As for last semester’s seminar, one of the two main threads, the one which Alessandro Valentino and I helped to organize, was a look at some of the material needed to approach Jacob Lurie’s paper on the classification of topological quantum field theories. The idea was for the research seminar to present the basic tools that are used in that paper to a larger audience, mostly of graduate students – enough to give a fairly precise statement, and develop the tools needed to follow the proof. (By the way, for a nice and lengthier discussion by Chris Schommer-Pries about this subject, which includes more details on much of what’s in this post, check out this video.)

So: the key result is a slightly generalized form of the Cobordism Hypothesis.

Cobordism Hypothesis

The sort of theory which the paper classifies are those which “extend down to a point”. So what does this mean? A topological field theory can be seen as a sort of “quantum field theory up to homotopy”, which abstract away any geometric information about the underlying space where the fields live – their local degrees of freedom.  We do this by looking only at the classes of fields up to the diffeomorphism symmetries of the space.  The local, geometric, information gets thrown away by taking this quotient of the space of solutions.

In spite of reducing the space of fields this way, we want to capture the intuition that the theory is still somehow “local”, in that we can cut up spaces into parts and make sense of the theory on those parts separately, and determine what it does on a larger space by gluing pieces together, rather than somehow having to take account of the entire space at once, indissolubly. This reasoning should apply to the highest-dimensional space, but also to boundaries, and to any figures we draw on boundaries when cutting them up in turn.

Carrying this on to the logical end point, this means that a topological quantum field theory in the fully extended sense should assign some sort of data to every geometric entity from a zero-dimensional point up to an -dimensional cobordism.  This is all expressed by saying it’s an -functor:

.

Well, once we know what this means, we’ll know (in principle) what a TQFT is.  It’s less important, for the purposes of Lurie’s paper, what is than what is.  The reason is that we want to classify these field theories (i.e. functors).  It will turn out that has the sort of structure that makes it easy to classify the functors out of it into any target -category .  A guess about what kind of structure is actually there was expressed by Baez and Dolan as the Cobordism Hypothesis.  It’s been slightly rephrased from the original form to get a form which has a proof.  The version Lurie proves says:

The -category is equivalent to the free symmetric monoidal -category generated by one fully-dualizable object.

The basic point is that, since is a free structure, the classification means that the extended TQFT’s amount precisely to the choice of a fully-dualizable object of (which includes a choice of a bunch of morphisms exhibiting the “dualizability”). However, to make sense of this, we need to have a suitable idea of an -category, and know what a fully dualizable object is. Let’s begin with the first.

-Categories

In one sense, the Cobordism Hypothesis, which was originally made about -categories at a time when these were only beginning to be defined, could be taken as a criterion for an acceptable definition. That is, it expressed an intuition which was important enough that any definition which wouldn’t allow one to prove the Cobordism Hypothesis in some form ought to be rejected. To really make it work, one had to bring in the “infinity” part of -categories. The point here is that we are talking about category-like structures which have morphisms between objects, 2-morphisms between morphisms, and so on, with -morphisms between -morphisms for every possible degree. The inspiration for this comes from homotopy theory, where one has maps, homotopies of maps, homotopies of homotopies, etc.

Nowadays, there are several possible concrete models for -categories (see this survey article by Julie Bergner for a summary of four of them). They are all equivalent definitions, in a suitable up-to-homotopy way, but for purposes of the proof, Lurie is taking the definition that an -category is an n-fold complete Segal space. One theme that shows up in all the definitions is that of simplicial methods. (In our seminar, we started with a series of two talks introducing the notions of simplicial sets, simplicial objects in a category, and Kan complexes. If you don’t already know this, essentially everything we need is nicely explained in here.)

One of the underlying ideas is that a category can be associated with a simplicial set, its nerve , where the set of -dimensional simplexes is just the set of composable -tuples of morphisms in . If is a groupoid (everything is invertible), then the simplicial set is a Kan complex – it satisfies some filling conditions, which ensure that any morphism has an inverse. Not every Kan complex is the nerve of a groupoid, but one can think of them as weak versions of groupoids – -groupoids, or -categories – where the higher morphisms may not be completely trivial (as with a groupoid), but where at least they’re all invertible. This leads to another desirable feature in any definition of -category, which is the Homotopy Hypothesis: that the -category of -categories, also called -groupoids, should be equivalent (in the same weak sense) to a category of Hausdorff spaces with some other nice properties, which we call for short. This is true of Kan complexes.

Thus, up to homotopy, specifying an -groupoid is the same as specifying a space.

The data which defines a Segal space (which was however first explicitly defined by Charlez Rezk) is a simplicial space : for each , there are spaces , thought of as the space of composable -tuples of morphisms. To keep things tame, we suppose that , the space of objects, is discrete – that is, we have only a set of objects. Being a simplicial space means that the come equipped with a collection of face maps , which we should think of as compositions: to get from an -tuple to an -tuple of morphisms, one can compose two morphisms together at any of positions in the tuple.

One condition which a simplicial space has to satisfy to be a Segal space has to do with the “weakening” which makes a Segal space a weaker notion than just a category lies in the fact that the cannot be arbitrary, but must be homotopy equivalent to the “actual” space of -tuples, which is a strict pullback . That is, in a Segal space, the pullback which defines these tuples for a category is weakened to be a homotopy pullback. Combining this with the various face maps, we therefore get a weakened notion of composition: . Because we start by replacing the space of -tuples with the homotopy-equivalent , the composition rule will only satisfy all the relations which define composition (associativity, for instance) up to homotopy.

To be complete, the Segal space must have a notion of equivalence for which agrees with that for Kan complexes seen as -groupoids. In particular, there is a sub-simplicial object , which we understand to consist of the spaces of invertible -morphisms. Since there should be nothing interesting happening above the top dimension, we ask that, for these spaces, the face and degeneracy maps are all homotopy equivalences: up to homotopy, the space of invertible higher morphisms has no new information.

Then, an -fold complete Segal space is defined recursively, just as one might define -categories (without the infinitely many layers of invertible morphisms “at the top”). In that case, we might say that a double category is just a category internal to : it has a category of objects, and a category of morphims, and the various maps and operations, such as composition, which make up the definition of a category are all defined as functors. That turns out to be the same as a structure with objects, horizontal and vertical morphisms, and square-shaped 2-cells. If we insist that the category of objects is discrete (i.e. really just a set, with no interesting morphisms), then the result amounts to a 2-category. Then we can define a 3-category to be a category internal to (whose 2-category of objects is discrete), and so on. This approach really defines an -fold category (see e.g. Chapter 5 of Cheng and Lauda to see a variation of this approach, due to Tamsamani and Simpson), but imposing the condition that the objects really amount to a set at each step gives exactly the usual intuition of a (strict!) -category.

This is exactly the approach we take with -fold complete Segal spaces, except that some degree of weakness is automatic. Since a C.S.S. is a simplicial object with some properties (we separately define objects of -tuples of morphisms for every , and all the various composition operations), the same recursive approach leads to a definition of an “-fold complete Segal space” as simply a simplicial object in -fold C.S.S.’s (with the same properties), such that the objects form a set. In principle, this gives a big class of “spaces of morphisms” one needs to define – one for every -fold product of simplexes of any dimension – but all those requirements that any space of objects “is just a set” (i.e. is homotopy-equivalent to a discrete set of points) simplifies things a bit.

Cobordism Category as -Category

So how should we think of cobordisms as forming an -category? There are a few stages in making a precise definition, but the basic idea is simple enough. One starts with manifolds and cobordisms embedded in some fixed finite-dimensional vector space , and then takes a limit over all . In each , the coordinates of the factor give ways of cutting the cobordism into pieces, and gluing them back together defines composition in a different direction. Now, this won’t actually produce a complete Segal space: one has to take a certain kind of completion. But the idea is intuitive enough.

We want to define an -fold C.S.S. of cobordisms (and cobordisms between cobordisms, and so on, up to -morphisms). To start with, think of the case : then the space of objects of consists of all embeddings of a -dimensional manifold into . The space of -simplexes (of -tuples of morphisms) consists of all ways of cutting up a -dimensional cobordism embedded in by choosing , where we think of the cobordism having been glued from two pieces, where at the slice , we have the object where the two pieces were composed. (One has to be careful to specify that the Morse function on the cobordisms, got by projection only , has its critical points away from the – the generic case – to make sure that the objects where gluing happens are actual manifolds.)

Now, what about the higher morphisms of the -category? The point is that one needs to have an -groupoid – that is, a space! – of morphisms between two cobordisms and . To make sense of this, we just take the space of diffeomorphisms – not just as a set of morphisms, but including its topology as well. The higher morphisms, therefore, can be thought of precisely as paths, homotopies, homotopies between homotopies, and so on, in these spaces. So the essential difference between the 1-category of cobordisms and the -category is that in the first case, morphisms are diffeomorphism classes of cobordisms, whereas in the latter, the higher morphisms are made precisely of the space of diffeomorphisms which we quotient out by in the first case.

Now, -categories, can have non-invertible morphisms between morphisms all the way up to dimension , after which everything is invertible. An -fold C.S.S. does this by taking the definition of a complete Segal space and copying it inside -fold C.S.S’s: that is, one has an -fold Complete Segal Space of -tuples of morphisms, for each , they form a simplicial object, and so forth.

Now, if we want to build an -category of cobordisms, the idea is the same, except that we have a simplicial object, in a category of simplicial objects, and so on. However, the way to define this is essentially similar. To specify an -fold C.S.S., we have to specify a whole collection of spaces associated to cobordisms equipped with embeddings into . In particular, for each tuple , we have the space of such embeddings, such that for each one has special points along the coordinate axis. These are the ways of breaking down a given cobordism into a composite of pieces. Again, one has to make sure that these critical points of the Morse functions defined by the projections onto these coordinate axes avoid these special which define the manifolds where gluing takes place. The composition maps which make these into a simplical object are quite natural – they just come by deleting special points.

