nLab
local object

Contents

Idea

Localizations of categories and higher categories in the sense of left adjoint functors L:CCL : C \to C' to inclusions CCC' \hookrightarrow C of full subcategories (as in particular for geometric embeddings) are characterized by the collection SMor(C)S \subset Mor(C) of morphisms of CC which are sent by LL to isomorphisms, or more generally to equivalences, as well as by the collection of objects which are local with respect to these morphisms, in that these morphisms behave as equivalences with respect to homming into objects.

Definition for ordinary categories

Local objects

Let CC be a category and SS a collection of morphisms in CC. Then an object cCc \in C is SS-local if the hom-functor

C(,c):C opSet C(-,c) : C^{op} \to Set

sends morphisms in SS to isomorphisms in Set, i.e. if for every s:abs : a \to b in SS, the function

C(s,c):C(b,c)C(a,c) C(s,c) : C(b,c) \to C(a,c)

is a bijection.

Local morphisms

Conversely, a morphism f:xyf : x \to y is SS-local if for every SS-local object cc the induced morphism

C(f,c):C(y,c)C(x,c) C(f,c) : C(y,c) \to C(x,c)

is an isomorphism.

Definition for (,1)(\infty,1)-categories

Local objects

Definition 5.5.4.1 in HTT

Let CC be an (∞,1)-category and SS a collection of morphisms in CC. Then an object cCc \in C is SS-local if the hom-functor

C(,c):C opTop C(-,c) : C^{op} \to \infty Top

evaluated on sSs \in S induces isomorphism in the homotopy category of Top.

Local morphisms

Conversely, a morphism f:xyf : x \to y is SS-local if for every SS-local object cc the induced morphism

C(f,c):C(y,c)C(x,c) C(f,c) : C(y,c) \to C(x,c)

induces an isomorphism in the homotopy category of Top.

Definition in model categories

Let CC be a model category (usefully but not necessarily a simplicial model category). And let SMor(C)S \subset Mor(C) be a collection of morphisms in CC.

Write RHom C(,):SSetSSet\mathbf{R}Hom_C(-,-) : SSet \to SSet for the derived hom space functor.

For instance if CC is a simplicial model category then this may be realized in terms of a cofibrant replacement functor Q:CCQ : C \to C and a fibrant replacement functor PP as

RHom C(X,Y)=C(QX,PY). \mathbf{R}Hom_C(X,Y) = C(Q X, P Y) \,.
Definition (local object, local weak equivalence)

An object cCc \in C is a SS-local object if for all s:abs : a \to b in CC the induced morphismm

RHom C(s,c):SSetSSet \mathbf{R}Hom_C(s,c) : SSet \to SSet

is a weak equivalence (in the standard model structure on simplicial sets);

A morphism f:xyf : x \to y in CC is an SS-local morphism or SS-equivalence if for every SS-local object cc the induced morphism

RHom C(f,c):SSetSSet \mathbf{R}Hom_C(f,c) : SSet \to SSet

is a weak equivalence.

An SS-localization of an object cc is an SS-local object c^\hat c and an SS-local equivalence cc^c \to \hat c.

An SS-localization of a morphism f:cdf : c \to d is a pair of SS-localizations cc^c \to \hat c and dd^d \to \hat d of objects, and a commuting square

c f d c^ d^. \array{ c &\stackrel{f}{\to}& d \\ \downarrow && \downarrow \\ \hat c &\to & \hat d } \,.

Properties

In left proper model categories there is an equivalent stronger characterization of SS-locality of cofibrations i:ABi : A \hookrightarrow B.

Proposition (characterization of SS-local cofibrations)

Let CC be a left proper simplicial model category and SMor(C)S \subset Mor(C) a collection of morphisms.

Then a cofibration i:ABi : A \hookrightarrow B is an SS-local weak equivalence precisely if for all fibrant SS-local objects XX the morphism

C(B,X)C(A,X) C(B,X) \to C(A,X)

is an acyclic fibration in the standard model structure on simplicial sets.

Remark

Notice that this is stronger than the statement that RHom(B,X)RHom(A,X)\mathbf{R}Hom(B,X) \to \mathbf{R}Hom(A,X) is a weak equivalence not only in that it asserts in addition a fibration, but also in that it deduces this without first passing to a cofibrant replacement of AA and BB.

Proof

This is HTT, lemma A.3.7.1.

