reflective factorization system

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Factorization systems
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[[!include factorization systems - contents]]
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* table of contents
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## Idea

A *reflective factorization system* is an [[orthogonal factorization system]] $(E,M)$ that is determined by the [[reflective subcategory]] $M/1$.

## Definition

Let $C$ be a [[category]] with a [[terminal object]] $1$.  If $(E,M)$ is an (orthogonal) [[orthogonal factorization system|factorization system]] on $C$, then the [[full subcategory]] $M/1 \subseteq C$ (consisting of those objects $X$ for which $X\to 1$ is in $M$) is [[reflective subcategory|reflective]].  The reflection of $Y\in C$ is obtained by the $(E,M)$-factorization $Y \xrightarrow{e} \ell Y \xrightarrow{m} 1$. (e.g. ([Rosicky-Tholen 08, 2.10](#RosickyTholen08)))

In fact, in this we do not need $(E,M)$ to be a factorization system; only a [[orthogonal factorization system#prefactorization_systems|prefactorization system]] with the property that any morphism with [[terminal object|terminal]] [[codomain]] admits an $(E,M)$-factorization.  For the nonce, let us call such a prefactorization system *favorable*.

Conversely, suppose that $A\hookrightarrow C$ is a [[reflective subcategory]], and define $E$ to be the class of morphisms inverted by the [[reflector]] $\ell\colon C\to A$, and define $M = E^\perp$.  Then $(E,M)$ is a favorable prefactorization system.  In this way we obtain an [[adjunction]]

$$ \Phi : \text{reflective subcategories} \; \rightleftarrows \; \text{favorable prefactorization systems} : \Psi. $$

Here subcategories form a (possibly [[large category|large]]) [[poset]] ordered by inclusion, and prefactorization systems form a poset ordered by inclusion of the right classes $M$.

The [[unit of an adjunction|unit]] of this adjunction is easily seen to be an [[isomorphism]].  That is, given a reflective subcategory $A$, if we construct $(E,M)$ from it as above, then $A \simeq M/1$.  Therefore, the adjunction allows us to identify reflective subcategories with certain favorable prefactorization systems.

The prefactorization systems arising in this way --- equivalently, those for which $(E,M) =  \Phi \Psi(E,M)$ --- are called the **reflective prefactorization systems**.  A **reflective factorization system** is a reflective prefactorization system which happens to be a factorization system.

More generally, for any favorable factorization system $(E,M)$, we have a reflective prefactorization system $ \Phi \Psi(E,M)$, called the **reflective interior** of $(E,M)$.  Dualizing, it also has a **coreflective closure**.

## Properties

### Characterization

The following is Theorem 2.3 in [CHK](#CHK).

+-- {: .num_theorem}
###### Theorem
Let $(E',M')$ be the reflective interior of $(E,M)$.  Then:

1. $f\in E'$ precisely when there exists a $g\in E$ such that $g f \in E$.
1. $(E,M)$ is reflective precisely when $g f\in E$ and $g\in E$ together imply $f\in E$.
=--
+-- {: .proof}
###### Proof
That (1) implies (2) is obvious, so we prove (1).

Since $E'$ is, by definition, the class of maps inverted by the reflector into $M/1$, it satisfies the [[2-out-of-3 property]].  Since $E\subseteq E'$, it follows that $f g\in E$ and $g\in E$ imply $f\in E'$.

Conversely, if $f\colon X\to Y$ is in $E'$, then we have $\eta_Y \circ f = \ell(f) \circ \eta_X$ by naturality, where $\ell$ is the reflector into $M/1$ and $\eta$ its unit.  But by construction of $\ell$, $\eta_Y$ and $\eta_X$ are in $E$, and by assumption, $\ell(f)$ is invertible; hence we can take $g = \eta_Y$.
=--

Note that the left class in any orthogonal factorization system is automatically closed under composition, contains the isomorphisms, and satisfies the property that $g f \in E$ and $f\in E$ together imply $g\in E$.  Therefore, $(E,M)$ is reflective precisely when $E$ is a system of [[category with weak equivalences|weak equivalences]].  See [Relation to Localization](#Localization), below.

### Construction of factorizations

The following is a slightly generalized version of Corollary 3.4 from [CHK](#CHK).

+-- {: .num_theorem}
###### Theorem
Suppose that $C$ is finitely complete and [[M-complete category|M-complete]] for some factorization system $(E,M)$, where $M$ consists of [[monomorphisms]] and contains the [[split monomorphism|split monics]].  Then any reflective prefactorization system on $C$ is a factorization system.
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+-- {: .proof}
###### Proof
This follows directly from [this theorem](/nlab/show/M-complete+category#OFSFromAdjunction) applied to the reflection adjunction.
=--

The following is a consequence of Theorems 4.1 and 4.3 from [CHK](#CHK).

+-- {: .num_theorem #SemiLeftExact}
###### Theorem
Suppose that $C$ is finitely complete and that $(E,M)$ is a reflective prefactorization system on $C$ such that $E$-morphisms are stable under pullback along $M$-morphisms.  Then $(E,M)$ is a factorization system.
=--
+-- {: .proof}
###### Proof
Write $\ell$ for the corresponding reflection.  Now given $f\colon A\to B$, let $m$ be the pullback of $\ell(f)$ along $\eta_B\colon B \to \ell B$:
$$\array{ Y & \overset{g}{\to} & \ell A \\
  ^m \downarrow & & \downarrow^{\ell(f)}\\
  B & \underset{\eta_B}{\to} & \ell B}$$
By closure properties of prefactorization systems, any morphism in $M/1$ lies in $M$, so in particular $\ell(f)\in M$.  Since $M$ is stable under pullback (being, again, the right class of a prefactorization system), we have $m\in M$.

But $f$ factors through $m$, by the universal property of the pullback applied to the naturality square for $\eta$ at $f$.  Thus we have $f = m e$ and it suffices to show $e\in E$.  However, we also have $g e = \eta_A$, where $\eta_A\in E$ by definition, and $g\in E$ by assumption (being the pullback of $\eta_B\in E$ along $\ell(f)\in M$).  By the characterization theorem above, since $(E,M)$ is reflective this implies $e\in E$, as desired.
=--

A reflection satisfying the condition of Theorem \ref{SemiLeftExact} is called **[[semi-left-exact reflection|semi-left-exact]]** (which see, for more equivalent characterizations).  Note that saying that $E$-morphisms are stable under *all* pullbacks is equivalent to saying that $\ell$ preserves all pullbacks, hence all finite limits---i.e. it is left-exact.  In this case the factorization system is called [[stable factorization system|stable]].  Thus the terminology "semi-left-exact" for the weaker assumption.

However, semi-left-exactness is not necessary for the "one-step" construction in the proof of Theorem \ref{SemiLeftExact} to work.  The necessary and sufficient condition is that the reflection is **simple**; one characterization of this is that a morphism $f$ is in $M$ if and only if its naturality square for the reflector $\ell:C\to A$ is a pullback.  For others, see [CHK, Theorem 4.1](#CHK).

### Relation to localizations {#Localization}

For any favorable prefactorization system $(E,M)$, it is easy to show that $M/1$ is the [[localization]] of $C$ at $E$.  If $(E',M')$ is the reflective interior of $(E,M)$, then since $E'$ is the class of maps inverted by the reflector into $M/1$, it is precisely the *saturation* of $E$ in the sense of localization (the class of maps inverted by the localization at $E$).

### Reflective stable factorization systems

A reflective factorization system on a finitely [[complete category]] is a [[stable factorization system]] if and only if its corresponding [[reflector]] preserves [[finite limits]].  A stable reflective factorization system is sometimes called **local**.

## Examples

Obviously, any [[reflective subcategory]] gives rise to a reflective factorization system.  Here are a few examples.

* The category of complete [[metric spaces]] is reflective in the category of all metric spaces; the reflector is completion.  In the corresponding factorization system, $E$ is the class of [[dense subspace|dense embeddings]].

* Given a small [[site]] $S$, the [[sheaf topos]] $Sh(S)$ is a reflective subcategory of the [[presheaf topos]] $Psh(S)$.  In the corresponding factorization system, $E$ is the class of [[local isomorphisms]].

On the other hand, many commonly encountered factorization systems are not reflective.

* The factorization system $(Epi, Mono)$ on [[Set]] is not reflective.  If  $(E',M')$ is its reflective interior, then $E'$ is the class of morphisms $e\colon X\to Y$ such that if $Y$ is [[inhabited set|inhabited]], so is $X$, while $M'$ is the class of morphisms $m\colon X\to Y$ such that if $X$ is inhabited, then $m$ is an isomorphism.

## Related concepts

* [[semi-left-exact reflection]]

* [[stable factorization system]]

* [[closure operator]]

* [[reflective localization]]

## References

The basic theory is developed in

* {#CHK} C. Cassidy, M. Hébert, [[Max Kelly]], _Reflective subcategories, localizations, and factorization systems_,  J. Austral. Math Soc. (Series A) **38** (1985) 287-329 [[doi:10.1017/S1446788700023624](https://doi.org/10.1017/S1446788700023624)]

* {#CJKP} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures **5** (1997) 1-58 [[doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444)]

Discussion of "simple" reflective factorization systems and of simultaneously reflective and coreflective factorization systems:

* {#RosickyTholen08} [[Jiri Rosicky]], [[Walter Tholen]], _Factorization, Fibration and Torsion_, Journal of Homotopy and Related Structures, **2** 2 (2007) 295-314  [[arXiv:0801.0063](http://arxiv.org/abs/0801.0063), [publisher](http://www.emis.de/journals/JHRS/volumes/2007/n2a14/)]

Discussion from the perspective of [[modalities]] and [[modal homotopy type theory]]:

* {#RSS} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], Section A.2 in: *Modalities in homotopy type theory*,  Logical Methods in Computer Science, **16** 1 (2020)  [[arXiv:1706.07526](https://arxiv.org/abs/1706.07526), doi:10.23638/LMCS-16(1:2)2020]

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[[!redirects coreflective prefactorization system]]
[[!redirects coreflective prefactorization systems]]
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reflective localization

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###Context###
#### Locality and descent
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[[!include descent and locality - contents]]
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#Contents#
* table of contents
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## Idea

A [[localization of a category]]/[[localization of an (∞,1)-category|of an (∞,1)-category]] is called **reflective** if its [[localization functor]] has a [[fully faithful functor|fully faithful]] [[right adjoint]], hence if it is the reflector of a [[reflective subcategory]]/[[reflective sub-(∞,1)-category]]-inclusion.

In fact every [[reflective subcategory]] inclusion exhibits a reflective localization (Prop. \ref{ReflectiveSubcategoriesAreLocalizations} below).

For reflective localizations the localized category has a particularly useful description (Prop. \ref{ReflectiveLocalizationGivenByLocalObjects} below): It is equivalent to the [[full subcategory]] of _[[local objects]]_ (Def. \ref{LocalObjects} below).

Therefore, sometimes reflective localizations at a class $S$ or morphism are understood as the default concept of localization, in fact often reflection onto the [[full subcategory]] of $S$-[[local objects]] (Def. \ref{LocalizationAtACollectionOfMorphisms} below) is understood by default.  Notably [[left Bousfield localizations]] are [[presentable (infinity,1)-category|presentations]] of reflective [[localizations of (∞,1)-categories]] in this sense.

These reflections onto $S$-local objects satisfy the universal property of an $S$-[[localization]] (only) for all _[[left adjoint]]_ functors that invert the class $S$ (Prop. \ref{ReflectionOntoLocalObjectsIsLocalizationWithRespectToLeftAdjoints} below).

