nLab
local object

Idea

Localizations of categories and higher categories in the sense of left adjoint functors $L:C\to C\prime$ to inclusions $C\prime \hookrightarrow C$ of full subcategories (as in particular for geometric embeddings) are characterized by the collection $S\subset \mathrm{Mor}(C)$ of morphisms of $C$ which are sent by $L$ 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 $C$ be a category and $S$ a collection of morphisms in $C$. Then an object $c\in C$ is $S$-local if the hom-functor

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

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

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

is an isomorphism.

Definition for $(\mathrm{\infty },1)$-categories

Local objects

Definition 5.5.4.1 in HTT

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

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

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

Local morphisms

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

$$C(f,c):C(y,c)\to C(x,c)$$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 $S\subset \mathrm{Mor}(C)$ be a collection of morphisms in $C$.

Write $R{\mathrm{Hom}}_C(-,-):\mathrm{SSet}\to \mathrm{SSet}$ 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

$$R{\mathrm{Hom}}_C(X,Y)=C(QX,PY).$$\mathbf{R}Hom_C(X,Y) = C(Q X, P Y) \,.

Definition (local object, local weak equivalence)

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

$$R{\mathrm{Hom}}_C(s,c):\mathrm{SSet}\to \mathrm{SSet}$$\mathbf{R}Hom_C(s,c) : SSet \to SSet

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

$$R{\mathrm{Hom}}_C(f,c):\mathrm{SSet}\to \mathrm{SSet}$$\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

$$\begin{array}{ccc}c& \stackrel{f}{\to }& d\\ \downarrow & & \downarrow \\ \hat{c}& \to & \hat{d}\end{array}.$$\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 $S\subset \mathrm{Mor}(C)$ 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

$$C(B,X)\to C(A,X)$$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 $R\mathrm{Hom}(B,A)\to R\mathrm{Hom}(A,X)$ 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 $\varnothing \hookrightarrow A\prime \stackrel{\simeq }{\to }A$ be a cofibrant replacement, factor $A\prime \to B$ as $A\prime \stackrel{i\prime }{\hookrightarrow }B\prime \stackrel{\simeq }{\to }B$ and consider the pushout diagram

$$\begin{array}{ccc}A\prime & \stackrel{i\prime }{\hookrightarrow }& B\prime \\ {\downarrow }^{f\in W}& & {\downarrow }_{g\in W}& {\searrow }^{f\prime \in W}\\ A& \stackrel{}{\hookrightarrow }& A\coprod _{A\prime }B& \stackrel{j\in W}{\to }& B\end{array}.$$\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 $i\prime$ 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

$$({i\prime }^{*}:C(B\prime ,X)\to C(A\prime ,X))\simeq (R\mathrm{Hom}(B,X)\to R\mathrm{Hom}(A,X))$$({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 $X$ is $S$-local, and also, as before, a fibration, since $i\prime$ 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

$$\begin{array}{ccc}C(A\coprod _{A\prime }B\prime ,X)& \to & C(B\prime ,X)\\ {}^q{\downarrow }^{\in (W\cap \mathrm{fib}{)}_{\mathrm{SSet}}}& & {}^{{i\prime }^{*}}{\downarrow }^{\in (W\cap \mathrm{fib}{)}_{\mathrm{SSet}}}\\ C(A,X)& \to & C(A\prime ,X)\end{array}$$\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

$$\begin{array}{ccc}& & A\\ & \swarrow & & \searrow \\ A\coprod _{A\prime }B\prime & & \stackrel{j}{\to }& & B\end{array}$$\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 $A\to A{\coprod }_{A\prime }B\prime$ as being the pushout of the cofibration $i\prime$) 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

$$\begin{array}{ccc}& & A\\ & \swarrow & \downarrow & {\searrow }^t\\ A\coprod _{A\prime }B\prime & \stackrel{j}{\to }& B& \to & X\end{array}$$\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…

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

• Jacob Lurie, Higher Topos Theory

Saturated class of morphisms

Every morphism in $S$ is $S$-local.

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

Remarks

• a reflective subcategory as well as a reflective (∞,1)-subcategory can be realized as the full ($(\mathrm{\infty },1)$-)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.