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\section*{closed monoidal category}

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

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{symmetric_closed_monoidal_category}{Symmetric closed monoidal category}\dotfill \pageref*{symmetric_closed_monoidal_category} \linebreak
\noindent\hyperlink{cartesian_closed_monoidal_category}{Cartesian closed monoidal category}\dotfill \pageref*{cartesian_closed_monoidal_category} \linebreak
\noindent\hyperlink{left_right_and_biclosed_monoidal_category}{Left-, right- and bi-closed monoidal category}\dotfill \pageref*{left_right_and_biclosed_monoidal_category} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{functor_categories}{Functor categories}\dotfill \pageref*{functor_categories} \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 closed monoidal category $C$ is a [[monoidal category]] that is also a [[closed category]], in a compatible way:

it has for each [[object]] $X$ a [[functor]] $(-) \otimes X : C \to C$ of forming the [[tensor product]] with $X$, as well as a functor $[X,-] : C \to C$ of forming the [[internal-hom]] with $X$, and these form a pair of [[adjoint functor]]s.

The strategy for formalizing the idea of a closed category, that ``the collection of morphisms from $a$ to $b$ can be regarded as an object of $C$ itself'', is to mimic the situation in [[Set]] where for any three objects (sets) $a$, $b$, $c$ we have an isomorphism

\begin{displaymath}
Hom(a \otimes b, c) \simeq Hom(a, [b,c])
  \,,
\end{displaymath}
naturally in all three arguments, where $\otimes = \times$ is the standard [[cartesian product]] of sets. This [[natural isomorphism]] is called [[currying]].

Currying can be read as a characterization of the [[internal hom]] $Hom(b,c)$ and is the basis for the following definition.

A closed monoidal category is a special case of the notion of [[closed pseudomonoid]] in a [[monoidal bicategory]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{symmetric_closed_monoidal_category}{}\subsubsection*{{Symmetric closed monoidal category}}\label{symmetric_closed_monoidal_category}

A [[symmetric monoidal category|symmetric]] [[monoidal category]] $C$ is \textbf{closed} if for all objects $b \in C_0$ the functor $- \otimes b : C \to C$ has a [[adjoint functor|right adjoint]] functor $[b,-] : C \to C$.

This means that for all $a,b,c \in C_0$ we have a [[natural bijection]]

\begin{displaymath}
Hom_C(a \otimes b, c) \simeq Hom_C(a, [b,c])
  \,,
\end{displaymath}
natural in all arguments.

The object $[b,c]$ is called the \textbf{[[internal hom]]} of $b$ and $c$. This is commonly also denoted by lower case $hom(b,c)$ (and then often underlined).

\hypertarget{cartesian_closed_monoidal_category}{}\subsubsection*{{Cartesian closed monoidal category}}\label{cartesian_closed_monoidal_category}

If the monoidal structure of $C$ is [[cartesian monoidal category|cartesian]] (and so in particular [[symmetric monoidal category|symmetric monoidal]]), then $C$ is called \textbf{cartesian closed}. In this case the internal hom is often called an \textbf{[[exponential object]]} and written $c^b$.

\hypertarget{left_right_and_biclosed_monoidal_category}{}\subsubsection*{{Left-, right- and bi-closed monoidal category}}\label{left_right_and_biclosed_monoidal_category}

If $C$ is [[monoidal category|monoidal]] not necessarily [[symmetric monoidal category|symmetric]], then left and right [[tensor product]] $-\otimes b$ and $b\otimes -$ may be non-equivalent functors, and either one or both may have an adjoint. The terminology here is less standard, but many people use \textbf{left closed}, \textbf{right closed}, and \textbf{biclosed} monoidal category to indicate, respectively, that the left tensor product, the right tensor product functors or both have [[right adjoints]]. (Other authors simply say \textbf{closed} instead of biclosed.) So in particular a symmetric closed monoidal category is automatically biclosed.

The analogue of exponential objects for monoidal categories are [[residual|left and right residuals]].

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

\begin{prop}
\label{TensorHomIsoInternalizes}\hypertarget{TensorHomIsoInternalizes}{}
For $(\mathcal{C}, \otimes, 1)$ a closed monoidal category with [[internal hom]] denoted $[-,-]$, then not only are there [[natural bijections]]

\begin{displaymath}
Hom_{\mathcal{C}}(X \otimes Y, Z)
  \simeq
   Hom_{\mathcal{C}}(X, [Y,Z])
\end{displaymath}
but these isomorphisms themselves ``internalize'' to isomorphisms in $\mathcal{C}$ of the form

\begin{displaymath}
[X \otimes Y, Z]
    \simeq
  [X,[Y,Z]]
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the external natural bijections there is for every $A \in \mathcal{C}$ a composite natural bijection

\begin{displaymath}
Hom_{\mathcal{C}}(A, [X \otimes Y, Z])
   \simeq
  Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z)
   \simeq
  Hom_{\mathcal{C}}((A \otimes X) \otimes Y, Z)
    \simeq
  Hom_{\mathcal{C}}(A \otimes X, [Y,Z])
    \simeq
  Hom_{\mathcal{C}}(A,[X,[Y,Z]])
  \,.
\end{displaymath}
Since this holds for every $A \in \mathcal{C}$, the [[Yoneda lemma]] (namely the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]]) implies that there is already an isomorphism

\begin{displaymath}
[X \otimes Y, Z] \simeq [X,[Y,Z]]
  \,.
\end{displaymath}
\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The tautological example is the category [[Set]] of sets with its [[Cartesian product]]: the collection of functions between any two sets is itself a set -- the [[function set]]. More generally, any [[topos]] is cartesian closed monoidal.


