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 be a functor with a site. The induced topology on by is the finest one such that is a continuous functor, i.e. such that the map takes sheaves on to sheaves on .
The classical comparison lemma (Verdier 1972):
Let be a small category, a site, and a fully faithful functor. Consider as a site with the topology induced by (Def. ). If every object has a covering by objects of , then induces an equivalence of categories of sheaves (of sets) .
Beilinson 2012 proves the following generalisation of the classical comparison lemma:
Let be an essentially small category and be an essentially small site. Suppose that is a faithful functor which exhibits (with the induced topology) as a dense subsite of , i.e. which satisfies the following condition:
Then:
The topology on induced by has the following simple description: a sieve is covering iff the sieve generated by the family is covering in .
The functor induces an equivalence of categories of sheaves (of sets) .
“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, -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 June 24, 2024 at 18:03:25. See the history of this page for a list of all contributions to it.