and in the string case one comes up with a gerbe; for the particle, this involves associated 2-cocycle, and for the string a 3-cocycle; for the particle, one ends up producing a twisted group algebra, and for the string, this is where one gets a “quasi-quantum group”.  The algebraic structures, as in the TQFT situation, come from, for instance, the “pants” cobordism which gives a multiplication and a comultiplication (by giving maps or the reverse, where is the object assigned to a circle).

There is some machinery along the way which I won’t describe in detail, except that it involves a tricomplex of forms – the gradings being form degree, the degree of a cocycle for group cohomology, and the number of overlaps.  As observed before, gerbes and their higher versions have transition functions on higher numbers of overlapping local neighborhoods than mere bundles.  (See the paper above for more)

Higher Gauge Theory

The talks I’ll summarize here touch on various aspects of higher-categorical connections or 2-groups (though at least one I’ll put off until later).  The division between this and the section on gerbes is a little arbitrary, since of course they’re deeply connected, but I’m making some judgements about emphasis or P.O.V. here.

Apart from giving lectures in the school sessions, John Huerta also spoke on “Higher Supergroups for String Theory”, which brings “super” (i.e. -graded) objects into higher gauge theory.  There are “super” versions of vector spaces and manifolds, which decompose into “even” and “odd” graded parts (a.k.a. “bosonic” and “fermionic” parts).  Thus there are “super” variants of Lie algebras and Lie groups, which are like the usual versions, except commutation properties have to take signs into account (e.g. a Lie superalgebra’s bracket is commutative if the product of the grades of two vectors is odd, anticommutative if it’s even).  Then there are Lie 2-algebras and 2-groups as well – categories internal to this setting.  The initial question has to do with whether one can integrate some Lie 2-algebra structures to Lie 2-group structures on a spacetime, which depends on the existence of some globally smooth cocycles.  The point is that when spacetime is of certain special dimensions, this can work, namely dimensions 3, 4, 6, and 10.  These are all 2 more than the real dimensions of the four real division algebras, , , and .  It’s in these dimensions that Lie 2-superalgebras can be integrated to Lie 2-supergroups.  The essential reason is that a certain cocycle condition will hold because of the properties of a form on the Clifford algebras that are associated to the division algebras.  (John has some related material here and here, though not about the 2-group case.)

Since we’re talking about higher versions of Lie groups/algebras, an important bunch of concepts to categorify are those in representation theory.  Derek Wise spoke on “2-Group Representations and Geometry”, based on work with Baez, Baratin and Freidel, most fully developed here, but summarized here.  The point is to describe the representation theory of Lie 2-groups, in particular geometrically.  They’re to be represented on (in general, infinite-dimensional) 2-vector spaces of some sort, which is chosen to be a category of measurable fields of Hilbert spaces on some measure space, which is called (intended to resemble, but not exactly be the same as, , the space of “functors into from the space , the way Kapranov-Voevodsky 2-vector spaces can be described as ).  The first work on this was by Crane and Sheppeard, and also Yetter.  One point is that for 2-groups, we have not only representations and intertwiners between them, but 2-intertwiners between these.  One can describe these geometrically – part of which is a choice of that measure space .

This done, we can say that a representation of a 2-group is a 2-functor , where is seen as a one-object 2-category.  Thinking about this geometrically, if we concretely describe by the crossed module , defines an action of on , and a map into the character group, which thereby becomes a -equivariant bundle.  One consequence of this description is that it becomes possible to distinguish not only irreducible representations (bundles over a single orbit) and indecomposible ones (where the fibres are particularly simple homogeneous spaces), but an intermediate notion called “irretractible” (though it’s not clear how much this provides).  An intertwining operator between reps over and can be described in terms of a bundle of Hilbert spaces – which is itself defined over the pullback of and seen as -bundles over .  A 2-intertwiner is a fibre-wise map between two such things.  This geometric picture specializes in various ways for particular examples of 2-groups.  A physically interesting one, which Crane and Sheppeard, and expanded on in that paper of [BBFW] up above, deals with the Poincaré 2-group, and where irreducible representations live over mass-shells in Minkowski space (or rather, the dual of ).

Moving on from 2-group stuff, there were a few talks related to 3-groups and 3-groupoids.  There are some new complexities that enter here, because while (weak) 2-categories are all (bi)equivalent to strict 2-categories (where things like associativity and the interchange law for composing 2-cells hold exactly), this isn’t true for 3-categories.  The best strictification result is that any 3-category is (tri)equivalent to a Gray category – where all those properties hold exactly, except for the interchange law for horizontal and vertical compositions of 2-cells, which is replaced by an “interchanger” isomorphism with some coherence properties.  John Barrett gave an introduction to this idea and spoke about “Diagrams for Gray Categories”, describing how to represent morphisms, 2-morphisms, and 3-morphisms in terms of higher versions of “string” diagrams involving (piecewise linear) surfaces satisfying some properties.  He also carefully explained how to reduce the dimensions in order to make them both clearer and easier to draw.  Bjorn Gohla spoke on “Mapping Spaces for Gray Categories”, but since it was essentially a shorter version of a talk I’ve already posted about, I’ll leave that for now, except to point out that it linked to the talk by Joao Faria Martins, “3D Holonomy” (though see also this paper with Roger Picken).

