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(∞,1)-comparison lemma

The -comparison lemma

Context

(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

The (,1)(\infty,1)-comparison lemma

Idea

The (∞,1)-comparison lemma says that, under certain conditions, a functor between (∞,1)-sites induces an equivalence between the categories of (∞,1)-sheaves on the sites.

In this paper, Lemma C.3, Hoyois proves the following comparison lemma.

Lemma

Let DD be a locally small (∞,1)-category, CC a small (∞,1)-category, and u:CDu : C \to D a fully faithful functor. Let τ\tau and ρ\rho be quasi-topologies on CC and DD, respectively. Suppose that:

a. Every τ\tau-sieve is generated by a cover {U iX}\{U_i \to X\} such that:

  1. the fiber products U i 0× X× XU i nU_{i_0} \times_X \cdots\times_X U_{i_n} exist and are preserved by uu;

  2. {u(U i)u(X)}\{u(U_i) \to u(X)\} is a ρ¯\bar\rho-cover.

b. For every XCX \in C and every ρ\rho-sieve Ru(X)R \hookrightarrow u(X), u *(R)Xu^*(R) \hookrightarrow X is a τ¯\bar\tau-sieve in CC.

c. Every XDX \in D admits a ρ¯\bar\rho-cover {U iX}\{U_i \to X\} such that the fiber products U i 0× X× XU i nU_{i_0} \times_X \cdots \times_X U_{i_n} exist and belong to the essential image of uu.

Then the adjunction u *u *u^* \dashv u_* restricts to an equivalence of ∞-categories Shv ρ(D)Shv τ(C)Shv_\rho(D) \simeq Shv_\tau(C).

Here, a quasi-topology is a collection of sieves closed under pullback and τ¯\bar\tau is the coarsest topology containing a quasi-topology τ\tau. The stability under pullback ensures that Shv τ(C)=Shv τ¯(C)Shv_\tau(C)=Shv_{\bar\tau}(C).

Generalization

It seems difficult to find a useful generalization not assuming the existence of some pullbacks. For the conclusion of the lemma, the following conditions (b is unchanged) are both necessary and sufficient:

a. For every τ\tau-sieve UXU \hookrightarrow X, a ρu !(UX)a_\rho u_!(U \to X) is an equivalence.

b. For every XCX \in C and every ρ\rho-sieve Ru(X)R \hookrightarrow u(X), u *(R)Xu^*(R) \hookrightarrow X is a τ¯\bar\tau-sieve in CC.

c. For every XDX \in D, its image in Shv ρ(D)Shv_\rho(D) belongs to the smallest subcategory generated by the image of CC under colimits.

We can take these conditions a and b to define, respectively, the notions of cover-preserving functor (continuous functor) and comorphism of sites (cocontinuous functor) for (∞,1)-sites.

References

Last revised on July 3, 2019 at 23:18:39. See the history of this page for a list of all contributions to it.