Finally, we take a limit over all (to get around limits to embeddings due to the dimension of ). So we know (at least abstractly) what the -category of cobordisms should be. The cobordism hypothesis claims it is equivalent to one defined in a free, algebraically-flavoured way, namely as the free symmetric monoidal -category on a fully-dualizable object. (That object is “the point” – which, up to the kind of homotopically-flavoured equivalence that matters here, is the only object when our highest-dimensional cobordisms have dimension ).

Dualizability

So what does that mean, a “fully dualizable object”?

First, to get the idea, let’s think of the 1-dimensional example.  Instead of “-category”, we would like to just think of this as a statement about a category.  Then is the 1-category of framed bordisms. For a manifold (or cobordism, which is a manifold with boundary), a framing is a trivialization of the tangent bundle.  That is, it amounts to a choice of isomorphism at each point between the tangent space there and the corresponding .  So the objects of are collections of (signed) points, and the morphisms are equivalence classes of framed 1-dimensional cobordisms.  These amount to oriented 1-manifolds with boundary, where the points (objects) on the boundary are the source and target of the cobordism.

Now we want to classify what TQFT’s live on this category.  These are functors .  We have two generating objects, and , the two signed points.  A TQFT must assign these objects vector spaces, which we’ll call and .  Collections of points get assigned tensor products of all the corresponding vector spaces, since the functor is monoidal, so knowing these two vector spaces determines what does to all objects.

What does do to morphisms?  Well, some generating morphsims of interest are cups and caps: these are lines which connect a positive to a negative point, but thought of as cobordisms taking two points to the empty set, and vice versa.  That is, we have an evaluation:This statement is what is generalized to say that -dimensional TQFT’s are classified by “fully” dualizable objects.

and a coevaluation:

Now, since cobordisms are taken up to equivalence, which in particular includes topological deformations, we get a bunch of relations which these have to satisfy.  The essential one is the “zig-zag” identity, reflecting the fact that a bent line can be straightened out, and we have the same 1-morphism in .  This implies that:

is the same as the identity.  This in turn means that the evaluation and coevaluation maps define a nondegenerate pairing between and .  The fact that this exists means two things.  First, is the dual of : .  Second, this only makes sense if both and its dual are finite dimensional (since the evaluation will just be the trace map, which is not even defined on the identity if is infinite dimensional).

On the other hand, once we know, , this determines up to isomorphism, as well as the evaluation and coevaluation maps.  In fact, this turns out to be enough to specify entirely.  The classification then is: 1-D TQFT’s are classified by finite-dimensional vector spaces .  Crucially, what made finiteness important is the existence of the dual and the (co)evaluation maps which express the duality.

In an -category, to say that an object is “fully dualizable” means more that the object has a dual (which, itself, implies the existence of the morphisms and ). It also means that and have duals themselves – or rather, since we’re talking about morphisms, “adjoints”. This in turn implies the existence of 2-morphisms which are the unit and counit of the adjunctions (the defining properties are essentially the same as those for morphisms which define a dual). In fact, every time we get a morphism of degree less than in this process, “fully dualizable” means that it too must have a dual (i.e. an adjoint).

This does run out eventually, though, since we only require this goes up to dimension : the -morphisms which this forces to exist (quite a few) aren’t required to have duals. This is good, because if they were, since all the higher morphisms available are invertible, this would mean that the dual -morphisms would actually be weak inverses (that is, their composite is isomorphic to the identity)… But that would mean that the dual -morphisms which forced them to exist would also be weak inverses (their composite would be weakly isomorphic to the identity)… and so on! In fact, if the property of “having duals” didn’t stop, then everything would be weakly invertible: we’d actually have a (weak) -groupoid!

Classifying TQFT

So finally, the point of the Cobordism Hypothesis is that a (fully extended) TQFT is a functor out of this into some target -category . There are various options, but whatever we pick, the functor must assign something in to the point, say , and something to each of and , as well as all the higher morphisms which must exist. Then functoriality means that all these images have to again satisfy the properties which make a fully dualizable object. Furthermore, since is the free gadget with all these properties on the single object , this is exactly what it means that is a functor. Saying that is fully dualizable, by implication, includes all the choices of morphisms like etc. which show it as fully dualizable. (Conceivably one could make the same object fully dualizable in more than one way – these would be different functors).

So an extended -dimensional TQFT is exactly the choice of a fully dualizable object , for some -category . This object is “what the TQFT assigns to a point”, but if we understand the structure of the object as a fully dualizable object, then we know what the TQFT assigns to any other manifold of any dimension up to , the highest dimension in the theory. This is how this algebraic characterization of cobordisms helps to classify such theories.

Since I moved to Hamburg,   Alessandro Valentino and I have been organizing one series of seminar talks whose goal is to bring people (mostly graduate students, and some postdocs and others) up to speed on the tools used in Jacob Lurie’s big paper on the classification of TQFT and proof of the Cobordism Hypothesis.  This is part of the Forschungsseminar (“research seminar”) for the working groups of Christoph Schweigert, Ingo Runkel, and Christoph Wockel.  First, I gave one introducing myself and what I’ve done on Extended TQFT. In our main series We’ve had a series of four so far – two in which Alessandro outlined a sketch of what Lurie’s result is, and another two by Sebastian Novak and Marc Palm that started catching our audience up on the simplicial methods used in the theory of -categories which it uses.  Coming up in the New Year, Nathan Bowler and I will be talking about first -categories, and then -categories.   I’ll do a few posts summarizing the talks around then.

Some people in the group have done some work on quantum field theories with defects, in relation to which, there’s this workshop coming up here in February!  The idea here is that one could have two regions of space where different field theories apply, which are connected along a boundary. We might imagine these are theories which are different approximations to what’s going on physically, with a different approximation useful in each region.  Whatever the intuition, the regions will be labelled by some category, and boundaries between regions are labelled by functors between categories.  Where different boundary walls meet, one can have natural transformations.  There’s a whole theory of how a 3D TQFT can be associated to modular tensor categories, in sort of the same sense that a 2D TQFT is associated to a Frobenius algebra. This whole program is intimately connected with the idea of “extending” a given TQFT, in the sense that it deals with theories that have inputs which are spaces (or, in the case of defects, sub-spaces of given ones) of many different dimensions.  Lurie’s paper describing the n-dimensional cobordism category, is very much related to the input to a theory like this.

Brno Visit

This time, I’d like to mention something which I began working on with Roger Picken in Lisbon, and talked about for the first time in Brno, Czech Republic, where I was invited to visit at Masaryk University.  I was in Brno for a week or so, and on Thursday, December 13, I gave this talk, called “Higher Gauge Theory and 2-Group Actions”.  But first, some pictures!

This fellow was near the hotel I stayed in:

Since this sculpture is both faceless and hard at work on nonspecific manual labour, I assume he’s a Communist-era artwork, but I don’t really know for sure.

The Christmas market was on in Náměstí Svobody (Freedom Square) in the centre of town.  This four-headed dragon caught my eye:

On the way back from Brno to Hamburg, I met up with my wife to spend a couple of days in Prague.  Here’s the Christmas market in the Old Town Square of Prague:

Anyway, it was a good visit to the Czech Republic.  Now, about the talk!

Moduli Spaces in Higher Gauge Theory

The motivation which I tried to emphasize is to define a specific, concrete situation in which to explore the concept of “2-Symmetry”.  The situation is supposed to be, if not a realistic physical theory, then at least one which has enough physics-like features to give a good proof of concept argument that such higher symmetries should be meaningful in nature.  The idea is that Higher Gauge theory is a field theory which can be understood as one in which the possible (classical) fields on a space/spacetime manifold consist of maps from that space into some target space .  For the topological theory, they are actually just homotopy classes of maps.  This is somewhat related to Sigma models used in theoretical physics, and mathematically to Homotopy Quantum Field Theory, which considers these maps as geometric structure on a manifold.  An HQFT is a functor taking such structured manifolds and cobordisms into Hilbert spaces and linear maps.  In the paper Roger and I are working on, we don’t talk about this stage of the process: we’re just considering how higher-symmetry appears in the moduli spaces for fields of this kind, which we think of in terms of Higher Gauge Theory.

Ordinary topological gauge theory – the study of flat connections on -bundles for some Lie group , can be looked at this way.  The target space is the “classifying space” of the Lie group – homotopy classes of maps in are the same as groupoid homomorphisms in .  Specifically, the pair of functors and relating groupoids and topological spaces are adjoints.  Now, this deals with the situation where is a homotopy 1-type, which is to say that it has a fundamental groupoid , and no other interesting homotopy groups.  To deal with more general target spaces , one should really deal with infinity-groupoids, which can capture the whole homotopy type of – in particular, all its higher homotopy groups at once (and various relations between them).  What we’re talking about in this paper is exactly one step in that direction: we deal with 2-groupoids.

We can think of this in terms of maps into a target space which is a 2-type, with nontrivial fundamental groupoid , but also interesting second homotopy group (and nothing higher).  These fit together to make a 2-groupoid , which is a 2-group if is connected.  The idea is that is the classifying space of some 2-group , which plays the role of the Lie group in gauge theory.  It is the “gauge 2-group”.  Homotopy classes of maps into correspond to flat connections in this 2-group.

