This page is about a general theorem in topos theory. For other meanings see e.g. comparison theorem (étale cohomology).
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.
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):
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:
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:
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 $u \colon B \to C$ induces an equivalence of categories of sheaves (of sets) $Sh(B) \to Sh(C)$.
“comparison functor” is mostly understood to refer to a different concept in categorical algebra
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]
Last revised on January 12, 2024 at 00:30:22. See the history of this page for a list of all contributions to it.