nLab comparison lemma

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

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.

Statement

Definition

Let $u \colon B \to C$ be a functor with $C$ a site. The induced topology on $B$ by $u$ is 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 $B$.

The classical comparison lemma (Verdier 1972):

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 $u$ (Def. ). If every object $x \in C$ has a covering $(u(a_\alpha) \to x)$ by objects of $B$, then $u \colon B \to C$ induces an equivalence of categories of sheaves (of sets) $B^\sim \to C^\sim$.

Beilinson 2012 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 $u \colon B \to C$ is a faithful functor which exhibits $B$ (with the induced topology) as a dense subsite of $C$, i.e. which satisfies the following condition:

• 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 $\Hom(b_\beta, a_\alpha) \hookrightarrow \Hom\big(u(b_\beta), u(a_\alpha)\big)$.

Then:

1. 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 $C$.

2. The functor $u \colon B \to C$ induces an equivalence of categories of sheaves (of sets) $Sh(B) \to Sh(C)$.

Related concepts

• (∞,1)-comparison lemma

• dense subsite

• “comparison functor” is mostly understood to refer to a different concept in categorical algebra

References

• Jean-Louis Verdier, Fonctorialité de catégories de faisceaux, in: Théorie des topos et cohomologie étale de schémas (SGA4), Tome 1, Lect. Notes in Math. 269, Springer (1972) 265-298.

• Anders Kock, Ieke Moerdijk, Section 2 of: Presentations of étendues. Cahiers Topologie Géom. Différentielle Catég. 32 2 (1991) 145–164 [EuDML]

• Peter Johnstone, Section C2.2 of: Sketches of an Elephant, Oxford University Press (2002)

• Alexander Beilinson, $p$-adic periods and derived de Rham cohomology, J. Amer. Math. Soc. 25 (2012) 715-738 [arXiv:1102.1294, doi:10.1090/S0894-0347-2012-00729-2, pdf]