### noncommutative geometry

So there’s a lot of preparations going on for the workshop HGTQGR coming up next week at IST, and the program(me) is much more developed – many of the talks are now listed, though the schedule has yet to be finalized.  This week we’ll be having a “pre-school school” to introduce the local mathematicans to some of the physics viewpoints that will be discussed at the workshop – Aleksandar Mikovic will be introducing Quantum Gravity (from the point of view of the loop/spin-foam approach), and Sebastian Guttenberg will be giving a mathematician’s introduction to String theory.

These are by no means the only approaches physicists have taken to the problem of finding a theory that incorporates both General Relativity and Quantum Field Theory.  They are, however, two approaches where lots of work has been done, and which appear to be amenable to using the mathematical tools of (higher) category theory which we’re going to be talking about at the workshop.  These are “higher gauge theory”, which very roughly is the analog of gauge theory (which includes both GR and QFT) using categorical groups, and TQFT, which is a very simple type of quantum field theory that has a natural description in terms of categories, which can be generalized to higher categories.

I’ll probably take a few posts after the workshop to write up these, and the many other talks and mini-courses we’ll be having, but right now, I’d like to say a little bit about another talk we had here recently.  Actually, the talk was in Porto, but several of us at IST in Lisbon attended by a videoconference.  This was the first time I’ve seen this for a colloquium-style talk, though I did once take a course in General Relativity from Eric Poisson that was split between U of Waterloo and U of Guelph.  I thought it was a great idea then, and it worked quite well this time, too.  This is the way of the future – and unfortunately it probably will be for some time to come…

Anyway, the talk in question was by Thomasz Brzezinski, about “Synthetic Non-Commutative Geometry” (link points to the slides).  The point here is to take two different approaches to extending differential geometry (DG) and combine the two insights.  The “Synthetic” part refers to synthetic differential geometry (SDG), which is a program for doing DG in a general topos.  One aspect of this is that in a topos where the Law of the Excluded Middle doesn’t apply, it’s possible for the real-numbers object to have infinitesimals: that is, elements which are smaller than any positive element, but bigger than zero.  This lets one take things which have to be treated in a roundabout way in ordinary DG, like $dx$, and take them at face value – as an infinitesimal change in $x$.  It also means doing geometry in a completely constructive way.

However, these aspects aren’t so important here.  The important fact about it here is that it’s based on building a theory that was originally defined in terms of sets, or topological spaces – that is, in the toposes $Sets$, or $Top$  – and transplanting it to another category.  This is because Brzezinski’s goal was to do something similar for a different extension of DG, namely non-commutative geometry (NCG).  This is a generalisation of DG which is based on the equivalence $CommAlg^{op} \simeq lCptHaus$ between the categories of commutative $C^{\star}$-algebras (and algebra maps, read “backward” as morphisms in $CommAlg^{op}$), and that of locally compact Hausdorff spaces (which, for objects, equates a space $X$ with the algebra $C(X)$ of continuous functions on it, and an algebra $A$ with its spectrum $Spec(A)$, the space of maximal ideals).  The generalization of NCG is to take structures defined for $lCptHaus$ that create DG, and make similar definitions in the category $Alg^{op}$, of not-necessarily-commutative $C^{\star}$-algebras.

This category is the one which plays the role of the topos $Top$.  It isn’t a topos, though: it’s some sort of monoidal category.  And this is what “synthetic NCG” is about: taking the definitions used in NCG and reproducing them in a generic monoidal category (to be clear, a braided monoidal category).

The way he illustrated this is by explaining what a principal bundle would be in such a generic category.

To begin with, we can start by giving a slightly nonstandard definition of the concept in ordinary DG: a principal $G$-bundle $P$ is a manifold with a free action of a (compact Lie) group $G$ on it.  The point is that this always looks like a “base space” manifold $B$, with a projection $\pi : P \rightarrow B$ so that the fibre at each point of $B$ looks like $G$.  This amounts to saying that $\pi$ is an equalizer:

$P \times G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B$

where the maps from $G\times P$ to $P$ are (a) the action, and (b) the projection onto $P$.  (Being an equalizer means that $\pi$ makes this diagram commute – has the same composite with both maps – and any other map $\phi$ that does the same factors uniquely through $\pi$.)  Another equivalent way to say this is that since $P \times G$ has two maps into $P$, then it has a map into the pullback $P \times_B P$ (the pullback of two copies of $P \stackrel{\pi}{\rightarrow} B$), and the claim is that it’s actually ismorphic.

The main points here are that (a) we take this definition in terms of diagrams and abstract it out of the category $Top$, and (b) when we do so, in general the products will be tensor products.

In particular, this means we need to have a general definition of a group object $G$ in any braided monoidal category (to know what $G$ is supposed to be like).  We reproduce the usual definition of a group objects so that $G$ must come equipped with a “multiplication” map $m : G \otimes G \rightarrow G$, an “inverse” map $\iota : G \rightarrow G$ and a “unit” map $u : I \rightarrow G$, where $I$ is the monoidal unit (which takes the role of the terminal object in a topos like $Top$, the unit for $\times$).  These need to satisfy the usual properties, such as the monoid property for multiplication:

$m \circ (m \otimes id_G) = m \circ (id_G \otimes m) : G \otimes G \otimes G \rightarrow G$

(usually given as a diagram, but I’m being lazy).

The big “however” is this: in $Sets$ or $Top$, any object $X$ is always a comonoid in a canonical way, and we use this implictly in defining some of the properties we need.  In particular, there’s always the diagonal map $\Delta : X \rightarrow X \times X$ which satisfies the dual of the monoid property:

$(id_X \times \Delta) \circ \Delta = (\Delta \times id_X) \circ \Delta$

There’s also a unique counit $\epsilon \rightarrow \star$, the map into the terminal object, which makes $(X,\Delta,\epsilon)$ a counital comonoid automatically.  But in a general braided monoidal category, we have to impose as a condition that our group object also be equipped with $\Delta : G \rightarrow G \otimes G$ and $\epsilon : G \rightarrow I$ making it a counital comonoid.  We need this property to even be able to make sense of the inverse axiom (which this time I’ll do as a diagram):

This diagram uses not only $\Delta$ but also the braiding map $\sigma_{G,G} : G \otimes G \rightarrow G \otimes G$ (part of the structure of the braided monoidal category which, in $Top$ or $Sets$ is just the “switch” map).  Now, in fact, since any object in $Set$ or $Top$ is automatically a comonoid, we’ll require that this structure be given for anything we look at: the analog of spaces (like $P$ above), or our group object $G$.  For the group object, we also must, in general, require something which comes for free in the topos world and therefore generally isn’t mentioned in the definition of a group.  Namely, the comonoid and monoid aspects of $G$ must get along.  (This comes for free in a topos essentially because the comonoid structure is given canonically for all objects.)  This means:

For a group in $Sets$ or $Top$, this essentially just says that the two ways we can go from $(x,y)$ to $(xy,xy)$ (duplicate, swap, then multiply, or on the other hand multiply then duplicate) are the same.

All these considerations about how honest-to-goodness groups are secretly also comonoids does explain why corresponding structures in noncommutative geometry seem to have more elaborate definitions: they have to explicitly say things that come for free in a topos.  So, for instance, a group object in the above sense in the braided monoidal category $Vect = (Vect_{\mathbb{F}}, \otimes_{\mathbb{F}}, \mathbb{F}, flip)$ is a Hopf algebra.  This is a nice canonical choice of category.  Another is the opposite category $Vect^{op}$ – this is a standard choice in NCG, since spaces are supposed to be algebras – this would be given the comonoid structure we demanded.

So now once we know all this, we can reproduce the diagrammatic definition of a principal $G$-bundle above: just replace the product $\times$ with the monoidal operation $\otimes$, the terminal object by $I$, and so forth.  The diagrams are understood to be diagrams of comonoids in our braided monoidal category.  In particular, we have an action $\rho : P \otimes G \rightarrow P$,which is compatible with the $\Delta$ maps – so in $Vect$ we would say that a noncommutative principal $G$-bundle $P$ is a right-module coalgebra over the Hopf algebra $G$.  We can likewise take this (in a suitably abstract sense of “algebra” or “module”) to be the definition in any braided monoidal category.

To have the “freeness” of the action, there needs to be an equalizer of:

$\rho, (id_P \otimes \epsilon) : P \otimes G \stackrel{\longrightarrow}{\rightarrow} P \stackrel{\pi}{\rightarrow} B$

The “freeness” condition for the action is likewise defined using a monoidal-category version of the pullback (fibre product) $P \times_B P$.

This was as far as Brzezinski took the idea of synthetic NCG in this particular talk, but the basic idea seems quite nice.  In SDG, one can define all sorts of differential geometric structures synthetically, that is, for a general topos: for example, Gonzalo Reyes has gone and defined the Einstein field equations synthetically.  Presumably, a lot of what’s done in NCG could also be done in this synthetic framework, and transplanted to other categories than the usual choices.

Brzezinski said he was mainly interested in the “usual” choices of category, $Vect$ and $Vect^{op}$ – so for instance in $Vect^{op}$, a “principal $G$-bundle” is what’s called a Hopf-Galois extension.  Roger Picken did, however, ask an interesting question about other possible candidates for the category to work in.  Given that one wants a braided monoidal category, a natural one to look at is the category whose morphisms are braids.  This one, as a matter of fact, isn’t quite enough (there’s no braid $m : n \otimes n \rightarrow n$, because this would be a braid with $2n$ strands in and $n$ strands out – which is impossible.  But some sort of category of tangles might make an interestingly abstract setting in which to see what NCG looks like.  So far, this doesn’t seem to have been done as far as I can see.

As I mentioned in my previous post, I’ve recently started out a new postdoc at IST – the Instituto Superior Tecnico in Lisbon, Portugal.  Making the move from North America to Europe with my family was a lot of work – both before and after the move – involving lots of paperwork and shifting of heavy objects.  But Lisbon is a good city, with lots of interesting things to do, and the maths department at IST is very large, with about a hundred faculty.  Among those are quite a few people doing things that interest me.

The group that I am actually part of is coordinated by Roger Picken, and has a focus on things related to Topological Quantum Field Theory.  There are a couple of postdocs and some graduate students here associated in some degree with the group, and elsewhere than IST Aleksandar Mikovic and Joao Faria Martins.   In the coming months there should be some activity going on in this group which I will get to talk about here, including a workshop which is still in development, so I’ll hold off on that until there’s an official announcement.

## Quantales

I’ve also had a chance to talk a bit with Pedro Resende, mostly on the subject of quantales.  This is something that I got interested in while at UWO, where there is a large contingent of people interested in category theory (mostly from the point of view of homotopy theory) as well as a good group in noncommutative geometry.  Quantales were originally introduced by Chris Mulvey – I’ve been looking recently at a few papers in which he gives a nice account of the subject – here, here, and here.
The idea emerged, in part, as a way of combining two different approaches to generalising the idea of a space.  One is the approach from topos theory, and more specifically, the generalisation of topological spaces to locales.  This direction also has connections to logic – a topos is a good setting for intuitionistic, but nevertheless classical, logic, whereas quantales give an approach to quantum logics in a similar spirit.

The other direction in which they generalize space is the $C^{\star}$-algebra approach used in noncommutative geometry.  One motivation of quantales is to say that they simultaneously incorporate the generalizations made in both of these directions – so that both locales and $C^{\star}$-algebras will give examples.  In particular, a quantale is a kind of lattice, intended to have the same sort of relation to a noncommutative space as a locale has to an ordinary topological space.  So to begin, I’ll look at locales.

A locale is a lattice which formally resembles the lattice of open sets for such a space.  A lattice is a partial order with operations $\bigwedge$ (“meet”) and $\bigvee$ (“join”).  These operations take the role of the intersection and union of open sets.  So to say it formally resembles a lattice of open sets means that the lattice is closed under arbitrary joins, and finite meets, and satisfies the distributive law:

$U \bigwedge (\bigvee_i V_i) =\bigvee_i (U \bigwedge V_i)$

Lattices like this can be called either “Frames” or “Locales” – the only difference between these two categories is the direction of the arrows.  A map of lattices is a function that preserves all the structure – order, meet, and join.   This is a frame morphism, but it’s also a morphism of locales in the opposite direction.  That is, $\mathbf{Frm} = \mathbf{Loc}^{op}$.

