nLab
topological localization

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

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 (,1)(C)PSh_{(\infty,1)}(C) is precisely a localization at Cech covers for a given Grothendieck topology on CC, yielding the corresponding (∞,1)-topos of (∞,1)-sheaves.

Sh (,1)(C)PSh (,1)(C) Sh_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)

and in fact equivalence classes of such topological localizations are in bijection with Grothendieck topologies on CC.

Notice that in general a topological localization is not a hypercomplete (∞,1)-topos. That in general requires localization further at hypercovers.

Definition

Recall that a reflective sub-(∞,1)-category DLCD \stackrel{\stackrel{L}{\leftarrow}}{\hookrightarrow} C is obtained by localizing at a collection SS of morphisms of CC.

The class S¯\bar S of all morphisms of CC that the left adjoint L:CDL : C \to D sends to equivalences is the strongly saturated class of morphisms generated by SS. By the recognition principle for exact localizations, the functor LL is exact if and only if S¯\bar S is stable under the formation of pullbacks.

We now define such localizations where the collection SS consists of monomorphisms .

Definition

Call a morphism f:XYf : X \to Y in an (∞,1)-category CC a monomorphism if it is a (-1)-truncated object in the overcategory X /YX_{/Y}.

Equivalently: if for every object ACA \in C the induced morphism in the homotopy category of ∞-groupoids

C(A,f):C(A,X)C(A,Y) C(A,f) : C(A,X) \to C(A,Y)

exhibits C(A,X)C(A,X) as a direct summand of C(A,Y)C(A,Y).

Equivalence classes of monomorphisms into an object XX form a poset Sub(X)Sub(X) of subobjects of XX.

This is HTT, p. 460

Example

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 (,1)(Diff)PSh_{(\infty,1)}(Diff) the (∞,1)-category of (∞,1)-presheaves on DiffDiff, which may be modeled by the global model structure on simplicial presheaves on DiffDiff.

For XDiffX \in Diff a manifold, let {U iX}\{U_i \hookrightarrow X\} be an open cover. Let C({U i})C(\{U_i\}) be the Cech nerve of this cover, the simplicial object of presheaves

C({U i})=( ijU iU j iU i). C(\{U_i\}) = \left( \cdots \coprod_{i j} U_i \cap U_j \stackrel{\to}{\to}\coprod_{i} U_i \right) \,.

which we may regard as a simplicial presheaf and hence as an object of PSh (,1)(Diff)PSh_{(\infty,1)}(Diff).

Then for VV any other manifold, we have that

PSh (,1)(V,C({U i})) PSh_{(\infty,1)}(V, C(\{U_i\}))

is the ∞-groupoid whose

  • objects are maps VXV \to X that factor through one of the U iU_i;

  • there is a unique morphism between two such maps precisely if they factor through a double intersection U iU jU_{i} \cap U_j;

  • and so on.

In the homtopy category of ∞-groupoids, this is equivalent to the 0-groupoid/set of those maps VXV \to X that factor through one of the U iU_i. Notice that this constitutes the sieve generated by the covering family {U iX}\{U_i \to X\}. This is a subset of the 0-groupoid/set PSh ()(V,X)=Hom Diff(V,X)PSh_{(\infty)}(V,X) = Hom_{Diff}(V,X), hence a direct summand.

Definition

topological localization

Let CC be a presentable (∞,1)-category.

A strongly saturated class S¯Mor(C)\bar S \subset Mor(C) of morphisms is called topological if

  • there is a subclass SS¯S \subset \bar S of monomorphisms that generates S¯\bar S;

  • under pullback in CC elements in S¯\bar S pull back to elements in S¯\bar S.

A reflective sub-(∞,1)-category

DLC D \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C

is called a topological localization if the class of morphisms S¯:=L 1(equiv)\bar S := L^{-1}(equiv) that LL sends to equivalences is topological.

This is HTT, def. 6.2.1.4

Definition

(,1)(\infty,1)-sheaves

Let CC be an (∞,1)-site.

Let SS be the collection of all monomorphisms UcU \to c to objects cYc \in Y (under Yoneda embedding) that correspond to covering sieves in CC. Say an object cPSh (,1)(C)c \in PSh_{(\infty,1)}(C) in the (∞,1)-category of (∞,1)-presheaves on CC is an (∞,1)-sheaf if it is an SS-local object (i.e. if it satisfies descent along all morphisms UcU \to c coming from covering sieves).

Write

Sh (,1)(C)PSh (,1)(X) Sh_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(X)

for the reflective sub-(∞,1)-category on these (,1)(\infty,1)-sheaves.

This is HTT, def. 6.2.2.6

Warning: A topological localization is, by definition, a left exact localization at a set of monomorphisms; left exactness is part of the definition. A general localization at a set of monomorphisms need not be left exact. For instance, the localization at one of the inclusions 11+11\to 1+1 is the (1)(-1)-truncation, which is not left exact.

Properties

Let throughout CC be a locally presentable (∞,1)-category.

General

Corollary

(topological localizations are exact)

Every topological localization is an exact localization in that the reflector L:CDL : C \to D preserves finite limits.

Proof

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.

Proposition

(generation from a small set of morphisms)

For every topological localization of CC at a strongly saturated class S¯\bar S there exists a small set of monomorphisms that generates S¯\bar S.

This is HTT, prop. 6.2.1.5.

Corollary

Every topological localization of CC 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.

Proposition

(localizations of presentable nn-categories are topological)

Let CC be a locall presentable (n,1)-category for nn \in \mathbb{N} finite with universal colimits. Then every left exact localization of CC is a topological localization

This is HTT, prop. 6.4.3.9.

Proof

This means that every (n,1)-topos of nn-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 nn-truncated for some nn, then it won’t “see” kk-bounded hypercovers for large enough kk 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 nn-toposes with finite nn there is no distinction between Cech descent and hyperdescent. The distinction becomes visible only for untruncated \infty-presheaves.

For (,1)(\infty,1)-presheaf (,1)(\infty,1)-categories

Let throughout CC be a small (∞,1)-category and write PSh (,1)(C)PSh_{(\infty,1)}(C) for the (∞,1)-category of (∞,1)-presheaves on CC.

Proposition

sheaves form a topological localization

If CC is endowed with a Grothendieck topology, the inclusion

Sh (,1)(C)PSh (,1)(C) Sh_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)

is a topological localization.

This is HTT, Prop. 6.2.2.7.

Proposition

All topological localizations of PSh (,1)(C)PSh_{(\infty,1)}(C) arise this way:

There is a bijection between Grothendieck topologies on CC and equivalence classes of topological localizations of PSh (,1)(C)PSh_{(\infty,1)}(C).

This is HTT, prop. 6.2.2.17.

Proof

Model category presentation

See at Čech model structure on simplicial sheaves.

References

Topological localizations are the topic of section 6.2, from def. 6.2.1.5 on, in

Revised on November 2, 2014 17:16:36 by Mike Shulman (107.201.31.189)