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\begin{document}

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\section*{reflective subcategory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{notions_of_subcategory}{}\paragraph*{{Notions of subcategory}}\label{notions_of_subcategory}

[[!include notions of subcategory]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{characterizations}{Characterizations}\dotfill \pageref*{characterizations} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{ExactReflectiveSubcategories}{Exact reflective subcategories}\dotfill \pageref*{ExactReflectiveSubcategories} \linebreak
\noindent\hyperlink{complete_reflective_subcategories}{Complete reflective subcategories}\dotfill \pageref*{complete_reflective_subcategories} \linebreak
\noindent\hyperlink{AccessibleReflectiveSubcategories}{Accessible reflective subcategories}\dotfill \pageref*{AccessibleReflectiveSubcategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{AsEilenbergMooreCategory}{As Eilenberg-Moore category of the idempotent monad}\dotfill \pageref*{AsEilenbergMooreCategory} \linebreak
\noindent\hyperlink{reflective_subcategories_of_locally_presentable_categories}{Reflective subcategories of locally presentable categories}\dotfill \pageref*{reflective_subcategories_of_locally_presentable_categories} \linebreak
\noindent\hyperlink{ReflectiveSubcategoriesOfCartesianClosedCategotries}{Reflective subcategories of cartesian closed categories}\dotfill \pageref*{ReflectiveSubcategoriesOfCartesianClosedCategotries} \linebreak
\noindent\hyperlink{reflective_and_coreflective_subcategories}{Reflective and coreflective subcategories}\dotfill \pageref*{reflective_and_coreflective_subcategories} \linebreak
\noindent\hyperlink{property_vs_structure}{Property vs structure}\dotfill \pageref*{property_vs_structure} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

A \emph{reflective subcategory} is a [[full subcategory]]

\begin{displaymath}
C \hookrightarrow D
\end{displaymath}
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 $d \to Td$ and the reflection of an object $c \in C$ is equipped with an [[isomorphism]] $T c \cong c$.

A canonical example is the inclusion

\begin{displaymath}
\mathrm{Ab} \hookrightarrow \mathrm{Grp}
\end{displaymath}
of the [[category of abelian groups]] into the [[category of groups]], whose reflector is the operation of \emph{[[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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A [[full subcategory]] $i : C \hookrightarrow D$ is \textbf{reflective} if the inclusion [[functor]] $i$ has a [[left adjoint]] $T$:

\begin{displaymath}
(T \dashv i) :  C \stackrel{\stackrel{T}{\leftarrow}}{\hookrightarrow}
  D
  \,.
\end{displaymath}
\end{defn}
The left adjoint is sometimes called the \textbf{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 \textbf{reflection}. Of course, there are dual notions of \textbf{[[coreflective subcategory]]}, \textbf{coreflector}, and \textbf{coreflection}.

\begin{remark}
\label{NonFullReflections}\hypertarget{NonFullReflections}{}
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 \href{http://www.tac.mta.ca/tac/volumes/30/41/30-41.pdf}{TAC paper} by Ad\'a{}mek and Rosick\'y{}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The components of the [[unit of an adjunction|unit]]

\begin{displaymath}
\itexarray{
    & \nearrow &\Downarrow^{\eta}& \searrow^{Id}
    \\
    D &\stackrel{T}{\to}& C &\hookrightarrow & D
  }
\end{displaymath}
of this [[adjunction]] ``reflect'' each object $d \in D$ into its image $T d$ in the reflective subcategory

\begin{displaymath}
\eta_d : d \to T  d
  \,.
\end{displaymath}
\end{remark}
This reflection is sometimes called a \emph{[[localization]]} (due to \href{reflective+localization#ReflectiveSubcategoriesAreLocalizations}{this Prop.} at \emph{[[reflective localization]]}), although sometimes this term is reserved for the case when the functor $T$ is [[exact functor|left exact]].

\begin{defn}
\label{}\hypertarget{}{}
If the reflector $T$ is [[faithful functor|faithful]], the reflection is called a \textbf{[[completion]]}.

\end{defn}
\hypertarget{characterizations}{}\subsection*{{Characterizations}}\label{characterizations}

\begin{prop}
\label{CharacterizationByLocalization}\hypertarget{CharacterizationByLocalization}{}
\textbf{(equivalent characterizations)}

Given any pair of [[adjoint functors]]

\begin{displaymath}
(Q^*\dashv Q_*)
  \;:\; 
  B
    \underoverset
      {\underset{Q_*}{\longrightarrow}}
      {\overset{Q^*}{\longleftarrow}} 
      {\bot}
  A
\end{displaymath}
the following are equivalent:

\begin{enumerate}%
\item The [[right adjoint]] $Q_*$ is [[full and faithful functor|fully faithful]]. (In this case $B$ is equivalent to its [[essential image]] in $A$ under $Q_*$, a full [[reflective subcategory]] of $A$.)


