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\section*{opposite category}

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

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

[[!include category theory - contents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - 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{in_category_theory}{In category theory}\dotfill \pageref*{in_category_theory} \linebreak
\noindent\hyperlink{opposite_functors}{Opposite functors}\dotfill \pageref*{opposite_functors} \linebreak
\noindent\hyperlink{opposite_natural_transformations}{Opposite natural transformations}\dotfill \pageref*{opposite_natural_transformations} \linebreak
\noindent\hyperlink{the_oppositization_2functor}{The oppositization 2-functor}\dotfill \pageref*{the_oppositization_2functor} \linebreak
\noindent\hyperlink{in_enriched_category_theory}{In enriched category theory}\dotfill \pageref*{in_enriched_category_theory} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classes_of_examples}{Classes of examples}\dotfill \pageref*{classes_of_examples} \linebreak
\noindent\hyperlink{OppositeGroup}{Opposite group}\dotfill \pageref*{OppositeGroup} \linebreak
\noindent\hyperlink{opposite_of_the_opposite}{Opposite of the opposite}\dotfill \pageref*{opposite_of_the_opposite} \linebreak
\noindent\hyperlink{coalgebraic_structures}{Co-algebraic structures}\dotfill \pageref*{coalgebraic_structures} \linebreak
\noindent\hyperlink{duality_2}{Duality}\dotfill \pageref*{duality_2} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{opposite_of__and_}{Opposite of $Set$ and $FinSet$}\dotfill \pageref*{opposite_of__and_} \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}

For a [[category]] $C$, its \emph{opposite category} $C^{op}$ is the category obtained by formally reversing the direction of all its [[morphisms]] (while retaining their original composition law).

Categories generalize (are a [[horizontal categorification]] of) [[monoids]], [[groups]] and [[algebras]], and forming the opposite category corresponds to forming the opposite of a group, of a monoid, of an algebra.

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

\hypertarget{in_category_theory}{}\subsubsection*{{In category theory}}\label{in_category_theory}

For a [[category]] $C$, the \textbf{opposite category} $C^{op}$ has the same [[objects]] as $C$, but a [[morphism]] $f : x \to y$ in $C^{op}$ is the same as a morphism $f : y \to x$ in $C$, and a composite of morphisms $g f$ in $C^{op}$ is defined to be the composite $f g$ in $C$.

More precisely, $C_{\mathrm{obj}}$ and $C_{\mathrm{mor}}$ are, respectively, the collections of [[objects]] and of [[morphisms]] of $C$, and if the structure maps of $C$ are

\begin{itemize}%
\item source and target: $s_C,t_C : C_{\mathrm{mor}} \to C_{\mathrm{obj}}$


\item identity-assignment: $i_C : C_{\mathrm{obj}} \to C_{\mathrm{mor}}$


\item composition: $\circ_C : C_{\mathrm{mor}} \times_{C_{\mathrm{obj}}} C_{\mathrm{mor}} \to C_{\mathrm{mor}}$



\end{itemize}
then $C^{op}$ is the category with

\begin{itemize}%
\item the same (isomorphic) collections of objects and morphisms

$(C^{op})_{\mathrm{obj}} := C_{\mathrm{obj}}$

$(C^{op})_{\mathrm{mor}} := C_{\mathrm{mor}}$


\item the same identity-assigning map

$i_{C^{op}} := i_C$


\item \emph{switched} source and target maps

$s_{C^{op}} := t_C$

$t_{C^{op}} := s_C$


\item the \emph{same} composition operation,

$\circ_{C^{op}} := \circ_{C}$.

or more precisely, the composition operation of $C^{op}$ is

\begin{displaymath}
\circ_{C^{op}}
  : 
  C_{\mathrm{mor}}^{op} {}_{s^{op}}\times_{t^{op}} C_{\mathrm{mor}}^{op}
  =
  C_{\mathrm{mor}} {}_{t} \times_{s} C_{\mathrm{mor}}
  \stackrel{\simeq}{\to}
  C_{\mathrm{mor}} {}_{s} \times_{t} C_{\mathrm{mor}}
  \stackrel{\circ}{\to}
  C_{\mathrm{mor}}
  \,,
\end{displaymath}
where the isomorphism in the middle is the unique one induced from the universality of the [[pullback]].



\end{itemize}
Notice that hence the composition law does \emph{not} change when passing to the opposite category. Only the interpretation of in which direction the arrows point does change. So forming the opposite category is a completely formal process. Nevertheless, due to the switch of source and target, the opposite category $C^{op}$ is usually far from being [[equivalence of categories|equivalent]] to $C$. See the examples below.