Another name for this sort of object is a “Heyting algebra”.  One of the great things about topos theory (of which this is a tiny starting point) is that it unifies topology and logic.  So, the “internal logic” of a topos has a Heyting algebra (i.e. a locale) of truth values, where the meet and join take the place of logical operators “and” and “or”.  The usual two-valued logic is the initial object in $\mathbf{Loc}$, so while it is special, it isn’t unique.  One vital fact here is that any topological space (via the lattice of open sets) produces a locale, and the locale is enough to identify the space – so $\mathbf{Top} \rightarrow \mathbf{Loc}$ is an embedding.  (For convenience, I’m eliding over the fact that the spaces have to be “sober” – for example, Hausdorff.)  In terms of logic, we could imagine that the space is a “state space”, and the truth values in the logic identify for which states a given proposition is true.  There’s nothing particularly exotic about this: “it is raining” is a statement whose truth is local, in that it depends on where and when you happen to look.

To see locales as a generalisation of spaces, it helps to note that the embedding above is full – if $A$ and $B$ are locales that come from topological spaces, there are no extra morphisms in $\mathbf{Loc}(A,B)$ that don’t come from continuous maps in $\mathbf{Top}(A,B)$.  So the category of locales makes the category of topological spaces bigger only by adding more objects – not inventing new morphisms.  The analogous noncommutative statement turns out not to be true for quantales, which is a little red-flag warning which Pedro Resende pointed out to me.

What would this statement be?  Well, the noncommutative analogue of the idea of a topological space comes from another embedding of categories.  To start with, there is an equivalence $\mathbf{LCptHaus}^{op} \simeq \mathbf{CommC}^{\star}\mathbf{Alg}$: the category of locally compact, Hausdorff, topological spaces is (up to equivalence) the opposite of the category of commutative $C^{\star}$-algebras.  So one simply takes the larger category of all $C^{\star}$-algebras (or rather, its opposite) as the category of “noncommutative spaces”, which includes the commutative ones – the original locally compact Hausdorff spaces.  The correspondence between an algebra and a space is given by taking the algebra of functions on the space.

So what is a quantale?  It’s a lattice which is formally similar to the lattice of subspaces in some $C^{\star}$-algebra.  Special elements – “right”, “left,” or “two-sided” elements – then resemble those subspaces that happen to be ideals.  Some intuition comes from thinking about where the two generalizations coincide – a (locally compact) topological space.  There is a lattice of open sets, of course.  In the algebra of continuous functions, each open set $O$ determines an ideal – namely, the subspace of functions which vanish on $O$.  When such an ideal is norm-closed, it will correspond to an open set (it’s easy to see that continuous functions which can be approximated by those vanishing on an open set will also do so – if the set is not open, this isn’t the case).

So the definition of a quantale looks much like that for a locale, except that the meet operation $\bigwedge$ is replaced by an associative product, usually called $\&$.  Note that unlike the meet, this isn’t assumed to be commutative – this is the point where the generalization happens.  So in particular, any locate gives a quantale with $\& = \bigwedge$.  So does any $C^{\star}$-algebra, in the form of its lattice of ideals.  But there are others which don’t show up in either of these two ways, so one might hope to say this is a nice all-encompassing generalisation of the idea of space.

Now, as I said, there was a bit of a warning that comes attached to this hope.  This is that, although there is an embedding of the category of $C^{\star}$-algebras into the category of quantales, it isn’t full.  That is, not only does one get new objects, one gets new morphisms between old objects.  So, given algebras $A$ and $B$, which we think of as noncommutative spaces, and a map of algebras between them, we get a morphism between the associated quantales – lattice maps that preserve the operations.  However, unlike what happened with locales, there are quantale morphisms that don’t correspond to algebra maps.  Even worse, this is still true even in the case where the algebras are commutative, and just come from locally compact Hausdorff spaces: the associated quantales still may have extra morphisms that don’t come from continuous functions.

There seem to be three possible attitudes to this situation.  First, maybe this is just the wrong approach to generalising spaces altogether, and the hints in its favour are simply misleading.  Second, maybe quantales are absolutely the right generalisation of space, and these new morphisms are telling us something profound and interesting.  The third attitude, which Pedro mentioned when pointing out this problem to me, seems most likely, and goes as follows.  There is something special that happens with $C^{\star}$-algebras, where the analytic structure of the norm makes the algebras more rigid than one might expect.  In algebraic geometry, one can take a space (algebraic variety or scheme) and consider its algebra of global functions.  To make sure that an algebra map corresponds to a map of schemes, though, one really needs to make sure that it actually respects the whole structure sheaf for the space – which describe local functions.  When passing from a topological space to a $C^{\star}$-algebra, there is a norm structure that comes into play, which is rigid enough that all algebra morphisms will automatically do this – as I said above, the structure of ideals of the algebra tells you all about the open sets.  So the third option is to say that a quantale in itself doesn’t quite have enough information, and one needs some extra data something like the structure sheaf for a scheme.  This would then pick out which are the “good” morphisms between two quantales – namely, the ones that preserve this extra data.  What, precisely, this data ought to be isn’t so clear, though, at least to me.

So there are some complications to treating a quantale as a space.  One further point, which may or may not go anywhere, is that this type of lattice doesn’t quite get along with quantum logic in quite the same way that locales get along with (intuitionistic) classical logic (though it does have connections to linear logic).

In particular, a quantale is a distributive lattice (though taking the product, rather than $\bigwedge$, as the thing which distributes over $\bigvee$), whereas the “propositional lattice” in quantum logic need not be distributive.  One can understand the failure of distributivity in terms of the uncertainty principle.  Take a statement such as “particle $X$ has momentum $p$ and is either on the left or right of this barrier”.  Since position and momentum are conjugate variables, and momentum has been determined completely, the position is completely uncertain, so we can’t truthfully say either “particle $X$ has momentum $p$ and is on the left or “particle $X$ has momentum $p$ and is on the right”.  Thus, the combined statement that either one or the other isn’t true, even though that’s exactly what the distributive law says: “P and (Q or S) = (P and Q) or (P and S)”.

The lack of distributivity shows up in a standard example of a quantum logic.  This is one where the (truth values of) propositions denote subspaces of a vector space $V$.  “And” (the meet operation $\bigwedge$) denotes the intersection of subspaces, while “or” (the join operation $\bigvee$) is the direct sum $\oplus$.  Consider two distinct lines through the origin of $V$ – any other line in the plane they span has trivial intersection with either one, but lies entirely in the direct sum.  So the lattice of subspaces is non-distributive.  What the lattice for a quantum logic should be is orthocomplemented, which happens when $V$ has an inner product – so for any subspace $W$, there is an orthogonal complement $W^{\bot}$.

Quantum logics are not very good from a logician’s point of view, though – lacking distributivity, they also lack a sensible notion of implication, and hence there’s no good idea of a proof system.  Non-distributive lattices are fine (I just gave an example), and very much in keeping with the quantum-theoretic strategy of replacing configuration spaces with Hilbert spaces, and subsets with subspaces… but viewing them as logics is troublesome, so maybe that’s the source of the problem.

Now, in a quantale, there may be a “meet” operation, separate from the product, which is non-distributive, but if the product is taken to be the analog of “and”, then the corresponding logic is something different.  In fact, the natural form of logic related to quantales is linear logic. This is also considered relevant to quantum mechanics and quantum computation, and as a logic is much more tractable.  The internal semantics of certain monoidal categories – namely, star-autonomous ones (which have a nice notion of dual) – can be described in terms of linear logic (a fairly extensive explanation is found in this paper by Paul-André Melliès).

Part of the point in the connection seems to be resource-limitedness: in linear logic, one can only use a “resource” (which, in standard logic, might be a truth value, but in computation could be the state of some memory register) a limited number of times – often just once.  This seems to be related to the noncommutativity of $\&$ in a quantale.  The way Pedro Resende described this to me is in terms of observations of a system.  In the ordinary (commutative) logic of a locale, you can form statements such as “A is true, AND B is true, AND C is true” – whose truth value is locally defined.  In a quantale, the product operation allows you to say something like “I observed A, AND THEN observed B, AND THEN observed C”.  Even leaving aside quantum physics, it’s not hard to imagine that in a system which you observe by interacting with it, statements like this will be order-dependent.  I still don’t quite see exactly how these two frameworks are related, though.

On the other hand, the kind of orthocomplemented lattice that is formed by the subspaces of a Hilbert space CAN be recovered in (at least some) quantale settings.  Pedro gave me a nice example: take a Hilbert space $H$, and the collection of all projection operators on it, $P(H)$.  This is one of those orthocomplemented lattices again, since projections and subspaces are closely related.  There’s a quantale that can be formed out of its endomorphisms, $End(P(H))$, where the product is composition.  In any quantale, one can talk about the “right” elements (and the “left” elements, and “two sided” elements), by analogy with right/left/two-sided ideals – these are elements which, if you take the product with the maximal element, $1$, the result is less than or equal to what you started with: $a \& 1 \leq a$ means $a$ is a right element.  The right elements of the quantale I just mentioned happen to form a lattice which is just isomorphic to $P(H)$.

So in this case, the quantale, with its connections to linear logic, also has a sublattice which can be described in terms of quantum logic.  This is a more complicated situation than the relation between locales and intuitionistic logic, but maybe this is the best sort of connection one can expect here.

In short, both in terms of logic and spaces, hoping quantales will be “just” a noncommutative variation on locales seems to set one up to be disappointed as things turn out to be more complex.  On the other hand, this complexity may be revealing something interesting.

Coming soon: summaries of some talks I’ve attended here recently, including Ivan Smith on 3-manifolds, symplectic geometry, and Floer cohomology.

It’s the last week of classes here at UWO, and things have been wrapping up. There have also been a whole series of interesting talks, as both Doug Ravenel and Paul Baum have been visiting members of the department. Doug Ravenel gave a colloquium explaining work by himself, and collaborators Mike Hopkins and Mike Hill, solving the “Kervaire Invariant One” problem – basically, showing that certain kinds of framed manifolds – and, closely related, certain kinds of maps between spectra – don’t exist (namely, those where the Kervaire invariant is nonzero). This was an interesting and very engaging talk, but as a colloqium it necessarily had to skip past some of the subtleties of stable homotopy theory involved, and since my understanding of this subject is limited, I don’t really know if I could do it justice.

In any case, I have my work cut out for me with what I am going to try to do (taking blame for any mistakes or imprecisions I introduce in here, BTW, since I may not be able to do this justice either). This is to discussing the first two of four talks which Paul Baum gave here last week, starting with an introduction to K-theory, and ending up with some discussion of the Baum-Connes Conjecture. This is a famous conjecture in noncommutative geometry which Baum and Alain Connes proposed in 1982 (and which Baum now seems to be fairly convinced is probably not true, though nobody knows a counterexample at the moment).

It’s a statement about (locally compact, Hausdorff, topological) groups $G$; it relates K-theory for a $C^{\star}$-algebra associated to $G$, with the equivariant K-homology of a space associated to $G$ (in fact, it asserts a certain map $\mu$, which always exists, is furthermore always an isomorphism). It implies a great many things about any case where it IS true, which includes a good many cases, such as when $G$ is commutative, or a compact Lie group. But to backtrack, we need to define those terms:

K-Theory

The basic point of K-theory, which like a great many things began with Alexandre Grothendieck, is that it defines some invariants – which happen to be abelian groups – for various entities. There is a topological and an algebraic version, so the “entities” in question are, in the first case, topological spaces, and in the second, algebras (and more classically, algebraic varieties). Part of Paul Baum’s point in his talk was to describe the underlying unity of these two – essentially, both correspond to particular kinds of algebras. Taking this point of view has the added advantage that it lets you generalize K-theory to “noncommutative spaces” quite trivially. That is: the category of locally compact topological spaces is equivalent to the opposite category of commutative $C^{\star}$-algebras – so taking the opposite of the category of ALL $C^{\star}$ algebras gives a noncommutative generalization of “space”. Defining K-theory in terms of $C^{\star}$ algebras extends the invariant to this new sort of space, and also somewhat unifies topological and algebraic K-theory.

