structures in a cohesive (∞,1)-topos
inside the (∞,1)-category of (∞,1)-presheaves on .
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).
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 covering sieves on in and equivalence class of monomorphisms in .
This is HTT, prop. 220.127.116.11.
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 18.104.22.168.
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.
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.
Section 6.2.2 of