For practical purposes, we use the fact that there are several equivalent ways of describing 2-groups.  Two very directly equivalent ways to define them are as group objects internal to , or as categories internal to – which have a group of objects and a group of morphisms, and group homomorphisms that define source, target, composition, and so on.  This second way is fairly close to the equivalent formulation as crossed modules .  The definition is in the slides, but essentially the point is that is the group of objects, and with the action , one gets the semidirect product which is the group of morphisms.  The map makes it possible to speak of and acting on each other, and that these actions “look like conjugation” (the precise meaning of which is in the defining properties of the crossed module).

The reason for looking at the crossed-module formulation is that it then becomes fairly easy to understand the geometric nature of the fields we’re talking about.  In ordinary gauge theory, a connection can be described locally as a 1-form with values in , the Lie algebra of .  Integrating such forms along curves gives another way to describe the connection, in terms of a rule assigning to every curve a holonomy valued in which describes how to transport something (generally, a fibre of a bundle) along the curve.  It’s somewhat nontrivial to say how this relates to the classic definition of a connection on a bundle, which can be described locally on “patches” of the manifold via 1-forms together with gluing functions where patches overlap.  The resulting categories are equivalent, though.

In higher gauge theory, we take a similar view. There is a local view of “connections on gerbes“, described by forms and gluing functions (the main difference in higher gauge theory is that the gluing functions related to higher cohomology).  But we will take the equivalent point of view where the connection is described by -valued holonomies along paths, and -valued holonomies over surfaces, for a crossed module , which satisfy some flatness conditions.  These amount to 2-functors of 2-categories .

The moduli space of all such 2-connections is only part of the story.  2-functors are related by natural transformations, which are in turn related by “modifications”.  In gauge theory, the natural transformations are called “gauge transformations”, and though the term doesn’t seem to be in common use, the obvious term for the next layer would be “gauge modifications”. It is possible to assemble a 2-groupoid , whose space of objects is exactly the moduli space of 2-connections, and whose 1- and 2-morphisms are exactly these gauge transformations and modifications.  So the question is, what is the meaning of the extra information contained in the 2-groupoid which doesn’t appear in the moduli space itself?

Our claim is that this information expresses how the moduli space carries “higher symmetry”.

2-Group Actions and the Transformation Double Category

What would it mean to say that something exhibits “higher” symmetry? A rudimentary way to formalize the intuition of “symmetry” is to say that there is a group (of “symmetries”) which acts on some object. One could get more subtle, but this should be enough to begin with. We already noted that “higher” gauge theory uses 2-groups (and beyond into -groups) in the place of ordinary groups.  So in this context, the natural way to interpret it is by saying that there is an action of a 2-group on something.

Just as there are several equivalent ways to define a 2-group, there are different ways to say what it means for it to have an action on something.  One definition of a 2-group is to say that it’s a 2-category with one object and all morphisms and 2-morphisms invertible.  This definition makes it clear that a 2-group has to act on an object of some 2-category . For our purposes, just as we normally think of group actions on sets, we will focus on 2-group actions on categories, so that is the 2-category of interest. Then an action is just a map:

The unique object of – let’s call it , gets taken to some object .  This object is the thing being “acted on” by .  The existence of the action implies that there are automorphisms for every morphism in (which correspond to the elements of the group of the crossed module).  This would be enough to describe ordinary symmetry, but the higher symmetry is also expressed in the images of 2-morphisms , which we might call 2-symmetries relating 1-symmetries.

What we want to do in our paper, which the talk summarizes, is to show how this sort of 2-group action gives rise to a 2-groupoid (actually, just a 2-category when the being acted on is a general category).  Then we claim that the 2-groupoid of connections can be seen as one that shows up in exactly this way.  (In the following, I have to give some credit to Dany Majard for talking this out and helping to find a better formalism.)

To make sense of this, we use the fact that there is a diagrammatic way to describe the transformation groupoid associated to the action of a group on a set .  The set of morphisms is built as a pullback of the action map, .

This means that morphisms are pairs , thought of as going from to .  The rule for composing these is another pullback.  The diagram which shows how it’s done appears in the slides.  The whole construction ends up giving a cubical diagram in , whose top and bottom faces are mere commuting diagrams, and whose four other faces are all pullback squares.

To construct a 2-category from a 2-group action is similar. For now we assume that the 2-group action is strict (rather than being given by a weak 2-functor).  In this case, it’s enough to think of our 2-group not as a 2-category, but as a group-object in – the same way that a 1-group, as well as being a category, can be seen as a group object in .  The set of objects of this category is the group of morphisms of the 2-category, and the morphisms make up the group of 2-morphisms.  Being a group object is the same as having all the extra structure making up a 2-group.

To describe a strict action of such a on , we just reproduce in the diagram that defines an action in :

The fact that is an action just means this commutes. In principle, we could define a weak action, which would mean that this commutes up to isomorphism, but we won’t be looking at that here.

Constructing the same diagram which describes the structure of a transformation groupoid (p29 in the slides for the talk), we get a structure with a “category of objects” and a “category of morphisms”.  The construction in gives us directly a set of morphisms, while itself is the set of objects. Similarly, in , the category of objects is just , while the construction gives a category of morphisms.

The two together make a category internal to , which is to say a double category.  By analogy with , we call this double category .

We take as the category of objects, as the “horizontal category”, whose morphisms are the horizontal arrows of the double category. The category of morphisms of shows up by letting its objects be the vertical arrows of the double category, and its morphisms be the squares.  These look like this:

The vertical arrows are given by pairs of objects , and just like the transformation 1-groupoid, each corresponds to the fact that the action of takes to . Each square (morphism in the category of morphisms) is given by a pair of morphisms, one from (given by an element in ), and one from .

The horizontal arrow on the bottom of this square is:

The fact that these are equal is exactly the fact that is a natural transformation.

The double category turns out to have a very natural example which occurs in higher gauge theory.

Higher Symmetry of the Moduli Space

The point of the talk is to show how the 2-groupoid of connections, previously described as , can be seen as coming from a 2-group action on a category – the objects of this category being exactly the connections. In the slides above, for various reasons, we did this in a discretized setting – a manifold with a decomposition into cells. This is useful for writing things down explicitly, but not essential to the idea behind the 2-symmetry of the moduli space.

The point is that there is a category we call , whose objects are the connections: these assign -holonomies to edges of our discretization (in general, to paths), and -holonomies to 2D faces. (Without discretization, one would describe these in terms of -valued 1-forms and -valued 2-forms.)

The morphisms of are one type of “gauge transformation”: namely, those which assign -holonomies to edges. (Or: -valued 1-forms). They affect the edge holonomies of a connection just like a 2-morphism in .  Face holonomies are affected by the -value that comes from the boundary of the face.

What’s physically significant here is that both objects and morphisms of describe nonlocal geometric information.  They describe holonomies over edges and surfaces: not what happens at a point.  The “2-group of gauge transformations”, which we call , on the other hand, is purely about local transformations.  If is the vertex set of the discretized manifold, then : one copy of the gauge 2-group at each vertex.  (Keeping this finite dimensional and avoiding technical details was one main reason we chose to use a discretization.  In principle, one could also talk about the 2-group of -valued functions, whose objects and morphisms, thinking of it as a group object in , are functions valued in morphisms of .)

Now, the way acts on is essentially by conjugation: edge holonomies are affected by pre- and post-multiplication by the values at the two vertices on the edge – whether objects or morphisms of .  (Face holonomies are unaffected).  There are details about this in the slides, but the important thing is that this is a 2-group of purely local changes.  The objects of are gauge transformations of this other type.  In a continuous setting, they would be described by -valued functions.  The morphisms are gauge modifications, and could be described by -valued functions.

The main conceptual point here is that we have really distinguished between two kinds of gauge transformation, which are the horizontal and vertical arrows of the double category .  This expresses the 2-symmetry by moving some gauge transformations into the category of connections, and others into the 2-group which acts on it.  But physically, we would like to say that both are “gauge transformations”.  So one way to do this is to “collapse” the double category to a bicategory: just formally allow horizontal and vertical arrows to compose, so that there is only one kind of arrow.  Squares become 2-cells.

So then if we collapse the double category expressing our 2-symmetry relation this way, the result is exactly equivalent to the functor category way of describing connections.  (The morphisms will all be invertible because is a groupoid and is a 2-group).

I’m interested in this kind of geometrical example partly because it gives a good way to visualize something new happening here.  There appears to be some natural 2-symmetry on this space of fields, which is fairly easy to see geometrically, and distinguishes in a fundamental way between two types of gauge transformation.  This sort of phenomenon doesn’t occur in the world of – a set has no morphisms, after all, so the transformation groupoid for a group action on it is much simpler.

In broad terms, this means that 2-symmetry has qualitatively new features that familiar old 1-symmetry doesn’t have.  Higher categorical versions – -groups acting on -groupoids, as might show up in more complicated HQFT – will certainly be even more complicated.  The 2-categorical version is just the first non-trivial situation where this happens, so it gives a nice starting point to understand what’s new in higher symmetry that we didn’t already know.

The talk that Derek gave was based on a project of his and Steffen Gielen’s, which has taken written form in a few papers (two shorter ones, “Spontaneously broken Lorentz symmetry for Hamiltonian gravity“, “Linking Covariant and Canonical General Relativity via Local Observers“, and a new, longer one called “Lifting General Relativity to Observer Space“).

The key idea behind this project is the notion of “observer space”, which is exactly what it sounds like: a space of all observers in a given universe.  This is easiest to picture when one has a spacetime – a manifold with a Lorentzian metric, – to begin with.  Then an observer can be specified by choosing a particular point in spacetime, as well as a unit future-directed timelike vector .  This vector is a tangent to the observer’s worldline at .  The observer space is therefore a bundle over , the “future unit tangent bundle”.  However, using the notion of a “Cartan geometry”, one can give a general definition of observer space which makes sense even when there is no underlying .