The point in Joao’s talk starts with the fact that we can describe holonomies for 3-connections on 3-bundles valued in Gray-groups (i.e. the maximally strict form of a general 3-group) in terms of Gray-functors .  Here, is the fundamental 3-groupoid of , which turns points, paths, homotopies of paths, and homotopies of homotopies into a Gray groupoid (modulo some technicalities about “thin” or “laminated”  homotopies) and is a gauge Gray-group.  Just as a 2-group can be represented by a crossed module, a Gray (3-)group can be represented by a “2-crossed module” (yes, the level shift in the terminology is occasionally confusing).  This is a chain of groups , where acts on the other groups, together with some structure maps (for instance, the Peiffer commutator for a crossed module becomes a lifting ) which all fit together nicely.  Then a tri-connection can be given locally by forms valued in the Lie algebras of these groups: in  .  Relating the global description in terms of and local description in terms of is a matter of integrating forms over paths, surfaces, or 3-volumes that give the various -morphisms of .  This sort of construction of parallel transport as functor has been developed in detail by Waldorf and Schreiber (viz. these slides, or the full paper), some time ago, which is why, thematically, they’re the next two speakers I’ll summarize.

Konrad Waldorf spoke about “Abelian Gauge Theories on Loop Spaces and their Regression”.  (For more, see two papers by Konrad on this)  The point here is that there is a relation between two kinds of theories – string theory (with -field) on a manifold , and ordinary gauge theory on its loop space .  The relation between them goes by the name “regression” (passing from gauge theory on to string theory on ), or “transgression”, going the other way.  This amounts to showing an equivalence of categories between [principal -bundles with connection on ] and [-gerbes with connection on ].  This nicely gives a way of seeing how gerbes “categorify” bundles, since passing to the loop space – whose points are maps means a holonomy functor is now looking at objects (points in ) which would be morphisms in the fundamental groupoid of , and morphisms which are paths of loops (surfaces in which trace out homotopies).  So things are shifted by one level.  Anyway, Konrad explained how this works in more detail, and how it should be interpreted as relating connections on loop space to the -field in string theory.

Urs Schreiber kicked the whole categorification program up a notch by talking about -Connections and their Chern-Simons Functionals .  So now we’re getting up into -categories, and particularly -toposes (see Jacob Lurie’s paper, or even book if so inclined to find out what these are), and in particular a “cohesive topos”, where derived geometry can be developed (Urs suggested people look here, where a bunch of background is collected). The point is that -topoi are good for talking about homotopy theory.  We want a setting which allows all that structure, but also allows us to do differential geometry and derived geometry.  So there’s a “cohesive” -topos called , of “sheaves” (in the -topos sense) of -groupoids on smooth manifolds.  This setting is the minimal common generalization of homotopy theory and differential geometry.

This is about a higher analog of this setup: since there’s a smooth classifying space (in fact, a Lie groupoid) for -bundles, , there’s also an equivalence between categories of -principal bundles, and (of functors into ).  Moreover, there’s a similar setup with for bundles with connection.  This can be described topologically, or there’s also a “differential refinement” to talk about the smooth situation.  This equivalence lives within a category of (smooth) sheaves of groupoids.  For higher gauge theory, we want a higher version as in described above.  Then we should get an equivalence – in this cohesive topos – of and a category of --gerbes.

Then the part about the  “Chern-Simons functionals” refers to the fact that CS theory for a manifold (which is a kind of TQFT) is built using an action functional that is found as an integral of the forms that describe some -connection over the manifold.  (Then one does a path-integral of this functional over all connections to find partition functions etc.)  So the idea is that for these higher -gerbes, whose classifying spaces we’ve just described, there should be corresponding functionals.  This is why, as Urs remarked in wrapping up, this whole picture has an explicit presentation in terms of forms.  Actually, in terms of Cech-cocycles (due to the fact we’re talking about gerbes), whose coefficients are taken in sheaves of complexes (this is the derived geometry part) of differential forms whose coefficients are in -algebroids (the -groupoid version of Lie algebras, since in general we’re talking about a theory with gauge -groupoids now).

Whew!  Okay, that’s enough for this post.  Next time, wrapping up blogging the workshop, finally.