The Stacks project

18.45 Sheaves of pointed sets

In this section we collect some facts about sheaves of pointed sets which we've previously mentioned only for abelian sheaves.

A pointed set is a pair $(S, 0)$ where $S$ is a set and $0 \in S$ is an element of $S$. A morphism $(S, 0) \to (S', 0')$ of pointed sets is simply a map of sets $S \to S'$ sending $0$ to $0'$. We'll abuse notation and say “let $S$ be a pointed set” to mean $S$ is endowed with a marked element $0 \in S$. A sheaf of pointed sets is the same thing as a sheaf of sets $\mathcal{F}$ endowed with a “marking” $0 : * \to \mathcal{F}$ where $*$ is the final sheaf (Sites, Example 7.10.2).

Given a morphism of sites or of topoi, there are pushforward and pullback functors on the categories of sheaves of pointed sets, see Sites, Section 7.44. These are constructed by taking the pushforward, resp. pullback of the underlying sheaf of sets and suitably marking it (using that the pullback of the final sheaf is the final sheaf).

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor between sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the morphism of topoi associated with $u$, see Sites, Lemma 7.21.1. Then $g^{-1}$ on sheaves of pointed sets has an left adjoint $g_!$ as well. The construction of this functor is entirely analogous to the construction of $g_!$ on abelian sheaves in Section 18.16.

Similarly, if $j : \mathcal{C}/U \to \mathcal{C}$ is as in Section 18.19 then there is a left adjoint $j_!$ to the functor $j^{-1}$ on sheaves of pointed sets

If we ever need these facts and constructions we will precisely state and prove here the corresponding lemmas.