The result is a surprising, relatively new physical intuition is that “spacetime” is a local and observer-dependent notion, which in some special cases can be extended so that all observers see the same spacetime.  This is somewhat related to the relativity of locality, which I’ve blogged about previously.  Geometrically, it is similar to the fact that a slicing of spacetime into space and time is not unique, and not respected by the full symmetries of the theory of Relativity, even for flat spacetime (much less for the case of General Relativity).  Similarly, we will see a notion of “observer space”, which can sometimes be turned into a bundle over an objective spacetime , but not in all cases.

So, how is this described mathematically?  In particular, what did I mean up there by saying that spacetime becomes observer-dependent?

Cartan Geometry

The answer uses Cartan geometry, which is a framework for differential geometry that is slightly broader than what is commonly used in physics.  Roughly, one can say “Cartan geometry is to Klein geometry as Riemannian geometry is to Euclidean geometry”.  The more familiar direction of generalization here is the fact that, like Riemannian geometry, Cartan is concerned with manifolds which have local models in terms of simple, “flat” geometries, but which have curvature, and fail to be homogeneous.  First let’s remember how Klein geometry works.

Klein’s Erlangen Program, carried out in the mid-19th-century, systematically brought abstract algebra, and specifically the theory of Lie groups, into geometry, by placing the idea of symmetry in the leading role.  It describes “homogeneous spaces”, which are geometries in which every point is indistinguishable from every other point.  This is expressed by the existence of a transitive action of some Lie group of all symmetries on an underlying space.  Any given point will be fixed by some symmetries, and not others, so one also has a subgroup .  This is the “stabilizer subgroup”, consisting of all symmetries which fix .  That the space is homogeneous means that for any two points , the subgroups and are conjugate (by a symmetry taking to ).  Then the homogeneous space, or Klein geometry, associated to is, up to isomorphism, just the same as the quotient space of the obvious action of on .

The advantage of this program is that it has a great many examples, but the most relevant ones for now are:

• -dimensional Euclidean space. the Euclidean group is precisely the group of transformations that leave the data of Euclidean geometry, lengths and angles, invariant.  It acts transitively on .  Any point will be fixed by the group of rotations centred at that point, which is a subgroup of isomorphic to .  Klein’s insight is to reverse this: we may define Euclidean space by .
• -dimensional Minkowski space.  Similarly, we can define this space to be .  The Euclidean group has been replaced by the Poincaré group, and rotations by the Lorentz group (of rotations and boosts), but otherwise the situation is essentially the same.
• de Sitter space.  As a Klein geometry, this is the quotient .  That is, the stabilizer of any point is the Lorentz group – so things look locally rather similar to Minkowski space around any given point.  But the global symmetries of de Sitter space are different.  Even more, it looks like Minkowski space locally in the sense that the Lie algebras give representations and are identical, seen as representations of .  It’s natural to identify them with the tangent space at a point.  de Sitter space as a whole is easiest to visualize as a 4D hyperboloid in .  This is supposed to be seen as a local model of spacetime in a theory in which there is a cosmological constant that gives empty space a constant negative curvature.
• anti-de Sitter space. This is similar, but now the quotient is – in fact, this whole theory goes through for any of the last three examples: Minkowski; de Sitter; and anti-de Sitter, each of which acts as a “local model” for spacetime in General Relativity with the cosmological constant, respectively: zero; positive; and negative.

Now, what does it mean to say that a Cartan geometry has a local model?  Well, just as a Lorentzian or Riemannian manifold is “locally modelled” by Minkowski or Euclidean space, a Cartan geometry is locally modelled by some Klein geometry.  This is best described in terms of a connection on a principal -bundle, and the associated -bundle, over some manifold .  The crucial bundle in a Riemannian or Lorenztian geometry is the frame bundle: the fibre over each point consists of all the ways to isometrically embed a standard Euclidean or Minkowski space into the tangent space.  A connection on this bundle specifies how this embedding should transform as one moves along a path.  It’s determined by a 1-form on , valued in the Lie algebra of .

Given a parametrized path, one can apply this form to the tangent vector at each point, and get a Lie algebra-valued answer.  Integrating along the path, we get a path in the Lie group (which is independent of the parametrization).  This is called a “development” of the path, and by applying the -values to the model space , we see that the connection tells us how to move through a copy of as we move along the path.  The image this suggests is of “rolling without slipping” – think of the case where the model space is a sphere.  The connection describes how the model space “rolls” over the surface of the manifold .  Curvature of the connection measures the failure to commute of the processes of rolling in two different directions.  A connection with zero curvature describes a space which (locally at least) looks exactly like the model space: picture a sphere rolling against its mirror image.  Transporting the sphere-shaped fibre around any closed curve always brings it back to its starting position. Now, curvature is defined in terms of transports of these Klein-geometry fibres.  If curvature is measured by the development of curves, we can think of each homogeneous space as a flat Cartan geometry with itself as a local model.

This idea, that the curvature of a manifold depends on the model geometry being used to measure it, shows up in the way we apply this geometry to physics.

Gravity and Cartan Geometry

MacDowell-Mansouri gravity can be understood as a theory in which General Relativity is modelled by a Cartan geometry.  Of course, a standard way of presenting GR is in terms of the geometry of a Lorentzian manifold.  In the Palatini formalism, the basic fields are a connection and a vierbein (coframe field) called , with dynamics encoded in the Palatini action, which is the integral over of , where is the curvature 2-form for .

This can be derived from a Cartan geometry, whose model geometry is de Sitter space .   Then MacDowell-Mansouri gravity gets and by splitting the Lie algebra as .  This “breaks the full symmetry” at each point.  Then one has a fairly natural action on the -connection:

Here, is the part of the curvature of the big connection.  The splitting of the connection means that , and the action above is rewritten, up to a normalization, as the Palatini action for General Relativity (plus a topological term, which has no effect on the equations of motion we get from the action).  So General Relativity can be written as the theory of a Cartan geometry modelled on de Sitter space.

The cosmological constant in GR shows up because a “flat” connection for a Cartan geometry based on de Sitter space will look (if measured by Minkowski space) as if it has constant curvature which is exactly that of the model Klein geometry.  The way to think of this is to take the fibre bundle of homogeneous model spaces as a replacement for the tangent bundle to the manifold.  The fibre at each point describes the local appearance of spacetime.  If empty spacetime is flat, this local model is Minkowski space, , and one can really speak of tangent “vectors”.  The tangent homogeneous space is not linear.  In these first cases, the fibres are not vector spaces, precisely because the large group of symmetries doesn’t contain a group of translations, but they are Klein geometries constructed in just the same way as Minkowski space. Thus, the local description of the connection in terms of -valued forms can be treated in the same way, regardless of which Klein geometry occurs in the fibres.  In particular, General Relativity, formulated in terms of Cartan geometry, always says that, in the absence of matter, the geometry of space is flat, and the cosmological constant is included naturally by the choice of which Klein geometry is the local model of spacetime.

Observer Space

The idea in defining an observer space is to combine two symmetry reductions into one.  The reduction from to gives de Sitter space, as a model Klein geometry, which reflects the “symmetry breaking” that happens when choosing one particular point in spacetime, or event.  Then, the reduction of to similarly reflects the symmetry breaking that occurs when one chooses a specific time direction (a future-directed unit timelike vector).  These are the tangent vectors to the worldline of an observer at the chosen point, so the model Klein geometry, is the space of such possible observers.  The stabilizer subgroup for a point in this space consists of just the rotations of space around the corresponding observer – the boosts in translate between observers.  So locally, choosing an observer amounts to a splitting of the model spacetime at the point into a product of space and time. If we combine both reductions at once, we get the 7-dimensional Klein geometry .  This is just the future unit tangent bundle of de Sitter space, which we think of as a homogeneous model for the “space of observers”

A general observer space , however, is just a Cartan geometry modelled on .  This is a 7-dimensional manifold, equipped with the structure of a Cartan geometry.  One class of examples are exactly the future unit tangent bundles to 4-dimensional Lorentzian spacetimes.  In these cases, observer space is naturally a contact manifold: that is, it’s an odd-dimensional manifold equipped with a 1-form , the contact form, which is such that the top-dimensional form is nowhere zero.  This is the odd-dimensional analog of a symplectic manifold.  Contact manifolds are, intuitively, configuration spaces of systems which involve “rolling without slipping” – for instance, a sphere rolling on a plane.  In this case, it’s better to think of the local space of observers which “rolls without slipping” on a spacetime manifold .

Now, Minkowski space has a slicing into space and time – in fact, one for each observer, who defines the time direction, but the time coordinate does not transform in any meaningful way under the symmetries of the theory, and different observers will choose different ones.  In just the same way, the homogeneous model of observer space can naturally be written as a bundle .  But a general observer space may or may not be a bundle over an ordinary spacetime manifold, .  Every Cartan geometry gives rise to an observer space as the bundle of future-directed timelike vectors, but not every Cartan geometry is of this form, in any natural way. Indeed, without a further condition, we can’t even reconstruct observer space as such a bundle in an open neighborhood of a given observer.

This may be intuitively surprising: it gives a perfectly concrete geometric model in which “spacetime” is relative and observer-dependent, and perhaps only locally meaningful, in just the same way as the distinction between “space” and “time” in General Relativity. It may be impossible, that is, to determine objectively whether two observers are located at the same base event or not. This is a kind of “Relativity of Locality” which is geometrically much like the by-now more familiar Relativity of Simultaneity. Each observer will reach certain conclusions as to which observers share the same base event, but different observers may not agree.  The coincident observers according to a given observer are those reached by a good class of geodesics in moving only in directions that observer sees as boosts.