## Definition

### Reflective localization
 {#ReflectiveLoc}

+-- {: .num_defn #CategoryWithWeakEquivalences}
###### Definition
**([[category with weak equivalences]])**

A _[[category with weak equivalences]]_ is

1. a [[category]] $\mathcal{C}$,

1. a [[subcategory]] $W \subset \mathcal{C}$

such that the [[morphisms]] in $W$

1. include all the [[isomorphisms]] of $\mathcal{C}$,

1. satisfy _[[two-out-of-three]]_:

   If for $g$, $f$ any two [[composition|composable]] [[morphisms]] in $\mathcal{C}$, two out of the set $\{g,\, f,\, g \circ f \}$ are in $W$, then so is the third.

   $$
     \array{
       & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
       \\
       && \underset{ g \circ f }{\longrightarrow}
     }
   $$

=--

+-- {: .num_defn #LocalizationOfACategory}
###### Definition
**([[localization of a category]])**

Let $W \subset \mathcal{C}$ be a [[category with weak equivalences]] (Def. \ref{CategoryWithWeakEquivalences}). Then the _[[localization of a category|localization]]_ of $\mathcal{C}$ at $W$ is, if it exists

1. a [[category]] $\mathcal{C}[W^{-1}]$

1. a [[functor]] $\gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]$

such that

1. $\gamma$ sends all morphisms in $W \subset \mathcal{C}$ to [[isomorphisms]],

1. $\gamma$ is [[universal property|universal with this property]]: If $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ is any functor with this property, then it factors through $\gamma$, up to [[natural isomorphism]]:

   $$
     F \;\simeq\;
     D F \circ \gamma
     \phantom{AAAAAAA}
     \array{
       \mathcal{C}
         && \overset{F}{\longrightarrow} &&
       \mathcal{D}
       \\
       & {}_{\mathllap{\gamma}}\searrow
         &{}^{\rho}\Downarrow_{\simeq}&
       \nearrow_{\mathrlap{D F}}
       \\
       && \mathcal{C}[W^{-1}]
     }
   $$

   and any two such factorizations $D F$ and $D^' F$ are related by a unique [[natural isomorphism]] $\kappa$ compatible with $\rho$ and $\rho^'$:

\[
  \label{CompatibleNaturalIsoForLocalization}
     \array{
       \mathcal{C}
         && \overset{F}{\longrightarrow} &&
       \mathcal{D}
       \\
       &
       {}_{\mathllap{\gamma}}\searrow
         &{}^{\rho}\Downarrow_{\simeq}&
       \nearrow_{\mathrlap{D F}}
       && \searrow^{\mathrlap{id}}
       \\
       && \mathcal{C}[W^{-1}]
       && {}_{\simeq}\seArrow^{\kappa} && \mathcal{D}
       \\
       && &  {}_{\mathllap{id}}\searrow && \nearrow_{\mathrlap{D^' F}}
       \\
       && && \mathcal{C}[W^{-1}]
     }
     \phantom{AAAA}
     =
     \phantom{AAAA}
     \array{
       \mathcal{C}
         && \overset{F}{\longrightarrow} &&
       \mathcal{D}
       \\
       & {}_{\mathllap{\gamma}}\searrow
         &{}^{\rho^'}\Downarrow_{\simeq}&
       \nearrow_{\mathrlap{D^' F}}
       \\
       && \mathcal{C}[W^{-1}]
     }
\]

Such a localization is called a _[[reflective localization]]_ if the localization functor has a [[fully faithful functor|fully faithful]] [[right adjoint]], exhibiting it as the reflection functor of a [[reflective subcategory]]-inclusion

$$
  \mathcal{C}[W^{-1}]
  \underoverset
    {\underset{\phantom{AAAA}}{\hookrightarrow}}
    {\overset{ \phantom{AA} \gamma \phantom{AA} }{\longleftarrow}}
    {\bot}
  \mathcal{C}
$$

=--

### Reflection onto local objects
 {#ReflectionOntoLocalObjects}

It turns out (Prop. \ref{ReflectiveLocalizationGivenByLocalObjects}) below, that reflective localizations at a collection $S$ of [[morphisms]] are, when they exist, [[reflective subcategory|reflections]] onto the [[full subcategory]] of _$S$-[[local objects]]_ (Def. \ref{LocalObjects} below). Often this reflection of $S$-[[local objects]] is what one is more interested in than the [[universal property]] of the $S$-[[localization]] according to (Def. \ref{CategoryWithWeakEquivalences}). This _reflection onto local objects_ (Def. \ref{LocalizationAtACollectionOfMorphisms} below) is what is often meant by default with "localization" (for instance in [[Bousfield localization]]).

+-- {: .num_defn #LocalObjects}
###### Definition
**([[local object]])**

Let $\mathcal{C}$ be a [[category]] and let $S \subset Mor_{\mathcal{C}}$ be a set of [[morphisms]]. Then an [[object]] $X \in \mathcal{C}$ is called an _$S$-[[local object]]_ if for all $A \overset{s}{\to} B \; \in S$ the [[hom-functor]] from $s$ into $X$ yields a [[bijection]]

$$
  Hom_{\mathcal{C}}(s,X)
  \;\colon\;
  Hom_{\mathcal{C}}(B,X)
   \overset{ \phantom{AA} \simeq \phantom{AA} }{\longrightarrow}
  Hom_{\mathcal{C}}(A,X)
  \,,
$$

hence if every morphism $A \overset{f}{\longrightarrow} X$ [[extension|extends]] uniquely along $s$ to $B$:

$$
  \array{
    A &\overset{\phantom{A}f\phantom{A}}{\longrightarrow}& X
    \\
    {}^{\mathllap{s}}\big\downarrow & \nearrow_{\mathrlap{ \exists! }}
    \\
    B
  }
$$

We write

\[
  \label{FullSubcategoryOfSLocalObjects}
  \mathcal{C}_S
  \overset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}
  \mathcal{C}
\]

for the [[full subcategory]] of $S$-local objects.

=--

+-- {: .num_defn #LocalizationAtACollectionOfMorphisms}
###### Definition
**([[reflective subcategory|reflection]] onto [[full subcategory]] of [[local objects]])**

Let $\mathcal{C}$ be a [[category]] and set $S \subset Mor_{\mathcal{C}}$ be a sub-[[class]] of its [[morphisms]]. Then the _reflection onto local $S$-objects_ (often called "localization at the collection $S$" is, if it exists, a [[left adjoint]] $L$ to the [[full subcategory]]-inclusion of the $S$-[[local objects]] (eq:FullSubcategoryOfSLocalObjects):

$$
  \mathcal{C}_S
    \underoverset
      {\underset{\iota}{\hookrightarrow}}
      {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}}
      {\bot}
  \mathcal{C}
  \,.
$$

=--

## Properties

+-- {: .num_prop #ReflectiveSubcategoriesAreLocalizations}
###### Proposition
**([[reflective subcategories]] are [[localizations]])**

Every [[reflective subcategory]]-inclusion

$$
  \mathcal{C}_{L}
  \underoverset
    {\underset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}}
    {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}}
    {\bot}
  \mathcal{C}
$$

is [[generalized the|the]] [[reflective localization]] at the class $W \coloneqq L^{-1}(Isos)$ of morphisms that are sent to isomorphisms by the reflector $L$.

=--

+-- {: .proof}
###### Proof

Let $F \;\colon\; \mathcal{C} \to \mathcal{D}$ be a [[functor]] which inverts morphisms that are inverted by $L$.

First we need to show that it factors through $L$, up to natural isomorphism. But consider the following [[whiskering]] $F(\eta)$ of the [[adjunction unit]] $\eta$ with $F$:

$$
  \array{
    \mathcal{C}
    && \overset{F}{\longrightarrow} &&
    \mathcal{D}
    \\
    & {}_{\mathllap{L}}\searrow &\Downarrow& \nearrow_{\mathrlap{D F}}
    \\
    && \mathcal{C}_L
  }
  \phantom{AA} \coloneqq \phantom{AA}
  \array{
    \mathcal{C}
      && \overset{id}{\longrightarrow} &&
    \mathcal{C}
      & \overset{F}{\longrightarrow}&
    \mathcal{D}
    \\
    & {}_{\mathllap{L}}\searrow &\Downarrow^{\eta}& \nearrow_{\mathrlap{\iota}}
    \\
    && \mathcal{C}_L
  }
$$

By [[idempotent monad|idempotency]], the components of the [[adjunction unit]] $\eta$ are inverted by $L$, and hence by assumption they are also inverted by $F$, so that on the right the [[natural transformation]] $F(\eta)$ is indeed a [[natural isomorphism]].

It remains to show that this factorization is unique up to unique natural isomorphism. So consider any other factorization $D^' F$ via a natural isomorphism $\rho$. Pasting this now with the [[adjunction counit]]

$$
  \array{
    &&
    \mathcal{C} && \overset{F}{\longrightarrow} &&  \mathcal{D}
    \\
    &
    {}^{\mathllap{\iota}}\nearrow
    &
    {}^{\epsilon}\Downarrow
    & {}_{\mathllap{L}}\searrow
    &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}}
    \\
    \mathcal{C}_L
    &&
      \underset{ id }{\longrightarrow}
    &&
    \mathcal{C}_L
  }
$$

exhibits a natural isomorphism $\epsilon \cdot \rho$ between $D F \simeq D^' F$. Moreover, this is compatible with $F(\eta)$ according to (eq:CompatibleNaturalIsoForLocalization), due to the [[triangle identity]]:

$$
  \array{
    \mathcal{C}
    &&
     \overset{id}{\longrightarrow}
    &&
    \mathcal{C} && \overset{F}{\longrightarrow} &&  \mathcal{D}
    \\
    &
      {}_{\mathllap{id}}\searrow
    &
    {}^{\mathllap{\eta}}\Downarrow
    &
    {}^{\mathllap{\iota}}\nearrow
    &
    {}^{\epsilon}\Downarrow
    & {}_{\mathllap{L}}\searrow
    &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}}
    \\
    &&
    \mathcal{C}_L
    &&
      \underset{ id }{\longrightarrow}
    &&
    \mathcal{C}_L
  }
  \phantom{AAAA}
  =
  \phantom{AAAA}
  \array{
    \mathcal{C} && \overset{F}{\longrightarrow} &&  \mathcal{D}
    \\
    & \searrow &\Downarrow^\rho& \swarrow
    \\
    && \mathcal{C}_L
  }
$$

Finally, since $L$ is [[essentially surjective functor]], by [[idempotent monad|idempotency]], it is clear that this is the unique such natural isomorphism.

=--

+-- {: .num_prop #ReflectiveLocalizationGivenByLocalObjects}
###### Proposition
**([[reflective localization]] [[reflective subcategory|reflects]] onto [[full subcategory]] of [[local objects]])**

Let $W \subset \mathcal{C}$ be a [[category with weak equivalences]] (Def. \ref{CategoryWithWeakEquivalences}). If its [[reflective localization]] (Def. \ref{LocalizationOfACategory}) exists

$$
  \mathcal{C}[W^{-1}]
    \underoverset
      {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}}
      {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}}
      {\bot}
  \mathcal{C}
$$

then $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is [[equivalence of categories|equivalently]] the inclusion of the [[full subcategory]] on the $W$-[[local objects]] (Def. \ref{LocalObjects}), and hence $L$ is equivalently reflection onto the $W$-local objects, according to Def. \ref{LocalizationAtACollectionOfMorphisms}.

=--

+-- {: .proof}
###### Proof

We need to show that

1. every $X \in \mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$ is $W$-[[local object|local]],

1. every $Y \in \mathcal{C}$ is $W$-[[local object|local]] precisely if it is [[isomorphism|isomorphic]] to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.

The first statement follows directly with the [[adjoint functor|adjunction isomorphism]]:

$$
  Hom_{\mathcal{C}}(w, \iota(X))
    \simeq
  Hom_{\mathcal{C}[W^{-1}]}(L(w), X)
$$

and the fact that the [[hom-functor]] takes [[isomorphisms]] to [[bijections]].

For the second statement, consider the case that $Y$ is $W$-local. Observe that then $Y$ is also local with respect to the class

$$
  W_{sat} \;\coloneqq\; L^{-1}(Isos)
$$

of _all_ morphisms that are inverted by $L$ (the "[[saturated class of morphisms]]"): For consider the [[hom-functor]] $\mathcal{C} \overset{Hom_{\mathcal{C}}(-,Y)}{\longrightarrow} Set^{op}$ to the [[opposite category|opposite]] of the [[category of sets]]. But assumption on $Y$ this takes elements in $W$ to isomorphisms. Hence, by the defining [[universal property]] of the [[localization]]-functor $L$, it factors through $L$, up to [[natural isomorphism]].

