Understanding the definition of stacks

Question

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it here.

I am trying to understand the notion of sheaves and stacks. Intuitively, sheaves are bit easy to understand as a gluing of compatible families of sets assign to opens sets of a topological space. In other words, it is a contravariant functor $\mathcal{F}:\mathbf{Open}(X)^{\operatorname{op}}\to\mathbf{Set}$ such that $$\mathcal{F}(U)=\lim\left(\prod_{i\in I}\mathcal{F}(U_i) \rightrightarrows \prod_{j,k\in I^2}\mathcal{F}(U_j\cap U_k)\right)$$ where $X$ is a topological space, $U$ is an open set of $X$ and $\{U_i\}_{i\in I}$ is any open cover of $U.$ Further, it is easy to imagine a sheaf as a étalé space over $X.$

Then I started reading about stacks using this notes and nlab as my primary sources. I learned that a stack is a contravariant functor $\mathcal{F}:\mathcal{C}^{\operatorname{op}}\to\mathbf{Grpd}$ satisfying a descent property and, categories fibered in groupoids over $\mathcal{C}$ is an intuitive way to think about stacks, where $\mathcal{C}$ is a site (category equipped with a coverage). Now I have following questions:

• How can I understand the descent property for stack? To be more specific, how can triple fiber products (intersections) appear in the equalizer fork diagram?

• What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks? Is this the correct analogue of étalé space of a stack?

Answer 1

A canonical example of a sheaf of sets on a topological space $X$ is the sheaf that sends an open subset $U$ of $X$ to the set of continuous real-valued functions on $U$. The gluing property then says that a continuous function on a union of open subsets $U_i$ of $X$ is the same thing as a collection of continuous functions $f_i: U_i \to \mathbb{R}$ such that $f_i$ and $f_j$ coincide on $U_i∩U_j$.

A canonical example of an ∞-sheaf (alias stack) of groupoids on a topological space $X$ is the ∞-sheaf that sends an open subset $U$ of $X$ to the groupoid of finite-dimensional continuous real vector bundles on $U$. (Isomorphisms in this groupoid are continuous fiberwise linear isomorphisms of vector bundles on $U$.) The gluing property then says that a vector bundle on a union of open subsets $U_i$ of $X$ is the same thing as a collection of vector bundles $V_i$ on $U_i$, together with isomorphisms $t_{i,j}: V_i→V_j$ of vector bundles restricted to $U_i∩U_j$, and such that $t_{j,k}t_{i,j}=t_{i,k}$ on $U_i∩U_j∩U_k$. This last condition is known as the cocycle condition and in some textbooks vector bundles are defined in this manner.

So the point of triple intersections is that isomorphisms of vector bundles over pairwise intersections must themselves satisfy a higher coherence identity. This condition is trivial for sheaves of sets because two functions can be equal in exactly one way, unlike vector bundles, which can be isomorphic in many different ways.

To answer the second question: the analog of the etale space of a sheaf of sets is the etale stack of an ∞-sheaf of groupoids. This stack is no longer an ∞-sheaf on the original topological space, but rather on the site of all topological spaces. (Some care must be taken when dealing with size issues here, since topological spaces do not form a small category, but I suppress these issues here for simplicity.) This etale stack can be constructed in many different ways. For example, there is a unique homotopy cocontinuous functor from ∞-sheaves of groupoids on a topological space $X$ to ∞-sheaves of groupoids on all topological spaces that sends a representable sheaf given by an open subset $U$ of $X$ to the representable sheaf of $U$ as an object in the site of all topological spaces. The image of a given ∞-sheaf $F$ of groupoids under this functor $E$ is the etale stack $E(F)$ of $F$, which is equipped with a canonical morphism (in the category of ∞-sheaves of groupoids on topological spaces) to the representable sheaf of $X$.

If we now take the ∞-sheaf of sections of the resulting map $E(F)→X$ of stacks, we recover the original ∞-sheaf $F$.

(There are many other constructions, of course. For example, one could work instead with topological groupoids, or rather, localic groupoids, instead of sheaves of groupoids on topological spaces.)