When one can reconstruct , two observers will agree whether or not they are coincident.  This extra condition which makes this possible is an integrability constraint on the action of the Lie algebra (in our main example, ) on the observer space .  In this case, the fibres of the bundle are the orbits of this action, and we have the familiar world of Relativity, where simultaneity may be relative, but locality is absolute.

Lifting Gravity to Observer Space

Apart from describing this model of relative spacetime, another motivation for describing observer space is that one can formulate canonical (Hamiltonian) GR locally near each point in such an observer space.  The goal is to make a link between covariant and canonical quantization of gravity.  Covariant quantization treats the geometry of spacetime all at once, by means of a Lagrangian action functional.  This is mathematically appealing, since it respects the symmetry of General Relativity, namely its diffeomorphism-invariance.  On the other hand, it is remote from the canonical (Hamiltonian) approach to quantization of physical systems, in which the concept of time is fundamental. In the canonical approach, one gets a Hilbert space by quantizing the space of states of a system at a given point in time, and the Hamiltonian for the theory describes its evolution.  This is problematic for diffeomorphism-, or even Lorentz-invariance, since coordinate time depends on a choice of observer.  The point of observer space is that we consider all these choices at once.  Describing GR in is both covariant, and based on (local) choices of time direction.

This is easiest to describe in the case of a bundle .  Then a “field of observers” to be a section of the bundle: a choice, at each base event in , of an observer based at that event.  A field of observers may or may not correspond to a particular decomposition of spacetime into space evolving in time, but locally, at each point in , it always looks like one.  The resulting theory describes the dynamics of space-geometry over time, as seen locally by a given observer.  In this case, a Cartan connection on observer space is described by to a -valued form.  This decomposes into four Lie-algebra valued forms, interpreted as infinitesimal transformations of the model observer by: (1) spatial rotations; (2) boosts; (3) spatial translations; (4) time translation.  The four-fold division is based on two distinctions: first, between the base event at which the observer lives, and the choice of observer (i.e. the reduction of to , which symmetry breaking entails choosing a point); and second, between space and time (i.e. the reduction of to , which symmetry breaking entails choosing a time direction).

This splitting, along the same lines as the one in MacDowell-Mansouri gravity described above, suggests that one could lift GR to a theory on an observer space .  This amount to describing fields on and an action functional, so that the splitting of the fields gives back the usual fields of GR on spacetime, and the action gives back the usual action.  This part of the project is still under development, but this lifting has been described.  In the case when there is no “objective” spacetime, the result includes some surprising new fields which it’s not clear how to deal with, but when there is an objective spacetime, the resulting theory looks just like GR.

Higher Structures in China III

So last year I had a bunch of discussions I had with Chenchang Zhu and Weiwei Pan, who at the time were both in Göttingen, about my work with Jamie Vicary, which I wrote about last time when the paper was posted to the arXiv.  In that, we showed how the Baez-Dolan groupoidification of the Heisenberg algebra can be seen as a representation of Khovanov’s categorification.  Chenchang and Weiwei and I had been talking about how these ideas might extend to other examples, in particular to give nice groupoidifications of categorified Lie algebras and quantum groups.

That is still under development, but I was invited to give a couple of talks on the subject at the workshop.  It was a long trip: from Lisbon, the farthest-west of the main cities of (continental) Eurasia all the way to one of the furthest-East.   (Not quite the furthest, but Changchun is in the northeast of China, just a few hours north of Korea, and it took just about exactly 24 hours including stopovers to get there).  It was a long way to go for a three day workshop, but as there were also three days of a big excursion to Changbai Mountain, just on the border with North Korea, for hiking and general touring around.  So that was a sort of holiday, with 11 other mathematicians.  Here is me with Dany Majard, in a national park along the way to the mountains:

Here’s me with Alex Hoffnung, on Changbai Mountain (in the background is China):

And finally, here’s me a little to the left of the previous picture, where you can see into the volcanic crater.  The lake at the bottom is cut out of the picture, but you can see the crater rim, of which this particular part is in North Korea, as seen from China:

Well, that was fun!

Anyway, the format of the workshop involved some talks from foreigners and some from locals, with a fairly big local audience including a good many graduate students from Jilin University.  So they got a chance to see some new work being done elsewhere – mostly in categorification of one kind or another.  We got a chance to see a little of what’s being done in China, although not as much as we might have. I gather that not much is being done yet that fit the theme of the workshop, which was part of the reason to organize the workshop, and especially for having a session aimed specially at the graduate students.

Categorified Algebra

This is a sort of broad term, but certainly would include my own talk.  The essential point is to show how the groupoidification of the Heisenberg algebra is a representation of Khovanov’s categorification of the same algebra, in a particular 2-category.  The emphasis here is on the fact that it’s a representation in a 2-category whose objects are groupoids, but whose morphisms aren’t just functors, but spans of functors – that is, composites of functors and co-functors.  This is a pretty conservative weakening of “representations on categories” – but it lets one build really simple combinatorial examples.  I’ve discussed this general subject in recent posts, so I won’t elaborate too much.  The lecture notes are here, if you like, though – they have more detail than my previous post, but are less technical than the paper with Jamie Vicary.

Aaron Lauda gave a nice introduction to the program of categorifying quantum groups, mainly through the example of the special case , somewhat along the same lines as in his introductory paper on the subject.  The story which gives the motivation is nice: one has knot invariants such as the Jones polynomial, based on representations of groups and quantum groups.  The Jones polynomial can be categorified to give Khovanov homology (which assigns a complex to a knot, whose graded Euler characteristic is the Jones polynomial) – but also assigns maps of complexes to cobordisms of knots.  One then wants to categorify the representation theory behind it – to describe actions of, for instance, quantum on categories.  This starting point is nice, because it can work by just mimicking the construction of and representations in terms of weight spaces: one gets categories which correspond to the “weight spaces” (usually just vector spaces), and the and operators give functors between them, and so forth.

Finding examples of categories and functors with this structure, and satisfying the right relations, gives “categorified representations” of the algebra – the monoidal categories of diagrams which are the “categorifications of the algebra” then are seen as the abstraction of exactly which relations these are supposed to satisfy.  One such example involves flag varieties.  A flag, as one might eventually guess from the name, is a nested collection of subspaces in some -dimensional space.  A simple example is the Grassmannian , which is the space of all 1-dimensional subspaces of (i.e. the projective space ), which is of course an algebraic variety.  Likewise, , the space of all -dimensional subspaces of is a variety.  The flag variety consists of all pairs , of a -dimensional subspace of , inside a -dimensional subspace (the case calls to mind the reason for the name: a plane intersecting a given line resembles a flag stuck to a flagpole).  This collection is again a variety.  One can go all the way up to the variety of “complete flags”, (where is -dimenisonal), any point of which picks out a subspace of each dimension, each inside the next.

The way this relates to representations is by way of geometric representation theory. One can see those flag varieties of the form as relating the Grassmanians: there are projections and , which act by just ignoring one or the other of the two subspaces of a flag.  This pair of maps, by way of pulling-back and pushing-forward functions, gives maps between the cohomology rings of these spaces.  So one gets a sequence , and maps between the adjacent ones.  This becomes a representation of the Lie algebra.  Categorifying this, one replaces the cohomology rings with derived categories of sheaves on the flag varieties – then the same sort of “pull-push” operation through (derived categories of sheaves on) the flag varieties defines functors between those categories.  So one gets a categorified representation.

Heather Russell‘s talk, based on this paper with Aaron Lauda, built on the idea that categorified algebras were motivated by Khovanov homology.  The point is that there are really two different kinds of Khovanov homology – the usual kind, and an Odd Khovanov Homology, which is mainly different in that the role played in Khovanov homology by a symmetric algebra is instead played by an exterior (antisymmetric) algebra.  The two look the same over a field of characteristic 2, but otherwise different.  The idea is then that there should be “odd” versions of various structures that show up in the categorifications of (and other algebras) mentioned above.

One example is the fact that, in the “even” form of those categorifications, there is a natural action of the Nil Hecke algebra on composites of the generators.  This is an algebra which can be seen to act on the space of polynomials in commuting variables, , generated by the multiplication operators , and the “divided difference operators” based on the swapping of two adjacent variables.  The Hecke algebra is defined in terms of “swap” generators, which satisfy some -deformed variation of the relations that define the symmetric group (and hence its group algebra).   The Nil Hecke algebra is so called since the “swap” (i.e. the divided difference) is nilpotent: the square of the swap is zero.  The way this acts on the objects of the diagrammatic category is reflected by morphisms drawn as crossings of strands, which are then formally forced to satisfy the relations of the Nil Hecke algebra.

The ODD Nil Hecke algebra, on the other hand, is an analogue of this, but the are anti-commuting, and one has different relations satisfied by the generators (they differ by a sign, because of the anti-commutation).  This sort of “oddification” is then supposed to happen all over.  The main point of the talk was to to describe the “odd” version of the categorified representation defined using flag varieties.  Then the odd Nil Hecke algebra acts on that, analogously to the even case above.

Marco Mackaay gave a couple of talks about the web algebra, describing the results of this paper with Weiwei Pan and Daniel Tubbenhauer.  This is the analog of the above, for , describing a diagram calculus which accounts for representations of the quantum group.  The “web algebra” was introduced by Greg Kuperberg – it’s an algebra built from diagrams which can now include some trivalent vertices, along with rules imposing relations on these.  When categorifying, one gets a calculus of “foams” between such diagrams.  Since this is obviously fairly diagram-heavy, I won’t try here to reproduce what’s in the paper – but an important part of is the correspondence between webs and Young Tableaux, since these are labels in the representation theory of the quantum group – so there is some interesting combinatorics here as well.

Algebraic Structures

Some of the talks were about structures in algebra in a more conventional sense.

