(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The structure of an -site on an (∞,1)-category is precisely the data encoding an (∞,1)-category of (∞,1)-sheaves
inside the (∞,1)-category of (∞,1)-presheaves on .
The notion is the analog in (∞,1)-category theory of the notion of a site in 1-category theory.
The definition of -sites parallels that of 1-categorical sites closely. In fact the structure of an -site on an -category is equivalent to that of a 1-categorical site on its homotopy category (see below).
(-Grothendieck topology)
A sieve in an (∞,1)-category is a full sub-(∞,1)-category which is closed under precomposition with morphisms in .
A sieve on an object is a sieve in the overcategory .
Equivalently, a sieve on is an equivalence class of monomorphisms in the (∞,1)-category of (∞,1)-presheaves , with the (∞,1)-Yoneda embedding. (See below for the proof of this equivalence).
For a sieve on and a morphism into , we take the pullback sieve on to be that spanned by all those morphisms into that become equivalent to a morphism in after postcomposition with .
A Grothendieck topology on the -category is the specification of a collection of sieves on each object of – called the covering sieves , subject to the following conditions:
the trivial sieve covers – For each object the overcategory regarded as a maximal subcategory of itself is a covering sieve on . Equivalently: the monomorphism covers.
the pullback of a sieve covers – If is a covering sieve on and a morphism, then the pullback sieve is a covering sieve on . Equivalently, the pullback
in is covering.
a sieve covers if its pullbacks cover – For a covering sieve on and any sieve on , if the pullback sieve for every is covering, then itself is covering.
An -category equipped with a Grothendieck topology is an -site.
A sieve on that contains a covering sieve is itself covering.
For every an object of , the pullback sieve equals the pullback sieve . So it covers by the second axiom on sieves. So by the third axiom itself is covering.
There is a natural bijection between sieves on in and equivalence class of monomorphisms in .
This is HTT, prop. 6.2.2.5.
First observe that equivalence classes of -truncated object of are in bijection with sieves on :
An -presheaf is -truncated if its value on any object is either the empty ∞-groupoid or a contractible -groupoid. The full subcategory of on those objects on which takes a contractible value is evidently a sieve (because there is no morphism from a contractible to the empty -groupoid). Conversely, given a sieve on we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value on those objects that are in , and otherwise.
Now, as described at Interaction of presheaves and overcategories we have an equivalence
Under this equivalence our bijection above maps to the statement that there is a bijection between sieves on and equivalence class of -truncated objects in . But such a (-1)-truncated object is precisely a monomorphism .
The set of Grothendieck topologies on an -category is in natural bijection with the set of Grothendieck topologies on its homotopy category.
This is HTT, remark 6.2.2.3.
Because picking full sub-1-categories as well as full sub--categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.
If the -category 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 -site on it is the same as the 1-categorical structure of a site on it.
Structures of -sites on an (∞,1)-category correspond bijectively to topological localizations of the (∞,1)-category of (∞,1)-presheaves to a (∞,1)-category of (∞,1)-sheaves. See there for more details.
If (∞,1)-categories are incarnated as simplicially enriched categories, then an -site appears as an
If -categories are presented by model categories, then the notion of -site appears as that of
The trivial Grothendieck-topology on an -category is that where the only covering sieve on each object is itself. Equivalently, where the only covering monomorphisms in are the equivalences.
The (∞,1)-category of (∞,1)-sheaves on this site is just the (∞,1)-category of (∞,1)-presheaves itself. The localization is an equivalence.
-site
internal site / internal (infinity,1)-site?
Jacob Lurie, Section 6.2.2 of: Higher Topos Theory (2009)
Raffael Stenzel, Notions of -sites and related formal structures [arXiv:2306.06619]
Raffael Stenzel, Higher sites and their higher categorical logic, talk at HoTT Electronic Seminar (18 November 2021) [video:YT, slides:pdf]
Last revised on June 13, 2023 at 08:38:10. See the history of this page for a list of all contributions to it.