Definition 7.14.1 (00X1)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.14: Morphisms of sites
Definition 7.14.1 (cite)

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Definition 7.14.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A morphism of sites $f : \mathcal{D} \to \mathcal{C}$ is given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$ such that the functor $u_ s$ is exact.

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Comment #7739 by Haohao Liu on September 07, 2022 at 09:26

A possibly stupid question: as $u_s:Sh(C)\to Sh(D)$ is a functor between the categories of sheaves of "sets", so in general $Sh(D)$ is not an abelian category. Then what does " $u_s$ is exact" mean here? Maybe it means the restriction of $u_s$ to the sheaves of abelian groups is exact?

Comment #7740 by Johan on September 07, 2022 at 17:35

Put "exact definition" in the search bar (without the quotes) and look at the first definition. It does take some getting used to, but it is a very useful notion!

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