nLab comparison lemma

The comparison lemma


Topos Theory

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In higher category theory


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

The comparison lemma


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:BCu \colon B \to C be a functor with CC a site. The induced topology on BB by uu is 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.

The classical comparison lemma (Verdier 1972):


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 (Def. ). 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 \colon B \to C induces an equivalence of categories of sheaves (of sets) B C B^\sim \to C^\sim.

Beilinson 2012 proves the following generalisation of the classical comparison lemma:


Let BB be an essentially small category and CC be an essentially small site. Suppose that u:BCu \colon B \to C is a faithful functor which exhibits BB (with the induced topology) as a dense subsite of CC, i.e. which satisfies the following condition:

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


  1. 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.

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


Last revised on January 12, 2024 at 00:30:22. See the history of this page for a list of all contributions to it.