Classically, anyway, Atiyah and Hirzebruch’s definition for K-theory (adapted to the topological case by Adams) gives an abelian group from a (topological or algebraic) space $X$, using the category of (respectively, topological or algebraic) vector bundles over $X$. The point is, from this category one naturally gets a set of isomorphism classes of bundles, with a commutative addition (namely, direct sum) – this is an abelian semigroup. One can turn any abelian semigroup $J$ (with or without zero) into an abelian group, by taking pairs – $J \oplus J$, and taking the quotient by the relation $(x,y) \sim (x',y')$ which holds when there is $z \in J$ with $x + y' + z = x' + y + z$. This is like taking “formal differences” (and any $(x,x)$ becomes zero, even if there was no zero originally). In fact, it does a little more, since if $x$ and $x'$ are not equal, but become equal upon adding some $z$, they’re forced to be equal (so an equivalence relation is being imposed on bundles as well as allowing formal inverses).

In fact, a definition equivalent to Atiyah and Hirzebruch’s (in terms of bundles) can be given in terms of the coordinate ring of a variety $X$, or ring of continuous complex-valued functions on a (compact, Hausdorff) topological space $X$. Given a ring $\Lambda$, one defines $J(\Lambda)$ to be the abelian semigroup of all idempotents (i.e. projections) in the rings of matrices $M_n(\Lambda)$ up to STABLE similarity. Two idempotent matrices $\alpha$ and $\beta$ are equivalent if they become similar – that is, conjugate matrices – possibly after adjoining some zeros by the direct sum $\oplus$. (In particular, this means we needn’t assume $\alpha$ and $\beta$ were the same size). Then $K_0^{alg}(\Lambda)$ comes from this $J(\Lambda)$ by the completion to a group as just described.

A class of idempotents (projections) in a matrix algebra over $\mathbb{C}$ is characterized by the image, up to similarity (so, really, the dimension). Since these are matrices over a ring of functions on a space, we’re then secretly talking about vector bundles over that space. However, defining things in terms of the ring $\Lambda$ is what allows the generalization to noncommutative spaces (where there is no literal space, and the “coordinate ring” is no longer commutative, but this construction still makes sense).

Now, there’s quite a bit more to say about this – it was originally used to prove the Hirzebruch-Riemann-Roch theorem, which for nice projective varieties $M$ defines an invariant from the alternating sum of dimensions of some sheaf-cohomology groups – roughly, cohomology where we look at sections of the aforementioned vector bundles over $M$ rather than functions on $M$. The point is that the actual cohomology dimensions depend sensitively on how you turn an underlying topological space into an algebraic variety, but the HRR invariant doesn’t. Paul Baum also talked a bit about some work by J.F. Adams using K-theory to prove some results about vector fields on spheres.

For the Baum-Connes conjecture, we’re looking at the K-theory of a certain $C^{\star}$-algebra. In general, given such an algebra $A$, the (level-j) K-theory $K_j(A)$ can be defined to be the $(j-1)$ homotopy group of $GL(A)$ – the direct limit of all the finite matrix algebras $GL_n(A)$, which have a chain of inclusions under extensions where $GL_n(A) \rightarrow GL_{n+1}(A)$ by direct sum with the 1-by-1 identity. This looks a little different from the algebraic case above, but they are closely connected – in particular, under this definition $K_0(A)$ is just the same as $K^{alg}_0(A)$ as defined above (so the norm and involution on $A$ can be ignored for the level-0 K-theory of a $C^{\star}$-algebra, though not for level-1).

You might also notice this appears to define $K_0(A)$ in terms of negative-one-dimensional homotopy groups. One point of framing the definition this way is that it reveals that there are only two levels which matter – namely the even and the odd – so $K_0(A) = K_2(A) = K_4(A) \dots$, and $K_1(A) = K_3(A) = \dots$, and this detail turns out not to matter. This is a result of Bott periodicity. Changing the level of homotopy groups amounts to the same thing as taking loop spaces. Specifically, the functor $\Omega$ that takes the space of loops $\Omega(X)$ of a space $X$ is right-adjoint to the suspension functor $S$ – and since $S(S^{n-1}) = S^n$, this means that $\pi_{j+1}(X) = [S(S^n),X] \cong [S^n,\Omega(X)]$. (Note that $[S^n,X]$ is the group of homotopy classes of maps from the $n$-sphere into $X$). On the other hand, Bott periodicity says that $\Omega^2(GL(A)) \sim GL(A)$ – taking the loop-space twice gives something homotopic to the original $GL(A)$. So the tower of homotopy groups repeats every two dimensions. (So, in particular, one may as well take that $j-1$ to be $j+1$, and just find $K_2$ for $K_0$).

Now, to get the other side of the map in the Baum-Connes conjecture, we need a different part of K-theory.

K-Homology

Now, as with homology and cohomology, there are two related functors in the world of K-theory from spaces (of whatever kind) into abelian groups. The one described above is contravariant (for “spaces”, not algebras – don’t forget this duality!). Thus, maps $f : X \rightarrow Y$ give maps $K^0(f) : K^0(Y) \rightarrow K^0(X)$, which is like cohomology. There is also a covariant functor $K_0$ (so $f$ gives $K_0(f) : K_0(X) \rightarrow K_0(Y)$), appropriately called K-homology. If the K-theory is described in terms of vector bundles on $X$, K-homology – in the case of algebraic varieties, anyway – is about coherent sheaves of vector spaces on $X$ – concretely, you can think of these as resembling vector bundles, without a local triviality condition (one thinks, for instance, of the “skyscraper sheaf” which assigns a fixed vector space $V$ to any open set containing a given point $x \in X$, and $0$ to any other, which is like a “bundle” having fibre $V$ at $x$, and $0$ everywhere else – generalizations putting a given fibre on a fixed subvariety – and of course one can add such examples. This image explains why any vector bundle can be interpreted as a coherent sheaf – so there is a map $K^0 \rightarrow K_0$. When the variety $X$ is not singular, this turns out to be an isomorphism (the groups one ends up constructing after all the identifications involved turn out the same, even though sheaves in general form a bigger category to begin with).

But to take $K_0$ into the topological setting, this description doesn’t work anymore. There are different ways to describe $K_0$, but the one Baum chose – because it extends nicely to the NCG world where our “space” is a (not necessarily commutative) $C^{\star}$-algebra $A$ – is in terms of generalized elliptic operators. This is to say, triples $(H, \psi, T)$, where $H$ is a (separable) Hilbert space, $\psi$ is a representation of $A$ in terms of bounded operators on $H$, and $T$ is some bounded operator on $H$ with some nice properties. Namely, $T$ is selfadjoint, and for any $a \in A$, both its commutator with $\psi(a)$ and $\psi(a)(I - T^2)$ land in $\mathcal{K}(H)$, the ideal of compact operators. (This is the only norm-closed ideal in $\mathcal{L}(H)$, the bounded operators – the idea being that for this purpose, operators in this ideal are “almost” zero).

These are “abstract” elliptic operators – but many interesting examples are concrete ones – that is, $H = L^2(S)$ for some space $S$, and $T$ is describing some actual elliptic operator on functions on $S$. (He gave the case where $S$ is the circle, and $T$ is a version of the Dirac operator $-i \partial/\partial \theta$ – normalized so all its nonzero eigenvalues are $\pm 1$ – then we’d be doing K-homology for the circle.)

Then there’s a notion of homotopy between these operators (which I’ll elide), and the collection of these things up to homotopy forms an abelian group, which is called $K^1(A)$. This is the ODD case – that is, there’s a tower of groups $K^j(A)$, but due to Bott periodicity they repeat with period 2, so we only need to give $K^0(A)$ and $K^1(A)$. The definition for $K^0(A)$ is similar to the one for $K^1(A)$, except that we drop the “self-adjoint” condition on $T$, which necessitates expanding the other two conditions – there’s a commutator for both $T$ and $T^*$, and the condition for $T^2$ becomes two conditions, for $TT^*$ and $T^* T$.  Now, all these $K^j(A)$ should be seen as the K-homology groups $K_j(X)$ of spaces $X$ (the sub/super script is denoting co/contra-variance).

Now, for the Baum-Connes conjecture, which is about groups, one actually needs to have an equivariant version of all this – that is, we want to deal with categories of $G$-spaces (i.e. spaces with a $G$-action, and maps compatible with the $G$-action). This generalizes to noncommutative spaces perfectly well – there are $G$-$C^{\star}$-algebras with suitable abstract elliptic operators (one needs a unitary representation of $G$ on the Hilbert space $H$ in the triple to define the compatibility – given by a conjugation action), $G$-homotopies, and so forth, and then there’s an equivariant K-homology group, $K^G_j(X)$ for a $G$-space $X$.  (Actually, for these purposes, one cares about proper $G$-actions – ones where $X$ and the quotient space are suitably nice).

Baum-Connes Conjecture

Now, suppose we have a (locally compact, Hausdorff) group $G$. The Baum-Connes conjecture asserts that a map $\mu$, which always exists, between two particular abelian groups found from K-theory, is always an isomorphism. In fact, this is supposed to be true for the whole complex of groups, but by Bott periodicity, we only need the even and the odd case. For simplicity, let’s just think about one of $j=0,1$ at a time.

So then the first abelian group associated to $G$ comes from the equivariant K-homology for $G$-spaces.  In particular, there is a classifying space $\underline{E}G$ – this is the terminal object in a category of (“proper”) $G$-spaces (that is, any other $G$-space has a $G$-map into $\underline{E}G$). The group we want is the equivariant K-homology of this space: $K_j^G(\underline{E}G)$.  Since $\underline{E}G$ is a terminal object among $G$-spaces, and $K_j$ is covariant, it makes sense that this group is a limit over $G$-spaces (with some caveats), so another way to define it is $K^G_j(\underline{E}G) = lim K^G_j(X)$, where the limit is over all ($G$-compact) $G$-spaces.  Now, being defined in this abstract way makes this a tricky thing to deal with computationally (which is presumably one reason the conjecture has resisted proof).  Not so for the second group:

The second group is $K_j(C^{\star}_r(G))$ the reduced $C^{\star}$-algebra of a (locally compact, Hausdorff topological) group $G$. To get this, you take the compactly supported continuous functions on $G$, with the convolution product, and then, thinking of these as acting on $L^2(G)$ by multiplication, take the completion in the algebra of all such operators. This is still closed under the convolution product. Then one takes the K-theory for this algebra at level $j$.

So then there is always a particular map $\mu : K^G_j(\underline{E}G) \rightarrow K_j(C^{\star}_r(G)$, which is defined in terms of index theory.  The conjecture is that this is always an isomorphism (which, if true, would make the equivariant K-homology much more tractable).  There aren’t any known counterexamples, and in fact this is known to be true for all finite groups, and compact Lie groups – but for infinite discrete groups, there’s no proof known.  Indeed, it’s not even known whether it’s true for some specific, not very complicated groups, notably $SL(3,\mathbb{Z})$ – the 3-by-3 integer matrices of determinant 1.

In fact, Paul Baum seemed to be pretty confident that the conjecture is wrong (that there is a counterexample $G$) – essentially because it implies so many things (the Kadison-Kaplansky conjecture, that groups with no torsion have group rings with no idempotents; the Novikov conjecture, that certain manifold invariants coming from $G$ are homotopy invariants; and many more) that it would be too good to be true.  However, it does imply all these things about each particular group it holds for.

Now, I’ve not learned much about K-theory in the past, but Paul Baum’s talks clarified a lot of things about it for me.  One thing I realized is that some invariants I’ve thought more about, in the context of Extended TQFT – which do have to do with equivariant coherent sheaves of vector spaces – are nevertheless not the same invariants as in K-theory (at least in general).  I’ve been asked this question several times, and on my limited understanding, I thought it was true – for finite groups, they’re closely related (the 2-vector spaces that appear in ETQFT are abelian categories, but you can easily get abelian groups out of them, and it looks to me like they’re the K-homology groups).  But in the topological case, K-theory can’t readily be described in these terms, and furthermore the ETQFT invariants don’t seem to have all the identifications you find in K-theory – so it seems in general they’re not the same, though there are some concepts in common. But it does inspire me to learn more about K-theory.

Coming up: more reporting on talks from our seminar on Stacks and Groupoids, by Tom Prince and Jose Malagon-Lopez, who were talking about stacks in terms of homotopical algebra and category theory.

