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

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos



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.


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).


((,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.


Of sieves


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


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.


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.


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


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


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.


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


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



Section 6.2.2 of

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