# nLab comparison lemma

The comparison lemma

This page is about a general theorem in topos theory. For other meanings see e.g. comparison theorem (étale cohomology).

topos theory

# The comparison lemma

## Idea

A functor from a category to a site induces a topology on the source category. The comparison lemma says that, under certain conditions, such a functor induces an equivalence between the categories of sheaves on the sites.

## Induced topology

Let $u : B \to C$ be a functor with $C$ a site. Recall that the topology induced on $B$ by $u$ is defined to be the finest one such that $u$ is a continuous functor, i.e. such that the map $G \mapsto G \circ u$ takes sheaves on $C$ to sheaves on $B$.

## Statement

###### Theorem

Let $B$ be a small category, $C$ a site, and $u : B \to C$ a fully faithful functor. Consider $B$ as a site with the topology induced by $u$. If every object $x \in C$ has a covering $(u(a_\alpha) \to x)$ by objects of $B$, then $u : B \to C$ induces an equivalence of categories of sheaves (of sets) $B^\sim \to C^\sim$.

In this paper, Beilinson proves the following generalisation of the classical comparison lemma.

###### Theorem

Let $B$ be an essentially small category and $C$ be an essentially small site. Suppose that $u : B \to C$ is a faithful functor which exhibits a dense subsite, hence satisfying the following condition:

(B) for every object $x \in C$ and finite family $(x \to u(a_\alpha))_\alpha$, with $a_\alpha \in B$, there exists a covering family $(u(b_\beta) \to x)_\beta$ of $x$ such that every composite $u(b_\beta) \to x \to u(a_\alpha)$ lies in the image of $\Hom(b_\beta, a_\alpha) \hookrightarrow \Hom(u(b_\beta), u(a_\alpha))$.

Then

• The topology on $B$ induced by $u$ has the following simple description: a sieve $(x_\gamma \to x)_\gamma$ is covering iff the sieve generated by the family $(u(x_\gamma) \to u(x))_\gamma$ is covering in $C$.

• The functor $u : B \to C$ induces an equivalence of categories of sheaves (of sets) $B^\sim \to C^\sim$.

• Jean-Louis Verdier, Fonctorialité de catégories de faisceaux, Théorie des topos et cohomologie étale de schémas (SGA 4), Tome 1, Lect. Notes in Math. 269, Springer-Verlag, 1972, pp. 265–298.

• Alexander Beilinson, p-adic periods and derived de Rham cohomology. arXiv:1102.1294

Section C2.2. of