nLab
(infinity,1)-site

Context

(,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

Contents

Idea

The structure of an (,1)(\infty,1)-site on an (∞,1)-category CC is precisely the data encoding an (∞,1)-category of (∞,1)-sheaves

Sh(C)PSh(C) Sh(C) \hookrightarrow PSh(C)

inside the (∞,1)-category of (∞,1)-presheaves on CC.

The notion is the analog in (∞,1)-category theory of the notion of a site in 1-category theory.

Definition

The definition of (,1)(\infty,1)-sites parallels that of 1-categorical sites closely. In fact the structure of an (,1)(\infty,1)-site on an (,1)(\infty,1)-category is equivalent to that of a 1-categorical site on its homotopy category (see below).

Definition

((,1)(\infty,1)-Grothendieck topology)

A sieve in an (∞,1)-category CC is a full sub-(∞,1)-category DCD \subset C which is closed under precomposition with morphisms in CC.

A sieve on an object cCc \in C is a sieve in the overcategory C /cC_{/c}.

Equivalently, a sieve on cc is an equivalence class of monomorphisms Uj(c)U \to j(c) in the (∞,1)-category of (∞,1)-presheaves PSh(C)PSh(C), with j:CPSh(C)j : C \to PSh(C) the (∞,1)-Yoneda embedding. (See below for the proof of this equivalence).

For SS a sieve on cc and f:dcf : d \to c a morphism into cc, we take the pullback sieve f *Sf^* S on dd to be that spanned by all those morphisms into dd that become equivalent to a morphism in SS after postcomposition with ff.

A Grothendieck topology on the (,1)(\infty,1)-category CC is the specification of a collection of sieves on each object of CC – called the covering sieves , subject to the following conditions:

  1. the trivial sieve covers – For each object cCc \in C the overcategory C /cC_{/c} regarded as a maximal subcategory of itself is a covering sieve on cc. Equivalently: the monomorphism Id:j(c)j(c)Id : j(c) \to j(c) covers.

  2. the pullback of a sieve covers – If SS is a covering sieve on cc and f:dcf : d \to c a morphism, then the pullback sieve f *Sf^* S is a covering sieve on dd. Equivalently, the pullback

    f *U U d f c \array{ f^* U &\to& U \\ \downarrow && \downarrow \\ d &\stackrel{f}{\to}& c }

    in PSh(C)PSh(C) is covering.

  3. a sieve covers if its pullbacks cover – For SS a covering sieve on cc and TT any sieve on cc, if the pullback sieve f *Tf^* T for every fSf \in S is covering, then TT itself is covering.

An (,1)(\infty,1)-category equipped with a Grothendieck topology is an (,1)(\infty,1)-site.

Properties

Of sieves

Lemma

A sieve SS' on cc that contains a covering sieve SSS \subset S' is itself covering.

Proof

For every f:dcf : d \to c an object of SC /cS \subset C_{/c}, the pullback sieve f *Sf^* S' equals the pullback sieve f *Sf^* S. So it covers dd by the second axiom on sieves. So by the third axiom SS' itself is covering.

Proposition

There is a natural bijection between sieves on cc in CC and equivalence class of monomorphisms Uj(C)U \to j(C) in PSh(C)PSh(C).

This is HTT, prop. 6.2.2.5.

Proof

First observe that equivalence classes of (1)(-1)-truncated object of PSh(C /c)PSh(C_{/c}) are in bijection with sieves on cc:

An (,1)(\infty,1)-presheaf FF is (1)(-1)-truncated if its value on any object is either the empty ∞-groupoid \emptyset or a contractible \infty-groupoid. The full subcategory of C /cC_{/c} on those objects on which FF takes a contractible value is evidently a sieve (because there is no morphism from a contractible to the empty \infty-groupoid). Conversely, given a sieve SS on cc we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value *=Δ[0]Grpd* = \Delta[0] \in \infty Grpd on those objects that are in SS, and \emptyset otherwise.

Now, as described at Interaction of presheaves and overcategories we have an equivalence

PSh(C /c)PSh(C) /j(c). PSh(C_{/c}) \simeq PSh(C)_{/j(c)} \,.

Under this equivalence our bijection above maps to the statement that there is a bijection between sieves on cc and equivalence class of (1)(-1)-truncated objects in PSh(C) /j(c)PSh(C)_{/j (c)}. But such a (-1)-truncated object is precisely a monomorphism Uj(c)U \to j(c).

Of coverages

Observation

The set of Grothendieck topologies on an (,1)(\infty,1)-category CC is in natural bijection with the set of Grothendieck topologies on its homotopy category.

This is HTT, remark 6.2.2.3.

Proof

Because picking full sub-1-categories as well as full sub-(,1)(\infty,1)-categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.

Corollary

If the (,1)(\infty,1)-category CC happens to be an ordinary category (for instance in its incarnation as a quasi-category it is the nerve of an ordinary category), then the structure of an (,1)(\infty,1)-site on it is the same as the 1-categorical structure of a site on it.

Of sites

Proposition

Structures of (,1)(\infty,1)-sites on an (∞,1)-category CC correspond bijectively to topological localizations of the (∞,1)-category of (∞,1)-presheaves to a (∞,1)-category of (∞,1)-sheaves. See there for more details.

Incarnations and models

If (∞,1)-categories are incarnated as simplicially enriched categories, then an (,1)(\infty,1)-site appears as an

If (,1)(\infty,1)-categories are presented by model categories, then the notion of (,1)(\infty,1)-site appears as that of

Examples

References

Section 6.2.2 of

Revised on March 29, 2014 13:26:04 by Anonymous Heru? (188.100.50.220)