### toposes

So Dan Christensen, who used to be my supervisor while I was a postdoc at the University of Western Ontario, came to Lisbon last week and gave a talk about a topic I remember hearing about while I was there.  This is the category $Diff$ of diffeological spaces as a setting for homotopy theory.  Just to make things scan more nicely, I’m going to say “smooth space” for “diffeological space” here, although this term is in fact ambiguous (see Andrew Stacey’s “Comparative Smootheology” for lots of details about options).  There’s a lot of information about $Diff$ in Patrick Iglesias-Zimmour’s draft-of-a-book.

Motivation

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

So, for instance, this is a good context for constructing loop spaces: a manifold $M$ is a smooth space, and so is its loop space $LM = M^{S^1} = Hom(S^1,M)$, the space of all maps of the circle into $M$.  This becomes important for talking about things like higher cohomology, gerbes, etc.  When starting with the category of manifolds, doing this requires you to go off and define infinite dimensional manifolds before $LM$ can even be defined.  Likewise, the irrational torus is hard to talk about as a manifold: you take a torus, thought of as $\mathbb{R}^2 / \mathbb{Z}^2$.  Then take a direction in $\mathbb{R}^2$ with irrational slope, and identify any two points which are translates of each other in $\mathbb{R}^2$ along the direction of this line.  The orbit of any point is then dense in the torus, so this is a very nasty space, certainly not a manifold.  But it’s a perfectly good smooth space.

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

Smooth Spaces

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

• A set $X$ (of “points”)
• A set $\{ f : U \rightarrow X \}$ (of “plots”) for every n and open $U \subset \mathbb{R}^n$ such that:
1. All constant maps are plots
2. If $f: U \rightarrow X$ is a plot, and $g : V \rightarrow U$ is a smooth map, $f \circ g : V \rightarrow X$ is a plot
3. If $\{ g_i : U_i \rightarrow U\}$ is an open cover of $U$, and $f : U \rightarrow X$ is a map, whose restrictions $f \circ g_i : U_i \rightarrow X$ are all plots, so is $f$

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

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

So, in particular, diffeologies can encode a little more than the charts of a manifold.  Just for one example, a diffeology can have “stop signs”, as Dan put it – points with the property that any smooth map from $I= [0,1]$ which passes through them must stop at that point (have derivative zero – or higher derivatives, if you like).  Along the same lines, there’s a nonstandard diffeology on $I$ itself with the property that any smooth map from this $I$ into a manifold $M$ must have all derivatives zero at the endpoints.  This is a better object for defining smooth fundamental groups: you can concatenate these paths at will and they’re guaranteed to be smooth.

As a Quasitopos

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

So remember that a subobject classifier is an object with a map $t : 1 \rightarrow \Omega$ from the terminal object, so that any monomorphism (subobject) $A \rightarrow X$ is the pullback of $t$ along some map $X \rightarrow \Omega$ (the classifying map).  In the topos of sets, this is just the inclusion of a one-element set $\{\star\}$ into a two-element set $\{T,F\}$: the classifying map for a subset $A \subset X$ sends everything in $A$ (i.e. in the image of the inclusion map) to $T = Im(t)$, and everything else to $F$.  (That is, it’s the characteristic function.)  So pulling back $T$

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

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

As a Model Category

The basic idea here is pretty well supported: naively, one can do a lot of the things done in homotopy theory in $Diff$: to start with, one can define the “smooth homotopy groups” $\pi_n^s(X;x_0)$ of a pointed space.  It’s a theorem by Dan and Enxin that several possible ways of doing this are equivalent.  But, for example, Iglesias-Zimmour defines them inductively, so that $\pi_0^s(X)$ is the set of path-components of $X$, and $\pi_k^s(X) = \pi_{k-1}^s(LX)$ is defined recursively using loop spaces, mentioned above.  The point is that this all works in $Diff$ much as for topological spaces.

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

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

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

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

Dan and Enxin are still working on nailing down some of the proofs, but it appears to be working.  Their strategy is based on the principle that, for purposes of homotopy, topological spaces act like simplicial complexes.  So they define an affine “simplex”, $\mathbb{A}^n = \{ (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} | \sum x_i = 1 \}$.  These aren’t literally simplexes: they’re affine planes, which we understand as smooth spaces – with the subspace diffeology from $\mathbb{R}^{n+1}$.  But they behave like simplexes: there are face and degeneracy maps for them, and the like.  They form a “cosimplicial object”, which we can think of as a functor $\Delta \rightarrow Diff$, where $\Delta$ is the simplex category).