It’s been a while since I posted here, partly because I was working on getting this paper ready for submission. Since I wrote about its subject in my previous post, about Derek Wise’s talk at Perimeter Institute, I’ll let that stand for now. In the meantime, we’ve had a few talks in the seminar on stacks and groupoids. Tom Prince gave a couple of interesting talks about stacks from the point of view of simplicial sheaves, explaining how they can be seen as certain categories of objects satisfying descent. Since I only have handwritten notes on this talk, and I still haven’t entirely digested it, I think I’ll talk about that at the same time as discussing the upcoming talk about descent and related stuff by José Malagon-Lopez. For right now, I’ll write about Enxin Wu’s talk on diffeological bundles and the irrational torus. (DVI notes here)  Some of the theory of diffeological spaces has been worked out by Souriau (originally) and then Patrick Iglesias-Zemmour.  Some of the categorical properties he discussed are explained by Baez and Hoffnung (Enxin’s notes give some references).  Enxin and Dan Christensen have looked a bit at diffeological spaces in the context of homotopy theory and model categories.

Part of the motivation for this seminar was to look at how groupoids and some related entities, namely stacks, and algebra in the form of noncommutative geometry (although we didn’t get as much on this as I’d hoped), can be treated as ways to expand the notion of “space”. One reason for doing this is to handle certain kinds of moduli problems, but another – more directly related to the motivation for noncommutative geometry (NCG) – is to deal with certain quotients. The irrational torus is one of these, and under the name “noncommutative torus” is a standard example in NCG. A brief introduction to it by John Baez can be found here, and more detailed discussion is in, for example, Ch3, section 2.β of Connes’ “Noncommutative Geometry“, which describes how to find its cyclic cohomology (a noncommutative anolog of cohomology of a space), which turns out to be 2-dimensional.

The point here should be to think of it as the quotient of a space by a group action. (Which gives a transformation groupoid, and from there a – noncommutative – groupoid $C^{\star}$-algebra). The space is a torus , and the group acting on it is $\mathbb{R}$ acting by translation parallel to a line with irrational slope. In particular, we can treat $T^2$ as a group $\{ (e^{ix},e^{iy}) | x,y \in \mathbb{R} \}$ with componentwise multiplication, and think of the irrational torus, given an irrational $\theta$, as the quotient $T^2/\mathbb{R}_{\theta}$ by the subgroup $\mathbb{R}_{\theta} = \{ (e^{ix},e^{i \theta x}) \}$.

Now, this is quite well-defined as a set, but as a space it’s quite horrible, even though both groups are quite nice Lie groups. In particular, the subgroup $\mathbb{R}_{\theta}$ is dense in $T^2$ – or, thought of in terms of a group acting on the torus, the orbit space of any given point is dense. So the quotient is not a manifold – in fact, it’s quite hard to visualize. This illustrates the point that smooth manifolds are badly behaved with respect to quotients. In his talk, Enxin told us about another way to approach this problem by moving to the category of diffeological spaces. As I mentioned in a previous post, this is one of a number of attempts to expand the category of smooth manifolds $\mathbf{Mfld}$, to get a category which has nice properties $\mathbf{Mfld}$ does not have, such as having quotient objects, mapping objects, and so on. Now, the category $\mathbf{Top}$ is such an example, but this loses all the information about which maps are smooth. The point is to find some intermediate generalization, which still carries information about geometry, not just topology.

A diffeological space can be defined as a concrete (i.e. $\mathbf{Set}$-valued) sheaf on the site whose objects are open neighborhoods of $\mathbb{R}^n$ (for all $n$) and whose morphisms are smooth maps, though this is sort of an abstract way to define a space. The point of it, however, is that this site gives a model of all the maps we want to call “smooth”. Defining the category $\mathbf{Diff}$ of diffeological spaces in terms of sheaves on sites helps to ensure it has nice categorical properties, but more intuitively, a smooth space $X$ is described by giving a set, and defining all the smooth maps into the space from neighborhoods of $\mathbb{R}^n$ (these are called plots, and the collection is a diffeology). This differs from a manifold, which is defined in terms of (ahem) an atlas of charts – which unlike plots are required to be local homeomorphisms into a topological space, which fit together in smooth ways. The smooth maps into $X$ also have to be compatible – which is what the condition of being a sheaf guarantees – but the point is that we no longer suppose $X$ locally looks just like $\mathbb{R}^n$, so it can include strange quotients like the irrational torus.

Now $\mathbf{Diff}$ has lots of good properties, some of which are listed in Enxin’s notes. For instance, it has all limits and colimits, and is cartesian closed. What’s more, there’s a pair of adjoint functors between $\mathbf{Top}$ and $\mathbf{Diff}$ – so there’s an “underlying topological space” for any diffeological space (a topology making all the plots continuous), and a free diffeology on any topological space (where any continuous map from a neighborhood in $\mathbb{R}^n$ is smooth). There’s also a natural diffeology on any manifold (the one generated by taking all the charts to be plots).

The real point, though, is that a lot of standard geometric constructions that are made for manifolds also make sense for diffeological spaces, so they “support geometry”. Some things which can be defined in the context of $\mathbf{Diff}$ include: dimension; tangent spaces; differential forms; cohomology; smooth homotopy groups.

Naturally, one can define a diffeological groupoid: this is just an internal groupoid in $\mathbf{Diff}$ – there are diffeological spaces $Ob$ and $Mor$ of objects and morphisms (and of course composable pairs, $Mor \times_{Ob} Mor$, which, being a limit, is also in $\mathbf{Diff}$), and the structure maps are all smooth. These are related to diffeological bundles (defined below) in that certain groupoids can be build from bundles. The resulting groupoids all have the property of being perfect, which means that $(s,t) : Mor \rightarrow Ob \times Ob$ is a subduction – i.e. is onto, and such that the product diffeology which is the natural one on $Ob \times Ob$ is also the minimal one making this map smooth.

In fact, we need this to even define diffeological bundles, which are particular kinds of surjective maps $f : X \rightarrow Y$ in $\mathbf{Diff}$. Specifically, one gets a groupoid $K_f$ whose objects are points of $Y$, and where the morphisms $hom(y,y')$ are just smooth maps from the fibre $f^{-1}(y)$ to the fibre $f^{-1}(y')$ (which, of course, are diffeological spaces because they are subsets of $X$). It’s when this groupoid is perfect that one has a bundle.

The point here is that, unlike for manifolds, we don’t have local charts, so we can’t use the definition that a bundle is “locally trivializable”, but we do have this analogous condition. In both cases, the condition implies that all the fibres are diffeomorphic to each other (in the relevant sense). Enxin also gave a few equivalent conditions, which amount to saying one gets locally trivial bundles over neighborhoods in $\mathbb{R}^n$ when pulling back $f$ along any plot.

So now we can at least point out that the irrational torus can be construed as a diffeological bundle – thinking of it as a quotient of a group by a subgroup, we can think of this as a bundle where $X = T^2$ is the total space, the base $Y$ is the space of orbits, and the fibres are all diffeomorphic to $F = \mathbb{R}_{\theta}$.

The punchline of the talk is to use this as an example which illustrates the theorem that there is a diffeological version of the long exact sequence of homotopy groups:

$\dots \rightarrow \pi_n^D(F) \rightarrow \pi_n^D(X) \rightarrow \pi_n^D(Y) \rightarrow \pi_{n-1}^D(F) \rightarrow \dots$

Using this long exact sequence, and the fact that the (diffeological) homotopy groups for manifolds (in this case, $X = T^2$ and $F = \mathbb{R}_{\theta}$ are the same as the usual ones, one can work out the homotopy groups for the base $Y$, which is the quotient $T^2/\mathbb{R}_{\theta}$. Whereas, for topological spaces, since $\mathbb{R}_{\theta}$ is dense in $T^2$, the usual homotopy groups are all zero, for diffeological spaces, we get a different answer. In particular, $\pi_1^D(Y) = \mathbb{Z} \oplus \mathbb{Z}$, a two-dimensional lattice.

It’s interesting that this essentially agrees with what noncommutative geometry tells us about the quotient, while keeping some of our plain intuitions about “space” intact – that is, without moving whole-hog into (the opposite of) a category of noncommutative algebras. It would be interesting to know how far one can push this correspondence.

While I’d like to write up right away a description of the talk which Derek Wise gave recently at the Perimeter Institute (mostly about some work of mine which is preliminary to a collaboration we’re working on), I think I’ll take this next post as a chance to describe a couple of talks given in the seminar on stacks, groupoids, and algebras which I’m organizing, namely mine on representation theory of groupoids (focusing on Morita equivalence), and Peter Oman’s, called Toposes and Groupoids about how any topos can be compared to sheaves on a groupoid (sort of!). So here we go:

Representations of Groupoids and Morita Equivalence

The motivation here is to address directly what Morita equivalence means for groupoids, and particularly Lie groupoids. (One of the main references I used to prepare on this was this paper by Klaas Landsman, which gives Morita equivalence theorems for a variety of bicategories). The classic description of a Morita equivalence of rings $R$ and $L$ is often described in terms of the existence of an $L$-$R$-bimodule $M$ having certain properties. But the point of this bimodule is that on can turn $R$-modules into $L$-modules by tensoring with it, and vice versa. Actually, it’s better than this, in that there are functors

$- \otimes_L M : Mod(L) \rightarrow Mod(R)$

and

$M \otimes_R - : Mod(R) \rightarrow Mod(L)$

And moreover, either composite of these is naturally isomorphic to the appropriate identity, so in particular one has $M \otimes_L M \cong R$ and $M \otimes_R M \cong L$ (since tensoring with the base ring is the identity for modules). But this just says that these two functors are actually giving an equivalence of the categories $L-Mod$ and $R-Mod$.

So this is the point of Morita equivalence. Suppose, for a class of algebraic gadget (ring, algebra, groupoid, etc.), one has the notion of a representation of such a gadget $R$ (as a module is the right idea of the representation of a ring), and all the representations of $R$ form a category $Rep(R)$. Then Morita equivalence is the equivalence relation induced by equivalence of the representation categories – gadgets $R$ and $L$ are Morita equivalent if there is an equivalence of the representation categories. For nice categories of gadgets – rings and von Neumann algebras, for instance, this occurs if and only if a condition like the existence of the bimodule $M$ above is true. In other cases, this is only a sufficient condition for Morita equivalence, not a necessary one.

I’ll comment here that there are therefore several natural notions of Morita equivalence, which a priori might be different, since categories like $Rep(R)$ carry quite a bit of structure. For example, there is a tensor product of representations that makes it a symmetric monoidal category; there is a direct sum of representations making it abelian. So we might want to ask that the equivalence between them be an equivalence of:

• categories
• abelian categories
• monoidal abelian categories
• symmetric monoidal abelian categories

(in principle we could also take the last two and drop “abelian”, for a total of six versions of the concept, but this progression is most natural in much the same way that “set – abelian group – ring” is a natural progression).

Reallly, what one wants is the strongest of these notions. Equivalence as abelian categories just means having the same number of irreducible representations (which are the generators). It’s less obvious that the “symmetric” qualifier is important, but there are examples where these are different.

So then one gets Morita equivalence for groupoids $\mathbf{G}$ from the categories $Rep(\mathbf{G})$ in this standard way. One point here is that, whereas representations of groups are actions on vector spaces, representations of groupoids are actions on vector bundles $E$ over the space of objects of $\mathbf{G}$ (call this $M$). So for A morphism from $x \in M$ to $y \in M$, the representation gives a linear map from the fibre $E_x$ to the fibre $E_y$ (which is necessarily iso).

The above paper by Landsman is nice in that it defines this concept for several different categories, and gives the corresponding versions of a theorem showing that this Morita equivalence is either the same as, or implied by (depending on the case) equivalence in a certain bicategory. For Lie groupoids, this bicategory $\mathbf{LG}$ has Lie groupoids for objects, certain bibundles as morphisms, and bibundle maps as 2-morphisms – the others are roughly analogous. The bibundles in question are called “Hilsum-Skandalis maps” (on this, I found Janez Mrcun’s thesis a useful place to look). This $\mathbf{LG}$ does in this context essentially what the bicategory of spans does for finite groupoids (many simplifying features about the finite case obscure what’s really going on, so in some ways it’s better to look at this case).

The general phenomenon here is the idea of “strong Morita equivalence” of rings/algebras/groupoids $R$ and $L$. What, precisely, this means depends on the setting, but generally it means there is some sort of interpolating object between $R$ and $L$. The paper by Landsman gives specifics in various cases – the interpolating object may be a bimodule, or bibundle of some sort (these Hilsum-Skandalis maps), and in the case of discrete groupoids one can derive this from a span. In any case, strong Morita equivalence appears to amount to an equivalence internal to a bicategory in which these are the morphisms (and the 2-morphisms are something natural, such as bimodule maps in the case of rings – just linear maps compatible with the left and right actions on two bimodules). In all cases, strong Morita equivalence implies Morita equivalence, but only in some cases (not including the case of Lie groupoids) is the converse true.

