###### tags: Hartshorne AG

# Colimits and Limits of Sheaves

*Reference : Hartshorne Exercise II.2.10,12*

Using some category theory - namely results (co)limits and results in the presheaf category - one can easily describe (co)limits in the category of sheaves.

Recall that in our note on presheaves, we have seen that (co)limits of presheaves are given levelwise. In the case of the category of sheaves, there are some analogous results.

Fix a topological space $X$. Fix a bicomplete category $\mathscr{C}$; here we are thinking $\mathscr{C}$ as something like the category of abelian groups, commutative rings, sets.

Given a system $\{\mathscr{F}_i\}_i$ of sheaves, it can also be regarded as a system of presheaves, so to distinguish whether we are taking the (co)limit of this system in $\mathbf{psh}(X,\mathscr{C})$ or $\mathbf{sh}(X,\mathscr{C})$, we specify using terminologies like presheaf (co)limits or sheaf (co)limits.

## Sheaf colimits are sheafified presheaf colimits
As sheafification is a left adjoint, it preserves colimits. If you have confusion with this statement, the following is a sketch of proof using Yoneda lemma.
:::spoiler
$$\begin{align}\operatorname{Hom}(L\varinjlim_iX_i,Y)\simeq&\operatorname{Hom}(\varinjlim_iX_i,RY)\simeq\varprojlim_i\operatorname{Hom}(X_i,RY)\\\simeq&\varprojlim_i\operatorname{Hom}(LX_i,Y)\simeq\operatorname{Hom}(\varinjlim_iLX_i,Y)\end{align}$$
:::

## Presheaf limit of sheaves is a sheaf that is also a sheaf limit

Recall that sheaves are defined using equalizers.

A presheaf $\mathscr{F}$ is a sheaf iff for each $\{U_i\}_i\in\operatorname{Cov}(U),U\in\operatorname{Op}(X)$, the following is an equalizer :

$$\mathscr{F}(U)\to\prod_{i}\mathscr{F}(U_i)\rightrightarrows \prod_{ij}\mathscr{F}(U_{ij})$$

Recall also the following the facts :
- Products and equalizers are limits (indexed either by a discrete category or $\bullet\rightrightarrows\bullet$)
- Limit of presheaves can be computed pointwise
- Limits commutes with limits (use something like $\varprojlim_{I}\varprojlim_{J}\simeq\varprojlim_{I\times J}\simeq\varprojlim_{J\times I}\simeq\varprojlim_{J}\varprojlim_{I}$)

Assembling these facts together, we see that a presheaf limit of sheaves is a sheaf.

As $\mathbf{sh}(X,\mathscr{C})$ is a full subcategory of $\mathbf{psh}(X,\mathscr{C})$, this presheaf limit is readily a sheaf limit.