Since by [[idempotent monad|idempotency]] the [[adjunction unit]] $\eta_Y$ is in $W_{sat}$, this implies that we have a [[bijection]] of the form

$$
  Hom_{\mathcal{C}}( \eta_Y, Y )
  \;\colon\;
  Hom_{\mathcal{C}}( \iota L(Y), Y )
    \overset{\simeq}{\longrightarrow}
  Hom_{\mathcal{C}}(Y, Y)
  \,.
$$

In particular the [[identity morphism]] $id_Y$ has a [[preimage]] $\eta_Y^{-1}$ under this function, hence a [[left inverse]] to $\eta$:

$$
  \eta_Y^{-1} \circ \eta_Y \;=\; id_Y
  \,.
$$

But by [[2-out-of-3]] this implies that $\eta_Y^{-1} \in W_{sat}$. Since the first item above shows that $\iota L(Y)$ is $W_{sat}$-local, this allows to apply this same kind of argument again,

$$
  Hom_{\mathcal{C}}( \eta^{-1}_Y, \iota L(Y) )
    \;\colon\;
  Hom_{\mathcal{C}}( Y, \iota L(Y) )
    \overset{\simeq}{\longrightarrow}
  Hom_{\mathcal{C}}( \iota L(Y) , \iota L(Y))
  \,,
$$

to deduce that also $\eta_Y^{-1}$ has a [[left inverse]] $(\eta_Y^{-1})^{-1} \circ \eta_Y^{-1}$. But since a [[left inverse]] that itself has a [[left inverse]] is in fact an [[inverse morphisms]] ([this Lemma](retract#LeftInverseWithLeftInverseIsLeftInverse)), this means that $\eta^{-1}_Y$ is an [[inverse morphism]] to $\eta_Y$, hence that $\eta_Y \;\colon\; Y \to \iota L (Y)$ is an [[isomorphism]] and hence that $Y$ is isomorphic to an object in $\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}$.

Conversely, if there is an [[isomorphism]] from $Y$ to a morphism in the image of $\iota$ hence, by the first item, to a $W$-local object, it follows immediatly that also $Y$ is $W$-local, since the [[hom-functor]] takes [[isomorphisms]] to [[bijections]] and since bijections satisfy [[2-out-of-3]].

=--

$\,$

* For a left exact reflective localization, the class of morphisms that is inverted forms a left-[[multiplicative system]]. For the moment see  at _[[geometric embedding]]_ for details on this.

+-- {: .num_prop #ReflectionOntoLocalObjectsIsLocalizationWithRespectToLeftAdjoints}
###### Proposition
**([[reflective subcategory|reflection]] onto [[local objects]] in [[localization]] with respect to [[left adjoints]])**

Let $\mathcal{C}$ be a [[category]] and let $S \subset Mor_{\mathcal{C}}$ be a [[class]] of [[morphisms]] in $\mathcal{C}$. Then the [[reflective subcategory|reflection]] onto the $S$-[[local objects]] (Def. \ref{LocalizationAtACollectionOfMorphisms})
satisfies, if it exists, the [[universal property]] of a [[localization of categories]] (Def. \ref{LocalizationOfACategory}) with respect to _[[left adjoint]]_ functors inverting $S$.

=--

+-- {: .proof}
###### Proof

Write

$$
  \mathcal{C}_S
    \underoverset
      {\underset{ \phantom{AA}\iota\phantom{AA} }{\hookrightarrow}}
      {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}}
      {\bot}
  \mathcal{C}
$$

for the [[reflective subcategory]]-inclusion of the $S$-[[local objects]].

Say that a [[morphism]] $f$ in $\mathcal{C}$ is an _$S$-[[local morphism]]_ if for every $S$-[[local object]] $A \in \mathcal{C}$ the [[hom-functor]] from $f$ to $A$ yields a [[bijection]] $Hom_{\mathcal{C}}(f,A)$.
Notice that, by the [[Yoneda embedding]] for $\mathcal{C}_S$, the $S$-[[local morphisms]] are precisely the morphisms that are taken to isomorphisms by the reflector $L$.

Now let

$$
  (F \dashv G)
  \;\colon\;
  \mathcal{C}
    \underoverset
      {\underset{G}{\longleftarrow}}
      {\overset{ \phantom{AA} F \phantom{AA} }{\longrightarrow}}
      {\bot}
   \mathcal{D}
$$

be a pair of [[adjoint functors]], such that the [[left adjoint]] $F$ inverts the morphisms in $S$. By the adjunction hom-isomorphism it follows that $G$ takes values in $S$-[[local objects]]. This in turn implies, now via the [[Yoneda embedding]] for $\mathcal{D}$, that $F$ inverts all $S$-[[local morphisms]], and hence all morphisms that are inverted by $L$.

Thus the essentially unique factorization of $F$ through $L$ now follows by Prop. \ref{ReflectiveSubcategoriesAreLocalizations}.

=--

## Related concepts

* [[reflective subcategory]], [[coreflective subcategory]]

* [[essential localization]]

  [[quintessential localization]]

  [[bireflective subcategory]]

* [[reflective factorization system]]

* [[Bousfield localization]], [[Quillen reflection]]

## References

The concept of reflective localization was originally highlighted in

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], _[[Calculus of fractions and homotopy theory]]_, Springer 1967 ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf))

A formalization in [[homotopy type theory]] of reflection onto local objects is discussed in

* {#RijkeShulmanSpitters17} [[Egbert Rijke]], [[Michael Shulman]], [[Bas Spitters]], _Modalities in homotopy type theory_ ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526))

[[!redirects reflective localizations]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
#### $(\infty,1)$-topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[reflective sub-(∞,1)-category]] such that the [[reflector]] preserves [[products]].

## Properties

When formutating "reflective sub-$(\infty,1)$-category" in the [[internal language of an (∞,1)-topos]] the respect for products is automatic. See [[reflective product-preserving sub-(∞,1)-category - internal formulation]].

[[!redirects reflective product-preserving sub-(∞,1)-categories]]

[[!redirects reflective product-preserving sub-(infinity,1)-category]]
[[!redirects reflective product-preserving sub-(infinity,1)-categories]]


reflective product-preserving sub-(∞,1)-category - internal formulation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
#### $(\infty,1)$-topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The external definition of [[reflective sub-(∞,1)-category]] via the [universal property of the reflector](http://ncatlab.org/nlab/show/reflective+sub-%28infinity%2C1%29-category#CharacterizationOfReflectors) has an immediate formulation in the [[internal language of an (∞,1)-topos]]. This internal formulation, however, automatically gives a [[reflective product-preserving sub-(∞,1)-category]].

## References

[[HoTT]]-[[Coq]] code for internal reflective subcategories is at

* _[HoTT/Coq/ReflectiveSubcategory.v](https://github.com/UrsSchreiber/HoTT/blob/master/Coq/ReflectiveSubcategory.v)_

The fact that in the internal formulation reflective subcategories are automatically product-preserving is mentioned on p. 5 of

* [[Eugenio Moggi]], _A note on SDT in Filter spaces_ ([pdf](http://www.disi.unige.it/person/MoggiE/ftp/filter2.pdf))


reflective sub-(infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of  _reflective $(\infty,1)$-subcategory_ is the generalization of the notion of [[reflective subcategory]] from [[category theory]] to [[(∞,1)-category]] theory.

## Definition

### Reflective sub-$(\infty,1)$-category

+-- {: .num_defn}
###### Definition
**(local objects, local equivalences)**

A [[full and faithful (∞,1)-functor]]

$$
  R : D \hookrightarrow C
$$

exhibits $D$ as a **reflective [[sub-(∞,1)-category]]** (of $C$) if it has a left [[adjoint (∞,1)-functor]] $L : C \to D$.

$$
  (L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}}
  C
  \,.
$$

If $L$ moreover is a left [[exact functor]] in that it preserves [[finite limit|finite]] [[(∞,1)-limits]], then the embedding is called **exact**.

=--

The [[(∞,1)-functor]] $R$ or its composite

$$
  Loc := R \circ L : C \to C
$$

may be understood as exhibiting a [[localization of an (∞,1)-category|localization]] of $C$ at those morphisms that $L$ sends to [[equivalence in an (∞,1)-category|equivalence]]s in $D$. If $L$ preserves [[finite limit]]s (is a left [[exact functor]]), then this is an **exact localization**

### Local objects and local morphisms {#LocalObjects}

One finds, as discussed [below](#Properties), that reflective subcategories may be entirely characterized by the class of morphisms that the localization functor $Loc : C \to C$ sends to weak equivalences.

+-- {: .num_defn}
###### Definition
**(local objects, local equivalences)**

Let $S \subset Mor(C)$ be a class of morphisms.

* An object $c \in C$ is called an **$S$-[[local object]]** if for all morphisms $f : x \to y$ in $S$ the induced morphism

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

  is an equivalence (of [[∞-groupoid]]s).

* An morphism $f : x \to y$ in $C$ is called an **$S$-[[local morphism]]** or **$S$-[[local equivalence]]** if for all $S$-[[local object]]s $c  \in C$ we have that

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

  is an equivalence (of [[∞-groupoid]]s).

=--

Notice that the class of $S$-equivalences always contains $S$ itself. Hence passing from a collection $S$ to its class $\bar S$ of $S$-equivalences is a kind of saturation procedure. This is formalized by the following definition, whose justification is given by the propositions [below](#Properties).

+-- {: .num_defn}
###### Definition
**(strongly saturated class of morphisms)**

For $C$ an [[(∞,1)-category]] with small [[(∞,1)-colimits]], a class $S \subset C_1$ of morphisms in $C$ is said to be **strongly saturated** if its satisfies the following three conditions

1. It is stable under [[pushout]]s along arbitrary morphisms of $C$;

1. The full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] $Func(\Delta[1], C)$ on $S$ has all [[(∞,1)-colimit]]s;

1. it satisfies the
   [[category with weak equivalences|2-out-of-3 property]].

=--

Notice that this definition has some immediate consequences:

The identity $Id_\emptyset$ on the [[initial object]] of $C$, which is the initial object in $Func(\Delta[1],C)$ is in $S$, since it is the colimit of the empty diagram. Moreover, every [[equivalence in an (∞,1)-category|equivalence]] is a [[pushout]] of $Id_{\emptyset}$ so

* A strongly saturated class contains every [[equivalence in an (∞,1)-category|equivalence]].

Given any collection $\{S_i\}_i$ of strongly saturated classes of morphisms in $C$, their intersection is clearly also strongly saturated. Therefore for every collection $S$ of morphisms, there is a smallest strongly saturated class $\bar S$ containing it. We say that $S$ **generates** the strongly saturated class $\bar S$. If $S$ is a [[small set]], then $\bar S$ is said to be **of small generation**.

The smallest strongly saturated class of morphism in $C$ is that containing only the equivalences of $C$.

Of importance are the strongly saturated classes arising as follows.

+-- {: .num_lemma}
###### Lemma

For $C$ and $D$ two $(\infty,1)$-categories that have small [[(∞,1)-colimit]]s, and for $F : C \to D$ an [[(∞,1)-functor]] that preserves small $(\infty,1)$-colimits, given a strongly saturated class of morphisms $S$ in $D$, its preimage $F^{-1}(S)$ is a strongly saturated class in $C$.

In particular the class of morphisms in $C$ sent to equivalences by $F$ is strongly saturated.

=--

+-- {: .num_lemma}
###### Lemma

The class of $S_0$-[[local equivalence]]s for $S_0$ any class of morphisms is strongly saturated.

=--

+-- {: .proof}
###### Proof

For each object $c \in C$ let $j(c) : C \to \infty Grpd^{op}$ be the functor [[representable functor|represented]] by $c$. Let $S_c$ be the class of morphisms sent by $j(c)$ to weak equivalences in [[∞Grpd]]. Since $j(c)$ preserves small colimits, this is a strongly saturated class, by the above lemma. Now observe that $S$ is the intersection $S = \cap_c S_c$ where $c$ ranges over the $S_0$-[[local object]]s.

=--

In the following this language of local morphisms is used to characterize reflective $(\infty,1)$-subcategories.

## Properties {#Properties}

### Characterization of reflectors
 {#CharacterizationOfReflectors}

The following proposition characterizes the _reflectors_ of a reflective $(\infty,1)$-subcategory. (You may read this proposition as the characterization of [[adjoint functors]] via [[universal arrows]] (via [this prop.](adjoint+functor#CollectionOfUniversalArrowsEquivalentToAdjointFunctor)), but maybe as a preparation for the proofs to come there is some value in looking at its concrete proof in this special case of an $(\infty,1)$-adjunction.)

+-- {: .num_lemma #UniversalityOfReflectionCounit}
###### Lemma
**(universality of reflection counit)**

Let $C$ be an [[(∞,1)-category]] and $D \hookrightarrow C$ a full [[sub-(∞,1)-category]]. Then this inclusion has a left [[adjoint (∞,1)-functor]] precisely if

* for every object $c \in C$ there is a _localization_ or _reflection_: a morphism $f : c \to \bar c$ such that $\bar c \in D\hookrightarrow C$ and such that for all $e \in D \hookrightarrow C$ we have that

  $$
    Hom_C(f,e) : Hom_C(\bar c,e) \to Hom_C(c,e)
  $$

  is an equivalence (of [[∞-groupoid]]s).

=--

This appears as [[Higher Topos Theory|HTT, prop. 5.2.7.8]].