There are more details on this in my slides, and in the references above, but now I’d like to move on…

Toposes and Groupoids

Peter Oman gave the most recent talk in our seminar, the motivation for which is to explain how the idea of a topos as a generalization of space fits in with the idea of a groupoid as a generalization of space. As a motivation, he mentioned a theorem of Butz and Moerdijk, that any topos $\mathcal{E}$ with “enough points” is equivalent to the topos of sheaves on some topological groupoid. The generalization drops the “enough points” condition, to say that any topos is equivalent to a topos of sheaves on a localic groupoid. Locales are a sort of point-free generalization of topological spaces – they are distributive lattices closed under meets and finite joins, just like the lattice of open sets in a topological space (the meet and join operations then are just unions and intersections). Actually, with the usual idea of a map of lattices (which are functors, since a lattice is a poset, hence a category), the morphisms point the wrong way, so one actually takes the opposite category, $\mathbf{Loc}$.

(Let me just add that as a generalization of space that isn’t essentially about points, this is nice, but in a rather “commutative” way. There is a noncommutative notion, namely quantales, which are related to locales in rather the same way lattices of subspaces of a Hilbert space $H$ relate to those of open sets in a space. It would be great if an analogous theorem applied there, but neither I nor Peter happen to know if this is so.)

Anyway, the main theorem (due to Joyal and Tierney, in “An Extension of the Galois Theory of Grothendieck” – though see this, for instance) is that one can represent any topos as sheaves on a localic groupoid – ie. internal groupoids in $\mathbf{Loc}$.

The essential bit of these theorems is localic reflection. This refers to an adjoint pair of functors between $\mathbf{Loc}$ and $\mathbf{Top}$. The functor $pt : \mathbf{Loc} \rightarrow \mathbf{Top}$ gives the space of points of a locale (i.e. atomic elements of the lattice – those with no other elements between them and the minimal elements which corresponds to the empty set in a topology). The functor $Loc : \mathbf{Top} \rightarrow \mathbf{Loc}$ gives, for any topological space, the locale which is its lattice of open sets. This adjunction turns out to give an equivalence when one restricts to “sober” spaces (for example, Hausdorff spaces are sober), and locales with “enough points” (having no other context for the term, I’ll take this to be a definition of “enough points” for the time being).

Now, part of the point is that locales are a generalization of topological space, and topoi generalize this somewhat further: any locale gives rise to a topos of sheaves on it (analogous to the sheaf of continuous functions on a space). A topos $\mathcal{E}$ may or may not be equivalent to a topos of sheaves on a locale: i.e. $\mathcal{E} \simeq Sh(X)$ might hold for some locale $X$. If so, the topos is “localic”. Localic reflection just says that $Sh$ induces an equivalence between hom-categories in the 2-categories $\mathbf{Loc}$ and $\mathbf{Topoi}$. Now, not every topos is localic, but, there is always some locale such that we can compare $\mathcal{E}$ to $Sh(X)$.

In particular, given a map of locales (or even more particularly, a continuous map of spaces) $f : X \rightarrow Y$, there’s an adjoint pair of inverse-image and direct-image maps $f^*: Sh(Y) \rightarrow Sh(X)$ and $f_* : Sh(X) \rightarrow Sh(Y)$ for passing sheaves back and forth. This gives the idea of a “geometric morphism” of topoi, which is just such an adjoint pair. The theorem is that given any topos $\mathcal{E}$, these is some “surjective” geometric morphism $Sh(X) \rightarrow \mathcal{E}$ (surjectivity amounts to the claim that the inverse image functor $f^*$ is faithful – i.e. ignores no part of $\mathcal{E}$). Of course, this $f$ might not be an equivalence (so $\mathbf{Topoi}$ is bigger than $\mathbf{Loc}$).

Now, the point, however, is that this comparison functor means that $\mathcal{E}$ can’t be TOO much more general than
sheaves on a locale. The point is, given this geometric morphism $f$, one can form the pullback of $f$ along itself, to get a “fibre product” of topoi: $Sh(X) \times_{\mathcal{E}} Sh(X)$ with the obvious projection maps to $Sh(X)$. Indeed, one can get $Sh(X) \times_{\mathcal{E}} Sh(X) \times_{\mathcal{E}} Sh(X)$, and so on. It turns out these topoi, and these projection maps (thought of, via localic reflection, as locales, and maps between locales) can be treated as the objects and structure maps for a groupoid internal to $\mathbf{Loc}$. So in particular, we can think of $Sh(X) \times_{\mathcal{E}} Sh(X)$ as the locale of morphisms in the groupoid, and $Sh(X) \times_{\mathcal{E}} Sh(X) \times_{\mathcal{E}} Sh(X)$ as the locale of composable pairs of morphisms.

The theorem, then, is that $\mathcal{E}$ is related to the topos of sheaves on this localic groupoid. More particularly, it is equivalent to the subcategory of objects which satisfy a descent condition. Descent, of course, is a huge issue – and one that’s likely to get much more play in future talks in this seminar, but for the moment, it’s probably sufficient to point to Peter’s slides, and observe that objects which satisfy descent are “global” in some sense (in the case of a sheaf of functions on a space, they correspond to sheaves in which locally defined functions which match on intersections of open sets can be “pasted” to form global functions).

So part of the point here is that locales generalize spaces, and toposes generalize locales, but only about as far as groupoids generalize spaces (by encoding local symmetry). There is also a more refined version (due to Moerdijk and Pronk) that has to do with ringed topoi (which generalize ringed spaces), giving a few conditions which amount to being equivalent to the topos of sheaves on an orbifold (which has some local manifold-like structure, and where the morphisms in the groupoid are fairly tame in that the automorphism groups at each point are finite).

Coming up in the seminar, Tom Prince will be talking about an approach to this whole subject due to Rick Jardine, involving simplicial presheaves.

I say this is about a “recent” talk, though of course it was last year… But to catch up: Ivan Dynov was visiting from York and gave a series of talks, mainly to the noncommutative geometry group here at UWO, about the problem of classifying von Neumann algebras. (Strictly speaking, since there is not yet a complete set of invariants for von Neumann algebras known, one could dispute the following is a “classification”, but here it is anyway).

The first point is that any von Neumann algebra $\mathcal{A}$ is a direct integral of factors, which are highly noncommutative in that the centre of a factor consists of just the multiples of the identity. The factors are the irreducible building blocks of the noncommutative features of $\mathcal{A}$.

There are two basic tools that provide what classification we have for von Neumann algebras: first, the order theory for projections; second, the Tomita-Takesaki theory. I’ve mentioned the Tomita flow previously, but as for the first part:

A projection (self-adjoint idempotent) is just what it sounds like, if you reprpsent $\mathcal{M}$ as an algebra of bounded operators on a Hilbert space. An extremal but informative case is $\mathcal{M} = \mathcal{B}(H)$, but in general not every bounded operator appears in $\mathcal{M}$.

In the case where $\mathcal{M} = \mathcal{B}(H)$, then a projection in $\mathcal{M}$ is the same thing as a subspace of $H$. There is an (orthomodular) lattice of them (in general, the lattice of projections is $\mathcal{P(M)}$). For subspaces, the dimension characterizes $H$ up to isomorphism – any any two subspaces of the same dimension are isomorphic by some operator in $\mathcal{B}(H)$ (but not necessarily in a general $\mathcal{M}$).

The idea is to generalize this to projections in a general $\mathcal{A}$, and get some characterization of $\mathcal{A}$. The kind of isomorphism that matters for subspaces is a partial isometry – a map $u$ which preserves the metric on some subspace, and otherwise acts as a projection. In fact, the corresponding projections are then conjugate by $u$. So we define, for a general $\mathcal{M}$, an equivalence relation on projections, which amounts to saying that $e \sim f$ if there’s a partial isometry $u \in \mathcal{M}$ with $e = u*u$, and $f = uu*$ (i.e. the projections are conjugate by $u$).

Then there’s an order relation on the equivalence classes of projections – which, as suggested above, we should think of as generalizing “dimension” from the case $\mathcal{M} = \mathcal{B}(H)$. The order relation says that $e \leq f$ if $e \sim e_0$ where $e_0 \leq f$ as a projection (i.e. inclusion thinking of a projection as its image subspace of $H$). But the fact that $\mathcal{M}$ may not be all of $\mathcal{B}(H)$ has some counterintuitive consequences. For example, we can define a projection $e \in \mathcal{M}$ to be finite if the only time $e \sim e_0 \leq e$ is when $e_0 = e$ (which is just the usual definition of finite, relativized to use only maps in $\mathcal{M}$). We can call $e \in \mathcal{M}$ a minimal projection if it is nonzero and $f \leq e$ imples $f = e$ or $f = 0$.

Then the first pass at a classification of factors (i.e. “irreducible” von Neumann algebras) says a factor $\mathcal{M}$ is:

• Type $I$: If $\mathcal{M}$ contains a minimal projection
• Type $II$: If $\mathcal{M}$ contains no minimal projection, but contains a (nontrivial) finite projection
• Type $III$: If $\mathcal{M}$ contains no minimal or nontrivial finite projection

We can further subdivide them by following the “dimension-function” analogy, which captures the ordering of projections for $\mathcal{M} = \mathcal{B}(H)$, since it’s a theorem that there will be a function $d : \mathcal{P(M)} \rightarrow [0,\infty]$ which has the properties of “dimension” in that it gets along with the equivalence relation $\sim$, respects finiteness, and “dimension” of direct sums. Then letting $D$ be the range of this function, we have a few types. There may be more than one function $d$, but every case has one of the types:

• Type $I_n$: When $D = \{0,1,\dots,n\}$ (That is, there is a maximal, finite projection)
• Type $I_\infty$: When $D = \{ 0, 1, \dots, \infty \}$ (If there is an infinite projection in $\mathcal{M}$
• Type $II_1$: When $D = [ 0 , 1 ]$ (The maximal projection is finite – such a case can always be rescaled so the maximum $d$ is $1$)
• Type $II_\infty$: When $D = [ 0 , \infty ]$ (The maximal projection is infinite – notice that this has the same order type as type $II_1$)
• Type $III_\infty$ \: When $D = [0,\infty]$ (An infinite maximal projection)
• Type $III$: $D = \{0,1\}$, (these are called properly infinite)

The type $I$ case are all just (equivalent to) matrix algebras on some countable or finite dimensional vector space – which we can think of as a function space like $l_2(X)$ for some set $X$. Types $II$ and $III$ are more interesting. Type $II$ algebras are related to what von Neumann called “continuous geometries” – analogs of projective geometry (i.e. geometry of subspaces), with a continuous dimension function.

(If we think of these algebras $\mathcal{M}$ as represented on a Hilbert space $H$, then in fact, thought of as subspaces of $H$, all the projections give infinite dimensional subspaces. But since the definition of “finite” is relative to $\mathcal{M}$, and any partial isometry from a subspace $H' \leq H$ to a proper subspace $H'' < H'$ of itself that may exist in $\mathcal{B}(H)$ is not in $M$.)

In any case, this doesn’t exhaust what we know about factors. In his presentation, Ivan Dynov described some examples constructed from crossed products of algebras, which is important later, but for the moment, I’ll finish describing another invariant which helps pick apart the type $III$ factors. This is related to Tomita-Takesaki theory, which I’ve mentioned in here before.

You’ll recall that the Tomita flow (associated to a given state $\phi$) is given by $\sigma^{\phi}_t(A) = e^{i \Delta t} A e^{-i \Delta t}$, where $\Delta$ is the self-adjoint part of the conjugation operator $S$ (which depends on the state $\phi$ because it refers to the GNS representation of $\mathcal{M}$ on a Hilbert space $H$). This flow is uninteresting for Type $I$ or $II$ factors, but for type $III$ factors, it’s the basis of Connes’ classification.

