So at the instigation of Dan Christensen, I started organizing a seminar at UWO’s math department this year (here is its website). The title I came up with, “Seminar on Stacks, Groupoids and Algebras” is less than self-explanatory, and it remains to be seen how accurate it is, but it was the best title I could think of to express the theme I wanted to investigate. This, roughly, is that there are several ways of looking at “spaces” which carry some built-in symmetry, which have various points of commonality (at least they can be used to describe similar situations) but are based on quite different viewpoints. The seminar is supposed to introduce them, and illuminate some of the points of similarity and difference.

I gave the first talk, the notes for which are here, entitled “Topological and Lie Groupoids, with relations to C*-Algebras and Stacks”. This was mainly about the point of view which I’m most comfortable with, namely groupoids, but also tries to give a brief overview of how groupoids are related to the C*-algebras used in noncommutative geometry to represent some generalizations of spaces (especially to give a nice way to represent quotient spaces for group actions with singular points and similar bad behaviour), and to stacks.

Now, a quotient groupoid (in $\mathbf{Set}$) comes from a group $G$ acting on a set $X$. The objects of this groupoid are just the elements of $X$, and the morphisms are pairs $(g,x) \in G \times X$, where $(g,x) : x \rightarrow g(x)$. Analogous statements apply to groupoids in $\mathbf{Top}$, or $\mathbf{Diff}$, or $\mathbf{Aff}$, the category of affine schemes which is important in algebraic geometry). The particular example that brought me to this subject came out of looking at ETQFT, and is a fairly particular class of quotient groupoids. Namely, the one in which $X$ is the space of flat $G$-bundles on a fixed (and let’s assume connected) manifold $M$. (That is, bundles equipped and $G$ is interpreted as the group of gauge transformations. As we’ll see, this is closely related to the usual motivating examples for stacks.

Most of the geometric substance of my talk dealt with Lie groupoids, which is to say groupoids which have smooth manifolds of both objects and morphisms, and that all the structure maps are morphisms in the category $\mathbf{Diff}$ of differentiable manifolds. In particular, I talked about Lie algebroids – an extension of the idea of Lie algebras to a more differential-geometric setting (a Lie algebroid is a vector bundle over a base manifold $M$ with some algebraic structure – and there is a canonical one associated to a Lie groupoid, which gives the usual Lie algebra of a Lie group $G$, thought of as a groupoid over the point $M = \star$. Sections of Lie groupoids are used in Alan Weinstein’s definition of a volume form on a Lie groupoid (or, indeed, a differentiable stack), which generalizes the differential-geometric idea of a volume form to this Lie-theory flavoured situation. Together these give a nice illustration of the interplay between the manifold-like and the Lie-group like properties of Lie groupoids.

I mentioned that Weinstein’s volume form is referred to STACKS. In this context, the usual definition of a stack is that it’s an equivalence class of groupoids. The equivalence relation is Morita equivalence (more on this in the next post, I think). Now, because I was mainly interested in groupoids in $\mathbf{Diff}$, I mainly paid attention to topological stacks (this concept had been around informally for some time, but as far as I can determine, was formally defined, and several of the important theorems proved, fairly recently – in any case, they are collected in the paper of Behrang Noohi which I cite in the notes). This isn’t, historically, the first use of the concept, though, or the original definition… The origins and motivation of stacks were the topic of the next talk in the seminar.

The second talk was by Ajneet Dhillon, (his slides are here), just entitled “Stacks”. Aji is coming from a background of algebraic geometry, which is the context in which stacks were first proposed. The idea originates with Alexandre Grothendieck, and was taken up and refined by Deligne and Mumford (in this paper), and by Artin. Aji began by explaining the original motivation, which, roughly, is to deal wth problems at arise when trying to describe moduli spaces for certain algebro-geometric structures, by relaxing ones’ goal from a moduli SPACE to a moduli STACK.

Now, a moduli space in this context is more than just a space which parametrizes some collection of objects – this might make sense if the objects are just sets, but if they are spaces (topological, algebraic, differentiable, or otherwise), one needs to be more careful. In particular, a moduli space for structures of some type $T$ should be a space $\mathcal{M}$, with a universal family of $T$-structures over it: $\mathcal{U} \rightarrow \mathcal{M}$, so that the fibre over a point $x \in \mathcal{M}$, $E_x$, is a structure of type $T$.

In particular, he gave the example of the (nonexistent!) moduli space in the case of smooth projective curves of genus 1 with a marked point (essentially, this is a torus $\mathbb{C}/\Lambda$, where $\Lambda$ is a full sublattice of $\mathbb{C}$ – i.e. one which spans $\mathbb{C}$ as a real vector space, where the marked point corresponds to the lattice points). Then the idea is that the fibres of $\mathcal{U} \rightarrow \mathcal{M}$ are exactly these curves.