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

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

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

Cohesive Infinity-Toposes

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

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

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

But this omits there are a couple of other things that apply to $Top$: $U$ has a right adjoint, $Codisc: Set \rightarrow Top$, where $Codisc(S)$ has only $S$ and $\emptyset$ as its open sets.  In $Codisc(S)$, all the points are “stuck together” in one piece.  On the other hand, $Disc$ itself has a left adjoint, $\Pi_0: Top \rightarrow Set$, which gives the set of connected components of a space.  $\Pi_0(X)$ is another kind of “underlying set” of a space.  So we call a topos $\mathcal{E}$ “cohesive” when the terminal geometric morphism extends to a chain of four adjoint functors in just this way, which satisfy a few properties that characterize what’s happening here.  (We can talk about “cohesive sites”, where this happens.)

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

Where the “infinity” part of “infinity-topos” comes in is the connection to homotopy theory.  Here, we replace the topos $Sets$ with the infinity-topos of infinity-groupoids.  Then the “underlying” functor captures not just the set of points of a space $X$, but its whole fundamental infinity-groupoid.  Its objects are points of $X$, its morphisms are paths, 2-morphisms are homotopies of paths, and so on.  All the homotopy groups of $X$ live here.  So a cohesive inifinity-topos is defined much like above, but with $\infty-Gpd$ playing the role of $Set$, and with that $\Pi_0$ functor replaced by $\Pi$, something which, implicitly, gives all the homotopy groups of $X$.  We might look for cohesive infinity-toposes to be given by the (infinity)-categories of simplicial sheaves on cohesive sites.

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

In the first week of November, I was in Montreal for the biannual meeting of the Philosophy of Science Association, at the invitation of Hans Halvorson and Steve Awodey.  This was for a special session called “Category Theoretical Reflections on the Foundations of Physics”, which also had talks by Bob Coecke (from Oxford), Klaas Landsman (from Radboud University in Nijmegen), and Gonzalo Reyes (from the University of Montreal).  Slides from the talks in this session have been collected here by Steve Awodey.  The meeting was pretty big, and there were a lot of talks on a lot of different topics, some more technical, and some less.  There were enough sessions relating to physics that I had a full schedule just attending those, although for example there were sessions on biology and cognition which I might otherwise have been interested in sitting in on, with titles like “Biology: Evolution, Genomes and Biochemistry”, “Exploring the Complementarity between Economics and Recent Evolutionary Theory”, “Cognitive Sciences and Neuroscience”, and “Methodological Issues in Cognitive Neuroscience”.  And, of course, more fundamental philosophy of science topics like “Fictions and Scientific Realism” and “Kinds: Chemical, Biological and Social”, as well as socially-oriented ones such as “Philosophy of Commercialized Science” and “Improving Peer Review in the Sciences”.  However, interesting as these are, one can’t do everything.

In some ways, this was a really great confluence of interests for me – physics and category theory, as seen through a philosophical lens.  I don’t know exactly how this session came about, but Hans Halvorson is a philosopher of science who started out in physics (and has now, for example, learned enough category theory to teach the course in it offered at Princeton), and Steve Awodey is a philosopher of mathematics who is interested in category theory in its own right.  They managed to get this session brought in to present some of the various ideas about the overlap between category theory and physics to an audience mostly consisting of philosophers, which seems like a good idea.  It was also interesting for me to get a view into how philosophers approach these subjects – what kind of questions they ask, how they argue, and so on.  As with any well-developed subject, there’s a certain amount of jargon and received ideas that people can refer to – for example, I learned the word and current usage (though not the basic concept) of supervenience, which came up, oh, maybe 5-10 times each day.

There are now a reasonable number of people bringing categorical tools to bear on physics – especially quantum physics.  What people who think about the philosophy of science can bring to this research is the usual: careful, clear thinking about the fundamental concepts involved in a way that tries not to get distracted by the technicalities and keep the focus on what is important to the question at hand in a deep way.  In this case, the question at hand is physics.  Philosophy doesn’t always accomplish this, of course, and sometimes get sidetracked by what some might call “pseudoquestions” – the kind of questions that tend to arise when you use some folk-theory or simple intuitive understanding of some subtler concept that is much better expressed in mathematics.  This is why anyone who’s really interested in the philosophy of science needs to learn a lot about science in its own terms.  On the whole, this is what they actually do.