+-- {: .proof}
###### Proof

We produce an evident [[cograph of a functor|cograph]] realization $K$ of the inclusion and check that it being also a [[coCartesian fibration]], hence exhibiting $R$ as a right adjoint, is equivalent to the second statement.

Let $K \subset C \times \Delta[1]$ be the full subcategory on those objects $(c,i)$ for which $c \in D$ if $i = 1$. Let $p : K \to \Delta[1]$ be the induced projection.  One checks that this is the correspondence which is associated to the inclusion functor $D \hookrightarrow C$.

Therefore by the properties of [[adjoint (∞,1)-functor]]s, we have that the inclusion functor has a left adjoint precisely if $p$ is not only a [[Cartesian fibration]] but also a [[coCartesian fibration]].

To see that this is the case precisely if every $c$ has a reflection $f : c \to d$, recall the characterization of [[coCartesian morphism]]s $\tilde f : (c,0) \to (d,1)$ as those making the squares

$$
  \array{
    Hom_K((d,1),(e,i)) &\stackrel{Hom_K(\tilde f,(e,i))}{\to} & Hom_K((c,0),(e,i))
    \\
    \downarrow && \downarrow
    \\
    Hom_{\Delta[1]}(1, i)
    &\stackrel{}{\to}&
    Hom_{\Delta[1]}(0, i)
  }
$$

being [[homotopy pullback]] squares, for all $(e,i) \in K$. Now in $\Delta[1]$ all [[hom-object in a quasi-category|hom-objects]] are either empty or are points, so that the bottom morphism becomes the identity on the point if $i = 1$. Since for $i = 0$ everything becomes entirely trivial we consider the case that $i =1$ and hence $e \in D$.

In that case the homotopy-pullback property is equivalent to the top morphism being an equivalence, hence to

$$
    Hom_C(d,e) \stackrel{Hom_C(f,e)}{\to}  Hom_C(c,a)
$$

being an equivalence. This way the reflectors are identified precisely with the coCartesian morphisms in $K \to \Delta[1]$ that exhibit the left [[adjoint (∞,1)-functor]] to the inclusion functor.

=--

The following proposition asserts that localizations are entirely determined by the corresponding [[local object]]s.

+-- {: .num_prop}
###### Proposition

Let

$$
  D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} C
$$

be a localization of the $(\infty,1)$-category $C$ and let

$$
  Loc : C \stackrel{L}{\to} D \overset{R}{\hookrightarrow} C
$$

be the corresponding localization [[(∞,1)-monad]]. Write $S \subset Mor(C)$ for the collection of morphisms that $Loc$ sends to [[equivalence in an (∞,1)-category|equivalences]].

Then

* an object $c \in C$ is an $S$-[[local object]] precisely if it is in the essential image of $Loc$ (equivalent to an object of the form $Loc x$);

* every $S$-[[local morphism]] is already in $S$.

=--

This is [[Higher Topos Theory|HTT, prop 5.5.4.2]].

+-- {: .proof}
###### Proof

The reasoning is entirely analogous to the 1-categorical case (see for instance [[localization]], [[reflective subcategory]] and [[geometric embedding]]).

First notice that because $D \hookrightarrow C$ is a [[full and faithful (∞,1)-functor]] we have that the counit $L R \stackrel{\simeq}{\to} Id_D$ is an equivalence. From this it follows that precomposition with the unit $i_z : z \to Loc z$ of morphisms in the image of $Loc$ is a weak equivalence: for all $z,x \in C$ we have

$$
    Hom_C(i_z, Loc z)
    :
    Hom_C(Loc z, Loc x)
    \stackrel{\simeq}{\to}
    Hom_C(z, Loc x)
  \,.
$$

If $z$ is itself in the image of $Loc$, then this means that precomposition with the unit $z \to Loc z$ is an isomorphism on [[hom-set]]s in the [[homotopy category of an (infinity,1)-category|homotopy category]] of $Loc C$, hence by the [[Yoneda lemma]] is itself an isomorphism in the homotopy category, hence $i_z : z \to Loc z$ is a weak equivalence if $z$ is itself in the image of $Loc$.

Applying this statement to the naturality square for the [[natural transformation]] $Id \to Loc$ on $i_s$

$$
  \array{
    s &\stackrel{i_s}{\to} & Loc s
    \\
    \downarrow^{\mathrlap{i_s}} && \downarrow^{\mathrlap{Loc i_s}}
    \\
    Loc s &\stackrel{i_{Loc s}}{\to}& Loc Loc s
  }
$$

we find that $Loc i_s \simeq i_{Loc s}$, hence that $Loc i_s$ is a weak equivalence, and hence that $i_s$ is in $S$, for all $s \in C$.

Now to show that for all $x \in X$ the object $Loc x$ is $S$-local, let $f : y \to z$ be in $S \subset Mor(C)$ and consider the induced square

$$
  \array{
    Hom_C(Loc z, Loc x) &\stackrel{\simeq}{\to}& Hom_C(Loc y, Loc x)
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
    \\
    Hom_C(z, Loc x) &\to& Hom_C(y, Loc x)
  }
  \,.
$$

Here the vertical morphisms are equivalences by the above remark, and the top morphism is an equivalence by the assumption that $f$ in in $S$. It follows that the bottom morphism is an equivalence. This says that $Loc x$ is $S$-local, for all $x \in C$.

Conversely, to show that for $s \in C$ an $S$-local object, we have that $s$ is in the essential image of $Loc$ use that since $i_s : s \to Loc s$ is in $S$, we have an equivalence  $Hom_C(i_s, s) : Hom_C(Loc s, s) \stackrel{\simeq}{\to} Hom_C(s,s)$.  The pre-image of the identity under this equivalence is hence a left-inverse $Loc s \to s$ of $s \to Loc s$. But this means that $Loc s \to s$ is itself in $S$ (since the morphisms in $S$ evidently satisfy [[category with weak equivalences|2-out-of-three]]), hence by applying the same argument again, we find that the left inverse $Loc s \to s$ has itself a left inverse. That implies that it is actually an inverse of $s \to Loc s$, hence that this is an equivalence. So this shows that the $S$-local $s$ is indeed in the essential iamge of $Loc$.

Finally, to show that every $S$-local morphism is already in $S$, let $f : x \to y$ be such an $S$-local morphism and consider the square

$$
  \array{
    x &\stackrel{f}{\to}& y
    \\
    \downarrow^{\mathrlap{i_x}} && \downarrow^{\mathrlap{i_y}}
    \\
    Loc x &\stackrel{Loc f}{\to}& Loc y
  }
  \,.
$$

By the above we know now that the vertical morphisms here are also $S$-local. It follows that the image of $Loc f : Loc x \to Loc y$ on the homotopy category of $Ho(Loc C)$ corepresents an isomorphism, hence by the [[Yoneda lemma]] that $Lof f$ is a weak equivalence. Hence $f$ is indeed in $S$.

=--

### Reflective localization at a set of morphisms

Above is discussed that every reflective subcategory is the localization at the collection local morphisms, those which the left adjoint functor inverts.
One can turn this around and define or construct reflective $(\infty,1)$-subcategories by specifying collections of local morphisms.

+-- {: .num_prop #LocalizationProposition}
###### Proposition
**(localization proposition)**

Let $C$ be a [[presentable (∞,1)-category]] and $S_0$  be a [[small set]] of morphisms of $C$.

Then the full [[sub-(∞,1)-category]]

$$
  R : D \hookrightarrow C
$$

on $S_0$-[[local object]]s is a reflective $(\infty,1)$-subcategory.

If $L : C \to D$ denotes the left [[adjoint (∞,1)-functor]] of the inclusion, then for $f \in Mor(C)$ a morphism, the following are equivalent

1. $f$ is an $S_0$-[[local equivalence]];

1. $f$ belongs to the strongly saturated class $S$ generated by $S_0$;

1. the morphism $L f$ is an equivalence.

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.4.15]].

The main ingredient in the proof of this assertion is the following lemma, whose proof we give below in [Proof of the localization lemma](#ProofOfLocalization).

+-- {: .num_prop}
###### Proposition
**(localization lemma)**

Let $C$ be a [[locally presentable (∞,1)-category]], and let $S \subset Mor(C)$ be a strongly saturated collection of morphisms, generated from a [[small set]] $S_0$.

Then for every object $c \in C$ there exists a reflector, i.e. a morphism  $f : c \to d$ such that $d$ is an $S$-[[local object]] and $f \in S$.

=--

With that in hand we look at the proof of the above proposition:

+-- {: .proof}
###### Proof
(localization proposition)

The localization lemma gives for each object $c \in C$ a reflector $f : c \to d$ with $d$ $S$-local. By lemma \ref{UniversalityOfReflectionCounit}, this already gives the reflective embedding

$$
  D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} C
$$

of the full subcateory of $S$-local objects in $C$.

It remains to prove the statements about the role of $S$ in the localization:

First, by one of the above lemmas, we have that the $S_0$-local equivalences are a strongly saturated class of morphisms containong $S_0$. Hence they in particular contain $S$. So the second claim implies the first.

That the first and the third condition are equivalent follows from noticing that for any local object $d \in D$  the morphism $Hom_C(f,R d)$ is an equivalence precisely if $Hom_D(L f, d)$ is and then applying the [[Yoneda lemma]] (for instance in the [[homotopy category of an (infinity,1)-category|homotopy category]]), which implies that if a morphism produces an equivalence when hommed into all objects, then it is itself an equivalence.

It remains to show that the third item implies the second. Let $f : c \to d$ be a morphism such that $L f : L c \to Ld$ is an equivalence. Consider the commuting triangle

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

Since every reflector is in $S$ and the reflectors are the units of the reflective adjunction constructed from them, we have that the vertical morphisms in this diagram are in $S$, and the bottom morphism is, since it is an equivalence by assumption. By applying the 2-out-of-3 property of $S$ twice it follows that $f$ is in $S$.

=--

### Proof of the localization lemma {#ProofOfLocalization}

We here spell out the proof of

+-- {: .num_prop}
###### Proposition
**(localization lemma)**

Let $C$ be a [[locally presentable (∞,1)-category]], and let $S \subset Mor(C)$ be a strongly saturated collection of morphisms, generated from a [[small set]] $S_0$.

Then for every object $c \in C$ there exists a reflector, i.e. a morphism  $f : c \to d$ such that $d$ is an $S$-[[local object]] and $f \in S$.

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.5.14]].

+-- {: .proof}
###### Proof

Regard all $(\infty,1)$-categories as [[quasi-categories]] for the purpose of this proof. Write $D \subset Func(\Delta[1], C)$ for the full sub-quasicategory on the elements of $S$.  Consider the [[pullback]] (in [[sSet]])

$$
  \array{
    D_c &\to& D
    \\
    \downarrow && \downarrow
    \\
    \{c\} &\to& Func(\{0\}, C)
  }
  \,.
$$

Since $S$ is by assumption closed under pushouts in $C$, we have for each morphism $x \to y$ in $D \simeq Func(\{0\}, C)$ and each lift

$$
  \array{
    x &\to& y
    \\
    \downarrow
    \\
    x'
  }
$$

of its source to $Func(\Delta[1], C)$ a lift of this morphism with this source, given by the the pushout square

$$
  \array{
    x &\to& y
    \\
    \downarrow && \downarrow
    \\
    x' &\to& x' \coprod_x y
  }
$$

in $C$, regarded as a morphism in $Func(\Delta[1], C)$. By the universality of the pushout, one finds that this is a [[coCartesian morphism|coCartesian lift]]. Hence $D \to Func(\{0\}, C) \simeq C$ is a [[coCartesian fibration]]. Moreover, by the [behaviour under pullback](http://ncatlab.org/nlab/show/Cartesian+fibration#BehaviourUnderPullback) of [[Cartesian fibration]]s it follows that the above diagram is a [[homotopy pullback]] diagram in the [[model structure for quasi-categories|Joyal model structure]] $sSet_{Joyal}$.

Use now that [[accessible (infinity,1)-category|accessible quasi-categories]] are stable under [[homotopy pullback]] to conclude that $D_c$ is accessible. Moreover, one can check that $D_c$ has all small colimits. Together this means that $D_c$ is a [[locally presentable (∞,1)-category]].  This implies in particular that $D_c$ also has all small [[(∞,1)-limits]] and hence contains a [[terminal object in a quasi-category|terminal object]], $f : c \to d$.