Answer 2

What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks?

A category fibered in groupoids over $\mathcal{C}$ is given by a functor $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$ satisfying certain conditions (I am not writing the definition as I assume you already know what is a category fibered in groupoids); look at Definition 4.2 in the paper Orbifolds as stacks?

Put a Grothendieck topology on the category $\mathcal{C}$; considering it as a site.

Given an object $U$ of the category $\mathcal{C}$, we consider its fiber; a category, denoted by $\mathcal{F}(U)$, defined as $$\text{Obj}(\mathcal{F}(U))=\{V\in \text{Obj}(\mathcal{F}):\pi_{\mathcal{F}}(V)=U\},$$ $$\text{Mor}_{\mathcal{F}(U)}(V_1,V_2)=\{(f:V_1\rightarrow V_2)\in \text{Mor}_{\mathcal{F}}(V_1,V_2):\pi_{\mathcal{F}}(f)=1_U\}.$$

Given a cover $\{U_\alpha\rightarrow U\}$ of the object $U$ (remember that we fixed a Grothendieck topology), we consider its descent category, denoted by $\mathcal{F}(\{U_\alpha\rightarrow U\})$. An object of the category $\mathcal{F}(\{U_\alpha\rightarrow U\})$ is given by the following data:

1. for each index $i\in \Lambda$, an object $a_i$ in the category $\mathcal{F}(U_i)$,
2. for each pair of indices $i,j\in \Lambda$, an isomorphism $\phi_{ij}:pr_2^*(a_j)\rightarrow pr_1^*(a_i)$ in the category $\mathcal{F}(U_i\times_{U}U_j)$

satisfying appropriate cocycle condition.

Now, given a categroy fibered in groupoids $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$, an object $U$ of $\mathcal{C}$ and a cover $\mathcal{U}(U)=\{U_\alpha\rightarrow U\}$ of $U$ in $\mathcal{C}$, there is an obvious functor $$p_{\mathcal{U}(U)}:\mathcal{F}(U)\rightarrow \mathcal{F}(\{U_\alpha\rightarrow U\})$$ (which you might have guessed already but let me say that), at the level of objects $$a\mapsto ((a|_{U_\alpha}),(\phi_{ij}))$$

A category fibered in groupoids $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$ is said to be a stack if, for each object $U$ of $\mathcal{C}$ and for each cover $\mathcal{U}(U)=\{U_\alpha\rightarrow U\}$, the functor $$p_{\mathcal{U}(U)}:\mathcal{F}(U)\rightarrow \mathcal{F}(\{U_\alpha\rightarrow U\})$$ is an equivalence of categories.

Now, you can ask what does equivalence of categories has anything to do with "sheaf like" properties? For a functor $\mathcal{D}\rightarrow \mathcal{C}$ to be an equivalence of categories, along other things, for each object $d\in \mathcal{D}$ we need an object $c\in \mathcal{C}$ such that, there is an isomorphism $F(c)\rightarrow d$.

Let $((a_\alpha),\{\phi_{\alpha\beta}\})$ be an object of $\mathcal{F}(\{U_\alpha\rightarrow U\})$. For this, by equivalence of categories, gives an element $a\in \mathcal{F}(U)$ that maps to $((a_\alpha),\{\phi_{\alpha\beta}\})$. That is, given an object $U$ of $\mathcal{C}$, an open cover $\{U_\alpha\rightarrow U\}$, for each collection of objects $\{a_\alpha\in \mathcal{F}(U_\alpha)\}$ that are compatible in some sense, there exists an object $a\in \mathcal{F}(U)$, such that, under appropriate restriction of $a$, you get the objects $a_{\alpha}$. This should remind the notion of sheaf on a topological space. This is how a stack is seen as a generalization of sheaf.

References :

1. Notes on Grothendieck topologies, fibered categories and descent theory.
2. Orbifolds as stacks?
3. How is a Stack the generalisation of a sheaf from a 2-category point of view?