And, of course, both mathematicians and physicists try to do this kind of thinking themselves, but in those fields it’s easy – and important! – to spend a lot of time thinking about some technical question, or doing extensive computations, or working out the fiddly details of a proof, and so forth.  This is the real substance of the work in those fields – but sometimes the bigger “why” questions, that address what it means or how to interpret the results, get glossed over, or answered on the basis of some superficial analogy.  Mind you – one often can’t really assess how a line of research is working out until you’ve been doing the technical stuff for a while.  Then the problem is that people who do such thinking professionally – philosophers – are at a loss to understand the material because it’s recent and technical.  This is maybe why technical proficiency in science has tended to run ahead of real understanding – people still debate what quantum mechanics “means”, even though we can use it competently enough to build computers, nuclear reactors, interferometers, and so forth.

Anyway – as for the substance of the talks…  In our session, since every speaker was a mathematician in some form, they tended to be more technical.  You can check out the slides linked to above for more details, but basically, four views of how to draw on category theory to talk about physics were represented.  I’ve actually discussed each of them in previous posts, but in summary:

• Bob Coecke, on “Quantum Picturalism”, was addressing the monoidal dagger-category point of view, which looks at describing quantum mechanical operations (generally understood to be happening in a category of Hilbert spaces) purely in terms of the structure of that category, which one can see as a language for handling a particular kind of logic.  Monoidal categories, as Peter Selinger as painstakingly documented, can be described using various graphical calculi (essentially, certain categories whose morphisms are variously-decorated “strands”, considered invariant under various kinds of topological moves, are the free monoidal categories with various structures – so anything you can prove using these diagrams is automatically true for any example of such categories).  Selinger has also shown that, for the physically interesting case of dagger-compact closed monoidal categories, a theorem is true in general if and only if it’s true for (finite dimensional) Hilbert spaces, which may account for why Hilbert spaces play such a big role in quantum mechanics.  This program is based on describing as much of quantum mechanics as possible in terms of this kind of diagrammatic language.  This stuff has, in some ways, been explored more through the lens of computer science than physics per se – certainly Selinger is coming from that background.  There’s also more on this connection in the “Rosetta Stone” paper by John Baez and Mike Stay,
• My talk (actually third, but I put it here for logical flow) fits this framework, more or less.  I was in some sense there representing a viewpoint whose current form is due to Baez and Dolan, namely “groupoidification”.  The point is to treat the category $Span(Gpd)$ as a “categorification” of (finite dimensional) Hilbert spaces in the sense that there is a representation map $D : Span(Gpd) \rightarrow Hilb$ so that phenomena living in $Hilb$ can be explained as the image of phenomena in $Span(Gpd)$.  Having done that, there is also a representation of $Span(Gpd)$ into 2-Hilbert spaces, which shows up more detail (much more, at the object level, since Tannaka-Krein reconstruction means that the monoidal 2-Hilbert space of representations of a groupoid is, at least in nice cases, enough to completely reconstruct it).  This gives structures in $2Hilb$ which “conceptually” categorify the structures in $Hilb$, and are also directly connected to specific Hilbert spaces and maps, even though taking equivalence classes in $2Hilb$ definitely doesn’t produce these.  A “state” in a 2-Hilbert space is an irreducible representation, though – so there’s a conceptual difference between what “state” means in categorified and standard settings.  (There’s a bit more discussion in my notes for the talk than in the slides above.)
• Klaas Landsman was talking about what he calls “Bohrification“, which, on the technical side, makes use of Topos theory.  The philosophical point comes from Niels Bohr’s “doctrine of classical concepts” – that one should understand quantum systems using concepts from the classical world.  In practice, this means taking a (noncommutative) von Neumann algebra $A$ which describes the observables a quantum system and looking at it via its commutative subalgebras.  These are organized into a lattice – in fact, a site.  The idea is that the spectrum of $A$ lives in the topos associated to this site: it’s a presheaf that, over each commutative subalgebra $C \subset A$, just gives the spectrum of $C$.  This is philosophically nice in that the “Bohrified” propositions actually behave in a logically sensible way.  The topos approach comes from Chris Isham, developed further with Andreas Doring. (Note the series of four papers by both from 2007.  Their approach is in some sense dual to that of Lansman, Heunen and Spitters, in the sense that they look at the same site, but look at dual toposes – one of sheaves, the other of cosheaves.  The key bit of jargon in Isham and Doring’s approach is “daseinization”, which is a reference to Heidegger’s “Being and Time”.  For some reason this makes me imagine Bohr and Heidegger in a room, one standing on the ceiling, one on the floor, disputing which is which.)
• Gonzalo Reyes talked about synthetic differential geometry (SDG) as a setting for building general relativity.  SDG is a way of doing differential geometry in a category where infinitesimals are actually available, that is, there is a nontrivial set $D = \{ x \in \mathbb{R} | x^2 = 0 \}$.  This simplifies discussions of vector fields (tangent vectors will just be infinitesimal vectors in spacetime).  A vector field is really a first order DE (and an integral curve tangent to it is a solution), so it’s useful to have, in SDG, the fact that any differentiable curve is, literally, infinitesimally a line.  Then the point is that while the gravitational “field” is a second-order DE, so not a field in this sense, the arguments for GR can be reproduced nicely in SDG by talking about infinitesimally-close families of curves following geodesics.  Gonzalo’s slides are brief by necessity, but happily, more details of this are in his paper on the subject.