We now complete the proof by showing that $f : c \to d$ being terminal in $D_c$ implies that $d$ is an $S$-local object. This is equivalent to showing that for $t : a \to b$ any element in $S$, composition with $t$ induces an equivalence

$$
  Hom_C(t,d) : Hom_C(b,d) \to Hom_C(a,d)
  \,.
$$

This in turn may be checked [by](http://ncatlab.org/nlab/show/fiber+sequence#CharOfEquivalences) checking that all its [[homotopy fiber]]s are [[contractible]]. By general statements about the [homotopy fiber of functor categories](http://ncatlab.org/nlab/show/fiber+sequence#OfFuncCats) the homotopy fiber of $Hom_C(t,d)$ over a point $g : a \to d$ of $Hom_C(a,d)$ is equivalent to the [[hom-object in a quasi-category|hom-object]] $Hom_{C_{a/}}(t,g)$ in the [[over quasi-category|under-quasi-category]] $C_{a/}$.

This in turn can be checked to be equivalent to $Hom_{C_{d/}}(g_* t, Id_d)$, where $g_* t$ is the $(\infty,1)$-categorical pushout

$$
  \array{
    a &\stackrel{t}{\to}& b
    \\
    \downarrow^{\mathrlap{g}} && \downarrow
    \\
    d &\underset{g_* t}{\to}& d \coprod_a b
  }
$$

in $C$. Notice that $g_* t$, being a pushout of $t \in S$, is itself in $S$.

Now pick a composite

$$
  \array{
    && d
    \\
    & {}^{\mathllap{f}}\nearrow &\Downarrow^\sigma& \searrow^{\mathrlap{g_* t}}
    \\
    c &&\underset{g_* t \circ f}{\to}&& d \coprod_a b
  }
$$

and observe that we have an isomorphism of simplicial sets

$$
  Hom_{C_{d/}}(g_* t, Id_{d})
  \simeq
  Hom_{C_{f/}}(\sigma, s_1(f))
$$

(where $s_1$ is the corresponding degeneracy map).

Applying the expression for [homotopy fibers of functor categories](http://ncatlab.org/nlab/show/fiber+sequence#OfFuncCats) once again, this is found to be the homotopy fiber of

$$
  Hom_{C_{c/}}(g_* t \circ f, f)
  \to
  Hom_{C_{c/}}(f,f)
  \,,
$$

because $Hom_{(C_{c/})_{f/}}(\sigma , s_1(f)) = Hom_{C_{f/}}(\sigma , s_1(f))$.

Finally we can use that $f$ is terminal in the full subcategory $D_c$ of $C_{c/}$ that contains $g_* t \circ f$. This implies that the above morphism goes between [[contractible]] $\infty$-groupoids and hence has contractible homotopy fibers.

=--

### Transport of reflective subcategories
 {#TransportOfReflectiveSubcategories}

+-- {: .num_prop #TransportOfReflectionAlongAdjunction}
###### Proposition

Let $f \colon \mathcal{C} \to \mathcal{D}$ be an [[(∞,1)-functor]] between [[presentable (∞,1)-categories]], and let $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ be a reflective sub-$(\infty,1)$-category. If $f$ has a [[right adjoint|right]] [[adjoint (∞,1)-functor]] $f^*$, then

$$
  \mathcal{D}^0 \coloneqq (f^*)^{-1}(\mathcal{C}^0) \hookrightarrow \mathcal{D}
$$

is also a reflective sub-$(\infty,1)$-category.

$$
  \array{
    \mathcal{C}^0 &\hookrightarrow& \mathcal{C}
    \\
    {}^{\mathllap{f^*}}\uparrow
   && {}^{\mathllap{f^*}}\uparrow\downarrow^{\mathrlap{f}}
    \\
    \mathcal{D}^0 &\hookrightarrow& \mathcal{D}
  }
  \,.
$$

=--

This is ([Lurie, lemma 5.5.4.17](#Lurie)).

+-- {: .proof}
###### Proof

By prop. \ref{LocalizationProposition}, $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ is the inclusion of the $S$-[[local objects]] for some class $S$ of morphisms of $\mathcal{C}$. By adjunction it follows that $\mathcal{D}^0$ is precisely the class of $f(S)$-[[local objects]], and hence is a reflective subcategory, again by prop. \ref{LocalizationProposition}.

=--

### Model category presentation
 {#ModelCategoryPresentation}

+-- {: .num_prop }
###### Proposition

Let $C$ be a [[left proper model category|left proper]] [[combinatorial simplicial model category]] which [[presentable (∞,1)-category|presents]] an [[(∞,1)-category]] $\mathcal{C} \simeq C^\circ$.

Then if $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ is an [[accessible (∞,1)-functor|accessible]] reflective inclusion with reflector $L \colon \mathcal{C} \to \mathcal{C}^0$, then there exists a corresponding [[Bousfield localization of model categories|left Bousfield localization]]

$$
  C' \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}
  C
$$

of the [[model category]] $C$ which presents this inclusion in that

1. an object in $C'$ is a [[fibrant object]] precisely if it is fibrant as an object of $C$ and in addition its image in the [[homotopy category]] $Ho(C) \simeq Ho(\mathcal{C})$ is in the inclusion $Ho(\mathcal{C}^0) \hookrightarrow Ho(\mathcal{C})$;

1. a morphism in $C'$ is a [[weak equivalence]] precisely if under $Ho(L) \colon Ho(C) \simeq Ho(\mathcal{C}) \to Ho(\mathcal{C}^0)$ is an [[isomorphism]].

=--

This is ([Lurie, prop. A.3.7.8](#Lurie)).

+-- {: .proof}
###### Proof

Use that by the above discussion $\mathcal{C}^0$ is the full subcategory on $S$-[[local objects]] for a [[small set]] of morphisms. By the discussion at [[Bousfield localization of model categories]] this presents precisely such localizations.

=--

## Extra conditions
 {#ExtraConditions}

Extra conditions on a reflective sub-$(\infty,1)$-category of relevance are

* _[Accessible reflective subcategories](#AccessibleReflectiveSubcategory)_

* _[Exact reflective subcategories](#ExactLocalizations)_

### Accessible reflective subcategories
 {#AccessibleReflectiveSubcategory}

The following proposition characterizes when a reflective subcategory of an [[accessible (∞,1)-category]] $C$ is accessible

+-- {: .num_prop #LocalizationAndAccessibility}
###### Proposition

Let $C$ be an [[accessible (∞,1)-category]] and

$$
  D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}}
  C
$$

a reflective subcategory.  Then the following conditions are equivalent:

1. $D$ is itself accessible;

1. The localization $Loc : R\circ L : C \to C$ is an [[accessible (∞,1)-functor]].

1. There exists a [[small set]] $S_0 \subset S := L^{-1}(equiv.)$ such that every $S$-[[local object]] is also $S_0$-local.

=--

This is ([Lurie, prop. 5.5.1.2 and prop. 5.5.4.2, part 3](#Lurie)).

+-- {: .proof}
###### Proof

This is work...

=--

### Exact localizations
 {#ExactLocalizations}

Recall that the reflective subcategory $D \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C$ is **exact** -- or $L$ an **exact localization** -- if $L$ is a left [[exact functor]] in that it preserves [[finite limit|finite]] [[limit in a quasi-category|limits]]. Accordinly we say:

+-- {: .num_defn}
###### Definition

An **exact reflective sub-$(\infty,1)$-category** is a reflective sub-$(\infty,1)$-category whose reflector is a [[left exact (∞,1)-functor]], hence preserves [[finite limit|finite]] [[(∞,1)-limits]].

=--

Recall also that by the above results, a reflective subcategory is characterized by the collection $S = L^{-1}(equiv) \subset Mor(C)$ of those morphisms, that $L$ sends to equivalences in $D$.

The following propositions say how the property that $L$ preserves finite limits is characterized by pullback-stability properties of $S$.

+-- {: .num_prop}
###### Proposition
**(recognition of exact localization)**

A reflective sub-$(\infty,1)$-category
$D \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C$
such that $C$ has all finite [[limit in a quasi-category|limits]]
is exact precisely if the collection $S := L^{-1}(equiv) \subset Mor(C)$ of morphisms that $L$ sends to equivalences is stable under [[pullback]].

So if for every pullback diagram

$$
  \array{
     X' &\to& X
     \\
     \downarrow^{\mathrlap{f'}} && \downarrow^{\mathrlap{f}}
     \\
     Y' &\to& Y
  }
$$

we have that if $L(f)$ is an equivalence then also $L(f')$ is
an equivalence.

=--

This is [[Higher Topos Theory|HTT, prop. 6.2.1.1]].

+-- {: .proof}
###### Proof

If $L$ preserves [[finite limit]]s, then it preserves [[pullback]]s,
so that $L(f')$ is a pullback of the equivalence $L(f)$, hence itself
an equivalence.

So it remains to check that, conversely, stability of $S$ under pullback
implies that $L$ preserves [[finite limits]]. By a general characterization of left [[exact functor]]s (see there) it suffices to check that $L$ preserves the [[terminal object]] and all [[pullback]]s.

Since the terminal object is evidently $S$-local, we have $L * \simeq *$.

Next we check that $L$ preserves [[product]]s, because we will need this to show that all binary pullbacks are preserved. For that it is sufficient to check that the morphism $L(x \times y) \to L(x) \times L(y)$ induced from the units $i_x : x \to L x$ and $i_y : y \to L y$ is in $S$. From inspection of the diagram

$$
  \array{
    x &\stackrel{i_x}{\to}& L x
    \\
    \uparrow && \uparrow
    \\
     x \times y &\stackrel{(i_x,Id)}{\to}& L x \times y
    \\
    \downarrow && \downarrow
    \\
    y &\stackrel{Id}{\to}& y
  }
$$

one finds that $x \times y \to L x \times L y$ is a [[pullback]] of $i_x$. Hence is in $S$, by assumption. Similarly in

$$
  \array{
    L x &\stackrel{Id}{\to}& L x
    \\
    \uparrow && \uparrow
    \\
    L x \times y &\to& L x \times L y
    \\
    \downarrow && \downarrow
    \\
    y &\stackrel{i_y}{\to}& L y
  }
$$

one see that $L x \times y \to L x \times L y$ is a pullback of $i_y$ and hence in $S$. The composite of these two morphisms is a morphism $x \times y \to L x \times L y$, which is in $S$ since $S$ is closed under composition. Applying $L$ hence yields an equivalence $L(x \times y) \stackrel{\simeq}{\to} L x \times L y$.

We now apply the same kind of argument to show that $L$ respects more generally pullbacks.

For that, first notice that for $x \to y \leftarrow z$ a diagram in $C$, the pullback $L x \times_{L_y} L_z$ of the image exists in $C$, by assumption, but is easily seen to be $S$-local and hence lands in $D$. Therefore to show that we have an equivalence $L(x \; \times_y z) \simeq L x \; \times_{L y} \times L z$ it is sufficient to show that the natural morphism, $x \times_y z \to L x \times_{L y} L z$ induced from the morphism of diagrams

$$
  \array{
    x &\stackrel{i_x}{\to}& L x
    \\
    \downarrow && \downarrow
    \\
    y &\stackrel{i_y}{\to}& L y
    \\
    \uparrow && \uparrow
    \\
    z &\stackrel{i_z}{\to}& L z
  }
$$

in $C$ with the adjunction unit morphism  on the horizonatals, is in $S$. By passing along these units one at a time

$$
  \array{
    x &\stackrel{Id}{\to}& x &\stackrel{i_x}{\to}& L x
    &\stackrel{Id}{\to}&
    L x
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{i_y f}} &&
    \downarrow^{L f}
    &&
    \downarrow^{L f}
    \\
    y &\stackrel{i_y}{\to}& L y &\stackrel{Id}{\to}& L y
    &\stackrel{Id}{\to}&
    L y
    \\
    \uparrow_{\mathllap{g}} && \uparrow_{\mathrlap{i_y g}}
    &&
    \uparrow_{\mathrlap{i_y g}} && \uparrow_{\mathrlap{L g}}
    \\
    z &\stackrel{Id}{\to}& z &\stackrel{Id}{\to}& z
    &\stackrel{i_z}{\to}&
    L z
  }
$$

this may be decomposed as a composite of three morphisms

$$
  x \times_y z \to x \times_{L y} z
  \to L x \times_{L y} z \to
  L x \times_{L y} L z
  \,.
$$

If we equivalently reformulate these [[pullback]]s as [[equalizer]]s then this is

$$
  \array{
    x \times_y z
    &\to&
    x \times z
    &
    \stackrel{\overset{g p_2}{\to}}{\underset{f p_1 }{\to}}
    &
    y
    \\
    \downarrow && \downarrow^{\mathrlap{Id}} &&
    \downarrow^{\mathrlap{i_y}}
    \\
    x \times_{L y} z
    &\to&
    x \times z
    &\stackrel{\overset{i_y f}{\to}}{\underset{i_y g}{\to}}&
    L y
    \\
    \downarrow && \downarrow^{\mathrlap{(i_x, Id)}} &&  \downarrow^{Id}
    \\
    L x \times_{L y}  z
    &\to&
    L x \times z
    &\stackrel{\overset{L f}{\to}}{\underset{i_y g}{\to}}&
    L y
    \\
    \downarrow && \downarrow^{\mathrlap{(Id, i_z)}}
    && \downarrow^{\mathrlap{Id}}
    \\
    L x \times_{L y} L z
    &\to&
    L x \times L z
    &\stackrel{\overset{L f}{\to}}{\underset{L g}{\to}}&
    L y
  }
$$

It is immediate to check that the two bottom left squares are [[pullback]] squares. So the two left vertical morphisms are pullbacks of $(Id, i_z)$ and $(i_x, Id)$, respectively. Of morphisms of this form we had seen above that they are in $S$. Hence by the assumed pullback-stability of $S$ also $x \times_{L y} z \to L x \times_{L y} z \to L x \times_{L y} L z$ is in $S$.