Jiang-Hua Lu: On a class of iterated Poisson polynomial algebras.  The starting point of this talk was to look at Poisson brackets on certain spaces and see that they can be found in terms of “semiclassical limits” of some associative product.  That is, the associative product of two elements gives a power series in some parameter (which one should think of as something like Planck’s constant in a quantum setting).  The “classical” limit is the constant term of the power series, and the “semiclassical” limit is the first-order term.  This gives a Poisson bracket (or rather, the commutator of the associative product does).  In the examples, the spaces where these things are defined are all spaces of polynomials (which makes a lot of explicit computer-driven calculations more convenient). The talk gives a way of constructing a big class of Poisson brackets (having some nice properties: they are “iterated Poisson brackets”) coming from quantum groups as semiclassical limits.  The construction uses words in the generating reflections for the Weyl group of a Lie group .

Li Guo: Successors and Duplicators of Operads – first described a whole range of different algebra-like structures which have come up in various settings, from physics and dynamical systems, through quantum field theory, to Hopf algebras, combinatorics, and so on.  Each of them is some sort of set (or vector space, etc.) with some number of operations satisfying some conditions – in some cases, lots of operations, and even more conditions.  In the slides you can find several examples – pre-Lie and post-Lie algebras, dendriform algebras, quadri- and octo-algebras, etc. etc.  Taken as a big pile of definitions of complicated structures, this seems like a terrible mess.  The point of the talk is to point out that it’s less messy than it appears: first, each definition of an algebra-like structure comes from an operad, which is a formal way of summing up a collection of operations with various “arities” (number of inputs), and relations that have to hold.  The second point is that there are some operations, “successor” and “duplicator”, which take one operad and give another, and that many of these complicated structures can be generated from simple structures by just these two operations.  The “successor” operation for an operad introduces a new product related to old ones – for example, the way one can get a Lie bracket from an associative product by taking the commutator.  The “duplicator” operation takes existing products and introduces two new products, whose sum is the previous one, and which satisfy various nice relations.  Combining these two operations in various ways to various starting points yields up a plethora of apparently complicated structures.

Dany Majard gave a talk about algebraic structures which are related to double groupoids, namely double categories where all the morphisms are invertible.  The first part just defined double categories: graphically, one has horizontal and vertical 1-morphisms, and square 2-morphsims, which compose in both directions.  Then there are several special degenerate cases, in the same way that categories have as degenerate cases (a) sets, seen as categories with only identity morphisms, and (b) monoids, seen as one-object categories.  Double categories have ordinary categories (and hence monoids and sets) as degenerate cases.  Other degenerate cases are 2-categories (horizontal and vertical morphisms are the same thing), and therefore their own special cases, monoidal categories and symmetric monoids.  There is also the special degenerate case of a double monoid (and the extra-special case of a double group).  (The slides have nice pictures showing how they’re all degenerate cases).  Dany then talked about some structure of double group(oids) – and gave a list of properties for double groupoids, (such as being “slim” – having at most one 2-cell per boundary configuration – as well as two others) which ensure that they’re equivalent to the semidirect product of an abelian group with the “bicrossed product”  of two groups and (each of which has to act on the other for this to make sense).  He gave the example of the Poincare double group, which breaks down as a triple bicrossed product by the Iwasawa decomposition:

( is certain group of matrices).  So there’s a unique double group which corresponds to it – it has squares labelled by , and the horizontial and vertical morphisms by elements of and respectively.  Dany finished by explaining that there are higher-dimensional analogs of all this – -tuple categories can be defined recursively by internalization (“internal categories in -tuple-Cat”).  There are somewhat more sophisticated versions of the same kind of structure, and finally leading up to a special class of -tuple groups.  The analogous theorem says that a special class of them is just the same as the semidirect product of an abelian group with an -fold iterated bicrossed product of groups.

Also in this category, Alex Hoffnung talked about deformation of formal group laws (based on this paper with various collaborators).  FGL’s are are structures with an algebraic operation which satisfies axioms similar to a group, but which can be expressed in terms of power series.  (So, in particular they have an underlying ring, for this to make sense).  In particular, the talk was about formal group algebras – essentially, parametrized deformations of group algebras – and in particular for Hecke Algebras.  Unfortunately, my notes on this talk are mangled, so I’ll just refer to the paper.

Physics

I’m using the subject-header “physics” to refer to those talks which are most directly inspired by physical ideas, though in fact the talks themselves were mathematical in nature.

Fei Han gave a series of overview talks intorducing “Equivariant Cohomology via Gauged Supersymmetric Field Theory”, explaining the Stolz-Teichner program.  There is more, using tools from differential geometry and cohomology to dig into these theories, but for now a summary will do.  Essentially, the point is that one can look at “fields” as sections of various bundles on manifolds, and these fields are related to cohomology theories.  For instance, the usual cohomology of a space is a quotient of the space of closed forms (so the cohomology, , is a quotient of the space of closed -forms – the quotient being that forms differing by a coboundary are considered the same).  There’s a similar construction for the -theory , which can be modelled as a quotient of the space of vector bundles over .  Fei Han mentioned topological modular forms, modelled by a quotient of the space of “Fredholm bundles” – bundles of Banach spaces with a Fredholm operator around.

The first two of these examples are known to be related to certain supersymmetric topological quantum field theories.  Now, a TFT is a functor into some kind of vector spaces from a category of -dimensional manifolds and -dimensional cobordisms

Intuitively, it gives a vector space of possible fields on the given space and a linear map on a given spacetime.  A supersymmetric field theory is likewise a functor, but one changes the category of “spacetimes” to have both bosonic and fermionic dimension.  A normal smooth manifold is a ringed space , since it comes equipped with a sheaf of rings (each open set has an associated ring of smooth functions, and these glue together nicely).  Supersymmetric theories work with manifolds which change this sheaf – so a -dimensional space has the sheaf of rings where one introduces some new antisymmetric coordinate functions , the “fermionic dimensions”:

Then a supersymmetric TFT is a functor:

(where is the category of supersymmetric topological vector spaces – defined similarly).  The connection to cohomology theories is that the classes of such field theories, up to a notion of equivalence called “concordance”, are classified by various cohomology theories.  Ordinary cohomology corresponds then to -dimensional extended TFT (that is, with 0 bosonic and 1 fermionic dimension), and -theory to a -dimensional extended TFT.  The Stoltz-Teichner Conjecture is that the third example (topological modular forms) is related in the same way to a -dimensional extended TFT – so these are the start of a series of cohomology theories related to various-dimension TFT’s.

Last but not least, Chris Rogers spoke about his ideas on “Higher Geometric Quantization”, on which he’s written a number of papers.  This is intended as a sort of categorification of the usual ways of quantizing symplectic manifolds.  I am still trying to catch up on some of the geometry This is rooted in some ideas that have been discussed by Brylinski, for example.  Roughly, the message here is that “categorification” of a space can be thought of as a way of acting on the loop space of a space.  The point is that, if points in a space are objects and paths are morphisms, then a loop space shifts things by one categorical level: its points are loops in , and its paths are therefore certain 2-morphisms of .  In particular, there is a parallel to the fact that a bundle with connection on a loop space can be thought of as a gerbe on the base space.  Intuitively, one can “parallel transport” things along a path in the loop space, which is a surface given by a path of loops in the original space.  The local description of this situation says that a 1-form (which can give transport along a curve, by integration) on the loop space is associated with a 2-form (giving transport along a surface) on the original space.

Then the idea is that geometric quantization of loop spaces is a sort of higher version of quantization of the original space. This “higher” version is associated with a form of higher degree than the symplectic (2-)form used in geometric quantization of .   The general notion of n-plectic geometry, where the usual symplectic geometry is the case , involves a -form analogous to the usual symplectic form.  Now, there’s a lot more to say here than I properly understand, much less can summarize in a couple of paragraphs.  But the main theorem of the talk gives a relation between n-plectic manifolds (i.e. ones endowed with the right kind of form) and Lie n-algebras built from the complex of forms on the manifold.  An important example (a theorem of Chris’ and John Baez) is that one has a natural example of a 2-plectic manifold in any compact simple Lie group together with a 3-form naturally constructed from its Maurer-Cartan form.

At any rate, this workshop had a great proportion of interesting talks, and overall, including the chance to see a little more of China, was a great experience!

I’ve given some talks about this project, but as we’ve been writing it up, it’s expanded considerably, including a lot of category-theoretic details which are going to be in the second paper in this series. But the basic point of this current paper is essentially visual and, in my opinion, fairly simple. The groupoidification of the QHO has a nice visual description, since it is all about the combinatorics of finite sets. This was described originally by Baez and Dolan, and in more detail in my very first paper. The other visual part here is the relation to Khovanov’s categorification of the Heisenberg algebra using a graphical calculus. (I wrote about this back when I first became aware of it.)

As a Representation

The scenario here actually has some common features with my last post. First, we have a monoidal category with duals, let’s say presented in terms of some generators and relations. Then, we find some concrete model of this abstractly-presented monoidal category with duals in a specific setting, namely .

Calling this “concrete” just refers to the fact that the objects in have some particular structure in terms of underlying sets and so on. By a “model” I just mean a functor (“model” and “representation” mean essentially the same thing in this context). In fact, for this to make sense, I think of as a 2-category with one object. Then a model is just some particular choices: a groupoid to represent the unique object, spans of groupoids to represent the generating morphisms, spans of spans to represent the generating 2-morphisms, all chosen so that the defining relations hold.

