local object

Reflective localizations of categories and higher categories in the sense of left adjoint functors $L : C \to C'$ to inclusions $C' \hookrightarrow C$ of full subcategories (as in particular for geometric embeddings) are characterized by the collection $S \subset Mor(C)$ 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.

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.

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.

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.

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.

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

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

**(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

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

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

**(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.

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 $B$.

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…

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

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

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

- 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.