Pursued By Stacks 2: Descent

16 May 2022

A stack, so I’m told, is a category fibered in groupoids satisfying some extra conditions. The stacks we’re interested in are geometric stacks, so this won’t be the last post on stacks, but with any luck we’ll see the definition in this post.

Let \(\mathsf{C}\) be a category fibered in groupoids over \(\mathbf{Top}\). We say an object \(\xi \in \mathsf{C}\) is over a space \(X\) if \(\pi(\xi) = X\). Both Lerman and Fantechi adopt the following convention: for every continuous map \(f\colon Y \to X\) and every object \(\xi\) over \(X\), we choose a lift \(f_\xi \colon f^*\xi \to \xi\) that we call “the” pullback of \(\xi\) along \(f\).

Notice that if we had another choice \(f'\colon \xi' \to \xi\) of pullback, then in \(\mathbf{Top}\) we have the following rather silly commutative diagram

\[\require{AMScd}\begin{CD} Y @>f>> X \\ @| @| \\ Y @>f>> X, \end{CD}\]

so by the second item in the definition of a category fibered in groupoids, there are unique maps \(g\colon f^*\xi \to \xi'\) and \(g'\colon \xi' \to f^*\xi\) such that, for example, the following diagram commutes

\[\begin{CD} f^*\xi @>f_\xi>> \xi \\ @VgVV @| \\ \xi' @>f'>> \xi. \end{CD}\]

It’s not hard to see that \(g\) and \(g'\) are therefore inverse isomorphisms. By the way, if \(f\) is the identity of \(X\), we see that this argument implies that the category of objects above \(X\) is a groupoid: every arrow is an isomorphism. Hence the name, I suppose. In fact, if \(f\colon Y \to X\) is a continuous map and \(\alpha\colon \xi \to \xi'\) is an arrow (necessarily an isomorphism) of objects over \(X\), a clever application of the second item again shows that there is a unique isomorphism \(f^*\alpha\colon f^*\xi \to f^*\xi'\) such that the following diagram commutes

\[\begin{CD} f^*\xi @>f_\xi>> \xi \\ @VVf^*\alpha V @VV\alpha V \\ f^*\xi' @>f_{\xi'}>> \xi'. \end{CD}\]

Notice as well that we do not require (and probably cannot in general) that \(g^*f^*\xi = (fg)^*\xi\), merely that they are canonically isomorphic.

Descent data

The idea of descent is that given, say, a vector bundle defined open set by open set on a space \(X\) satisfying certain gluing conditions, this data descends to a vector bundle on \(X\).

More formally, let \(\mathcal{U} = \{U_i\}_{i\in I}\) be an open cover of \(X\), and for each \(i\), let \(\iota_i \colon U_i \to X\) be the inclusion. We have inclusions \(\iota_{ji}\colon (U_j \cap U_i) \to U_j\) and \(\iota_{ij}\colon (U_j \cap U_i) \to U_i\) for each double intersection and \(\iota^i_{ijk}\), \(\iota^j_{ijk}\) and \(\iota^k_{ijk}\) for inclusions from the triple intersection \(U_i \cap U_j \cap U_k\) to the various double intersections, where the superscript indicates the omitted subscript in the target of the inclusion. A descent datum for \(\mathsf{C}\) over \(X\) is the following.

\[(\iota_{ijk}^j)^*\alpha_{ki} = (\iota_{ijk}^i)^*\alpha_{kj} \circ (\iota_{ijk}^k)^*\alpha_{ji}\]

The descent datum is effective if there exists an object \(\xi\) over \(X\) with isomorphisms \(\alpha_i \colon \iota_i^*\xi \to \xi_i\) in the fiber over \(U_i\) with the property that \(\alpha_{ji} = \iota_{ji}^*\alpha_j\circ (\iota_{ij}^*\alpha_i)^{-1}\).

In plainer words, a descent datum is that open set by open set construction of something you’d like to exist in \(\mathsf{C}\), and saying that the descent datum is effective says that the wished-for object really exists and is moreover obtained by gluing together it patch by patch.

Finally, we say that isomorphisms are a sheaf for \(\mathsf{C}\) if, roughly, isomorphisms between elements of \(\mathsf{C}\) over any space \(X\) are determined uniquely by their restriction to any open cover of \(X\). More formally, we require that for any space \(X\), every pair of objects \(\xi\) and \(\xi'\) in the fiber over \(X\), every open cover \(\{U_i\}_{i \in I}\) of \(X\) and every collection of isomorphisms \(\alpha_i\colon \iota_i^*\xi \to \iota_i^*\xi'\) with the property that \(\iota_{ij}^*\alpha_i = \iota_{ji}^*\alpha_j\) on \(U_i \cap U_j\), there exists a unique isomorphism \(\alpha\colon \xi \to \xi'\) such that \(\iota_i^*\alpha = \alpha_i\).

Our category fibered in groupoids \(\mathsf{C}\) is a stack if isomorphisms are a sheaf and every descent datum is effective.

A map of stacks \(\mathsf{C} \to \mathsf{D}\) over \(\mathbf{Top}\) is a functor \(F\colon \mathsf{C} \to \mathsf{D}\) with the property that \(\pi F = \pi\), where we abuse notation by writing both \(\pi\colon \mathsf{C} \to \mathbf{Top}\) and \(\pi\colon \mathsf{D} \to \mathbf{Top}\). A natural transformation of maps \(F\), \(G\colon \mathsf{C} \to \mathsf{D}\) is a natural transformation \(\eta\colon F \Rightarrow G\) with the property that for each \(\xi \in \mathsf{C}\), we have \(\pi(\eta_\xi) = 1_{\pi(\xi)}\). Notice that because \(F\xi\) and \(G\xi\) are in the fiber over \(\pi(\xi)\), the map \(\eta_\xi\) is automatically an isomorphism, so all natural transformations are natural isomorphisms. An isomorphism of stacks is an equivalence of categories over \(\mathbf{Top}\), i.e. \(F\colon \mathsf{C} \to \mathsf{D}\) is full, faithful and essentially surjective. Since we love the axiom of choice around here, this is equivalent to the existence of an inverse functor up to natural isomorphism.