Among the talks given in our seminar on stacks and groupoids, there have been a few which I haven’t posted about yet – two by Tom Prince about stacks and homotopy theory, and one by José Malagon-Lopez comparing different characterizations of stacks. Tom is a grad student, and José is a postdoc, and they both work with Rick Jardine, who has done a lot of important work in homotopy theory, notably from the simplicial point of view. There was some overlap, since José was comparing the different characterizations for stacks that had been used by different people through the seminar, including Tom, but there’s still quite a lot to say here. I’ll try to cover the main points as I understand them, focusing on what I personally find relevant.

A major theme for both of them is the use of descent, which in general is a way to talk about the objects of a category in terms of another category. A standard example of descent would be the case of sheaves. First, though, what is it that’s being described in terms of descent?

Well, there are two opposite points of view on stacks – as categories fibred in groupoids (CFG’s), and as sheaves of groupoids. (I’ve found this book by Behrend et al. on algebraic stacks handy in parsing through some of the definitions here, and Jose recommended Vistoli’s notes on sites, fibred categories, and descent) One of the things Jose summarized in his talk was how these are related (which was a key bit of Aji’s earlier talk, blogged here). A CFG over $\mathcal{S}$ is a functor $p: \mathcal{X} \rightarrow \mathcal{S}$ where the preimage over $(x,1_x)$ is a groupoid (that is, all the morphisms mapping to an identity are invertible).

Now, given such $p : \mathcal{X} \rightarrow \mathcal{S}$ one gets a (weak) functor from $\mathcal{S}$ into groupoids (the “fibre-selecting” functor, which, among other conditions, gives the groupoid $p^{-1}(x,1_x)$ for each object $x$. Specifying this and showing it is a weak functor takes a little work. But in particular, there are properties on CFG’s a stack is such a functor into $Gpd$ with the extra property that descent data are effective. This is a weak version of the condition for a sheaf.

Stacks and Descent

The classical setting for descent questions is sheaf theory. To begin with, we have some category $\mathcal{S}$ of spaces – this might be $Top$ (topological spaces), or $Sch$ (affine schemes), or something else – the classical version has $\mathcal{S} = \mathcal{O}(X)$, the category of open sets on a topological space. The main thing is that $\mathcal{S}$ must be a Grothendieck site; in particular, there is a notion of covering for an object $X \in \mathcal{S}$. This is a collection $\underline{U} = \{ f_{\alpha} : U_{\alpha} \rightarrow X \}$ of arrows satisfying some conditions that capture the intuitive idea of “open cover”.

So, just to recall: the idea of describing a space as a sheaf on a site involves a little shift of perspective, but it’s the idea behind diffeological spaces (as I described in my post on Enxin Wu’s talk in our seminar, and which, for me, is a good example to help understand this viewpoint). A diffeological space is determined by giving the set of all “smooth” maps into it from each object in a certain site. Now, any space $S \in \mathcal{S}$ can also be represented in $Hom(\mathcal{S}^{op},Set)$ (by the Yoneda embedding) as the sheaf $Hom(-,S)$ which gives, for each space $X$, the set of maps in $\mathcal{S}$ (topological, algebraic, or whatever) into $S$ – but one can get objects in a bigger category, namely that of sheaves, which is a way of describing them in terms of the objects in the site $\mathcal{S}$. In the case of diffeological spaces, the site in question is just the one consisting of neighborhoods in $\mathbb{R}^n$ for any $n$, with smooth maps, and the obvious idea of a cover. So representable ones are just Euclidean neighborhoods, and general ones are defined by smooth maps out of these: the sheaf condition is just a way to state the natural compatibility condition for these maps. Similar thinking applies to any site $\mathcal{S}$.

The point of this condition is to ask when we can take a cover of an object $S$, and describe global objects (functions on $S$) in terms of local objects (functions on elements in the cover), which are compatible. Descent is the gluing condition for a sheaf $F$: given a cover – a bunch of maps $f_i : U_i \rightarrow S$ which satisfy some conditions that capture the intuitive idea of covering $S$ – a descent datum is a collection of $x_i \in F(U_i)$, and isomorphisms between the restrictions (by $F(\leq)$) to overlaps $U_i \cap U_j$, where the isomorphisms satisfy some cocycle condition ensuring that restrictions to $U_i \cap U_j \cap U_k$ are equal. The datum is effective if all there is a “global” object $x \in F(S)$ where $x_i$ is the restriction of $x$. (I find this easiest to see when $\mathcal{S}=\mathcal{O}(X)$, where it says we can glue functions on local patches that agree on overlaps, and find that they must have come by restricting a global function on $X$.)

This all makes sense if $F$ has values in $Set$ (or some other 1-category), but the point for stacks is that we have a weak functor $G : \mathcal{S}^{op} \rightarrow Gpd$. That is, the values are in groupoids, which naturally form a 2-category. So the descent can be weakened – instead of an equality in the cocycle condition, we get an isomorphism, which has to be coherent. Part of the point of describing stacks as “sheaves of groupoids” is as a weakening this way of describing a space, to an “up to equivalence” kind of condition.

One point which Jose made, and which Tom made use of, is that this description of a Grothendieck topology really gives too much information – that is, the category of sheaves on a site (taken up to equivalence) doesn’t uniquely determine the site. Instead of coverings, one should talk about sieves – these are, one might say, one-sided ideals of maps into $S$. In particular, subfunctors $R \subset Hom(-,S)$ – that is, for each space $V$, a subset of all maps $V \rightarrow S$, in a way that gets along with composition of maps (which is how they resemble ideals). Any covering defines a seive – as the subfunctor of maps which factor through the covering maps – but more than one covering might define the same sieve (rather the same way an ideal can be presented in terms of different generators).