In particular, the we can understand the Tomita flow in terms of eigenvalues of $\Delta$, since it comes from exponentials of $\Delta$. Moreover, as I commented last time, the really interesting part of the flow is independent of which state we pick. So we are interested in the common eigenvalues of the $\Delta$ associated to different states $\phi$, and define

$S(\mathcal{M}) = \cap_{\phi \in W} Spec(\Delta_{\phi})$

(where $W$ is the set of all states on $\mathcal{M}$, or actually “weights”)

Then $S(\mathcal{M}) - \{ 0 \}$, it turns out, is always a multiplicative subgroup of the positive real line, and the possible cases refine to these:

• $S(\mathcal{M}) = \{ 1 \}$ : This is when $\mathcal{M}$ is type $I$ or $II$
• $S(\mathcal{M}) = [0, \infty )$ : Type $III_1$
• $S(\mathcal{M}) = \{ 0 \} \cup \{ \lambda^n : n \in \mathbb{Z}, 0 < \lambda < 1 \}$ : Type $III_{\lambda}$ (for each $\lambda$ in the range $(0,1)$, and
• $S(\mathcal{M}) = \{ 0 , 1 \}$ : Type $III_0$

(Taking logarithms, $S(\mathcal{M}) - \{ 0 \}$ gives an additive subgroup of $\mathbb{R}$, $\Gamma(\mathcal{M})$ which gives the same information). So roughly, the three types are: $I$ finite and countable matrix algebras, where the dimension function tells everything; $II$ where the dimension function behaves surprisingly (thought of as analogous to projective geometry); and $III$, where dimensions become infinite but a “time flow” dimension comes into play.  The spectra of $\Delta$ above tell us about how observables change in time by the Tomita flow:  high eigenvalues cause the observable’s value to change faster with time, while low ones change slower.  Thus the spectra describe the possible arrangements of these eigenvalues: apart from the two finite cases, the types are thus a continuous positive spectrum, and a discrete one with a single generator.  (I think of free and bound energy spectra, for an analogy – I’m not familiar enough with this stuff to be sure it’s the right one).

This role for time flow is interesting because of the procedures for constructing examples of type $III$, which Ivan Dynov also described to us. These are examples associated with dynamical systems. These show up as crossed products. See the link for details, but roughly this is a “product” of an algebra by a group action – a kind of von Neumann algebra equivalent of the semidirect product of groups $H \rtimes K$ incorporating an action of $K$ on $H$. Indeed, if a (locally compact) group $K$ acts on group $H$ then the crossed product of algebras is just the von Neumann algebra of the semidirect product group.

In general, a ($W*$)-dynamical system is $(\mathcal{M},G,\alpha)$, where $G$ is a locally compact group acting by automorphisms on the von Neumann algebra $\mathcal{M}$, by the map $\alpha : G \rightarrow Aut(\mathcal{M})$. Then the crossed product $\mathcal{M} \rtimes_{\alpha} G$ is the algebra for the dynamical system.

A significant part of the talks (which I won’t cover here in detail) described how to use some examples of these to construct particular type $III$ factors. In particular, a theorem of Murray and von Neumann says $\mathcal{M} = L^{\infty}(X,\mu) \rtimes_{\alpha} G$ is a factor if the action of discrete group $G$ on a finite measure space $X$ is ergodic (i.e. has no nontrivial proper invariant sets – roughly, each orbit is dense). Another says this factor is type $III$ unless there’s a measure equivalent to (i.e. absolutely continuous with) $\mu$, and which is equivariant. Some clever examples I won’t reconstruct gave some factors like this explicitly.

He concluded by talking about some efforts to improve the classification: the above is not a complete set of invariants, so a lot of work in this area is improving the completeness of the set. One set of results he told us about do this somewhat for the case of hyperfinite factors (i.e. ones which are limits of finite ones), namely that if they are type $III$, they are crossed products of with a discrete group.

At any rate, these constructions are interesting, but it would take more time than I have here to look in detail – perhaps another time.

When I made my previous two posts about ideas of “state”, one thing I was aiming at was to say something about the relationships between states and dynamics. The point here is that, although the idea of “state” is that it is intrinsically something like a snapshot capturing how things are at one instant in “time” (whatever that is), extrinsically, there’s more to the story. The “kinematics” of a physical theory consists of its collection of possible states. The “dynamics” consists of the regularities in how states change with time. Part of the point here is that these aren’t totally separate.

Just for one thing, in classical mechanics, the “state” includes time-derivatives of the quantities you know, and the dynamical laws tell you something about the second derivatives. This is true in both the Hamiltonian and Lagrangian formalism of dynamics. The Hamiltonian function, which represents the concept of “energy” in the context of a system, is based on a function $H(q,p)$, where $q$ is a vector representing the values of some collection of variables describing the system (generalized position variables, in some configuration space $X$), and the $p = m \dot{q}$ are corresponding “momentum” variables, which are the other coordinates in a phase space which in simple cases is just the cotangent bundle $T*X$. Here, $m$ refers to mass, or some equivalent. The familiar case of a moving point particle has “energy = kinetic + potential”, or $H = p^2 / m + V(q)$ for some potential function $V$. The symplectic form on $T*X$ can then be used to define a path through any point, which describes the evolution of the system in time – notably, it conserves the energy $H$. Then there’s the Lagrangian, which defines the “action” associated to a path, which comes from integrating some function $L(q, \dot{q})$ living on the tangent bundle $TX$, over the path. The physically realized paths (classically) are critical points of the action, with respect to variations of the path.

This is all based on the view of a “state” as an element of a set (which happens to be a symplectic manifold like $T*X$ or just a manifold if it’s $TX$), and both the “energy” and the “action” are some kind of function on this set. A little extra structure (symplectic form, or measure on path space) turns these functions into a notion of dynamics. Now a function on the space of states is what an observable is: energy certainly is easy to envision this way, and action (though harder to define intuitively) counts as well.

But another view of states which I mentioned in that first post is the one that pertains to statistical mechanics, in which a state is actually a statisticial distribution on the set of “pure” states. This is rather like a function – it’s slightly more general, since a distribution can have point-masses, but any function gives a distribution if there’s a fixed measure $d\mu$ around to integrate against – then a function like $H$ becomes the measure $H d\mu$. And this is where the notion of a Gibbs state comes from, though it’s slightly trickier. The idea is that the Gibbs state (in some circumstances called the Boltzmann distribution) is the state a system will end up in if it’s allowed to “thermalize” – it’s the maximum-entropy distribution for a given amount of energy in the specified system, at a given temperature $T$. So, for instance, for a gas in a box, this describes how, at a given temperature, the kinetic energies of the particles are (probably) distributed. Up to a bunch of constants of proportionality, one expects that the weight given to a state (or region in state space) is just $exp(-H/T)$, where $H$ is the Hamiltonian (energy) for that state. That is, the likelihood of being in a state is inversely proportional to the exponential of its energy – and higher temperature makes higher energy states more likely.

Now part of the point here is that, if you know the Gibbs state at temperature $T$, you can work out the Hamiltonian
just by taking a logarithm – so specifying a Hamiltonian and specifying the corresponding Gibbs state are completely equivalent. But specifying a Hamiltonian (given some other structure) completely determines the dynamics of the system.

This is the classical version of the idea Carlo Rovelli calls “Thermal Time”, which I first encountered in his book “Quantum Gravity”, but also is summarized in Rovelli’s FQXi essay “Forget Time“, and described in more detail in this paper by Rovelli and Alain Connes. Mathematically, this involves the Tomita flow on von Neumann algebras (which Connes used to great effect in his work on the classification of same). It was reading “Forget Time” which originally got me thinking about making the series of posts about different notions of state.

Physically, remember, these are von Neumann algebras of operators on a quantum system, the self-adjoint ones being observables; states are linear functionals on such algebras. The equivalent of a Gibbs state – a thermal equilibrium state – is called a KMS (Kubo-Martin-Schwinger) state (for a particular Hamiltonian). It’s important that the KMS state depends on the Hamiltonian, which is to say the dynamics and the notion of time with respect to which the system will evolve. Given a notion of time flow, there is a notion of KMS state.

One interesting place where KMS states come up is in (general) relativistic thermodynamics. In particular, the effect called the Unruh Effect is an example (here I’m referencing Robert Wald’s book, “Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics”). Physically, the Unruh effect says the following. Suppose you’re in flat spacetime (described by Minkowski space), and an inertial (unaccelerated) observer sees it in a vacuum. Then an accelerated observer will see space as full of a bath of particles at some temperature related to the acceleration. Mathematically, a change of coordinates (acceleration) implies there’s a one-parameter family of automorphisms of the von Neumann algebra which describes the quantum field for particles. There’s also a (trivial) family for the unaccelerated observer, since the coordinate system is not changing. The Unruh effect in this language is the fact that a vacuum state relative to the time-flow for an unaccelerated observer is a KMS state relative to the time-flow for the accelerated observer (at some temperature related to the acceleration).

The KMS state for a von Neumann algebra with a given Hamiltonian operator has a density matrix $\omega$, which is again, up to some constant factors, just the exponential of the Hamiltonian operator. (For pure states, $\omega = |\Psi \rangle \langle \Psi |$, and in general a matrix becomes a state by $\omega(A) = Tr(A \omega)$ which for pure states is just the usual expectation value value for A, $\langle \Psi | A | \Psi \rangle$).

Now, things are a bit more complicated in the von Neumann algebra picture than the classical picture, but Tomita-Takesaki theory tells us that as in the classical world, the correspondence between dynamics and KMS states goes both ways: there is a flow – the Tomita flow – associated to any given state, with respect to which the state is a KMS state. By “flow” here, I mean a one-parameter family of automorphisms of the von Neumann algebra. In the Heisenberg formalism for quantum mechanics, this is just what time is (i.e. states remain the same, but the algebra of observables is deformed with time). The way you find it is as follows (and why this is right involves some operator algebra I find a bit mysterious):

First, get the algebra $\mathcal{A}$ acting on a Hilbert space $H$, with a cyclic vector $\Psi$ (i.e. such that $\mathcal{A} \Psi$ is dense in $H$ – one way to get this is by the GNS representation, so that the state $\omega$ just acts on an operator $A$ by the expectation value at $\Psi$, as above, so that the vector $\Psi$ is standing in, in the Hilbert space picture, for the state $\omega$). Then one can define an operator $S$ by the fact that, for any $A \in \mathcal{A}$, one has

$(SA)\Psi = A^{\star}\Psi$

That is, $S$ acts like the conjugation operation on operators at $\Psi$, which is enough to define $S$ since $\Psi$ is cyclic. This $S$ has a polar decomposition (analogous for operators to the polar form for complex numbers) of $S = J \Delta$, where $J$ is antiunitary (this is conjugation, after all) and $\Delta$ is self-adjoint. We need the self-adjoint part, because the Tomita flow is a one-parameter family of automorphisms given by:

$\alpha_t(A) = \Delta^{-it} A \Delta^{it}$

An important fact for Connes’ classification of von Neumann algebras is that the Tomita flow is basically unique – that is, it’s unique up to an inner automorphism (i.e. a conjugation by some unitary operator – so in particular, if we’re talking about a relativistic physical theory, a change of coordinates giving a different $t$ parameter would be an example). So while there are different flows, they’re all “essentially” the same. There’s a unique notion of time flow if we reduce the algebra $\mathcal{A}$ to its cosets modulo inner automorphism. Now, in some cases, the Tomita flow consists entirely of inner automorphisms, and this reduction makes it disappear entirely (this happens in the finite-dimensional case, for instance). But in the general case this doesn’t happen, and the Connes-Rovelli paper summarizes this by saying that von Neumann algebras are “intrinsically dynamic objects”. So this is one interesting thing about the quantum view of states: there is a somewhat canonical notion of dynamics present just by virtue of the way states are described. In the classical world, this isn’t the case.

Now, Rovelli’s “Thermal Time” hypothesis is, basically, that the notion of time is a state-dependent one: instead of an independent variable, with respect to which other variables change, quantum mechanics (per Rovelli) makes predictions about correlations between different observed variables. More precisely, the hypothesis is that, given that we observe the world in some state, the right notion of time should just be the Tomita flow for that state. They claim that checking this for certain cosmological models, like the Friedman model, they get the usual notion of time flow. I have to admit, I have trouble grokking this idea as fundamental physics, because it seems like it’s implying that the universe (or any system in it we look at) is always, a priori, in thermal equilibrium, which seems wrong to me since it evidently isn’t. The Friedman model does assume an expanding universe in thermal equilibrium, but clearly we’re not in exactly that world. On the other hand, the Tomita flow is definitely there in the von Neumann algebra view of quantum mechanics and states, so possibly I’m misinterpreting the nature of the claim. Also, as applied to quantum gravity, a “state” perhaps should be read as a state for the whole spacetime geometry of the universe – which is presumably static – and then the apparent “time change” would then be a result of the Tomita flow on operators describing actual physical observables. But on this view, I’m not sure how to understand “thermal equilibrium”.  So in the end, I don’t really know how to take the “Thermal Time Hypothesis” as physics.

