Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit? Let $X$ be a topological space and $\{U_i\}$ and open cover for $X$. Suppose we have sheaves $\mathcal{F}_i$ on $U_i$ and for each $i,j$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} \to \mathcal{F}_j|_{U_i \cap U_j}$ such that $\varphi_{ii} = id$ and $\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$ on $U_i \cap U_j \cap U_k$. I want to define some kind of sheaf $\mathcal{F}$ on $X$; to do this we consider 
$$\mathcal{F}(W) := \Big\{(s_i)_{i \in I}, s_i \in \mathcal{F}_i(W \cap U_i) : \varphi_{ij}(s_i|_{W \cap U_i \cap U_j}) = s_j|_{W \cap U_i \cap U_j}  \Big\}.$$


My question is: Can we realise $\mathcal{F}(W)$ as some kind of limit of a diagram, or yet as an equalizer  of two maps? I ask this because I want to show that this $\mathcal{F}$ is a sheaf. At the moment it seems to me that $\mathcal{F}(W)$ is very close to being some kind of ``inverse limit", but I don't know exactly what it is.


 A: To answer your question, the sheaf $\mathscr{F}$ you are trying to construct can be described in the category $\mathbf{Sh}(X)$ as the limit of a certain diagram. We will consider sheaves on an open subset $U \subseteq X$ as sheaves on $X$ by the usual direct image construction. 


*

*The diagram has one vertex for every ordered pair of elements in the open cover; the object at the vertex $(i, j)$ is the sheaf $\mathscr{F}_i |_{U_i \cap U_j}$. 

*We have an edge $(i, i) \to (i, j)$ for every ordered pair $(i, j)$; the corresponding morphism $\mathscr{F}_i \to \mathscr{F}_i |_{U_i \cap U_j}$ is the restriction map.

*We have an edge $(i, j) \to (j, i)$ for every ordered pair $(j, i)$; the corresponding morphism $\mathscr{F}_i |_{U_i \cap U_j} \to \mathscr{F}_j |_{U_j \cap U_i}$ is the isomorphism $\varphi_{i,j}$.


The cocycle condition guarantees that all the triangles appearing in the diagram commute. Thus we have a well-defined diagram in $\mathbf{Sh}(X)$ and we can take its limit.
Amusingly, there is another sense in which gluing sheaves is a limit construction: the category $\mathbf{Sh}(X)$ itself is a bicategorical limit of a (pseudo)commutative diagram involving all the categories $\mathbf{Sh}(U_i)$, $\mathbf{Sh}(U_i \cap U_j)$, and $\mathbf{Sh}(U_i \cap U_j \cap U_k)$. In fancy language, we say that $\mathbf{Sh}(-)$ is a stack for the canonical topology on the site of open subsets of $X$.