So the view of stacks as sheaves $G$ (of groupoids) satisfying descent is then rephrased by saying that, for any covering sieve $R$ of an object $S \in \mathcal{S}$, there is an equivalence of functors between $Hom(E_S, G)$ and $Hom(E_R,G)$, where $E_S$ and $E_R$ are some sheaves on $\mathcal{S}$ constructed in a fairly natural way from the object $S$ itself, and from the sieve $R$. The point is that $Hom(E_S,G) = G(S)$ is a groupoid. The functor $E_R$ ends up such that $Hom(E_R,G)$ can be described in terms of covers $\{ U_i \rightarrow S \}$ as having objects which are compatible collections of objects from $U_i$ and isomorphisms between their restrictions – that is, descent data – and morphisms being compatible maps. So equivalence of these (2-)functors ends up being the stack condition.

One of Tom’s objectives was to look at all this from the point of view of simplicial sheaves – and here we need to think about homotopy-theoretic ideas of “equivalence”, instead of just the equivalence of categories we just used.

Model Structure

One of the major tools in homotopical algebra is the notion of a model structure (these slides by Peter May give the basic concepts). These show up throughout higher category theory because homotopies-between-homotopies-…-between-maps give a natural model of higher morphisms.

Model categories axiomatize three special kinds of maps one is interested in when talking about maps between spaces, up to homotopy. “Weak equivalence” generalizes a “homotopy equivalence” $f : X \rightarrow Y$ – a map which induces isomorphisms between homotopy groups of $X$ and $Y$ (as far as homotopy theory can detect, $X$ and $Y$ are “the same”). “Fibration” and “cofibration” are defined in homotopy theory by a lifting property (and its dual) – essentially, that if a map can be lifted along $f$, so can a homotopy of the map.  Fibrations generalize (“nice”) surjections, and cofibrations generalize (“nice”) inclusions.

In particular, Tom was making use of a notion of descent where the equations that define the descent conditions are just required to be weak equivalences. The point is that we can talk about sheaves of various kinds of things – sets, groupoids, or simplicial sets were the examples he gave. The relevant notion of equivalence for sets is isomorphism (the usual way of stating descent), but for groupoids it’s equivalence, and for simplicial sets, it’s another notion of weak equivalence (from the Joyal-Tierney model structure). When talking about stacks, we’re dealing with groupoids.

On the other hand, groupoids can be described in terms of simplicial sets, using the construction known as the simplicial nerve. In particular the classifying spaces of groupoids have no interesting homotopy groups above the first – so this ends up giving another way to state the weakened form of descent mentioned above. This type of construction – using the fact that simplicial sets are very versatile (can describe categories, or reasonable spaces, one $\infty$-categories, for instance), is what makes the study of simplicial presheaves, which is the basis of a lot of work by Rick Jardine (see the book Simplicial Homotopy Theory for a whole lot more that I can touch on here).

This gives another characterization of stacks: a sheaf of groupoids $G$ is a stack if and only if $BG$ (sheaf of classfying spaces), satisfies descent in that it is “pointwise” (that is, section-wise) weakly-equivalent to a certain kind of “globally fibrant replacement”. This is like the description of descent in terms of an equivalence of categories, as above – but in general is weaker. In fact, when the simplicial sets we’re talking about are classifying spaces for groupoids, then by construction these are just the same. This kind of replacement accomplishes for stacks roughly what “sheafification” does for sheaves – i.e. turns “prestacks” into “stacks”. This is done by taking a limit over all sieves – the universal property of the limit, then, is what ensures the existence of all the global objects that descent requires must exist. This is always a “local” weak equivalence, but only if we started with a stack is it one “pointwise” (i.e. in terms of sections).

Cocycles

As an aside: one thing which Tom talked about as a preliminary, but which I found particularly helpful from where I was coming from, had to do with “cocycle categories”. This is a somewhat unusual use of the term “cocycle”: here, a cocycle from $X$ to $Y$ is a certain kind of span – namely, a pair of maps from $Z$:

$X \stackrel{f}{\leftarrow} Z \stackrel{g}{\rightarrow} Y$

where $f$ is a “weak equivalence”. A morphism between cocycles is just a map $Z \rightarrow Z'$ which commutes with those in the cocycle. These form a category $H(X,Y)$. The point of introducing this is to say that there is a correspondence between components in this category – that is, $\pi_0(H(X,Y))$ and homotopy classes of maps from $X$ to $Y$ (the collection of which is denoted $[X,Y]$ in homotopy theory).

One way to think about this is that cocycles stand in relation to functions, roughly, as spans stand to relations. If we are in $Sets$, where weak equivalence is isomorphism, then $Z$ can be thought of as the graph of a function from $X$ to $Y$ – since $f$ is bijective, $Z$ can stand as a substitute for $X$. Moving to spaces, we weaken the requirement so that $Z$ is only a replacement for $X$ “up to homotopy” – thus, cocycles are adequate replacements for homotopy classes of functions. This business of replacing objects with other, nicer objects (say, “fibrant replacement”) is a recurring theme in homotopy theory. This digression on cocycles helped me understand why. Part of the point is that the equivalence classes of these “cocycles” is easier to calculate directly than, but equivalent to, homotopy classes of maps.

In any case, there’s more I could say about these talks, but I’ll leave off for now.

Over the next week, I’ll be visiting Derek Wise at UC Davis, to talk about some stuff having to do with ETQFT’s , but soon enough I’ll also do a writeup of Emre Coskun’s talks in the seminar about gerbes, which started today and continue tomorrow.