The other sessions I went to were mostly given by philosophers, rather than physicists or mathematicians, though with exceptions.  I’ll briefly present my own biased and personal highlights of what I attended.  They included sessions titled:

Quantum Physics“: Edward Slowik talked about the “prehistory of quantum gravity”, basically revisiting the debate between Newton and Leibniz on absolute versus relational space, suggesting that Leibniz’ view of space as a classification of the relation of his “monads” is more in line with relational theories such as spin foams etc.  M. Silberstein and W. Stuckey – gave a talk about their “relational blockworld” (described here) which talks about QFT as an approximation to a certain discrete theory, built on a graph, where the nodes of the graph are spacetime events, and using an action functional on the graph.

Meinard Kuhlmann gave an interesting talk about “trope bundles” and AQFTTrope ontology is an approach to “entities” that doesn’t assume there’s a split between “substrates” (which have no properties themselves), and “properties” which they carry around.  (A view of ontology that goes back at least to Aristotle’s “substance” and “accident” distinction, and maybe further for all I know).  Instead, this is a “one-category” ontology – the basic things in this ontology are “tropes”, which he defined as “individual property instances” (i.e. as opposed to abstract properties that happen to have instances).  “Things” then, are just collections of tropes.  To talk about the “identity” of a thing means to pick out certain of the tropes as the core ones that define that thing, and others as peripheral.  This struck me initially as a sort of misleading distinction we impose (say, “a sphere” has a core trope of its radial symmetry, and incidental tropes like its colour – but surely the way of picking the object out of the world is human-imposed), until he gave the example from AQFT.  To make a long story short, in this setup, the key entites are something like elementary particles, and the core tropes are those properties that define an irreducible representation of a $C^{\star}$-algebra (things like mass, spin, charge, etc.), whereas the non-core tropes are those that identify a state vector within such a representation: the attributes of the particle that change over time.

I’m not totally convinced by the “trope” part of this (surely there are lots of choices of the properties which determine a representation, but I don’t see the need to give those properties the burden of being the only ontologically primaries), but I also happen to like the conclusions because in the 2Hilbert picture, irreducible representations are states in a 2-Hilbert space, which are best thought of as morphisms, and the state vectors in their components are best thought of in terms of 2-morphisms.  An interpretation of that setup says that the 1-morphism states define which system one’s talking about, and the 2-morphism states describe what it’s doing.

New Directions Concerning Quantum Indistinguishability“: I only caught a couple of the talks in this session, notably missing Nick Huggett’s “Expanding the Horizons of Quantum Statistical Mechanics”.  There were talks by John Earman (“The Concept of Indistinguishable Particles in Quantum
Mechanics”), and by Adam Caulton (based on work with Jeremy Butterfield) on “On the Physical Content of the Indistinguishability Postulate”.  These are all about the idea of indistinguishable particles, and the statistics thereof.  Conventionally, in QM you only talk about bosons and fermions – one way to say what this means is that the permutation group $S_n$ naturally acts on a system of $n$ particles, and it acts either trivially (not altering the state vector at all), or by sign (each swap of two particles multiplies the state vector by a minus sign).  This amounts to saying that only one-dimensional representations of $S_n$ occur.  It is usually justified by the “spin-statistics theorem“, relating it to the fact that particles have either integer or half-integer spins (classifying representations of the rotation group).  But there are other representations of $S_n$, labelled by Young diagrams, though they are more than one-dimensional.  This gives rise to “paraparticle” statistics.  On the other hand, permuting particles in two dimensions is not homotopically trivial, so one ought to use the braid group $B_n$, rather than $S_n$, and this gives rise again to different statistics, called “anyonic” statistics.