\hypertarget{opposite_functors}{}\subsubsection*{{Opposite functors}}\label{opposite_functors}

Given [[categories]] $C$ and $D$, the \textbf{opposite functor} of a [[functor]] $F:C\to D$ is the functor $F^{op}:C^{op}\to D^{op}$ such that $F^{op}_{obj}=F_{obj}$ and $F^{op}_{mor}=F_{mor}$.

In the literature, $F^{op}$ is often confused with $F$. This is unfortunate, since (for example) [[natural transformations]] $F^{op}\to G^{op}$ (of functors $C^{op}\to D^{op}$) can be identified with natural transformations $G\to F$ (and not $F\to G$).

\hypertarget{opposite_natural_transformations}{}\subsubsection*{{Opposite natural transformations}}\label{opposite_natural_transformations}

Given [[categories]] $C$ and $D$, [[functors]] $F,G:C\to D$, the \textbf{opposite natural transformation} of a [[natural transformation]] $t:F\to G$ is the natural transformation $t^{op}:G^{op}\to F^{op}$, induced by the same map $C_{obj}\to D_{mor}$ as $t$.

Again, in the literature $t^{op}$ is often confused with $t$.

\hypertarget{the_oppositization_2functor}{}\subsubsection*{{The oppositization 2-functor}}\label{the_oppositization_2functor}

The above three constructions of the opposite category, opposite functor, and opposite natural transformation combine together into the \textbf{oppositization 2-functor}

\begin{displaymath}
Cat^{co}\to Cat,
\end{displaymath}
where $Cat^{co}$ denotes the 2-cell dual of the [[2-category]] $Cat$, with the direction of 2-morphisms reversed and the direction of 1-morphisms preserved.

\hypertarget{in_enriched_category_theory}{}\subsubsection*{{In enriched category theory}}\label{in_enriched_category_theory}

The definition has a direct generalization to [[enriched category theory]].

For $V$ a [[symmetric monoidal category]] and $C$ a $V$-[[enriched category]] the \textbf{opposite $V$-enriched category} $C^{op}$ is defined to be the $V$-enriched category with the same objects as $C$ and with

\begin{displaymath}
C^{op}(c,d) := C(d,c)
\end{displaymath}
and composition given by

\begin{displaymath}
C^{op}(b,c)\otimes C^{op}(a,b)
  :=
  C(c,b) \otimes C(b,a)
  \stackrel{\sigma}{\to}
  C(b,a) \otimes C(c,b)
  \stackrel{\circ_C}{\to} 
  C_{c,a}
  =:
  C^{op}(a,c)
  \,.
\end{displaymath}
The unit maps $j_a : I \to C^{op}(a,a)$ are those of $C$ under the identification $C^{op}(a,a) = C(a,a)$.

Note that the [[braiding]] of $V$ is used in defining composition for $C^{op}$. So, we cannot define the opposite of a $V$-enriched category if $V$ is merely a [[monoidal category]], though $V$-enriched categories still make perfect sense in this case. If $V$ is a [[braided monoidal category]] there are (at least) two ways to define ``$C^{op}$'', resulting in two different ``opposite categories'': we can use either the braiding or the inverse braiding. If $V$ is [[symmetric monoidal category|symmetric]] these two definitions coincide.

The opposite category can be regarded as a [[dual object]] of $C$ in the [[monoidal bicategory]] $V Prof$ of $V$-categories and $V$-[[profunctors]]. (Note that this does not characterize $C^{op}$ up to equivalence, but only up to [[Morita equivalence]], i.e. up to [[Cauchy completion]].) When $V$ is symmetric, then $V Prof$ is also symmetric monoidal, so there is only one notion of dual object. When $V$ is braided, then $V Prof$ is not symmetric and has two notions of dual: a left dual and a right dual. These are exactly the two different opposite categories referred to above (the ``left opposite'' and ``right opposite'').

\hypertarget{in_higher_category_theory}{}\subsubsection*{{In higher category theory}}\label{in_higher_category_theory}

See

\begin{itemize}%
\item [[opposite 2-category]]


\item [[opposite (∞,1)-category]]



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

The [[nerve]] $N(C^{op})$ of $C^{op}$ is the [[simplicial set]] that is degreewise the same as $N(C)$, but in each degree with the order of the face and the order of the degeneracy maps reversed. See [[opposite quasi-category]] for more details.

