First of all I should apologies if this question does not count as a research level one. I asked the [same question](https://math.stackexchange.com/questions/3583228/understanding-sheaves-and-stacks) 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](https://ncatlab.org/nlab/show/%C3%A9tal%C3%A9+space) over $X.$


Then I started reading about stacks using [this notes](http://www.maths.qmul.ac.uk/~noohi/papers/quick.pdf) 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?



Understanding the definition of stacks
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.)