One recurring idea is that, to deal with paraparticle statistics, one needs to change the formalism of QM a bit, and expand the idea of a “state vector” (or rather, ray) to a “generalized ray” which has more dimensions – corresponding to the dimension of the representation of $S_n$ one wants the particles to have.  Anyons can be dealt with a little more conventionally, since a 2D system may already have them.  Adam Caulton’s talk described how this can be seen as a topological phenomenon or a dynamical one – making an analogy with the Bohm-Aharonov effect, where the holonomy of an EM field around a solenoid can be described either dynamically with an interacting Lagrangian on flat space, or topologically with a free Lagrangian in space where the solenoid has been removed.

Quantum Mechanics“: A talk by Elias Okon and Craig Callender about QM and the Equivalence Principle, based on this.  There has been some discussion recently as to whether quantum mechanics is compatible with the principle that relates gravitational and inertial mass.  They point out that there are several versions of this principle, and that although QM is incompatible with some versions, these aren’t the versions that actually produce general relativity.  (For example, objects with large and small masses fall differently in quantum physics, because though the mean travel time is the same, the variance is different.  But this is not a problem for GR, which only demands that all matter responds dynamically to the same metric.)  Also, talks by Peter Lewis on problems with the so-called “transactional interpretation” of QM, and Bryan Roberts on time-reversal.

Why I Care About What I Don’t Yet Know“:  A funny name for a session about time-asymmetry, which is the essentially philosophical problem of why, if the laws of physics are time-symmetric (which they approximately are for most purposes), what we actually experience isn’t.  Personally, the best philosophical account of this I’ve read is Huw Price’s “Time’s Arrow“, though Reichenbach’s “The Direction of Time” has good stuff in it also, and there’s also Zeh’s more technical “The Physical Basis of the Direction of Time“. In the session, Chris Suhler and Craig Callender gave an account of how, given causal asymmetry, our subjective asymmetry of values for the future and the past can arise (the intuitively obvious point being that if we can influence the future and not the past, we tend to value it more).  Mathias Frisch talked about radiation asymmetry (the fact that it’s equally possible in EM to have waves converging on a source than spreading out from it, yet we don’t see this).  Owen Maroney argued that “There’s No Route from Thermodynamics to the Information Asymmetry” by describing in principle how to construct a time-reversed (probabilisitic) computer.  David Wallace spoke on “The Logic of the Past Hypothesis”, the idea inspired by Boltzmann that we see time-asymmetry because there is a point in what we call the “past” where entropy was very low, and so we perceive the direction away from that state as “forward” it time because the world tends to move toward equilibrium (though he pointed out that for dynamical reasons, the world can easily stay far away from equilibrium for a long time).  He went on to discuss the logic of this argument, and the idea of a “simple” (i.e. easy-to-describe) distribution, and the conjecture that the evolution of these will generally be describable in terms of an evolution that uses “coarse graining” (i.e. that repeatedly throws away microscopic information).

The Emergence of Spacetime in Quantum Theories of Gravity“:  This session addressed the idea that spacetime (or in some cases, just space) might not be fundamental, but could emerge from a more basic theory.  Christian Wüthrich spoke about “A-Priori versus A-Posteriori” versions of this idea, mostly focusing on ideas such as LQG and causal sets, which start with discrete structures, and get manifolds as approximations to them.  Nick Huggett gave an overview of noncommutative geometry for the philosophically minded audience, explaining how an algebra of observables can be treated like space by means of all the concepts from geometry which can be imported into the theory of $C^{\star}$-algebras, where space would be an approximate description of the algebra by letting the noncommutativity drop out of sight in some limit (which would be described as a “large scale” limit).  Sean Carroll discussed the possibility that “Space is Not Fundamental – But Time Might Be”, pointing out that even in classical mechanics, space is not a fundamental notion (since it’s possible to reformulate even Hamiltonian classical mechanics without making essential distinctions between position and momentum coordinates), and suggesting that space arises from the dynamics of an actual physical system – a Hamiltonian, in this example – by the principle “Position Is The Thing In Which Interactions Are Local”.  Finally, Sean Maudlin gave an argument for the fundamentality of time by showing how to reconstruct topology in space from a “linear structure” on points saying what a (directed!) path among the points is.