structures in a cohesive (∞,1)-topos
An -category of -sheaves is a reflective sub-(∞,1)-category
of an (∞,1)-category of (∞,1)-presheaves such that the following equivalent conditions hold
This is HTT, def. 18.104.22.168.
An -category of -sheaves is an (∞,1)-topos.
Typically here is the Cech nerve
The (n,1)-categorical counting is:
We reproduce the proof that the two characterization in def. 1 above are indeed equivalent.
For an (∞,1)-site, the full sub-(∞,1)-category of on local objects with respect to the covering monomorphisms in is indeed a topological localization, and hence is indeed an exact reflective sub-(∞,1)-category of and hence an (∞,1)-topos.
This is HTT, Prop. 22.214.171.124
where the colimit is taken over all covering sieves of ; this is called the plus construction. It defines a functor and there is an obvious morphism natural in .
It is clear that the construction preserves finite (∞,1)-limits, since filtered (∞,1)-colimits do, and it is easy to see that the map becomes an equivalence in . Given an ordinal , let be the -iteration of the plus construction applied to the presheaf . Then the functor preserves finite limits and the canonical map becomes an equivalence in . In particular, if is a sheaf, then . Thus, it suffices to show that there exists an ordinal such that, for every , is a sheaf.
Fix and a covering sieve of . Given a presheaf , we define an auxiliary presheaf by the formula
Restriction maps induce a morphism . Since we clearly have for , the functor is idempotent in the sense that and are (equivalent) equivalences.
By definition, is a sheaf if and only if for every and every covering sieve of . Our goal is therefore to find an ordinal (depending only on the (∞,1)-site ) such that, for every , the map
is an equivalence.
The morphism in factors as
Applying to this factorization, we get a commutative diagram
in which the map is an equivalence since is idempotent. By cofinality, the colimit of the maps as is an equivalence. Applying this to , we get
This almost means that is a sheaf. The problem is that the filtered colimit on the right-hand side need not commute with the limit appearing in the definition of , that is, the canonical map
need not be an equivalence. To solve this problem, we choose a cardinal such that for every and every covering sieve of , the functor preserves -filtered colimits. This is possible because is small and each of these functors, being the composition of the restriction functor and the limit functor , has a left adjoint (∞,1)-functor and is therefore accessible (see HTT Prop. 126.96.36.199). Then the above map with replaced by is an equivalence. For every ordinal , applying the above to shows that
Since is a limit ordinal, we deduce that is a sheaf by cofinality.
(equivalence of site structures and categories of sheaves)
This is HTT, prop. 188.8.131.52.
For a small (∞,1)-site and the corressponding reflective inclusion of (∞,1)-sheaves into (∞,1)-presheaves on we have that the image under of a sub--functor of a representable is covering precisely if is an equivalence.
This is HTT, lemma 184.108.40.206.
Since is the reflectuive localization of at covering monomorphisms, it is clear that if is covering, then is an equivalence.
To see the converse, form the 0-truncation of and conclude as for ordinary sheaves on the homotopy catgegory of .
We have seen in (…) that for every structure of an -site on we obtain a topological localization of the presheaf category, and that this is an injective map from site structures to equivalence classes of sheaf categories. It remains to show that it is also a surjective map, i.e. that every topological localization of comes from the structure of an (∞,1)-site on .
So consider a strongly saturated class of morphisms which s topological (closed under pullbacks). Write for the subcalss of those that are monomorphisms of the form .
Observe that then is indeed generated by (is the smallest strongly saturated class containing) : since by the co-Yoneda lemma every object is a colimit over representables. It follows that every monomorphism is a colimit (in ) of those of the form : for consider the pullback diagram
where the equivalence is due to the fact that we have universal colimits in . This realizes as a colimit over morphisms of the form that are each a pullback of a monomorphism. Since monomorphisms are stable under pullback (see monomorphism in an (∞,1)-category for details), all these component morphisms are themselves monomorphisms.
So every monomorphism in is generated from , but by the assumption that is topological, it is itself entirely generated from monomorphisms, hence is generated from .
So far this establishes that evry topological localization of is a localization at a collection of sieves/ subfunctors of representables. It remains to show that this collection of subfunctors is indeed an Grothendieck topology and hence exhibits on the structure of an (∞,1)-site. We check the necessary three axioms:
equivalences cover – The equivalences belong to and are monomorphisms, hence belong to .
pullback of a cover is covering - Since monomorphisms are stable under pullback, we haave for every in and every that also the pullback
is a monomorphism and in , hence in .
if restriction of a sieve to a cover is covering, then the sieve is covering – Let be an arbitrary monomorphism and in . Write and consider the pullback
where again we made use of the universal colimits in . Now notice that
is in by assumption;
is by pullback stability of ;
each of the is in by assumption, hence is by the fact that is strongly saturated.
so by commutativity is in .
finally by 2-out-of-3 this means that is in .
We discuss how -sheaves over a paracompact topological space are equivalent to topological spaces over . This is the analogue of the 1-categorical statement that sheaves on are equivalent to etale spaces over : an etale space over is one whose fibers are discrete topological space, hence 0-truncated spaces. The n-category analogy has homotopy n-types as fibers.
This is a relative version of the singular simplicial complex functor.
There is a model category structure on the over category with weak equivalences and fibration precisely those morphisms over such that for each the induced morphism is a weak equivalence or fibration, respectively, in the standard model structure on simplicial sets.
This is HTT, prop 220.127.116.11.
This is HTT, corollary 18.104.22.168.
The (∞,1)-toposes that are -categories of sheaves, i.e. that arise by topological localization from an (∞,1)-category of (∞,1)-presheaves, enjoy a number of special properties over other classes of -toposes, such as notably hypercomplete (∞,1)-toposes.
The following lists these properties. (HTT, section 6.5.4.)
The construction of (∞,1)-sheaf (∞,1)-toposes on a given locale is singled out over the construction of other kinds of -toposes (such as hypercomplete (∞,1)-toposes) by the following universal property:
This is HTT, lemma 22.214.171.124.
See also n-localic (∞,1)-topos.
This is HTT, prop. 126.96.36.199.
This statement is false for the hypercompletion of , in general.
For a topological space, let
This is HTT, theorem 188.8.131.52.
The topological localizations of an (∞,1)-category of (∞,1)-presheaves are presented by the left Bousfield localization of the global model structure on simplicial presheaves at the set of Cech covers.
Detailed discussion of this model category presentation is at
The study of simplicial presheaves apparently goes back to
This was later conceived in terms of a model structure on simplicial presheaves and on simplicial sheaves by Joyal and Jardine. Toën summarizes the situation and emphasizes the interpretation in terms of ∞-stacks living in -categories for instance in
B. Toën, Higher and derived stacks: a global overview (arXiv) .
This concerns mostly hypercomplete -sheaves, though.
localization -functors (-sheafification for the present purpose) are discussed in section 5.2.7;
local objects (-sheaves for the present purpose) and local isomorphisms are discussed in section 5.5.4;
the definition of -topoi of -sheaves is then definition 184.108.40.206 in section 6.1;
the characterization of -sheaves in terms of descent is in section 6.1.3
the relation between the Brown–Joyal–Jardine model and the general story is discussed at length in section 6.5.4
An overview is in