\item The [[counit of an adjunction|counit]] $\varepsilon : Q^* Q_*\to 1_B$ of the [[adjunction]] is a [[natural isomorphism]] of functors.


\item The [[monad]] $(Q_* Q^*,Q_*\varepsilon Q^*,\eta)$ associated with the adjunction is [[idempotent monad|idempotent]], the right adjoint $Q_*$ is [[conservative functor|conservative]], and the left adjoint $Q^*$ is [[essentially surjective functor|essentially surjective on objects]].


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



\end{enumerate}
\end{prop}
This is originally due to (\hyperlink{GabrielZisman67}{Gabriel-Zisman 67, prop. 1.3, page 7}).

\begin{proof}
The equivalence of 1) and 2) is \href{adjoint+functor#FullyFaithfulAndInvertibleAdjoints}{this prop.}. The equivalence of 1) and 3) is \href{idempotent+monad#EquivalentConditions}{this Prop.}.

For the last item see at \emph{[[reflective localization]]}.

\end{proof}
This is a well-known set of equivalences concerning [[idempotent monads]]. The essential point is that a reflective subcategory $i: B \to A$ is [[monadic functor|monadic]], 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]].

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

\hypertarget{ExactReflectiveSubcategories}{}\subsubsection*{{Exact reflective subcategories}}\label{ExactReflectiveSubcategories}

If the reflector (which as a [[left adjoint]] always preserves all [[colimit]]s) in addition preserves [[finite limits]], then the embedding is called \emph{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]]

\begin{displaymath}
Sh(C) \stackrel{\overset{sheafify}{\leftarrow}}{\hookrightarrow}
  PSh(C)
  \,.
\end{displaymath}
The reflector in that case is the [[sheafification]] functor.

\begin{theorem}
\label{}\hypertarget{}{}
If $X$ 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 product]].

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

\end{theorem}
This appears for instance as (\hyperlink{Johnstone}{Johnstone, A4.3.1}). See also at \emph{[[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.

\hypertarget{complete_reflective_subcategories}{}\subsubsection*{{Complete reflective subcategories}}\label{complete_reflective_subcategories}

When the unit of the reflector is a [[monomorphism]], a reflective category is often thought of as a full subcategory of \emph{complete} objects in some sense; the reflector takes each object in the ambient category to its completion. Such reflective subcategories are sometimes called \emph{mono-reflective}. One similarly has \emph{epi-reflective} (when the unit is an [[epimorphism]]) and \emph{bi-reflective} (when the unit is a [[bimorphism]]).

In the last case, note that if the unit is an \emph{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 \emph{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 \textbf{reflective over $Set$}.

\hypertarget{AccessibleReflectiveSubcategories}{}\subsubsection*{{Accessible reflective subcategories}}\label{AccessibleReflectiveSubcategories}

\begin{defn}
\label{AccessibleReflection}\hypertarget{AccessibleReflection}{}
A reflection

\begin{displaymath}
\mathcal{C}
  \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}}
  \mathcal{D}
\end{displaymath}
is called \textbf{accessible} if $\mathcal{D}$ is an [[accessible category]] and the reflector $R\circ L \colon \mathcal{D} \to \mathcal{D}$ is an [[accessible functor]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
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]].

\end{prop}
In this explicit form this appears as (\hyperlink{Lurie}{Lurie, prop. 5.5.1.2}). From (\hyperlink{AdamekRosicky}{Adamek-Rosick\'y{}}) 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).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

A reflective subcategory is always closed under [[limit|limits]] which exist in the ambient category (because the full inclusion is monadic, as noted above), and inherits [[colimit|colimits]] from the larger category by application of the reflector \hyperlink{Riehl}{Riehl, Prop 4.5.15}. 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.

\hypertarget{AsEilenbergMooreCategory}{}\subsubsection*{{As Eilenberg-Moore category of the idempotent monad}}\label{AsEilenbergMooreCategory}

\begin{prop}
\label{}\hypertarget{}{}
Any reflective subcategory is recovered as the [[Eilenberg-Moore category]] of [[algebra over a monad|algebras]] over its associated [[idempotent monad]].

\end{prop}
See for instance (\hyperlink{Borceux}{Borceux, vol 2, cor. 4.2.4}) and see at \emph{\href{idempotent+monad#AlgebrasForAnIdempotentMonad}{idempotent monad -- Properties -- Algebras for an idempotent monad and localization}}.

\hypertarget{reflective_subcategories_of_locally_presentable_categories}{}\subsubsection*{{Reflective subcategories of locally presentable categories}}\label{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:

\begin{theorem}
\label{}\hypertarget{}{}
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.

\end{theorem}
This is \hyperlink{AdamekRosicky}{AdamekRosicky, theorem 6.28}

\begin{theorem}
\label{}\hypertarget{}{}
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.