In any case, the idea that the right notion of time should be state-dependent does make some intuitive sense. The only physically, empirically accessible referent for time is “what a clock measures”: in other words, there is some chosen system which we refer to whenever we say we’re “measuring time”. Different choices of system (that is, different clocks) will give different readings even if they happen to be moving together in an inertial frame – atomic clocks sitting side by side will still gradually drift out of sync. Even if “the system” means the whole universe, or just the gravitational field, clearly the notion of time even in General Relativity depends on the state of this system. If there is a non-state-dependent “god’s-eye view” of which variable is time, we don’t have empirical access to it. So while I can’t really assess this idea confidently, it does seem to be getting at something important.

I’ve been looking over the last little bit at quantum groupoids, and how they can be used to deform the 2-linearization 2-functor $\Lambda : Span(Gpd) \rightarrow 2Vect$ (or into $2Hilb$) which I’ve discussed in here.

First a little motivation: that functor was part of the way I constructed extended TQFT’s. The inclusion $nCob_2 \rightarrow CoSpan_2(Man)$ realized cobordisms (with corners) in terms of spans of manifolds. Looking at fundamental groupoids using the 2-functor $[\Pi_1(-),G]$ allows us to think about these in terms of the bicategory $Span(Gpd)$, and then applying $\Lambda$ gave 2-Hilbert spaces and 2-linear maps for manifolds and cobordisms (and then natural transformations for cobordisms with corners). Since I made the claim that, with gauge group $G=SU(2)$ – and a suitably infinitary version of $\Lambda$, the extended TQFT gives a theory equivalent to the Ponzano-Regge model of quantum gravity, a reasonable question is: what about the Turaev-Viro model? The PR model is based on labelling edges of a triangulation with representations of $SU(2)$, and the TV model, with representations of $SU_q(2)$.

Now, the groupoids that show up in the above – groupoids of $G$-connections on a manifold, modulo gauge transformations – are quite closely related to this. In particular, the groupoid of connections for a circle (the basic 1-dimensional manifold that the 3-dimensional theory builds from) is $G//Ad G$, the transformation groupoid produced from the action of $G$ on itself by conjugation. (That is: the objects are elements of $G$, and the morphisms are all the conjugacy relations.) Applying $\Lambda$ gives the representation category of this, namely $hom(G // Ad G , Vect)$, so in particular, at the identity of $G$, one has $Rep(SU(2))$ as a sub-2-vector space. (The “states” in the 2-Hilbert space for the circle in the ETQFT are labelled by “masses and spins” – the mass=0 case is what gives the representations of $SU(2)$, and for nonzero mass, one has $Rep(U(1))$.)

More broadly: one can describe the state space of a gauge theory – or many other kinds of theory, in terms of transformation groupoids given by symmetries (gauge transformations, say) acting on states (connections, in that case). Is there a way of doing the same for systems whose symmetries are described by quantum groups? If so, then instead of getting 2-vector spaces which are representation categories of groupoids, we should get some which are representation categories of quantum groupoids.

This paper by Ping Xu describes quantum groupoids – or rather, quantum universal enveloping algebras. They’re described here as a “unification of quantum groups and star products” (star products being the partially-defined composition found in groupoids). This paper by Nikshych and Vainerman describes finite quantum groupoids and some applications – in particular, quantum transformation groupoids, which is the immediately relevant application.

First off, quantum groups: these are Hopf algebras, which in particular are bialgebras – they have both a product

$m : H \otimes H \rightarrow H$

and “coproduct”

$\Delta : H \rightarrow H \otimes H$.

This is because the point here is that we’re following the pattern in which spaces are replaced by algebras: in some simple examples, these are the algebras of functions on a space. The point of noncommutative geometry is that there’s a (contravariant) equivalence between the category of locally compact Hausdorff spaces and the category of commutative algebras, so generalizing to noncommutative algebras (and taking the opposite category) gives a generalization of “locally compact Hausdorff space”. Topological groups like Lie groups are group objects in this category of spaces – and quantum groups are group objects in $Alg^{op}$. So in particular, the group operation shows up as the coproduct $\Delta$, and the inverse operation is the antipode

$S : H \rightarrow H$.

Of course there are also the unit

$\eta : k \rightarrow H$

and co-unit

$\epsilon : H \rightarrow k$

(where $k$ is the base field, say $\mathbb{C}$). The co-unit is of course the “unit” map for the group object. These maps all satisfy some obvious relations.

Now what about quantum groupoids? These are “groupoid objects” – or rather, models of the theory of groupoids – in $Alg^{op}$. We can’t quite say “groupoid objects”, since a groupoid internal to a category $C$ consists of two objects in $C$. For example, a Lie groupoid is a groupoid in $Man$, the category of manifolds. It has a base manifold $B$ and a total manifold $M$, and two maps $s,t : M \rightarrow B$, and so forth. The interpretation is that there is a set (or manifold, or what-have-you) of objects, and a set (etc.) of morphisms. There is a (partially-defined) composition operation allowing morphisms to be composed if the source of one is the target of the other, and so forth.