The proof makes use of the following general construction: for f:ABf : A \to B any morphism let AA\emptyset \hookrightarrow A' \stackrel{\simeq}{\to} A be a cofibrant replacement, factor ABA' \to B as AiBBA' \stackrel{i'}{\hookrightarrow} B' \stackrel{\simeq}{\to} B and consider the pushout diagram

A i B fW gW fW A A AB jW B. \array{ A' &\stackrel{i'}{\hookrightarrow}& B' \\ \downarrow^{\mathrlap{f \in W}} && \downarrow_{\mathrlap{g\in W}} & \searrow^{\mathrlap{f' \in W}} \\ A &\stackrel{}{\hookrightarrow}& A \coprod_{A'} B &\stackrel{j \in W}{\to}& B } \,.

By left properness the pushout gg of the weak equivalence ff along the cofibration ii' is again a weak equivalence and by 2-out-of-3 the morphism jj is a weak equivalence.

Now assume that ii is an SS-local equivalence. We need to show that i *:C(B,X)C(A,X)i^* : C(B,X) \to C(A,X) is an acyclic Kan fibration for all fibrant SS-local XX. By the very definition of enriched model category it follows from ii being a cofibration and XX being fibrant that this is a Kan fibration. So it remains to show that it is a weak homotopy equivalence of simplicial sets. We know that the corresponding induced morphism

(i *:C(B,X)C(A,X))(RHom(B,X)RHom(A,X)) ({i'}^* : C(B',X) \to C(A',X)) \simeq (\mathbf{R}Hom(B,X) \to \mathbf{R}Hom(A,X))

on the cofibrant replacement is a weak equivalence, by the assumption that XX is SS-local, and also, as before, a fibration, since ii' is still a cofibration.

By homming the entire diagram above into XX, and using that the hom-functor C(,X)C(-,X) sends colimits to limits, we find the pullback diagram

C(A AB,X) C(B,X) q (Wfib) SSet i * (Wfib) SSet C(A,X) C(A,X) \array{ C(A \coprod_{A'} B', X) &\to& C(B',X) \\ {}^{q}\downarrow^{\mathrlap{\in (W\cap fib)_{SSet}}} && {}^{{i'}^*}\downarrow^{\mathrlap{\in (W\cap fib)_{SSet}}} \\ C(A,X) &\to& C(A',X) }

in SSet, which shows that qq is an acyclic fibration, being the pullback of an acyclic fibration.

To show that i *:C(B,X)C(A,X)i^*: C(B,X) \to C(A,X) is a weak equivalence it suffices to show that all its fibers (i *) 1)(t)(i^*)^{-1})(t) over elements t:AXt : A \to X are contractible Kan complexes. These fibers map to the corresponding fibers q 1(t)q^{-1}(t) by precomposition with jj. By the fact that jj, regarded as a morphism

A A AB j B \array{ && A \\ & {}\swarrow && \searrow \\ A \coprod_{A'} B' &&\stackrel{j}{\to}&& B }

in the model structure on the undercategory A/CA/C is a weak equivalence between cofibrant objects (because ABA \hookrightarrow B is a cofibration by assumption and AA ABA \to A \coprod_{A'} B' as being the pushout of the cofibration ii') we have that precomposition C(j,X)C(j,X) with jj is the image under the SSet-enriched hom-functor of a weak equivalence between cofibrant objects mapping into a fibrant object

A t A AB j B X \array{ && A \\ & \swarrow & \downarrow & \searrow^{t} \\ A \coprod_{A'} B' &\stackrel{j}{\to}& B &\to& X }

and hence, by the general properties of enriched homs between cofibrant/fibrant objects a weak equivalence. j *:(i *) 1(t)q 1(t)j^* : (i^*)^{-1}(t) \stackrel{\simeq}{\to} q^{-1}(t), so that indeed (i *) 1(t)(i^*)^{-1}(t) is contractible.

This proves the first part of the statement. For the converse statement, assume now that…

References

A classical textbook reference is section 3.2 of

  • Hirschhorn, Model categories and their localization

A useful reference with direct ties to the (∞,1)-category story in the background is section A.3.7 of

Saturated class of morphisms

Every morphism in SS is SS-local.

The collection SS of morphisms is called saturated if the collection of SS-local morphisms coincides with SS.

Remarks

Revised on November 17, 2014 07:25:59 by Tim Porter (2.31.52.143)