In my previous post, was a category of cobordisms, but in this case, it’s essentially Khovanov’s monoidal category whose objects are (oriented) dots and whose morphisms are certain classes of diagrams. The nice fact about the particular model we get is that the reasons these relations hold are easy to see in terms of a the combinatorics of sets. This is why our title describes what we got as “a combinatorial representation” Khovanov’s category of diagrams, for which the ring of isomorphism classes of objects is the integral form of the algebra. This uses that is not just a monoidal category: it can be a monoidal 2-category. What’s more, the monoidal category “is” also a 2-category – with one object. The objects of are really the morphisms of this 2-category.

So is in some sense a universal theory (because it’s defined freely in terms of generators and relations) of what a categorification of the Heisenberg algebra must look like. Baez-Dolan groupoidification of the QHO then turns out to be a representation or model of it. In fact, the model is faithful, so that we can even say that it provides a combinatorial interpretation of that category.

The Combinatorial Model

Between the links above, you can find a good summary of the situation, so I’ll be a bit cursory. The model is described in terms of structures on finite sets. This is why our title calls this a “combinatorial representation” of Khovanov’s categorification.

This means that the one object of (as a 2-category) is taken to the groupoid of finite sets and bijections (which we just called in the paper for brevity). This is the “Fock space” object. For simplicity, we can take an equivalent groupoid, which has just one -element set for each .

Now, a groupoid represents a system, whose possible configurations are the objects and whose symmetries are the morphisms. In this case, the possible configurations are the different numbers of “quanta”, and the symmetries (all set-bijections) show that all the quanta are interchangeable. I imagine a box containing some number of ping-pong balls.

A span of groupoids represents a process. It has a groupoid whose objects are histories (and morphisms are symmetries of histories). This groupoid has a pair of maps: to the system the process starts in, and to the system it ends in. In our model, the most important processes (which generate everything else) are the creation and annihilation operators, and – and their categorified equivalents, and . The spans that represent them are very simple: they are processes which put a new ball into the box, or take one out, respectively. (Algebraically, they’re just a way to organize all the inclusions of symmetric groups .)

The “canonical commutation relation“, which we write without subtraction thus:

is already understood in the Baez-Dolan story: it says that there is one more way to remove a ball from a box after putting a new one into it (one more history for the process ) than to remove a ball and then add a new one (histories for ). This is fairly obvious: in the first instance, you have one more to choose from when removing the ball.

But the original Baez-Dolan story has no interesting 2-morphisms (the actual diagrams which are the 1-morphisms in ), whereas these are absolutely the whole point of a categorification in the sense Khovanov gets one, since the 1-morphisms of determine what the isomorphism classes of objects even are.

So this means that we need to figure out what the 2-morphisms in need to be – first in general, and second in our particular representation of .

In general, a 2-morphism in is a span of span-maps. You’ll find other people who take it to be a span-map. This would be a functor between the groupoids of histories: roughly, a map which assigns a history in the source span to a history in the target span (and likewise for symmetries), in a way that respects how they’re histories. But we don’t want just a map: we want a process which has histories of its own. We want to describe a “movie of processes” which change one process into another. These can have many histories of their own.

In fact, they’re not too complicated. Here’s one of Khovanov’s relation in which forms part of how the commutation relation is expressed (shuffled to get rid of negatives, which we constantly need to do in the combinatorial model since we have no negative sets):

We read an upward arrow as “add a ball to the box”, and a downward arrow as “remove a ball”, and read right-to-left.  Both processes begin and end with“add then remove”. The right-hand side just leaves this process alone: it’s the identity.

The left-hand side shows a process-movie whose histories have two different cases. Suppose we begin with a history for which we add and then remove . The first case is that : we remove the same ball we put in. This amounts to doing nothing, so the first part of the movie eliminates all the adding and removing. The second part puts the add-remove pair back in.

The second case ensures that , since it takes the initial history to the history (of a different process!) in which we remove and then add (impossible if , since we can’t remove this ball before adding it). This in turn is taken to the history (of the original process!) where we add and then remove ; so this relates every history to itself, except for the case that . Overall the sum of these relations give the identity on histories, which is the right hand side.

This picture includes several of the new 2-morphisms that we need to add to the Baez-Dolan picture: swapping the order of two generators, and adding or removing a pair of add/remove operations. Finding spans of spans which accomplish this (and showing they satisfy the right relations) is all that’s needed to finish up the combinatorial model.  So, for instance, the span of spans which adds a “remove-then-add” pair is this one:

If this isn’t clear, well, it’s explained in more detail in the paper.  (Do notice, though, that this is a diagram in groupoids: we need to specify that there are identity 2-cells in the span, rather than some other 2-cells.)

So this is basically how the combinatorial model works.

Adjointness

But in fact this description is (as often happens) chronologically backwards: what actually happened was that we had worked out what the 2-morphisms should be for different reasons. While trying to to understand what kind of structure this produced, we realized (thanks to Marco Mackaay) that the result was related to , which in turn shed more light on the 2-morphisms we’d found.

So far so good. But what makes it possible to represent the kind of monoidal category we’re talking about in this setting is adjointness. This is another way of saying what I meant up at the top by saying we start with a monoidal category with duals.  This means morphisms each have a partner – a dual, or adjoint – going in the opposite direction.  The representations of the raising and lowering operators of the Heisenberg algebra on the Hilbert space for the QHO are linear adjoints. Their categorifications also need to be adjoints in the sense of adjoint 1-morphisms in a 2-category.

This is an abstraction of what it means for two functors and to be adjoint. In particular, it means there have to be certain 2-cells such as the unit and counit satisfying some nice relations. In fact, this only makes a left adjoint and a right adjoint – in this situation, we also have another pair which makes a right adjoint and a left one. That is, they should be “ambidextrous adjoints”, or “ambiadjoints” for short. This is crucial if they’re going to represent any graphical calculus of the kind that’s involved here (see the first part of this paper by Aaron Lauda, for instance).

So one of the theorems in the longer paper will show concretely that any 1-morphism in has an ambiadjoint – which happens to look like the same span, but thought of as going in the reverse direction. This is somewhat like how the adjoint of a real linear map, expressed as a matrix relative to well-chosen bases, is just the transpose of the same matrix. In particular, and are adjoints in just this way. The span-of-span-maps I showed above is exactly the unit for one side of this ambi-adjunction – but it is just a special case of something that will work for any span and its adjoint.

Finally, there’s something a little funny here. Since the morphisms of aren’t functors or maps, this combinatorial model is not exactly what people often mean by a “categorified representation”. That would be an action on a category in terms of functors and natural transformations. We do talk about how to get one of these on a 2-vector space out of our groupoidal representation toward the end.

In particular, this amounts to a functor into – the objects of being categories of a particular kind, and the morphisms being functors that preserve all the structure of those categories. As it turns out, the thing about this setting which is good for this purpose is that all those functors have ambiadjoints. The “2-linearization” that takes into is a 2-functor, and this means that all the 2-cells and equations that make two morphisms ambiadjoints carry over. In , it’s very easy for this to happen, since all those ambiadjoints are already present. So getting representations of categorified algebras that are made using these monoidal categories of diagrams on 2-vector spaces is fairly natural – and it agrees with the usual intuition about what “representation” means.

Anything I start to say about this is in danger of ballooning, but since we’re already some 40 pages into the second paper, I’ll save the elaboration for that…

Factoring TQFT

The older version of this paper talked about the untwisted version of the Dijkgraaf-Witten (DW for short) model, which is a certain kind of TQFT based on a gauge theory with a finite gauge group.  (Freed and Quinn put it as: “Chern-Simons theory with finite gauge group”).  The new version gets the general – that is, the twisted – form in the same way: factoring the theory into two parts. So, the DW model, which was originally described by Dijkgraaf and Witten in terms of a state-sum, is a functor

The “twisting” is the point of their paper, “Topological Gauge Theories and Group Cohomology”.  The twisting has to do with the action for some physical theory. Now, for a gauge theory involving flat connections, the kind of gauge-theory actions which involve the curvature of a connection make no sense: the curvature is zero.  So one wants an action which reflects purely global features of connections.  The cohomology of the gauge group is where this comes from.

Now, the machinery I describe is based on a point of view which has been described in a famous paper by Freed, Hopkins, Lurie and Teleman (FHLT for short – see further discussion here) in terms in which the two stages are called the “classical field theory” (which has values in groupoids), and the “quantization functor”, which takes one into Hilbert spaces.

Actually, we really want to have an “extended” TQFT: a TQFT gives a Hilbert space for each 2D manifold (“space”), and a linear map for a 3D cobordism (“spacetime”) between them. An extended TQFT will assign (higher) algebraic data to lower-dimension boundaries still.  My paper talks only about the case where we’ve extended down to codimension 2, whereas FHLT talk about extending “down to a point”. The point of this first stopping point is to unpack explicitly and computationally what the factorization into two parts looks like at the first level beyond the usual TQFT.

In the terminology I use, the classical field theory is:

This depends on a cohomology class . The “quantization functor” (which in this case I call “2-linearization”):

The middle stage involves the monoidal 2-category I call .  (In FHLT, they use different terminology, for instance “families” rather than “spans”, but the principle is the same.)

Freed and Quinn looked at the quantization of the “extended” DW model, and got a nice geometric picture. In it, the action is understood as a section of some particular line-bundle over a moduli space. This geometric picture is very elegant once you see how it works, which I found was a little easier in light of a factorization through .

This factorization isolates the geometry of this particular situation in the “classical field theory” – and reveals which of the features of their setup (the line bundle over a moduli space) are really part of some more universal construction.

In particular, this means laying out an explicit definition of both and .