So it remains to show that $x \times_y z \to x \times_{L y} z$ is in $S$.
We claim that this morphism in turn may be expressed as a pullback

$$
  \array{
    x \times_y z &\to& y
    \\
    \downarrow && \downarrow
    \\
    x \times_{L y} z &\to& \times_{L y} y
  }
$$

of the diagonal $y \to y \times_{L y} y$. To see this notice that cones $q$ over the corresponding pullback diagram are equivalently diagrams

$$
  \array{
     &&  q
     \\
     & \swarrow &\downarrow& \searrow
     \\
     x &\to& y &\leftarrow& z
     \\
     & \searrow &\downarrow& \swarrow
     \\
     && L y
  }
  \,.
$$

So now we need to show that the diagonal $y \to y \times_{L y} y$ is in $S$.

To see this, notice that it has a left inverse $y \times_{L y} y \to y$, given by any one of the two projections. So if finally we show that _this_ is in $S$, we are done, since $S$ satisfies 2-out-of-3. But this follows now from pullback stability of $S$, because this projection is the pullback of $y \to L y$ along itself.

=--

+-- {: .num_prop}
###### Proposition
**(accessibility of exact localizations)**

Let $C$ be a [[locally presentable (∞,1)-category]] with [[universal colimits]]. Assume moreover that finite limits commute with filtered colimits in $C$ (this holds for example if $C$ is an [[(∞,1)-topos]]). Let $S_0 \subset Mor(C)$ be a small set of morphisms, and $S$ the smallest strongly saturated class containing $S_0$ and stable under pullbacks. Then $S$ is strongly generated by a small set of morphisms.

=--

This is [[Higher Topos Theory|HTT, Prop. 6.2.1.2]]. This proposition is used to construct pullbacks of [[(∞,1)-topoi]], c.f. [[HTT|HTT Prop. 6.3.4.6]].

## Examples {#Examples}

### Inclusion of the terminal object

If $C$ has a [[terminal object in a quasi-category|terminal object]], then the full subcategory on terminal objects is a reflective subcategory of $C$.

### coCartesian fiber over a reflective subcategory

+-- {: .num_prop}
###### Proposition

let $p : C_1 \to C_0$ be a [[coCartesian fibration]] and $D_0 \stackrel{\leftarrow}{\hookrightarrow} C_0$ a reflective $(\infty,1)$-subcategory of the base.

The restriction $D_1 \hookrightarrow C_1$ of $C_1$ over $D_0$, i.e. the strict (say in [[sSet]] if everything is modeled by [[quasi-categories]]) pullback

$$
  \array{
    D_1 & \stackrel{}{\hookrightarrow} & C_1
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    D_0 &\hookrightarrow& C_0
  }
$$

is itself a reflective $(\infty,1)$-subcategory of $C_1$.

=--

+-- {: .proof}
###### Proof

By the above proposition on reflectors, it is sufficient to produce for every $c \in C_1$ there is a reflection morphism $f : c \to d$ with $d \in D_1$.

Such $f$ is obtained by choosing any [[coCartesian morphism|coCartesian lift]] of a reflector $p(f) : p(c) \to \bar d$.

To see this, consider for every object $e \in D_1$ the diagram

$$
  \array{
    Hom_{C_1}(d,e)
     &\stackrel{Hom_{C_1}(f,e)}{\to}&
     Hom_{C_1}(c,e)
    \\
    \downarrow && \downarrow
    \\
    Hom_{C_0}(p(d), p(e))
     &\stackrel{Hom_{C_0}(p(f) ,p(e))}{\to}&
    Hom_{C_0}(p(c), p(e))
  }
  \,.
$$

By assumption $p(f)$ is a reflector, hence the bottom morphism is an equivalence. By one of the characterizations of [[coCartesian morphism]]s, the fact that $f$ is a coCartesian lift means that this diagram is a (homotopy) pullback diagram.  This means that also the top horizontal morphism is an equivalence.

=--

### Locally presentable $(\infty,1)$-categories

If $C = PSh_{(\infty,1)}(K)$ is the [[(∞,1)-category of (∞,1)-presheaves]] on some small $(\infty,1)$-category $K$, then _accessibly embedded_ reflective subcategory

$$
  D \stackrel{\leftarrow}{\hookrightarrow}
  PSh_{(\infty,1)}(K)
$$

(i.e. one where the inclusion is an [[accessible (∞,1)-functor]]) is a [[locally presentable (∞,1)-category]].

### $(\infty,1)$-Toposes

If $C = PSh_{(\infty,1)}(K)$ is the [[(∞,1)-category of (∞,1)-presheaves]] on some small $(\infty,1)$-category $K$, then an [[accessible (∞,1)-functor|accessibly embedded]] _exact_ reflective subcategory

$$
  Sh_{(\infty,1)}(K) \stackrel{\overset{L}{\leftarrow}}{\underset{}{\hookrightarrow}}
  PSh_{(\infty,1)}(K)
$$

is an [[(∞,1)-category of (∞,1)-sheaves]] on $K$ -- an [[(∞,1)-topos]]. We have:

* the collection of morphism $S = L^{-1}(equiv.)$ that are sent to weak equivalences are the analog of **[[local isomorphism]]s** of ordinary [[sheaf and topos theory|sheaf theory]];

* the $S$-[[local object]]s are the **[[∞-stack]]s** ;

* the localizaton functor $Loc = R \circ L : PSh_{(\infty,1)}(K) \to PSh_{(\infty,1)}(K)$ is **[[∞-stackification]]** .

### $\infty$-Lie algebroids inside all $\infty$-Lie groupoids

Let $K = Alg_k^{op}$ be the opposite of the category of $k$-[[associative algebra]]s, regarded as a site with the [[fpqc-topology]]. Then an object in $Sh_{(\infty,1)}(Alg_k^{op})$ may be regarded as an algebraic $\infty$-groupoid. The [[infinitesimal object|infinitesimal]] version is an [[Lie ∞-algebroid]], which may be identified with an object in $(Alg_k^\Delta)^{op} \simeq (dgAlg_k)^{op}$ -- the opposite of the category of [[cosimplicial algebra]]s. The [[simplicial model category|simplicial]] [[model structure on cosimplicial algebras]], [[locally presentable (∞,1)-category|presents]] this as an $(\infty,1)$-category $(Alg_k^\Delta)^\circ$

The [[Yoneda embedding]] induces an inclusion

$$
  ((Alg_k^\Delta)^{op})^\circ
  \stackrel{}{\hookrightarrow}
  Sh_{(\infty,1)}(Alg_k^{op})
  \,.
$$

which is a reflective embedding. It exhibits localization at $A^1$-[[cohomology]], where $A^1 = Spec k[x]$ is the algebraic line object.

This is discussed at [[rational homotopy theory in an (∞,1)-topos]].

## Related concepts

* [[idempotent (∞,1)-monad]]

* [[reflective subuniverse]]

* [[Quillen reflection]]

## References

The general theory is discussed in section 5.2.7 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
 {#Lurie}

A [[Coq]]-formalization of left-exact reflective sub-$(\infty,1)$-categories in [[homotopy type theory]] is in

* [[Mike Shulman]], _[HoTT/Coq/Subcategories/LexReflectiveSubcategory.v](https://github.com/mikeshulman/HoTT/blob/master/Coq/Subcategories/LexReflectiveSubcategory.v)_
 {#Shulman}

[[!redirects reflective localization of an (∞,1)-category]]
[[!redirects reflective localisation of an (∞,1)-category]]

[[!redirects reflective sub-(∞,1)-category]]
[[!redirects reflective sub-(∞,1)-categories]]
[[!redirects reflective sub-(infinity,1)-categories]]

[[!redirects reflective (infinity,1)-subcategory]]
[[!redirects reflective (infinity,1)-subcategories]]

[[!redirects reflective (∞,1)-subcategory]]
[[!redirects reflective (∞,1)-subcategories]]

[[!redirects coreflective sub-(∞,1)-category]]
[[!redirects coreflective sub-(∞,1)-categories]]
[[!redirects coreflective sub-(infinity,1)-category]]
[[!redirects coreflective sub-(infinity,1)-categories]]

[[!redirects coreflective (∞,1)-subcategory]]
[[!redirects coreflective (∞,1)-subcategories]]
[[!redirects coreflective (infinity,1)-subcategory]]
[[!redirects coreflective (infinity,1)-subcategories]]


reflective sub-(infinity,1)-category - internal formulation

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The following gives a characterization of [[reflective sub-(∞,1)-categories]] of an [[(∞,1)-topos]] entirely in terms of its [[internal logic of an (∞,1)-topos|internal logic]]/[[homotopy type theory]].

## Background and subtleties

(...)

## Statement
 {#Statement}

(...)

## References

[[Coq]]-code for lex reflective subcategories in [[homotopy type theory]] is at

* [[Mike Shulman]], _[ReflectiveSubcategory.v](https://github.com/mikeshulman/HoTT/blob/master/Coq/ReflectiveSubcategory.v
)_

[[!redirects reflective sub-(∞,1)-category - internal formulation]]


reflective sub-2-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analog in [[2-category theory]] of the notion of _[[reflective subcategory]]_ in [[category theory]].

## Definition

A [[full and faithful 2-functor]] $C \hookrightarrow D$ is said to exhibit $C$ as a **reflective sub-2-category** of $D$ if it has a [[left adjoint|left]] [[adjoint 2-functor]].

[[!redirects reflective sub-2-categories]]


reflective subcategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A _reflective subcategory_ is a [[full subcategory]]

$$
  C \hookrightarrow D
$$

such that [[objects]] $d$ and [[morphisms]] $f \colon d \to d'$ in $D$ have "reflections" $T d$ and $T f \colon T d \to T d'$ in $C$. Every object in $D$ looks at its own reflection via a morphism $d \to T d$ and the reflection of an object $c \in C$ is equipped with an [[isomorphism]] $T c \cong c$.

A canonical example is the inclusion

$$
  \mathrm{Ab} \hookrightarrow \mathrm{Grp}
$$

of the [[category of abelian groups]] into the [[category of groups]], whose reflector is the operation of _[[abelianization]]_.

A useful property of reflective subcategories is that the inclusion $C \hookrightarrow D$ [[created limit|creates all limits]] of $D$ and $C$ has all [[colimits]] which $D$ admits.

## Definition

+-- {: .num_defn }
###### Definition

A [[full subcategory]] $i : C \hookrightarrow D $ is **reflective** if the inclusion [[functor]] $i$ has a [[left adjoint]] $T$:

$$
  (T \dashv i) :  C \stackrel{\stackrel{T}{\leftarrow}}{\hookrightarrow}
  D
  \,.
$$

=--

The left adjoint is sometimes called the __reflector__, and a functor which is a reflector (or has a fully faithful right adjoint, which is the same up to equivalence) is called a __reflection__.  Of course, there are dual notions of __[[coreflective subcategory]]__, __coreflector__, and __coreflection__.

+-- {: .num_remark #NonFullReflections}
###### Remark

A few sources (such as [[Categories Work]]) do not require a reflective subcategory to be full.  However, in light of the fact that non-full subcategories are not [[principle of equivalence|invariant under equivalence]], consideration of non-full reflective subcategories seems of limited usefulness.  The general consensus among category theorists nowadays seems to be that "reflective subcategory" implies fullness.  Examples for non-full subcategories and their behaviour can be found in a [TAC paper](http://www.tac.mta.ca/tac/volumes/30/41/30-41.pdf) by Adámek and Rosický.

