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).

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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

Statement

Theorem

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

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

Theorem

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

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

Then

  • The topology on BB induced by uu has the following simple description: a sieve (x γx) γ(x_\gamma \to x)_\gamma is covering iff the sieve generated by the family (u(x γ)u(x)) γ(u(x_\gamma) \to u(x))_\gamma is covering in CC.

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

References

  • 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

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Last revised on July 3, 2019 at 23:17:56. See the history of this page for a list of all contributions to it.