nLab local object

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Idea

For a class SS of morphisms in a category (or higher category) CC, we can consider the localization functor into the category of fractions L:CC[S 1]L: C\to C[S^{-1}] which is universal with respect to all functors inverting SS; this defines its (external) saturation, the class SMor(C)S'\subset Mor(C) of all morphisms which are sent to isomorphisms by LL, or more generally to equivalences. One also defined a collection of objects cc which are local with respect to these morphisms, in that these morphisms behave as isos/equivalences with respect to homming into cc. If L:CC[S 1]L: C\to C[S^{-1}] has a right adjoint then it is a reflective localization, and in particular the right adjoint is fully faithful; we can abstractly consider reflective localizations of the form L:CCL: C\to C' and start with internally saturated class SS of all morphisms inverted by LL. Then the collection of SS-local objects characterizes the localization. Moreover, SS can itself be enlarged to its external saturation, the collection of SS-local morphisms, that is all morphisms which are isomorphisms with respect to homing into all SS-local objects. In the reflective case, external and internal saturation coincide and in general external saturation is a subclass of internal saturation.

Gabriel-Zisman used the term “left-closed object for SS”. This terminology is still widely used in the context of abelian categories with or without “left”; JJ-closed is then closed in the sense of any kind of localization data JJ (for example, JJ being a torsion theory or a Gabriel filter). Sheaves and stacks are examples of closed objects in presheaf categories on sites/spaces.

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

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.

This is 5.5.4.1 in HTT

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(,):C op×CSSet\mathbf{R}Hom_C(-,-) : C^{op}\times C \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 SS the induced morphism

RHom C(s,c):RHom C(b,c)RHom C(a,c) \mathbf{R}Hom_C(s,c) : \mathbf{R}Hom_C(b,c) \to \mathbf{R}Hom_C(a,c)

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…

Saturated class of morphisms

Every morphism in SS is SS-local.

There are two notions of saturation in the literature. Following Casacuberta, Frei 2000 we distinguish them as internal and external. Internal saturation of a class SS of morphisms is simply the class of all SS-local morphisms. External saturation of a class SS of morphisms is the class of all morphisms which are inverted in the category of fractions (localization) at class SS.

In both variants, the collection SS of morphisms is called saturated if SS coincides with its saturation.

SS is contained in its external saturation which is in turn contained in its internal saturation; the external and internal saturation coincide if the localization in SS has either left or right adjoint (coreflective or reflective case).

Remarks

References

Local objects are treated, under the name of left-closed objects, in I.4.1 of

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

1-categorical picture

  • Carles Casacuberta, John Frei, On saturated classes of morphisms, Theory and Applications of Categories 7:4 (2000) 43–46 pdf

  • Carles Casacuberta, Georg Peschke, Markus Pfenniger, On orthogonal pairs in categories and localization, J. Pure & Applied Algebra 142:1 (1999) 25–33

Relation to the categorical shape theory

  • Luciano Stramaccia, Orthogonality, saturation and shape, Glasnik Matematički 2007, pdf

Last revised on August 29, 2024 at 01:37:40. See the history of this page for a list of all contributions to it.