(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A topological localization is a left exact localization of an (∞,1)-category – in the sense of passing to a reflective sub-(∞,1)-category – at a collection of morphisms that are monomorphisms.
A topological localization of an (∞,1)-category of (∞,1)-presheaves is precisely a localization at Cech covers for a given Grothendieck topology on , yielding the corresponding (∞,1)-topos of (∞,1)-sheaves.
and in fact equivalence classes of such topological localizations are in bijection with Grothendieck topologies on .
Notice that in general a topological localization is not a hypercomplete (∞,1)-topos. That in general requires localization further at hypercovers.
Recall that a reflective sub-(∞,1)-category is obtained by localizing at a collection of morphisms of .
The class of all morphisms of that the left adjoint sends to equivalences is the strongly saturated class of morphisms generated by . By the recognition principle for exact localizations, the functor is exact if and only if is stable under the formation of pullbacks.
We now define such localizations where the collection consists of monomorphisms .
Call a morphism in an (∞,1)-category a monomorphism if it is a (-1)-truncated object in the overcategory .
Equivalently: if for every object the induced morphism in the homotopy category of ∞-groupoids
exhibits as a direct summand of .
Equivalence classes of monomorphisms into an object form a poset of subobjects of .
This is HTT, p. 460
The standard example to keep in mind is that of a Cech nerve. In fact, as the propositions below will imply, this is for the purposes of localizations of an (∞,1)-category of (∞,1)-presheaves the only kind of example.
Let Diff be the category of smooth manifolds and the (∞,1)-category of (∞,1)-presheaves on , which may be modeled by the global model structure on simplicial presheaves on .
For a manifold, let be an open cover. Let be the Cech nerve of this cover, the simplicial object of presheaves
which we may regard as a simplicial presheaf and hence as an object of .
Then for any other manifold, we have that
is the ∞-groupoid whose
objects are maps that factor through one of the ;
there is a unique morphism between two such maps precisely if they factor through a double intersection ;
and so on.
In the homtopy category of ∞-groupoids, this is equivalent to the 0-groupoid/set of those maps that factor through one of the . Notice that this constitutes the sieve generated by the covering family . This is a subset of the 0-groupoid/set , hence a direct summand.
topological localization
Let be a presentable (∞,1)-category.
A strongly saturated class of morphisms is called topological if
there is a subclass of monomorphisms that generates ;
under pullback in elements in pull back to elements in .
A reflective sub-(∞,1)-category
is called a topological localization if the class of morphisms that sends to equivalences is topological.
This is HTT, def. 6.2.1.4
-sheaves
Let be an (∞,1)-site.
Let be the collection of all monomorphisms to objects (under Yoneda embedding) that correspond to covering sieves in . Say an object in the (∞,1)-category of (∞,1)-presheaves on is an (∞,1)-sheaf if it is an -local object (i.e. if it satisfies descent along all morphisms coming from covering sieves).
Write
for the reflective sub-(∞,1)-category on these -sheaves.
This is HTT, def. 6.2.2.6
Let throughout be a locally presentable (∞,1)-category.
(topological localizations are exact)
Every topological localization is an exact localization in that the reflector preserves finite limits.
At Properties of exact localizations it is shown that a reflective localization is exact precisely if the class of morphisms that it inverts is stable under pullback. This is the case for topological localizations by definition.
(generation from a small set of morphisms)
For every topological localization of at a strongly saturated class there exists a small set of monomorphisms that generates .
This is HTT, prop. 6.2.1.5.
Every topological localization of is necessarily accessible and exact.
This is HTT, cor. 6.2.1.6
The following proposition asserts that for the construction of (n,1)-toposes the notion of topological localization is empty: if colimits commute with products, then already every localization is topological. Accordingly, also the notion of hypercompletion is relevant only for (∞,1)-toposes.
(localizations of presentable -categories are topological)
Let be a locall presentable (n,1)-category for finite with universal colimits. Then every left exact localization of is a topological localization
This is HTT, prop. 6.4.3.9.
…
This means that every (n,1)-topos of -sheaves is a localization at Cech nerves of covers.
Remark Notice in this context the statement found for instance in
that a simplicial presheaf that satisfies descent on all Cech covers already satisfies descent for all bounded hypercovers. If the simplicial presheaf is -truncated for some , then it won’t “see” -bounded hypercovers for large enough anyway, and hence it follows that truncated simplicial presheaves that satisfy Cech descent already satisfy hyperdescent.
This is in line with the above statement that for -toposes with finite there is no distinction between Cech descent and hyperdescent. The distinction becomes visible only for untruncated -presheaves.
Let throughout be a small (∞,1)-category and write for the (∞,1)-category of (∞,1)-presheaves on .
sheaves form a topological localization
If is endowed with a Grothendieck topology, the inclusion
is a topological localization.
This is HTT, Prop. 6.2.2.7.
All topological localizations of arise this way:
There is a bijection between Grothendieck topologies on and equivalence classes of topological localizations of .
This is HTT, prop. 6.2.2.17.
…
Topological localizations are the topic of section 6.2, from def. 6.2.1.5 on, in