=--

+-- {: .num_remark }
###### Remark

The components of the [[unit of an adjunction|unit]]

$$
  \array{
    & \nearrow &\Downarrow^{\eta}& \searrow^{Id}
    \\
    D &\stackrel{T}{\to}& C &\stackrel{i}{\hookrightarrow} & D
  }
$$

of this [[adjunction]] "reflect" each object $d \in D$ into its image $T d$ in the reflective subcategory

$$
  \eta_d : d \to T  d
  \,.
$$

=--

This reflection is sometimes called a _[[localization]]_ (due to [this Prop.](reflective+localization#ReflectiveSubcategoriesAreLocalizations) at _[[reflective localization]]_), although sometimes this term is reserved for the case when the functor $T$ is [[exact functor|left exact]].

+-- {: .num_defn }
###### Definition

If the reflector $T$ is [[faithful functor|faithful]], the reflection is called a **[[completion]]**.

=--

## Characterizations

+-- {: .num_prop #CharacterizationByLocalization}
###### Proposition
**(equivalent characterizations)**

Given any pair of [[adjoint functors]]

$$
  (L \dashv R)
  \;:\;
  B
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\bot}
  A
$$

the following are equivalent:

1. The [[right adjoint]] $R$ is [[full and faithful functor|fully faithful]]. (In this case $B$ is equivalent to its [[essential image]] in $A$ under $R$, a full [[reflective subcategory]] of $A$.)

2. The [[counit of an adjunction|counit]] $\varepsilon : L R \to 1_B$ of the [[adjunction]] is a [[natural isomorphism]] of functors.

3.  There exists some natural isomorphism $L R \to 1_B$.

4. The [[monad]] $(R L, R\varepsilon L,\eta)$ associated with the adjunction is [[idempotent monad|idempotent]], the right adjoint $R$ is [[conservative functor|conservative]], and the left adjoint $L$ is [[essentially surjective functor|essentially surjective on objects]].

5. If $S$ is the set of morphisms $s$ in $A$ such that $L(s)$ is an [[isomorphism]] in $B$, then $L \colon A \to B$ realizes $B$ as the (nonstrict) [[localization]] of $A$ with respect to the class $S$.

6. The [[left adjoint]] $L$ is [[dense functor|dense]].

=--

The equivalence of statements (1), (2), (4) and (5) are originally due to ([Gabriel-Zisman 67, Prop. 1.3, page 7](#GabrielZisman67)). The equivalence of (1) and (6) is due to ([Ulmer, Theorem 1.13](#Ulmer68)).  The equivalence of (2) and (3) is ([Johnstone, Lemma A1.1.1](#Johnstone)).

+-- {: .proof}
###### Proof

The equivalence of (1) and (2) is [this proposition](adjoint+functor#FullyFaithfulAndInvertibleAdjoints). The equivalence of (1) and (4) is [this Prop.](idempotent+monad#EquivalentConditions). For (5) see _[[reflective localization]]_. The equivalence of (1) and (6) can be seen by observing that $lan_L L \cong L lan_L id \cong L R$, which is pointwise, since $lan_L id$ is absolute, and is isomorphic to the identity if and only if $R$ is fully faithful.

To prove that (3) implies (2), the argument is to transfer the comonad structure on $L R$ across the isomorphism to a comonad structure on $1_B$, and observe that for any comonad structure on $1_B$ the counit is inverse to the comultiplication; thus the counit $\varepsilon$ of the original comonad structure on $L R$ must have been invertible. The same argument shows that for a comonad in any 2-category the counit $\varepsilon : L R \to 1_B$ is an isomorphism iff $L R$ is isomorphic to $1_B$.

=--

This is a well-known set of equivalences concerning [[idempotent monads]]. The essential point is that a reflective subcategory $i \colon B \to A$ is [[monadic functor|monadic]] (Prop. \ref{ReflectiveSubcategoryInclusionIsMonadic}), i.e., realizes $B$ as the category of [[algebra over a monad|algebras for the monad]] $i r$ on $A$, where $r: A \to B$ is the reflector.

See also the related discussion at [[reflective sub-(infinity,1)-category]].

## Special cases

### Exact reflective subcategories
 {#ExactReflectiveSubcategories}

If the reflector (which as a [[left adjoint]] always preserves all [[colimit]]s) in addition preserves [[finite limits]], then the embedding is called _exact_. If the categories are [[topos]]es then such embeddings are called [[geometric embedding]]s.

In particular, every [[sheaf topos]] is an exact reflective subcategory of a [[category of presheaves]]

$$
  Sh(C) \stackrel{\overset{sheafify}{\leftarrow}}{\hookrightarrow}
  PSh(C)
  \,.
$$

The reflector in that case is the [[sheafification]] functor.

+-- {: .num_theorem}
######Theorem

If $C$ is a reflective subcategory of a [[cartesian closed category]], then it is an [[exponential ideal]] if and only if its [[reflector]] $D\to C$ preserves [[finite products]].

In particular, $C$ is then also cartesian closed.

=--

This appears for instance as ([Johnstone, A4.3.1](#Johnstone)). See also at _[[reflective subuniverse]]_.

So in particular if $C$ is an exact reflective subcategory of a cartesian closed category $D$, then $C$ is an [[exponential ideal]] of $D$.

See [[Day's reflection theorem]] for a more general statement and proof.

### Complete reflective subcategories

When the unit of the reflector is a [[monomorphism]], a reflective category is often thought of as a [[full subcategory]] of *complete* objects in some sense; the reflector takes each object in the ambient category to its completion.  Such reflective subcategories are sometimes called _mono-reflective_.  One similarly has _epi-reflective_ (when the unit is an [[epimorphism]]) and _bi-reflective_ (when the unit is both a monomorphism and an epimorphism).

In the last case, note that if the unit is an *iso*morphism, then the inclusion functor is an [[equivalence of categories]], so nontrivial bireflective subcategories can occur only in non-[[balanced categories]].  Also note that 'bireflective' here does *not* mean reflective and [[coreflective subcategory|coreflective]].  One sees this term often in discussions of [[concrete categories]] (such as [[topological categories]]) where really something stronger holds: that the reflector lies over the [[identity functor]] on [[Set]].  In this case, one can say that we have a subcategory that is __reflective over $Set$__.

### Accessible reflective subcategories
 {#AccessibleReflectiveSubcategories}

+-- {: .num_defn #AccessibleReflection}
###### Definition

A reflection

$$
  \mathcal{C}
  \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}}
  \mathcal{D}
$$

is called **accessible** if $\mathcal{D}$ is an [[accessible category]] and  the reflector $R\circ L \colon \mathcal{D} \to \mathcal{D}$ is an [[accessible functor]].

=--

+-- {: .num_prop}
###### Proposition

A reflective subcategory $\mathcal{C} \hookrightarrow \mathcal{D}$ of an [[accessible category]] is accessible, def. \ref{AccessibleReflection}, precisely if $\mathcal{C}$ is an [[accessible category]].

=--

In this explicit form this appears as ([Lurie, prop. 5.5.1.2](#Lurie)).
From ([Adamek-Rosický](#AdamekRosicky)) the "only if"-direction follows immediately from 2.53  there (saying that an accessibly embedded subcategory of an accessible category is accessible iff it is cone-reflective), while the "if"-direction follows immediately from 2.23 (saying any left or right adjoint between accessible categories is accessible).

## Properties

### General

A reflective subcategory is always closed under [[limit|limits]] which exist in the ambient category (because the full inclusion is monadic, by Prop. \ref{ReflectiveSubcategoryInclusionIsMonadic}, [hence](monadic+functor#MonadicFunctorsCreateLimits) creates limits, as noted above), and inherits [[colimit|colimits]] from the larger category by application of the reflector [Riehl, Prop 4.5.15](#Riehl). In particular, if the ambient category is complete and cocomplete then so is the reflective subcategory.

A morphism in a reflective subcategory is monic iff it is monic in the ambient category. A reflective subcategory of a well-powered category is well-powered.

### As Eilenberg-Moore category of the idempotent monad
 {#AsEilenbergMooreCategory}

\begin{proposition}
\label{ReflectiveSubcategoryInclusionIsMonadic}
Every reflective subcategory inclusion is a [[monadic functor]], exhibiting the reflective subcategory as the [[Eilenberg-Moore category]] of [[algebra over a monad|modules]] for its [induced](monad#RelationBetweenAdjunctionsAndMonads) [[idempotent monad]]. Conversely, the [[Eilenberg-Moore category]] of an [[idempotent monad]] is a reflective subcategory
\end{proposition}
A proof is spelled out for instance in [Borceux 1994, vol 2, cor. 4.2.4](#Borceux94b). A [[formal proof]] in [[cubical Agda]] is given in [1Lab](#1Lab). See also Prop. \ref{CharacterizationByLocalization} and see at _[idempotent monad -- Properties -- Algebras for an idempotent monad and localization](idempotent+monad#AlgebrasForAnIdempotentMonad)_.

### Reflective subcategories of locally presentable categories

Both the weak and strong versions of [[Vopěnka's principle]] are equivalent to fairly simple statements concerning reflective subcategories of locally presentable categories:

+-- {: .num_theorem}
###### Theorem

The weak [[Vopěnka's principle]] is equivalent to the statement:

For $C$ a [[locally presentable category]],  every [[full subcategory]] $D \hookrightarrow C$ which is closed under [[limit]]s is a reflective subcategory.

=--

This is [AdamekRosicky, theorem 6.28](#AdamekRosicky)

+-- {: .num_theorem}
###### Theorem
The strong [[Vopěnka's principle]] is equivalent to:

For $C$ a [[locally presentable category]],  every [[full subcategory]] $D \hookrightarrow C$ which is closed under [[limit]]s is a reflective subcategory; further on, $D$ is then also locally presentable.
=--
(Remark after corollary 6.24 in [Adamek-Rosicky book](#AdamekRosicky)).

### Reflective subcategories of cartesian closed categories
 {#ReflectiveSubcategoriesOfCartesianClosedCategotries}

In showing that a given category is [[cartesian closed category|cartesian closed]], the following theorem is often useful (cf. A4.3.1 in the [[Elephant]]):

+-- {: .num_theorem}
###### Theorem
If $C$ is cartesian closed, and $D\subseteq C$ is a [[reflective subcategory]], then the reflector $L\colon C\to D$ preserves finite [[products]] if and only if $D$ is an [[exponential ideal]] (i.e. $Y\in D$ implies $Y^X\in D$ for any $X\in C$).  In particular, if $L$ preserves finite products, then $D$ is cartesian closed.
=--

### Reflective and coreflective subcategories

+-- {: .num_theorem}
###### Theorem

A subcategory of a [[category of presheaves]] $[A^{op}, Set]$ which is both reflective and coreflective is itself a category of presheaves $[B^{op}, Set]$, and the inclusion is induced by a functor $A \to B$.

=--

This is shown in ([BashirVelebil](#BashirVelebil)).

### Property vs structure

Whenever $C$ is a full subcategory of $D$, we can say that objects of $C$ are objects of $D$ with some extra [[property, structure, stuff|property]].  But if $C$ is reflective in $D$, then we can turn this around and (by thinking of the left adjoint as a [[forgetful functor]]) think of objects of $D$ as objects of $C$ with (if we\'re lucky) some [[extra structure]] or (in any case) some [[extra stuff]].

This can always be made to work by brute force, but sometimes there is something insightful about it.  For example, a metric space is a complete metric space equipped with a dense subset.  Or, an [[integral domain]] is a [[field]] equipped with numerator and denominator functions.

## Examples
 {#Examples}

+-- {: .num_example }
###### Example

Complete [[metric spaces]] are mono-reflective in metric spaces; the reflector is called _completion_.

=--

+-- {: .num_example }
###### Example

The [[category of sheaves]] on a [[site]] $S$ is a reflective subcategory of the [[category of presheaves]] on $S$; the reflector is called _[[sheafification]]_. In fact, categories of sheaves are precisely those accessible reflective subcategories, def. \ref{AccessibleReflection},  of presheaf categories for which the reflector is left [[exact functor|exact]]. This makes the inclusion functor precisely a [[geometric inclusion]] of [[toposes]].

=--

+-- {: .num_example }
###### Example

A category of [[concrete presheaves]] inside a [[category of presheaves]] on a [[concrete site]] is a reflective subcategory.

=--

+-- {: .num_example }
###### Example

In a [[recollement]] situation, we have several reflectors and coreflectors. We have a reflective and coreflective subcategory $i_*: A' \hookrightarrow A$ with reflector $i^*$ and coreflector $i^!$. The functor $j^*$ is both a reflector for the reflective subcategory $j_*: A'' \hookrightarrow A$, and a coreflector for the coreflective subcategory $j_!: A'' \hookrightarrow A$.