2-Linearization Recalled

While I’ve talked about it before, it’s worth a brief recap of how 2-linearization works with a view to what happens when you twist it via groupoid cohomology. Here we have a 2-category , whose objects are groupoids (, , etc.), whose morphisms are spans of groupoids:

and whose 2-morphisms are spans of span-maps (taken up to isomorphism), which look like so:

(And, by the by: how annoying that WordPress doesn’t appear to support xypic figures…)

These form a (symmetric monoidal) 2-category, where composition of spans works by taking weak pullbacks.  Physically, the idea is that a groupoid has objects which are configurations (in the cause of gauge theory, connections on a manifold), and morphisms which are symmetries (gauge transformations, in this case).  Then a span is a groupoid of histories (connections on a cobordism, thought of as spacetime), and the maps pick out its starting and ending configuration.  That is, is the groupoid of flat -connections on a manifold , and is the groupoid of flat -connections on some cobordism , of which is part of the boundary.  So any such connection can be restricted to the boundary, and this restriction is .

Now 2-linearization is a 2-functor:

It gives a 2-vector space (a nice kind of category) for each groupoid .  Specifically, the category of its representations, .  Then a span turns into a functor which comes from “pulling” back along (the restricted representation where acts by first applying then the representation), then “pushing” forward along (to the induced representation).

What happens to the 2-morphisms is conceptually more complicated, but it depends on the fact that “pulling” and “pushing” are two-sided adjoints. Concretely, it ends up being described as a kind of “sum over histories” (where “histories” are the objects of ), which turns out to be exactly the path integral that occurs in the TQFT.

Or at least, it’s the path integral when the action is trivial! That is, if , so that what’s integrated over paths (“histories”) is just . So one question is: is there a way to factor things in this way if there’s a nontrivial action?

Cohomological Twisting

The answer is by twisting via cohomology. First, let’s remember what that means…

We’re talking about groupoid cohomology for some groupoid (which you can take to be a group, if you like).  “Cochains” will measure how much some nice algebraic fact, such as being a homomorphism, or being associative, “fails to occur”.  “Twisting by a cocycle” is a controlled way to force some such failure to happen.

So, an -cocycle is some function of composable morphisms of (or, if there’s only one object, “group elements”, which amounts to the same thing).  It takes values in some group of coefficients, which for us is always . 

The trivial case where is actually slightly subtle: a 0-cocycle is an invariant function on the objects of a groupoid. (That is, it takes the same value on any two objects related by an (iso)morphism. (Think of the object as a sequence of zero composable morphisms: it tells you where to start, but nothing else.)

The case is maybe a little more obvious. A 1-cochain  can measure how a function on objects might fail to be a 0-cocycle. It is a -valued function of morphisms (or, if you like, group elements).  The natural condition to ask for is that it be a homomorphism:

This condition means that a cochain is a cocycle. They form an abelian group, because functions satisfying the cocycle condition are closed under pointwise multiplication in . It will automatically by satisfied for a coboundary (i.e. if comes from a function on objects as ). But not every cocycle is a coboundary: the first cohomology is the quotient of cocycles by coboundaries. This pattern repeats.

It’s handy to think of this condition in terms of a triangle with edges , , and .  It says that if we go from the source to the target of the sequence with or without composing, and accumulate -values, our gives the same result.  Generally, a cocycle is a cochain satisfying a “coboundary” condition, which can be described in terms of an -simplex, like this triangle. What about a 2-cocycle? This describes how composition might fail to be respected.

So, for instance, a twisted representation of a group is not a representation in the strict sense. That would be a map into , such that .  That is, the group composition rule gets taken directly to the corresponding rule for composition of endomorphisms of the vector space .  A twisted representation only satisfies this up to a phase:

where is a function that captures the way this “representation” fails to respect composition.  Still, we want some nice properties: is a “cocycle” exactly when this twisting still makes respect the associative law:

Working out what this says in terms of , the cocycle condition says that for any composable triple we have:

So – the second group-cohomology group of – consists of exactly these which satisfy this condition, which ensures we have associativity.

Given one of these maps, we get a category of all the -twisted representations of . It behaves just like an ordinary representation category… because in fact it is one! It’s the category of representations of a twisted version of the group algebra of , called . The point is, we can use to twist the convolution product for functions on , and this is still an associative algebra just because satisfies the cocycle condition.

The pattern continues: a 3-cocycle captures how some function of 2 variable may fail to be associative: it specifies an associator map (a function of three variables), which has to satisfy some conditions for any four composable morphisms. A 4-cocycle captures how a map might fail to satisfy this condition, and so on. At each stage, the cocycle condition is automatically satisfied by coboundaries. Cohomology classes are elements of the quotient of cocycles by coboundaries.

So the idea of “twisted 2-linearization” is that we use this sort of data to change 2-linearization.

Twisted 2-Linearization

The idea behind the 2-category is that it contains , but that objects and morphisms also carry information about how to “twist” when applying the 2-linearization .  So in particular, what we have is a (symmetric monoidal) 2-category where:

• Objects consist of , where is a groupoid and $\theta \in Z^2(A,U(1))$
• Morphisms from to consist of a span from to , together with
• 2-Morphisms from to consist of a span from , together with

The cocycles have to satisfy some compatibility conditions (essentially, pullbacks of the cocycles from the source and target of a span should land in the same cohomology class).  One way to see the point of this requirement is to make twisted 2-linearization well-defined.

One can extend the monoidal structure and composition rules to objects with cocycles without too much trouble so that is a subcategory of . The 2-linearization functor extends to :

• On Objects: , the category of -twisted representation of
• On Morphisms: comes by pulling back a twisted representation in to one in , pulling it through the algebra map “multiplication by “, and pushing forward to
• On 2-Morphisms: For a span of span maps, one uses the usual formula (see the paper for details), but a sum over the objects picks up a weight of at each object

When the cocycles are trivial (evaluate to 1 always), we get back the 2-linearization we had before. Now the main point here is that the “sum over histories” that appears in the 2-morphisms now carries a weight.

So the twisted form of 2-linearization uses the same “pull-push” ideas as 2-linearization, but applied now to twisted representations. This twisting (at the object level) uses a 2-cocycle. At the morphism level, we have a “twist” between “pull” and “push” in constructing . What the “twist” actually means depends on which cohomology degree we’re in – in other words, whether it’s applied to objects, morphisms, or 2-morphisms.

The “twisting” by a 0-cocycle just means having a weight for each object – in other words, for each “history”, or connection on spacetime, in a big sum over histories. Physically, the 0-cocycle is playing the role of the Lagrangian functional for the DW model. Part of the point in the FHLT program can be expressed by saying that what Freed and Quinn are doing is showing how the other cocycles are also the Lagrangian – as it’s seen at higher codimension in the more “local” theory.

For a TQFT, the 1-cocycles associated to morphisms describe how to glue together values for the Lagrangian that are associated to histories that live on different parts of spacetime: the action isn’t just a number. It is a number only “locally”, and when we compose 2-morphisms, the 0-cocycle on the composite picks up a factor from the 1-morphism (or 0-morphism, for a horizontal composite) where they’re composed.

This has to do with the fact that connections on bits of spacetime can be glued by particular gauge transformations – that is, morphisms of the groupoid of connections. Just as the gauge transformations tell how to glue connections, the cocycles associated to them tell how to glue the actions. This is how the cohomological twisting captures the geometric insight that the action is a section of a line bundle – not just a function, which is a section of a trivial bundle – over the moduli space of histories.

So this explains how these cocycles can all be seen as parts of the Lagrangian when we quantize: they explain how to glue actions together before using them in a sum-over histories. Gluing them this way is essential to make sure that is actually a functor. But if we’re really going to see all the cocycles as aspects of “the action”, then what is the action really? Where do they come from, that they’re all slices of this bigger thing?

Twisting as Lagrangian

Now the DW model is a 3D theory, whose action is specified by a group-cohomology class . But this is the same thing as a class in the cohomology of the classifying space: . This takes a little unpacking, but certainly it’s helpful to understand that what cohomology classes actually classify are… gerbes. So another way to put a key idea of the FHLT paper, as Urs Schreiber put it to me a while ago, is that “the action is a gerbe on the classifying space for fields“.

What does this mean?

This map is given as a path integral over all connections on the space(-time) , which is actually just a sum, since the gauge group is finite and so all the connections are flat.  The point is that they’re described by assigning group elements to loops in :

But this amounts to the same thing as a map into the classifying space of :

This is essentially the definition of , and it implies various things, such as the fact that is a space whose fundamental group is , and has all other homotopy groups trivial. That is, is the Eilenberg-MacLane space . But the point is that the groupoid of connections and gauge transformations on just corresponds to the mapping space . So the groupoid cohomology classes we get amount to the same thing as cohomology classes on this space. If we’re given , then we can get at these by “transgression” – which is very nicely explained in a paper by Simon Willerton.

The essential idea is that a 3-cocycle (representing the class ) amounts to a nice 3-form on which we can integrate over a 3-dimentional submanifold to get a number. For a -dimensional , we get such a 3-manifold from a -dimensional submanifold of : each point gives a copy of in . Then we get a -cocycle on whose values come from integrating over this image. Here’s a picture I used to illustrate this in my talk:

Now, it turns out that this gives 2-cocycles for 1-manifolds (the objects of , 1-cocycles on 2D cobordisms between them, and 0-cocycles on 3D cobordisms between these cobordisms. The cocycles are for the groupoid of connections and gauge transformations in each case. In fact, because of Stokes’ theorem in , these have to satisfy all the conditions that make them into objects, morphisms, and 2-morphisms of . This is the geometric content of the Lagrangian: all the cocycles are really “reflections” of as seen by transgression: pulling back along the evaluation map from the picture. Then the way you use it in the quantization is described exactly by .

What I like about this is that is a fairly universal sort of thing – so while this example gets its cocycles from the nice geometry of which Freed and Quinn talk about, the insight that an action is a section of a (twisted) line bundle, that actions can be glued together in particular ways, and so on… These presumably can be moved to other contexts.

(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.)

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.