\end{theorem}
(Remark after corollary 6.24 in \hyperlink{AdamekRosicky}{Adamek-Rosicky book}).

\hypertarget{ReflectiveSubcategoriesOfCartesianClosedCategotries}{}\subsubsection*{{Reflective subcategories of cartesian closed categories}}\label{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]]):

\begin{theorem}
\label{}\hypertarget{}{}
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.

\end{theorem}
\hypertarget{reflective_and_coreflective_subcategories}{}\subsubsection*{{Reflective and coreflective subcategories}}\label{reflective_and_coreflective_subcategories}

\begin{theorem}
\label{}\hypertarget{}{}
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 $A \to B$.

\end{theorem}
This is shown in (\hyperlink{BashirVelebil}{BashirVelebil}).

\hypertarget{property_vs_structure}{}\subsubsection*{{Property vs structure}}\label{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.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{}\hypertarget{}{}
Complete [[metric spaces]] are mono-reflective in metric spaces; the reflector is called \emph{completion}.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The [[category of sheaves]] on a [[site]] $S$ is a reflective subcategory of the [[category of presheaves]] on $S$; the reflector is called \emph{[[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]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
A category of [[concrete presheaves]] inside a [[category of presheaves]] on a [[concrete site]] is a reflective subcategory.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
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 $j_!: A'' \hookrightarrow A$.

\end{example}
\begin{example}
\label{TheReflectiveSubcategoriesOfSet}\hypertarget{TheReflectiveSubcategoriesOfSet}{}
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|Lawvere?Tierney topology]] on $Set$.

\end{example}
\begin{example}
\label{AffineVarieties}\hypertarget{AffineVarieties}{}
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 \emph{[[function algebras on infinity-stacks]]}. The analogue in [[noncommutative algebraic geometry]] is in (\hyperlink{Rosenberg98}{Rosenberg 98, prop 4.4.3}).

\end{example}
\begin{example}
\label{NonUnitalRings}\hypertarget{NonUnitalRings}{}
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}.

\end{example}
\begin{remark}
\label{RemarkOnNonUnitalRings}\hypertarget{RemarkOnNonUnitalRings}{}
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).

\end{remark}
\begin{example}
\label{CategoryOfCategories}\hypertarget{CategoryOfCategories}{}
The subcategory

\begin{displaymath}
Cat \hookrightarrow sSet
\end{displaymath}
of the [[category of categories]] into the [[sSet|category of simplical sets]] is a reflective subcategory \hyperlink{Riehl}{Riehl, example 4.5.14 (vi)}. 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]].

\end{example}
\begin{example}
\label{ModelsOfALawvereTheory}\hypertarget{ModelsOfALawvereTheory}{}
For any [[Lawvere theory]] $T$, its category of models is the category

\begin{displaymath}
Prod(T, Set)
\end{displaymath}
of product preserving functors into $Set$ and natural transformations between them. The inclusion

\begin{displaymath}
Prod(T, Set) \hookrightarrow [T, Set]
\end{displaymath}
is a reflective subcategory \hyperlink{Buckley}{Buckley, theorem 5.2.1}. 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.

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[localization]], [[locally presentable category]]


\item [[reflective localization]]


\item [[Quillen reflection]], [[left Bousfield localization]]


\item [[reflective sub-(infinity,1)-category]]


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


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



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Pierre Gabriel]], [[Michel Zisman]], \emph{[[Calculus of fractions and homotopy theory]]}, Springer 1967 (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf}{pdf})


\item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]]}, vol.1, p. 196.


\item [[Jiri Adamek|Jiri Adamek]], [[Jiří Rosický]], \emph{[[Locally presentable and accessible categories]]} London Mathematical Society Lecture Note Series 189


\item Springer eom: \href{http://eom.springer.de/r/r080550.htm}{reflective subcategory}


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


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


\item Mitchell Buckley, \emph{Lawvere Theories, 2008 \href{http://web.science.mq.edu.au/~street/MitchB.pdf}{pdf}}



\end{itemize}
The relation of exponential ideals to [[reflective subcategories]] is discussed in section A4.3.1 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
Reflective and coreflective subcategories of presheaf categories are discussed in

\begin{itemize}%
\item R. Bashir, J. Velebil, \emph{Simultaneously reflective and coreflective subcategories of presheaves}, Theory and Applications of Categories, Vol 10. No. 16. (2002) (\href{http://www.emis.de/journals/TAC/volumes/10/16/10-16.pdf}{pdf}).

\end{itemize}
Related discussion of [[reflective sub-(∞,1)-categories]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
The example of affine schemes in [[noncommutative algebraic geometry]] is in

\begin{itemize}%
\item [[Alexander Rosenberg]], \emph{Noncommutative schemes}, Comp. Math. 112, 93--125 (1998)

\end{itemize}
[[!redirects reflector]] [[!redirects reflectors]]

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



\end{document}