=--

+-- {: .num_example #TheReflectiveSubcategoriesOfSet}
###### Example

Assuming classical logic, the category [[Set]] has exactly three reflective (and [[replete subcategory|replete]]) subcategories: the full subcategory containing all [[singleton set|singleton sets]]; the full subcategory containing all [[subsingletons]]; and $Set$ itself.

In [[constructive mathematics]], there are potentially more reflective subcategories, for instance the subcategory of $j$-sheaves for any [[Lawvere-Tierney topology]] on $Set$.

=--

+-- {: .num_example #AffineVarieties}
###### Example

The category of [[affine schemes]] is a reflective subcategory of the category of [[schemes]], with the reflector given by $X \mapsto Spec \Gamma(X,\mathcal{O}_X)$.

The generalization of this example to [[homotopy theory]] is discussed at _[[function algebras on infinity-stacks]]_. The analogue in [[noncommutative algebraic geometry]] is in ([Rosenberg 98, prop 4.4.3](#Rosenberg98)).

=--

+-- {: .num_example #NonUnitalRings}
###### (Counter)Example
The non-full inclusion of unital [[rings]] into non-unital rings has a left adjoint (with monic units), whose reflector formally adjoins an [[identity element]].  However, we do not call it a reflective subcategory, because the "inclusion" is not full; see remark \ref{NonFullReflections}.
=--

+-- {: .num_remark #RemarkOnNonUnitalRings}
###### Remark

Notice that for $R \in Ring$ a ring with unit, its reflection $L R$ in the above example is not in general isomorphic to $R$, but is much larger.
But an object in a reflective subcategory is necessarily isomorphic to its image under the reflector only if the reflective subcategory is full.  While the inclusion $\mathbf{Ring} \hookrightarrow \mathbf{Ring}$' does have a [[left adjoint]] (as any [[forgetful functor]] between varieties of algebras, by the [[adjoint lifting theorem]]), this inclusion is not full (an arrow in $\mathbf{Ring}$' need not preserve the identity).

=--

+-- {: .num_example #CategoryOfCategories}
###### Example

The subcategory
$$ Cat \hookrightarrow sSet$$
of the [[category of categories]] into the [[sSet|category of simplical sets]] is a reflective subcategory [Riehl, example 4.5.14 (vi)](#Riehl). The reflection is given by the [[homotopy category|homotopy category functor]]. This implies that [[Cat]] is complete and cocomplete because it inherits all limits and colimits from [[sSet]].
=--

+-- {: .num_example #ModelsOfALawvereTheory}
###### Example

For any [[Lawvere theory]] $T$, its category of models is the category
$$Prod(T, Set)$$
of product preserving functors into $Set$ and natural transformations between them. The inclusion
$$Prod(T, Set) \hookrightarrow [T, Set]$$
is a reflective subcategory [Buckley, theorem 5.2.1](#Buckley). Therefore, because $[T,Set]$ is complete and cocomplete (limits and colimits are computed pointwise), so is $Prod(T, Set)$. This implies that many familar algebraic categories such as [[Grp]], [[Mon]], [[Ring]], etc. are complete and cocomplete as a special case.
=--

## Related concepts

* [[localization]], [[locally presentable category]]

* [[reflective localization]]

* [[Quillen reflection]], [[left Bousfield localization]]

* [[reflective sub-(infinity,1)-category]]

* [[coreflective subcategory]]

* [[bireflective subcategory]]

* [[adjoint cylinder]], describing the situation when the reflector has a further left adjoint

* [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]

## References

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], _[[Calculus of fractions and homotopy theory]]_, Springer 1967 ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/GZ.pdf))

* {#Ulmer68} [[Friedrich Ulmer]], _Properties of Dense and Relative Adjoint Functors_, Journal of Algebra 1968

* [[Saunders MacLane]], §IV.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Borceux94a} [[Francis Borceux]], p. 196 in *Basic Category Theory*,  vol.1 of: *[[Handbook of Categorical Algebra]]*, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

* {#Borceux94b} [[Francis Borceux]], Cor. 2.4.2 in: *Categories and Structures*, *[[Handbook of Categorical Algebra]]*, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

* {#AdamekRosicky} [[Jiri Adamek|Jiri Adamek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_ London Mathematical Society Lecture Note Series 189

* Springer eom: [reflective subcategory](http://eom.springer.de/r/r080550.htm)

* cf. the notion of $Q^\circ$-category in the entry [[Q-category]]

* {#Riehl} [[Emily Riehl]], _[[Category Theory in Context]], p. 142, Courier Dover Publications 2017 ([pdf](http://www.math.jhu.edu/~eriehl/context.pdf))

* {#Buckley} Mitchell Buckley, _Lawvere Theories, 2008 [pdf](http://web.science.mq.edu.au/~street/MitchB.pdf)

The relation of exponential ideals to [[reflective subcategories]] is discussed in section A4.3.1 of

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

Reflective and coreflective subcategories of presheaf categories are discussed in

* {#BashirVelebil}R. Bashir, J. Velebil, _Simultaneously reflective and coreflective subcategories of presheaves_, Theory and  Applications of Categories, Vol 10. No. 16. (2002) ([pdf](http://www.emis.de/journals/TAC/volumes/10/16/10-16.pdf)).

Related discussion of [[reflective sub-(∞,1)-categories]] is in

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

The example of affine schemes in [[noncommutative algebraic geometry]] is in

* {#Rosenberg98} [[Alexander Rosenberg]], _Noncommutative schemes_, Comp. Math. 112, 93–125 (1998)

Formalization in [[cubical Agda]]:

* {#1Lab} [[1lab]], *[Reflective Subcategories](https://1lab.dev/Cat.Functor.Adjoint.Reflective.html)*

See also:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On reflective and coreflective hulls*, [[Cahiers|Cahiers Topologie Géométrie Différentielle Catégorique]] **56** (2015) 162–208 [[pdf](https://www.math.mcgill.ca/barr/papers/biref.pdf), [[BarrKennisonRaphael-Reflective.pdf:file]]]

[[!redirects reflector]]
[[!redirects reflectors]]

[[!redirects reflective subcategories]]
[[!redirects reflexive subcategory]]


reflective subuniverse

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[reflective sub-(infinity,1)-category|reflective]] sub-[[universe]]. If closed under [[dependent sum]], then this is the universe of [[modal types]] for a [[modal operator]].

For the relation to [[modal type theory]] see [Rijke, Shulman, Spitters](#RSS).

## Definition

+-- {: .num_defn}
###### Definition

=--

([HoTTBook, theorem 7.7.1](#HoTTBook))

## Properties

([HoTTBook, theorem 7.7.2](#HoTTBook))

...in [[categorical semantics]] this means that it  is an [[exponential ideal]], hence that the reflector preserves [[finite products]]

## Related concepts

* [[modal homotopy type theory]]

## References

* {#HoTTBook} [Univalent Foundations Project](http://ncatlab.org/nlab/show/UF-IAS-2012), section 7.7 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

* {#RSS} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], _[[Modalities in homotopy type theory]]_ ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526))

[[!redirects reflective subuniverses]]


reflexive Banach space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

Throughout, we work in the category $Bant$ whose objects are Banach spaces and where the morphisms are continuous linear maps. References below to the unit ball suggest that it _might_ be premature to cast everything in terms of a certain subcategory of TVS.

## Duality
Following the lead of Mac Lane (2nd ed.) Section IV.2, which does this for vector spaces) but with slight changes in notation: let $\overline{D}:Bant\to Bant$ be the contravariant functor which takes a Banach space to its [[dual vector space|dual space]], and sends a continuous linear map to its adjoint/dual map.

This gives rise, in a straightforward way, to two functors $D_L: Bant \to Bant^{op}$
 and $D_R: Bant^{op}\to Bant$. $D_L$ is the left adjoint of $D_R$, that is

$$ Bant^{op}(D_L X, Y) \cong Bant(X, D_R Y) $$

In general
$$ Bant^{op}(Y, D_L X) \not\cong Bant(D_R Y, X) $$
so that $D_L$ is not a right adjoint of $D_R$.
For example: take $Y$ to be the ground field $K$ and $X$ to be $c_0$ with the usual supremum norm.

**Not-a-proof-yet** of this claim: we have $D_R(K)\cong K$ and the $Bant$-morphisms from $K$ to $c_0$ are just vectors in $c_0$; but the $Bant$-morphisms from $D_L(c_0)\cong\ell^1$ to $K$ correspond to the vectors in $\ell^\infty$. (It would seem from this example that even in [[dream mathematics]] one doesn't get $D_L$ being a right adjoin of $D_R$.)

**Unit and counit.**
The unit of this adjunction is the canonical map $\kappa_X: X\to (X^*)^*$ from a Banach space $X$ to its second dual $X^{**}$. In the presence of
[[axiom of choice|Choice]], the [[Hahn–Banach theorem]] ensures that $\kappa_X$ is an isometry.

To get things to run smoothly, we seem to need more than $\kappa_X$ being monic in $Bant$; but I (YC) am not sure which of the usual variants -- extreme, regular, strong, strict -- is the key one.

The counit map $\veps_X: X^{***} \to X^*$ is sometimes known as the Dixmier projection from the third dual of a Banach space to (the canonical image of) its first dual; note that this map is weak-star-to-weak-star continuous. (It is a projection in the sense of vector spaces, by the triangle identity for the adjunction.)

## Definition

A [[Banach space]] $X$ is **reflexive** if $\kappa_X$ is an [[isomorphism]] in $Bant$. If we furthermore grant ourselves Hahn-Banach, then $\kappa_X$ will even be an [[isometric isomorphism]]: an isomorphism in the category of Banach spaces and [[short linear maps]].

## Properties

(...)

If two [[Banach space]]s are isomorphic as TVSes (but not necessarily isometrically isomorphic), then either both are reflexive or both are non-reflexive.

There is a nice proof that closed subspaces of reflexive [[Banach space]]s are reflexive (due to Linton? [^the details and citation are in the book of Cigler-Losert-Michor but the author of this sentence has no access to a copy at time of typing])
using naturality of $\kappa_X$.

It turns out that if $X$ is a [[Banach space]], then it is reflexive if and only if its (norm-)closed unit ball is compact in the $\sigma(X,X^*)$-topology. In particular, if $X$ is reflexive and $E\to X$, $X\to F$ are bounded linear operators, then the composition $E\to F$ is [[weakly compact]] as a linear operator.

A theorem of Davis-Figiel-Johnson-Pelczynski (1974) tells us that the converse is true:  every [[weakly compact]] linear operator between Banach spaces factors through some reflexive Banach space. The intermediate space is constructed by real interpolation and (at least as usually presented) does not seem to be canonical in any way.

## Examples and counterexamples

By the [[Riesz duality theorem]], every [[separable space|separable]] [[Hilbert space]] is reflexive.

The [[Lebesgue space]] $l^1$ is reflexive in [[dream mathematics]], but in [[classical mathematics]] it is not.

In [[dream mathematics]] $l^\infty$ is reflexive; but its closed subspace $c_0$ is not. On the other hand, if one works in a setting where $\kappa_X$ is a monomorphism, then closed subspaces of reflexive Banach spaces are reflexive (see above)

James (1950) constructed a separable non-reflexive Banach space which is isomorphic as a TVS to its second dual; nowadays this is known as the __James space__ $J$. More is true: $\kappa_J(J)J$ is a codimension-$1$ subspace of $J^{**}$.

One can renorm $J$ to obtain a non-reflexive Banach space which is isometrically isomorphic as a Banach space to its second dual (James, 1951).

## References

* [MR0039915](http://www.ams.org/mathscinet-getitem?mr=39915) (12,616b)
 James, Robert C. Bases and reflexivity of Banach spaces.
[Ann. of Math. (2) 52, (1950). 518–527.](http://dx.doi.org/10.2307/1969430)

* [MR0044024](http://www.ams.org/mathscinet-getitem?mr=44024) (13,356d)
James, Robert C. A non-reflexive Banach space isometric with its second conjugate space.
Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 174–177.

* [MR0355536](http://www.ams.org/mathscinet-getitem?mr=0355536) (50 #8010)
Davis, W. J.; Figiel, T.; Johnson, W. B.; Pełczyński, A.
Factoring weakly compact operators.
J. Functional Analysis 17 (1974), 311–327.

[[!redirects reflexive Banach space]]
[[!redirects reflexive Banach spaces]]