Why should the target category of a sheaf be complete?

Question

The categorical definition of sheaf given on Wikipedia (https://en.wikipedia.org/wiki/Sheaf_(mathematics)#Complements) requires that the target category $\mathcal{C}$ of the functor $\mathcal{F}:\mathcal{O}(X)^{\textrm{op}}\to \mathcal{C}$ admit all limits, meaning that it should be complete, and then goes on to ask that for all open covers $\{U_i\}$ of $U$ a certain diagram $$ \mathcal{F}(U)\to \prod_i F(U_i) \rightrightarrows \prod_{i,j}F(U_i\cap U_j) $$ is an equalizer diagram. I understand that we want $\mathcal{C}$ to contain all products because products are involved in the equalizer diagram. I also know that being complete is equivalent to the existence of all products and equalizers. Why does this definition ask that $\mathcal{C}$ be complete? Is it important for some reason that we get equalizers?

Answer 1

The equalizer of products $F(U)\to\prod_i F(U_i)\rightrightarrows\prod_{i,j}F(U_i\cap U_j)$ is actually the realizing $F(U)$ as the limit of a certain diagram of images of $F$. More precisely, it asserts that $F$ sends a certain cocone under a diagram in $O(X)$ to a limiting cone over the image of the diagram. The diagram in question is indexed by the category whose objects are the inclusion morphisms $U_i\hookrightarrow U$ and their pairwise intersections $U_i\cap U_j\hookrightarrow U$, and whose morphisms are just the inclusions $U_i\cap U_j\hookrightarrow U_i$ and $U_i\cap U_j\hookrightarrow U_j$.

Note that the indexing category of the diagram is a subcategory of the slice category $C/U$ over $U$ whose objects are morphisms $V\to U$ and whose mosphisms from $W\to U$ to $V\to U$ are morphisms $W\to V$ such that $W\to U$ factors as $W\to V\to U$. The latter has a domain functor $C/U\to C$ sending $V\to U$ to $V$ and simply giving the morhpism $W\to V$ in $C$ that underlies the morphism from $W\to U$ to $V\to U$. Moreover, there is a tautological cocone under this diagram with the injection morphism $V\to U$ corresponding to the object $V\to U$ of $C/U$ being $V\to U$ itself.

Thus, the diagram is the domain functor applied to the subcategory, and the cocone in question is the restriction of the tautological cocone on the slice category to the subcategory.

Now in practice, almost all sheaves that occur are valued in categories that are complete, hence ones for which the sheaf condition can be expressed as the above fork being an equalizer. There is an important generalization in a different direction, however. Namely, for the case where the domains of the sheaves are not topological spaces, i.e. not the category of open subsets of a topological space, but more general sites. That is to say, you can ask for a contravariant functor to satisfy a sheaf condition relative to a family of morphism $U_i\to U$ in an arbitrary category.

In that case, the sheaf condition again amounts to asserting that a certain cocone under a diagram is sent to a limiting cone, but the diagram will be more complicated if the site is not a poset, or does not have pullbacks.What works in general is enlarging the indexing category of the diagram with all morphisms $V\to U$ that factor as $V\to U_i\to U$ for some $i$. This full subcategory of $C/U$ is known as the sieve generated by the $U_i\to U$, and the more general theory of sheaves and the sites on which they are defined is often expressed in terms of these sieves.