# Pursued by Stacks 2: Descent

May 16, 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

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

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

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.

- An object $\xi_i$ over $ U_i$ for each $ i$ .
- For each pair $i$ and $ j$ , an isomorphism $\alpha_{ji} \colon \iota_{ij}^*\xi_i \to \iota_{ji}^*\xi_j$ in the fiber over $U_i \cap U_j$ .
- The isomorphisms above satisfy the
*cocycle condition*(so named for reasons of Čech cohomology probably, so way over my head)

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.