Gluing sheaves as direct limit. What is the ordering of this directed system.

Question

This is theorem 5.76 (Gluing Lemma) in Rotman, Homological Algebra.

Let $(U_i)_{i\in I}$ be an open cover of topological space and $U_{ij}=U_i\cap U_j$. Suppose for all $i\in I, F_i$ is a sheaf of abelian groups over $U_i$ and for each $i,j\in I$ there is sheaf isomorphism $\theta_{ij}:F_j\vert_{U_{ij}}\to F_i\vert_{U_{ij}}$ with $\theta_{ii}=1_{F_i}$ and cocycle condition for $i,j,k\in I,\theta_{ik}=\theta_{ij}\theta_{jk}$. Then there exists a unique sheaf $F$ over $X$ and isomorphism $\eta_i:F\vert_{U_i}\to F_i$ with $\eta_i\eta_j^{-1}=\theta_{ij}$ over $U_{ij}$ for all $i,j$.

In the proof, existence of $F(V)=\lim F_i(V\cap U_i)$ as directed limit.

1. What is the ordering of directed system here and the morphisms $F_i(V\cap U_i)\to F_j(V\cap U_j)$?

2. I am aware that there is $F_i(V\cap U_i)\to F_i(V\cap U_i\cap U_j)\cong F_j(V\cap U_j\cap U_i)$. How should I describe this directed system without knowing this is directed limit of some diagrams?

3. I have seen people saying this is inverse limit. Is this directed limit or inverse limit? If I think $F_i:U_i\to R$ being sheaves of continuous function of $U_i\subset R^m$, obviously this has to be cokernel of some exact sequence by identifying the part in agreement. Since it is cokernel, it had better be directed limit.

Answer 1

The system here is not a directed system, and we are taking a limit, not a colimit. Rotman's description of what's going on here is quite sloppy.

Here's the correct description of the gluing. Consider the category $\mathcal{I}$ which has an object for each $i\in I$ and for each pair $(i,j)\in I\times I$ and morphisms $i\to (i,j)$ and $j\to (i,j)$ for each $i$ and $j$. For each open set $V$, there is a functor $G_V:\mathcal{I}\to Ab$ such that $G_V(i)=\mathcal{F}_i(V\cap U_i)$ and $G_V(i,j)=\mathcal{F}_i(V\cap U_{ij})$, with $G_V$ sending the morphism $i\to (i,j)$ to the restriction map of the sheaf $\mathcal{F}_i$ and $G_V$ sending the morphism $j\to (i,j)$ to the restriction map $\mathcal{F}_j(V\cap U_j)\to \mathcal{F}_j(V\cap U_{ij})$ composed with the isomorphism $\mathcal{F}_j(V\cap U_{ij})\to\mathcal{F}_i(V\cap U_{ij})$ given by $\theta_{ij}$.

The sheaf $\mathcal{F}$ is then defined by taking $\mathcal{F}(V)$ to be the limit of this functor $G_V$. Explicitly, this means an element of $\mathcal{F}(V)$ consists of an element of $\mathcal{F}_i(V\cap U_i)$ for each $i$, such that their restrictions agree on $V\cap U_{ij}$ when you identify $\mathcal{F}_i(V\cap U_{ij})$ with $\mathcal{F}_j(V\cap U_{ij})$ via $\theta_{ij}$.