nLab local object

Idea

For a class $S$ of morphisms in a category (or higher category) $C$, we can consider the localization functor into the category of fractions $L: C\to C[S^{-1}]$ which is universal with respect to all functors inverting $S$; this defines its (external) saturation, the class $S'\subset Mor(C)$ of all morphisms which are sent to isomorphisms by $L$, or more generally to equivalences. One also defined a collection of objects $c$ which are local with respect to these morphisms, in that these morphisms behave as isos/equivalences with respect to homming into $c$. If $L: C\to C[S^{-1}]$ 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 $L: C\to C'$ and start with internally saturated class $S$ of all morphisms inverted by $L$. Then the collection of $S$-local objects characterizes the localization. Moreover, $S$ can itself be enlarged to its external saturation, the collection of $S$-local morphisms, that is all morphisms which are isomorphisms with respect to homing into all $S$-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 $S$”. This terminology is still widely used in the context of abelian categories with or without “left”; $J$-closed is then closed in the sense of any kind of localization data $J$ (for example, $J$ 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 $C$ be a category and $S$ a collection of morphisms in $C$. Then an object $c \in C$ is $S$-local if the hom-functor

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

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

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

is a bijection.

Local morphisms

Conversely, a morphism $f : x \to y$ is $S$-local if for every $S$-local object $c$ the induced morphism

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

is an isomorphism.

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

Local objects

Definition

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

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

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

This is 5.5.4.1 in HTT

Local morphisms

Conversely, a morphism $f : x \to y$ is $S$-local if for every $S$-local object $c$ the induced morphism

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

induces an isomorphism in the homotopy category of Top.

Definition in model categories

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

Write $\mathbf{R}Hom_C(-,-) : C^{op}\times C \to SSet$ for the derived hom space functor.

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

$$\mathbf{R}Hom_C(X,Y) = C(Q X, P Y) \,.$$

Definition

(local object, local weak equivalence)

An object $c \in C$ is a $S$-local object if for all $s : a \to b$ in $S$ the induced morphism

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

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

A morphism $f : x \to y$ in $C$ is an $S$-local morphism or $S$-equivalence if for every $S$-local object $c$ the induced morphism

$$\mathbf{R}Hom_C(f,c) : SSet \to SSet$$

is a weak equivalence.

An $S$-localization of an object $c$ is an $S$-local object $\hat c$ and an $S$-local equivalence $c \to \hat c$.

An $S$-localization of a morphism $f : c \to d$ is a pair of $S$-localizations $c \to \hat c$ and $d \to \hat d$ of objects, and a commuting square

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

Properties

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

Proposition

(characterization of $S$-local cofibrations)

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

Then a cofibration $i : A \hookrightarrow B$ is an $S$-local weak equivalence precisely if for all fibrant $S$-local objects $X$ the morphism

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

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

Remark

Notice that this is stronger than the statement that $\mathbf{R}Hom(B,X) \to \mathbf{R}Hom(A,X)$ 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 $A$ and $B$.

Proof

This is HTT, lemma A.3.7.1.

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

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

By left properness the pushout $g$ of the weak equivalence $f$ along the cofibration $i'$ is again a weak equivalence and by 2-out-of-3 the morphism $j$ is a weak equivalence.

Now assume that $i$ is an $S$-local equivalence. We need to show that $i^* : C(B,X) \to C(A,X)$ is an acyclic Kan fibration for all fibrant $S$-local $X$. By the very definition of enriched model category it follows from $i$ being a cofibration and $X$ 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

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

on the cofibrant replacement is a weak equivalence, by the assumption that $X$ is $S$-local, and also, as before, a fibration, since $i'$ is still a cofibration.

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

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

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

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

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

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

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

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

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

Saturated class of morphisms

Every morphism in $S$ is $S$-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 $S$ of morphisms is simply the class of all $S$-local morphisms. External saturation of a class $S$ of morphisms is the class of all morphisms which are inverted in the category of fractions (localization) at class $S$.

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

$S$ 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 $S$ has either left or right adjoint (coreflective or reflective case).

Remarks

• a reflective subcategory as well as a reflective (∞,1)-subcategory can be realized as the full ($(\infty,1)$-)subcategory on $S$-local objects, where $S$ is the collection of morphisms sent by the corresponding localization of an (∞,1)-category to equivalences. For details on this see the discussion at geometric embedding.

References

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

• Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967) [doi:10.1007/978-3-642-85844-4, pdf]

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

• Jacob Lurie, Higher Topos Theory

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