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.