(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 $PSh_{(\infty,1)}(C)$ is precisely a localization at Cech covers for a given Grothendieck topology on $C$, yielding the corresponding (∞,1)-topos of (∞,1)-sheaves.
and in fact equivalence classes of such topological localizations are in bijection with Grothendieck topologies on $C$.
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 $D \stackrel{\stackrel{L}{\leftarrow}}{\hookrightarrow} C$ is obtained by localizing at a collection $S$ of morphisms of $C$.
The class $\bar S$ of all morphisms of $C$ that the left adjoint $L : C \to D$ sends to equivalences is the strongly saturated class of morphisms generated by $S$. By the recognition principle for exact localizations, the functor $L$ is exact if and only if $\bar S$ is stable under the formation of pullbacks.
We now define such localizations where the collection $S$ consists of monomorphisms .
Call a morphism $f : X \to Y$ in an (∞,1)-category $C$ a monomorphism if it is a (-1)-truncated object in the overcategory $X_{/Y}$.
Equivalently: if for every object $A \in C$ the induced morphism in the homotopy category of ∞-groupoids
exhibits $C(A,X)$ as a direct summand of $C(A,Y)$.
Equivalence classes of monomorphisms into an object $X$ form a poset $Sub(X)$ of subobjects of $X$.
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 $PSh_{(\infty,1)}(Diff)$ the (∞,1)-category of (∞,1)-presheaves on $Diff$, which may be modeled by the global model structure on simplicial presheaves on $Diff$.
For $X \in Diff$ a manifold, let $\{U_i \hookrightarrow X\}$ be an open cover. Let $C(\{U_i\})$ 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 $PSh_{(\infty,1)}(Diff)$.
Then for $V$ any other manifold, we have that
is the ∞-groupoid whose
objects are maps $V \to X$ that factor through one of the $U_i$;
there is a unique morphism between two such maps precisely if they factor through a double intersection $U_{i} \cap U_j$;
and so on.
In the homtopy category of ∞-groupoids, this is equivalent to the 0-groupoid/set of those maps $V \to X$ that factor through one of the $U_i$. Notice that this constitutes the sieve generated by the covering family $\{U_i \to X\}$. This is a subset of the 0-groupoid/set $PSh_{(\infty)}(V,X) = Hom_{Diff}(V,X)$, hence a direct summand.
topological localization
Let $C$ be a presentable (∞,1)-category.
A strongly saturated class $\bar S \subset Mor(C)$ of morphisms is called topological if
there is a subclass $S \subset \bar S$ of monomorphisms that generates $\bar S$;
under pullback in $C$ elements in $\bar S$ pull back to elements in $\bar S$.
A reflective sub-(∞,1)-category
is called a topological localization if the class of morphisms $\bar S := L^{-1}(equiv)$ that $L$ sends to equivalences is topological.
This is HTT, def. 6.2.1.4
$(\infty,1)$-sheaves
Let $C$ be an (∞,1)-site.
Let $S$ be the collection of all monomorphisms $U \to c$ to objects $c \in Y$ (under Yoneda embedding) that correspond to covering sieves in $C$. Say an object $c \in PSh_{(\infty,1)}(C)$ in the (∞,1)-category of (∞,1)-presheaves on $C$ is an (∞,1)-sheaf if it is an $S$-local object (i.e. if it satisfies descent along all morphisms $U \to c$ coming from covering sieves).
Write
for the reflective sub-(∞,1)-category on these $(\infty,1)$-sheaves.
This is HTT, def. 6.2.2.6
Let throughout $C$ be a locally presentable (∞,1)-category.
(topological localizations are exact)
Every topological localization is an exact localization in that the reflector $L : C \to D$ 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 $C$ at a strongly saturated class $\bar S$ there exists a small set of monomorphisms that generates $\bar S$.
This is HTT, prop. 6.2.1.5.
Every topological localization of $C$ 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 $n$-categories are topological)
Let $C$ be a locall presentable (n,1)-category for $n \in \mathbb{N}$ finite with universal colimits. Then every left exact localization of $C$ is a topological localization
This is HTT, prop. 6.4.3.9.
…
This means that every (n,1)-topos of $n$-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 $n$-truncated for some $n$, then it won’t “see” $k$-bounded hypercovers for large enough $k$ 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 $n$-toposes with finite $n$ there is no distinction between Cech descent and hyperdescent. The distinction becomes visible only for untruncated $\infty$-presheaves.
Let throughout $C$ be a small (∞,1)-category and write $PSh_{(\infty,1)}(C)$ for the (∞,1)-category of (∞,1)-presheaves on $C$.
sheaves form a topological localization
If $C$ is endowed with a Grothendieck topology, the inclusion
is a topological localization.
This is HTT, Prop. 6.2.2.7.
All topological localizations of $PSh_{(\infty,1)}(C)$ arise this way:
There is a bijection between Grothendieck topologies on $C$ and equivalence classes of topological localizations of $PSh_{(\infty,1)}(C)$.
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