Lemma 7.14.4 (03CB)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.14: Morphisms of sites
Lemma 7.14.4 (cite)

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Composition of site functors respects continuity of site functors.

Lemma 7.14.4. Let $\mathcal{C}_ i$ , $i = 1, 2, 3$ be sites. Let $u : \mathcal{C}_2 \to \mathcal{C}_1$ and $v : \mathcal{C}_3 \to \mathcal{C}_2$ be continuous functors which induce morphisms of sites. Then the functor $u \circ v : \mathcal{C}_3 \to \mathcal{C}_1$ is continuous and defines a morphism of sites $\mathcal{C}_1 \to \mathcal{C}_3$ .

Proof. It is immediate from the definitions that $u \circ v$ is a continuous functor. In addition, we clearly have $(u \circ v)^ p = v^ p \circ u^ p$ , and hence $(u \circ v)^ s = v^ s \circ u^ s$ . Hence functors $(u \circ v)_ s$ and $u_ s \circ v_ s$ are both left adjoints of $(u \circ v)^ s$ . Therefore $(u \circ v)_ s \cong u_ s \circ v_ s$ and we conclude that $(u \circ v)_ s$ is exact as a composition of exact functors. $\square$

Comments (1)

Comment #7345 by Alejandro González Nevado on May 08, 2022 at 20:18

SS: Composition of site functors respects continuity of site functors.

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