\hypertarget{classes_of_examples}{}\subsection*{{Classes of examples}}\label{classes_of_examples}

\hypertarget{OppositeGroup}{}\subsubsection*{{Opposite group}}\label{OppositeGroup}

For $G = (S, \cdot)$ a [[group]] (or [[monoid]] or [[associative algebra]], etc.) with product operation

\begin{displaymath}
\cdot_G : S \times S \to S
\end{displaymath}
the \textbf{opposite group} $G^{op}$ is the group whose underlying [[set]] (underlying object, underlying vector space, etc.) is the same as that of $G$

\begin{displaymath}
S^{op} := S
\end{displaymath}
but whose product operation is that of $G$ but combined with a switch of the order of the arguments:

\begin{displaymath}
\cdot_{G^{op}} : S \times S \stackrel{\sigma}{\to} S \times S \stackrel{\cdot_S}{\to} S
  \,.
\end{displaymath}
So for $g,h \in S$ two elements we have that their product in the opposite group is

\begin{displaymath}
g \cdot_{G^{op}} h := h \cdot_{G} g
  \,.
\end{displaymath}
Now, the group $G$ may be thought of as the pointed one-object [[delooping]] [[groupoid]] $\mathbf{B}G$ which is the groupoid with a single object, with $S$ as its set of morphisms, and with $\cdot_G$ its composition operation.

Under this identification of groups with one-object categories, passing to the opposite category corresponds precisely to passing to the opposite group

\begin{displaymath}
(\mathbf{B}G)^{op} = \mathbf{B}(G^{op})
  \,.
\end{displaymath}
\hypertarget{opposite_of_the_opposite}{}\subsubsection*{{Opposite of the opposite}}\label{opposite_of_the_opposite}

The opposite of an opposite category is the original category:

\begin{displaymath}
(C^{op})^{op} = C
  \,.
\end{displaymath}
This is also true for $V$-enriched categories when $V$ is symmetric monoidal, but not when $V$ is merely braided. However, in the latter case we can say $(C^{op1})^{op2} = C = (C^{op2})^{op1}$, i.e. the two different notions of ``opposite category'' are inverse to each other (as is always the case for left and right dualization operations in a non-symmetric monoidal (bi)category).

\hypertarget{coalgebraic_structures}{}\subsubsection*{{Co-algebraic structures}}\label{coalgebraic_structures}

Every algebraic structure in a category, for instance the notion of [[monoid]] in a [[monoidal category]] $C$, has a co-version, where in the original definition the direction of all morphisms is reversed -- for instance the co-version of a monoid is a [[comonoid]]. Of an [[algebra]] its a [[coalgebra]], etc.

One may express this succinctly by saying that a co-structure in $C$ is an original structure in $C^{op}$. For instance a [[comonoid]] in $C$ is a [[monoid]] in $C^{op}$.

\hypertarget{duality_2}{}\subsubsection*{{Duality}}\label{duality_2}

Passing to the opposite category is a realization of abstract [[duality]].

This goes as far as \emph{defining} some entities as objects in an opposite category--in particular, all generalizations of [[geometry]] which characterize [[space and quantity|spaces]] in terms of [[algebra|algebras]]. The idea of [[noncommutative geometry]] is essentially to define a category of \emph{spaces} as the opposite category of a category of algebras. Similarly, a [[locale]] is opposite to a [[frame]].

Are there examples where algebras are defined as dual to spaces?

Another example is the definition of the category of $L_\infty$-[[Lie infinity-algebroid|algebroids]] as that opposite to quasi-free differential graded algebras, identifying every $L_\infty$-algebra with its [[duality|dual]] [[Chevalley-Eilenberg algebra]].

\hypertarget{specific_examples}{}\subsection*{{Specific examples}}\label{specific_examples}

\hypertarget{opposite_of__and_}{}\subsubsection*{{Opposite of $Set$ and $FinSet$}}\label{opposite_of__and_}

The [[power set]]-[[functor]]

\begin{displaymath}
\mathcal{P} \;\colon\; Set^{op} \to Bool
\end{displaymath}
constitutes an [[equivalence of categories]] from the opposite category of [[Set]] to that of [[complete atomic Boolean algebras]]. See at \emph{\href{Set#OppositeCategory}{Set -- Properties -- Opposite category and Boolean algebras}}

Restricted to [[finite sets]] this says that the opposite of the category [[FinSet]] of [[finite sets]] is [[equivalence of categories|equivalent]] to the category of finite [[boolean algebras]]

\begin{displaymath}
FinSet^{op} \simeq FinBoolAlg
  \,.
\end{displaymath}
See at \emph{\href{FinSet#OppositeCategory}{FinSet -- Properties -- Opposite category}}. See at \emph{[[Stone duality]]} for more.

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

\begin{itemize}%
\item [[opposite 2-category]]


\item [[opposite model category]]


\item [[opposite (infinity,1)-category]]



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

For the definition in enriched category theory see page 12 of

\begin{itemize}%
\item [[Max Kelly]], \emph{Basic concepts of enriched category theory} (\href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf}{pdf})

\end{itemize}
[[!redirects opposite categories]] [[!redirects dual category]] [[!redirects dual categories]]



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