Saying this family needs to be universal means that for any other family of such curves is a pullback of it. That is, given any family $\mathcal{C} \rightarrow \mathcal{X}$ whose fibres are these projective curves, there is a map $f : \mathcal{X} \rightarrow \mathcal{M}$ such that $\mathcal{C}$ is the pullback of $\mathcal{U}$ along $f$. The most obvious case is when $\mathcal{X}$ is a sub-space (in this algebraic context, a subscheme) of $\mathcal{M}$ and $f$ is the inclusion, in which case $\mathcal{C}$ is just the restriction of $\mathcal{U}$, and the fibre over any point $x \in X$ is the same as in $\mathcal{U}$.

So to show that such a moduli space doesn’t exist for the projective curves of genus 1, the idea is to construct a family of them which can’t be a pullback. The example is the family of curves $\mathbf{C}$, the collection of triples of complex numbers $(x,y,z)$ satisfying $y^2 = x(x-1)(x-z)$, parametrized by $z$ – that is, with a map into the affine plane taking each triple to the corresponding $z$, so that the fibre over $z$ is a projective curve. Now, the fibre over $z = \frac{1}{2}$ has an extra automorphism the other curves don’t – the corresponding lattice is just $\mathbb{Z}[i]$, and the isomorphism givern by $i \mapsto -i$ (which corresponds to the action of the map $z \mapsto (1-z)$. If there were a universal family, we would have to be able to extend this automorphism, but we can’t. So a larger category than schemes is needed to define a “moduli space”. Similar sorts of arguments apply for other sorts of spaces, such as those in $\mathbf{Top}$ or $\mathbf{Diff}$, though the algebraic setting has some extra complications. Whichever category of spaces we want, call it $\mathbf{S}$. The main thing is that $\mathbf{S}$ have a Grothendieck topology (essentially, a well-behaved rule to tell if a set of morphisms into $X \in \mathbf{S}$ is a “cover”).

Then the point is, there is a larger category in which $\mathbf{S}$ embeds, namely $Fun(\mathbf{S}^{op},\mathbf{Gpd}$, the category of (contravariant) weak functors from $\mathbf{S}$ into $\mathbf{Gpd}$ (note: some say “lax” here, but with the target $\mathbf{Gpd}$, these amount to the same thing). This is by an analog of the Yoneda embedding of $\mathbf{S}$ into $Fun(\mathbf{S}^{op},\mathbf{Sets})$ taking $S \in \mathbf{S}$ to the functor $Hom(-,S)$ (in fact, since a set is a trivial groupoid, this is a special case).

So in fact a stack is just a lack functor $F : \mathbf{S}^{op} \rightarrow \mathbf{Gpd}$ which satisfies some conditions. Some examples of lax functors that happen to be stacks are:

• the moduli stack of elliptic curves
• given a group $G$ (in Aji’s context this is an algebraic group), the “classifying stack” $BG$ of all principal $G$-bundles
• the relative version of this: given $G$ and a space $X$, the stack $Bun_X$ of principal $G$-bundles on $X$

The stack conditions are a kind of categorification of the defining conditions for a sheaf, to wit (roughly):

1. Morphisms can be “glued” – namely, for $X \in \mathbf{S}$ and $x,y \in F(X)$, the mapping $Isom(x,y)$ which takes a morphism $f: y \rightarrow x$ to the set of all isomorphisms from $f*x$ to $f*y$ is a sheaf on $latek \mathcal{S}/X$.
2. Objects can be “glued” – namely, “all descent data are effective”

Condition 2 means the following. Given a covering of $X$, say $\{ f_i: U_i \rightarrow X \}$, a descent datum is a collection of objects $x_i \in F(U_i) \in \mathbf{Gpd}$, along with isomorphisms $\phi_{ij}$ between restructions to the intersections $U_{ij} = U_i \times_X U_j$. So $\phi_{ij} : x_i|_{U_{ij}} \rightarrow x_j|_{U_{ij}}$ (restricting an object $x_j$ means pulling it back along the map from $U_{ij})$. To say that a descent datum $(\{ x_i \}, \{ \phi_{ij} \} )$ is effective means that all the $x_i$ come from pulling back an object in $F(X)$ to $U_i$ along $f_i$.

This is a weaker condition than the sheaf condition, which insists all the $\phi_{ij}$ are equalities. (In the sheaf case, a paradigm example is the sheaf of continuous functions on a topological space – functions on open sets which agree on their overlap can be “glued” to form a function on the union. In the stack case, a paradigm example is the stack of $G$-bundles – which can likewise be glued, even when bundles only agree up to isomorphism on an overlap).

Now, as is the way with blog entries, this one is a bit delayed, and Aji gave his second talk today. This included, along with several examples, a description of what properties distinguishes an “algebraic” stack (when $\mathbf{S} = \mathbf{Aff}$ one has to say a little more), an explanation the correspondence between stacks and groupoids (up to equivalence), and an explanation of the definition of a stack as a category fibred in groupoids, but I’ll put off writing that up for the time being…