So (a slightly tweaked version of) the definition of a quantum groupoid given by Xu has it consisting of $(H, R, \alpha, \beta, m, \Delta, \epsilon, S)$. These unpack in pretty natural ways: it helps to compare to both the definition of, say, a Lie groupoid, and a quantum group. $H$ is the “total algebra$and $R$ the “base algebra”, and they correspond to the “noncommutative spaces” of morphisms and objects of a groupoid, respectively. Just as a group can be seen as a groupoid with just one object, a quantum group would be a quantum groupoid where the base algebra $R$ is just the base field $k$. But then, if $R$ is not $k$, we need some nontrivial $\alpha, \beta : R \rightarrow H$ – the source and target maps respectively, which replace the unit map to $k$. Notice they go from the base $R$ to the total algebra $H$, not the other way around, because everything works as usual in $Alg^{op}$. The other maps are likewise dual to those in the definition of a groupoid. The major difference is that we need the equivalent of a partially defined multiplication/composition $m$ and the dual “co-multiplication”/”co-composition” $\Delta$. This works because using $\alpha$ and $\beta$, we get left and right actions of the base $R$ on $H$, which is thus an $(R,R)$-bimodule, hence we can form the bimodule product $H \otimes_R H$, and thus: $m : H \otimes_R H \rightarrow H$ and $\Delta : H \rightarrow H \otimes_R H$ The obvious analog of the unit $\eta : R \rightarrow H$ we had for quantum groups is hidden in Xu’s definition (it seems like it should take the place of the requirement that $H$ be unital), but the co-unit $\epsilon : H \rightarrow R$ is the dual way of describing the “identity” function $x \mapsto 1_x$. The antipode $S : H \rightarrow H$ plays the role of the inverse map for morphisms $g \mapsto g^{-1}$ in groupoids. All these maps have to satisfy various identities which are implied by saying this is a model of the theory of groupoids – check out either of the above papers to see them all explicitly. (A final observation about the definition: a groupoid is a category which has an inverse map from morphisms to morphisms. If we relax the assumption that we have an antipode $S$, we end up with just the definition of a bialgebroid (having $S$ makes it a “Hopf” algebroid). So “bialgebroid” would seem to be the natural “quantum” version of the concept of a general category…) So how might one construct such a “quantum action groupoid”? This is addressed (at least in the finite case) in the paper by Nikshych and Vainerman, in their section 2.6. This is generalizing the action groupoid arising from a group acting on a set. The set $S$ is replaced by an algebra $B$ (which must be separable, for them – the equivalent of a finite set – and thought of as a “quantum space”). The group $G$ is replaced by a quantum group (or, generally, Hopf algebra) $H$. The equivalent of having action of the group on the set is that $B$ is a (right) $H$-module. Now, the action groupoid for a $G$ action on $S$ has for objects the elements of $S$, and for morphisms, all relations $g(s) = s'$, which we can write as morphisms $g_s$, with source $s$ and target $s' = g(s)$. The action quantum groupoid associated to the $H$-module $B$ is the double crossed product $B^{op} \lhd H \rhd B$, with multiplication, co-multiplication, etc. defined in fairly natural ways. (Note: those triangles should be semidirect products, but I can’t seem to make that symbol appear here.) So finally, I seem to be claiming that a such a quantum groupoid, let’s call it $Q=(H,R,\alpha,\beta,m,\Delta,\epsilon,S)$ is the right “classical” state space (if that’s not too blatant a contradiction in terminology) for a theory having quantum-group symmetry – at least in the categorified picture. No doubt in many cases there is additional structure, capturing the equivalent of, say, symplectic structure, that should also be included (such things certainly can be found in NCG, but I’m still absorbing how exactly). Then the 2-vector space for the quantized version of such a theory is the category $Rep(Q)$, and a “2-state” just an object in here – a representation of $Q$. One thing that’s not quite clear to me just now is how this relates to the usual idea of “state” in NCG – a state for a “quantum space” (which is an algebra) being a linear functional on that algebra. Not necessarily a character (i.e. a homomorphism into $\mathbb{C}$), mind you – that would be a 1-dimensional representation, but just a functional. A couple of posts ago, I mentioned Max Jammer’s book “Concepts of Space” as a nice genealogy of that concept, with one shortcoming from my point of view – namely, as the subtitle suggests, it’s a “History of Theories of Space in Physics”, and since physics tends to use concepts out of mathematics, it lags a bit – at least as regards fundamental concepts. Riemannian geometry predates Einstein’s use of it in General Relativity by fifty some years, for example. Heisenberg reinvented matrices and matrix multiplication (which eventually led to wholesale importation of group theory and representation theory into physics). More examples no doubt could be found (String Theory purports to be a counterexample, though opinions differ as to whether it is real physics, or “merely” important mathematics; until it starts interacting with experiments, I’m inclined to the latter, though of course contra Hardy, all important mathematics eventually becomes useful for something). What I said was that it would be nice to see further investigation of concepts of space within mathematics, in particular Grothendieck’s and Connes’. Well, in a different context I was referred to this survey paper by Pierre Cartier from a few years back, “A Mad Day’s Work: From Grothendieck To Connes And Kontsevich, The Evolution Of Concepts Of Space And Symmetry”, which does at least some of that – it’s a fairly big-picture review that touches on the relationship between these new ideas of space. It follows that stream of the story of space up to the end of the 20th century or so. There’s also a little historical/biographical note on Alexander Grothendieck – the historical context is nice to see (one of the appealing things about Jammer’s book). In this case, much of the interesting detail is more relevant if you find recent European political history interesting – but I do, so that’s okay. In fact, I think it’s helpful – maybe not mathematically, but in other ways – to understand the development of mathematical ideas in the context of history. This view seems to be better received the more ancient the history in question. On the scientific end, Cartier tries to explain Grothendieck’s point of view of space – in particular what we now call topos theory – and how it developed, as well as how it relates to Connes’. Pleasantly enough, a key link between them turns out to be groupoids! However, I’ll pass on commenting on that at the moment. Instead, let me take a bit of a tangent and jump back to Jammer’s book. I’ll tell you something from his chapter “Emancipation from Aristotelianism” which I found intriguing. This would be an atomistic theory of space – an idea that’s now beginning to make something of a comeback, in the guise of some of the efforts toward a quantum theory of gravity (EDIT: but see comments below). Loop quantum gravity, for example, deals with space in terms of observables, which happen to take the form of holonomies of connections around loops. Some of these observables have interpretations in terms of lengths, areas, and volumes. It’s a prediction of LQG that these measurements should have “quantized”, which is to say integer, values: states of LQG are “spin networks”, which is to say graphs with (quantized) labels on the edges, interpreted as areas (in a dual cell complex). (Notice this is yet again another, different, view of space, different from Grothendieck’s or Connes’, but shares with Connes especially the idea of probing space in some empirical way. Grothendieck “probes” space mainly via cohomology – how “empirical” that is depends on your point of view.) The atomistic theory of space Jammer talks about is very different, but it does also come from trying to reconcile a discrete “quantum” theory of matter with a theory linking matter to space. In particular, the medieval Muslim philosophical school known as al Kalam tried to reconcile the Koran and Islamic theology with Greek philosophy (most of the “Hellenistic” world conquered by Alexander the Great, not least Egypt, is inside Dar al Islam, which is why many important Greek texts came into Europe via Arabic translations). Though they were, as Jammer says, “Emancipating” themselves from Aristotle, they did share some of his ideas about space. For Aristotle, space meant “place” – the answer to the questions “where is it?” and “what is its shape and size?”. In particular, it was first and foremost an attribute of some substance. All “where?” questions are about some THING. The answer is defined in terms of other things: my cat is on the ground, under the tree, beside the house. The “place” of an object was literally the inner shell of the containing body that held it (which was contained by some other body, and so on – there being no vacuum in Aristotle). So my “place” is defined by (depending how you look at it) my skin, my clothes, or the walls of the room I’m in. This is a relational view of space, though more hard-headed than, say, Leibniz’s. The philosophers of the Kalam had a similar relational view of space, but they didn’t accept Aristotle’s view of “substances”, where each thing has its own essential identity, on which attributes are hung like hats. Instead, they believed in atomism, following Democritus and Leucippus: bodies were made out of little indivisible nuggets called “atoms”. Macroscopic things were composites of atoms, and their attributes resulted from how the atoms were put together. Here’s Jammer’s description: The atoms of the Kalam are indivisible particles, equal to each other and devoid of all extension. Spatial magnitude can be attributed only to a combination of atoms forming a body. Although a definite position (hayyiz) belongs to each individual atom, it does not occupy space (makan). It is rather the set of these positions – one is almost tempted to say, the system of relations – that constitutes spatial extension…. In the Kalam, these rather complicated and surprisingly abstract ideas were deemed necessary in order to meet Aristotle’s objections against atomism on the ground that a spatial continuum cannot be constituted by, or resolved into, indivisibles nor can two points be continuous or contiguous with one another. So like people who prefer a “background independent” quantum theory of gravity, they wanted to believe that space (geometry) derives from matter, and that matter is discrete, but space was commonly held to be continuous. Also alike, they resolved the problem by discarding the assumption of continuous space, and, by consideration of motion, to discrete time. There are some differences, though. The most obvious is that the nodes of the graph in a spin network state don’t represent units of matter, or “atoms”. For that matter, quantum field theory doesn’t really have “atoms” in the sense of indivisible units which don’t break apart or interact. Everything interacts in QFT. (In some sense, interactions are more fundamental units in QFT than “particles” are – particles only (sic!) serve to connect one interaction with another.) Another key difference is how space relates to matter. In Aristotle, and in the Kalam, space is defined directly by matter: two bits of matter “define” the space between them. In General Relativity (the modern theory with the “relational” view of space), there’s still room for space as an actor in its own right, like Newton’s absolute space-as-independent-variable – in other words, room for a vacuum, which Aristotle categorically denied could even conceivably exist. In GR, what matter determines is the curvature of space (more precisely the Einstein tensor of the curvature). Well, so the differences are probably more informative than the similarities, (Edit: To emphasize a key difference glossed over before… It was coupling to quantum matter which suggested quantizing the picture of space. Discreteness of the spectrum of various observables is a logically separate prediction in each case. Either matter or space(time) could have had continuous spectrum for the relevant observables and still been quantized – discrete matter would have given discreteness for some observed quantities, but not area, length, and so on. So in the modern setting, the link is much less direct.) but the fact that theories of related discreteness in matter, space, and time, have been around for a thousand years or more is intriguing. The idea of empty space as an independent entity – in the modern form only about three hundred years old – appears to be the real novel part. One of the nice intuitions in Carlo Rovelli’s book on Quantum Gravity, for me at least, was to say that, rather than there being a separate “space”, we have a theory of fields defined on other fields as background – one of which, the “gravitational field” has customarily been taken for “space”. So spatial geometry is a field, and it has some propagating (through space!) degrees of freedom – the particle associated to this field is a graviton. Nobody’s ever seen one, mind you – but supposing they exist makes many of things easier. To re-state a previous point: I think this is a nice aspect of categorification for dealing with space. Extending the “stuff/structure/properties” trichotomy to allow space to resemble both “stuff” and relations between stuff leaves room for both points of view. I mention this because tomorrow I leave London (Ontario) for London (England), and thence to Nottingham, for the Quantum Gravity and Quantum Geometry Conference. It’s been a while since I worked much on quantum gravity, per se, but this conference should be interesting because it seems to be a confluence of mathematically and physically inclined people, as the name suggests. I read on the program, for example, that Jerzy Lewandowski is speaking on QFT in Quantum Curved Spacetime, and suddenly remember that, oh yes, I did a Masters thesis (viz) on QFT in curved (classical) spacetime… but that was back in the 20th century! It’s been a while, and I only made a small start at it before, but that whole area of physics is quite pretty. Anyway, it should be interesting, and there are a number of people I’m looking forward to talking to. In the past couple of weeks, Masoud Khalkhali and I have been reading and discussing this paper by Marcolli and Al-Yasry. Along the way, I’ve been explaining some things I know about bicategories, spans, cospans and cobordisms, and so on, while Masoud has been explaining to me some of the basic ideas of noncommutative geometry, and (today) K-theory and cyclic cohomology. I find the paper pretty interesting, especially with a bit of that background help to identify and understand the main points. Noncommutative geometry is fairly new to me, but a lot of the material that goes into it turns out to be familiar stuff bearing unfamiliar names, or looked at in a somewhat different way than the one I’m accustomed to. For example, as I mentioned when I went to the Groupoidfest conference, there’s a theme in NCG involving groupoids, and algebras of $\mathbb{C}$-linear combinations of “elements” in a groupoid. But these “elements” are actually morphisms, and this picture is commonly drawn without objects at all. I’ve mentioned before some ideas for how to deal with this (roughly: $\mathbb{C}$ is easy to confuse with the algebra of $1 \times 1$ matrices over $\mathbb{C}$), but anything special I have to say about that is something I’ll hide under my hat for the moment. I must say that, though some aspects of how people talk about it, like the one I just mentioned, seem a bit off, to my mind, I like NCG in many respects. One is the way it ties in to ideas I know a bit about from the physics end of things, such as algebras of operators on Hilbert spaces. People talk about Hamiltonians, concepts of time-evolution, creation and annihilation operators, and so on in the algebras that are supposed to represent spaces. I don’t yet understand how this all fits together, but it’s definitely appealing. Another good thing about NCG is the clever elegance of Connes’ original idea of yet another way to generalize the concept “space”. Namely, there was already a duality between spaces (in the usual sense) and commutative algebras (of functions on spaces), so generalizing to noncommutative algebras should give corresponding concepts of “spaces” which are different from all the usual ones in fairly profound ways. I’m assured, though I don’t really know how it all works, that one can do all sorts of things with these “spaces”, such as finding their volumes, defining derivatives of functions on them, and so on. They do lack some qualities traditionally associated with space – for instance, many of them don’t have many, or in some cases any, points. But then, “point” is a dubious concept to begin with, if you want a framework for physics – nobody’s ever seen one, physically, and it’s not clear to me what seeing one would consist of… (As an aside – this is different from other versions of “pointless” topology, such as the passage from ordinary topologies to, sites in the sense of Grothendieck. The notion of “space” went through some fairly serious mutations during the 20th century: from Einstein’s two theories of relativity, to these and other mathematicians’ generalizations, the concept of “space” has turned out to be either very problematic, or wonderfully flexible. A neat book is Max Jammer’s “Concepts of Space“: though it focuses on physics and stops in the 1930′s, you get to appreciate how this concept gradually came together out of folk concepts, went through several very different stages, and in the 20th century started to be warped out of all recognition. It’s as if – to adapt Dan Dennett – “their word for milk became our word for health”.I would like to see a comparable history of mathematicians’ more various concepts, covering more of the 20th century. Plus, one could probably write a less Eurocentric genealogy nowadays than Jammer did in 1954.) Anyway, what I’d like to say about the Marcolli and Al-Yasry paper at the moment has to do with the setup, rather than the later parts, which are also interesting. This has to do with the idea of a correspondence between noncommutative spaces. Masoud explained to me that, related to the matter of not having many points, such “spaces” also tend to be short on honest-to-goodness maps between them. Instead, it seems that people often use correspondences. Using that duality to replace spaces with algebras, a recurring idea is to think of a category where morphism from algebra $A$ to algebra $B$ is not a map, but a left-right $(A,B)$-bimodule, $_AM_B$. This is similar to the business of making categories of spans. Let me describe briefly what Marcolli and Al-Yasry describe in the paper. They actually have a 2-category. It has: Objects: An object is a copy of the 3-sphere $S^3$ with an embedded graph $G$. Morphisms: A morphism is a span of branched covers of 3-manifolds over $S^3$: $G_1 \subset S^3 \stackrel{\pi_1}{\longleftarrow} M \stackrel{\pi_2}{\longrightarrow} S^3 \supset G_2$ such that each of the maps $\pi_i$ is branched over a graph containing $G_i$ (perhaps strictly). In fact, as they point out, there’s a theorem (due to Alexander) proving that ANY 3-manifold $M$ can be realized as a branched cover over the 3-sphere, branched at some graph (though perhaps not including a given $G$, and certainly not uniquely). 2-Morphisms: A 2-morphism between morphisms $M_1$ and $M_2$ (together with their $\pi$ maps) is a cobordism $M_1 \rightarrow W \leftarrow M_2$, in a way that’s compatible with the structure of the$lateux M_i\$ as branched covers of the 3-sphere. The $M_i$ are being included as components of the boundary $\partial W$ – I’m writing it this way to emphasize that a cobordism is a kind of cospan. Here, it’s a cospan between spans.

This is somewhat familiar to me, though I’d been thinking mostly about examples of cospans between cospans – in fact, thinking of both as cobordisms. From a categorical point of view, this is very similar, except that with spans you compose not by gluing along a shared boundary, but taking a fibred product over one of the objects (in this case, one of the spheres). Abstractly, these are dual – one is a pushout, and the other is a pullback – but in practice, they look quite different.

However, this higher-categorical stuff can be put aside temporarily – they get back to it later, but to start with, they just collapse all the $hom$-categories into $hom$-sets by taking morphisms to be connected components of the categories. That is, they think about taking morphisms to be cobordism classes of manifolds (in a setting where both manifolds and cobordisms have some branched-covering information hanging around that needs to be respected – they’re supposed to be morphisms, after all).

So the result is a category. Because they’re writing for noncommutative geometry people, who are happy with the word “groupoid” but not “category”, they actually call it a “semigroupoid” – but as they point out, “semigroupoid” is essentially a synonym for (small) “category”.

Apparently it’s quite common in NCG to do certain things with groupoids $\mathcal{G}$ – like taking the groupoid algebra $\mathbb{C}[\mathcal{G}]$ of $\mathbb{C}$-linear combinations of morphisms, with a product that comes from multiplying coefficients and composing morphisms whenever possible. The corresponding general thing is a categorical algebra. There are several quantum-mechanical-flavoured things that can be done with it. One is to let it act as an algebra of operators on a Hilbert space.

This is, again, a fairly standard business. The way it works is to define a Hilbert space $\mathcal{H}(G)$ at each object $G$ of the category, which has a basis consisting of all morphisms whose source is $G$. Then the algebra acts on this, since any morphism $M'$ which can be post-composed with one $M$ starting at $G$ acts (by composition) to give a new morphism $M' \circ M$ starting at $G$ – that is, it acts on basis elements of $\mathcal{H}(G)$ to give new ones. Extending linearly, algebra elements (combinations of morphisms) also act on $\mathcal{H}(G)$.

So this gives, at each object $G$, an algebra of operators acting on a Hilbert space $\mathcal{H}(G)$ – the main components of a noncommutative space (actually, these need to be defined by a spectral triple: the missing ingredient in this description is a special Dirac operator). Furthermore, the morphisms (which in this case are, remember, given by those spans of branched covers) give correspondences between these.

Anyway, I don’t really grasp the big picture this fits into, but reading this paper with Masoud is interesting. It ties into a number of things I’ve already thought about, but also suggests all sorts of connections with other topics and opportunities to learn some new ideas. That’s nice, because although I still have plenty of work to do getting papers written up on work already done, I was starting to feel a little bit narrowly focused.