\item The category [[Ab]] of [[abelian groups]] with its [[tensor product of abelian groups]] is closed: for any two abelian groups $A, B$ the set of homomorphisms $A \to B$ carries (pointwise defined) abelian group structure.


\item A [[discrete category|discrete]] monoidal category (i.e., a [[monoid]]) is left closed iff it is right closed iff every object has an inverse (i.e., it is a [[group]]).


\item The [[delooping]] $\mathbf{B}M$ of a [[commutative monoid]] $M$ is a closed monoidal (in fact, [[compact closed]]) category with one object, with both the tensor and internal hom defined (on morphisms) using the multiplication operation $f \otimes g = [f,g] = f g$. Conversely, any closed monoidal category with one object must be isomorphic to one constructed from a commutative monoid. (See \hyperlink{EilenbergKelly65}{Eilenberg and Kelly (1965)}, IV.3, p.553.)


\item Certain [[nice category of spaces|nice categories]] of [[topological spaces]] are cartesian closed: for any two nice enough topological spaces $X$, $Y$ the set of continuous maps $X \to Y$ can be equipped with a topology to become a nice topological space itself.


\item Certain nice categories of \emph{[[pointed object|pointed/based]]} topological spaces are closed symmetric monoidal. The monoidal structure is the [[smash product]] and the internal-hom is the set of basepoint-preserving maps with topology induced from the space of unbased ones.


\item The category [[Cat]] is cartesian closed: the internal-hom is the [[functor category]] of functors and natural transformations.


\item The category $2 Cat$ of [[strict 2-category|strict 2-categories]] and strict 2-functors is closed symmetric monoidal under the [[Gray tensor product]]. The internal-hom is the 2-category of strict 2-functors, \emph{pseudo} natural transformations, and modifications.


\item The category of strict $\omega$-[[strict omega-category|categories]] is also biclosed monoidal, under the [[Crans-Gray tensor product]].


\item If $M$ is a monoidal category and $Set^{M^{op}}$ is endowed with the tensor product given by the induced [[Day convolution]] product, then the [[category of presheaves]] $Set^{M^{op}}$ is biclosed monoidal.


\item The category of [[species]], with the monoidal structure given by substitution product of species, is closed monoidal (each functor $- \circ G$ admits a right adjoint) but not biclosed monoidal.


\item The category of [[modules]] over any [[Hopf monoid]] in a closed monoidal category, or more generally algebras for any [[Hopf monad]], is again a closed monoidal category. In particular, the category of modules over any [[group object]] in a [[cartesian closed category]] is (cartesian) closed monoidal. For more on this phenomenon see at \emph{[[Tannaka duality]]}.


\item The category of [[locally convex topological vector spaces]] with the [[inductive tensor product]] and internal hom the space of continuous linear maps with the topology of pointwise convergence is symmetric closed monoidal.



\end{itemize}
\hypertarget{functor_categories}{}\subsubsection*{{Functor categories}}\label{functor_categories}

\begin{theorem}
\label{}\hypertarget{}{}
Let $C$ be a [[complete category|complete]] closed monoidal category and $I$ any [[small category]]. Then the [[functor category]] $[I,C]$ is closed monoidal with the pointwise tensor product, $(F\otimes G)(x) = F(x) \otimes G(x)$.

\end{theorem}
\begin{proof}
Since $C$ is complete, the category $[I,C]$ is [[comonadic functor|comonadic]] over $C^{ob I}$; the comonad is defined by right [[Kan extension]] along the inclusion $ob I \hookrightarrow I$. Now for any $F\in [I,C]$, consider the following square:

\begin{displaymath}
\itexarray{[I,C] & \overset{F\otimes - }{\to} & [I,C] \\
  \downarrow && \downarrow\\
  C^{ob I}& \underset{F_0 \otimes -}{\to} & C^{ob I}}
\end{displaymath}
This commutes because the tensor product in $[I,C]$ is pointwise (here $F_0$ means the family of objects $F(x)$ in $C^{ob I}$). Since $C$ is closed, $F_0 \otimes -$ has a right adjoint. Since the vertical functors are comonadic, the (dual of the) [[adjoint lifting theorem]] implies that $F\otimes -$ has a right adjoint as well.

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

\begin{itemize}%
\item [[monoidal category]], [[monoidal (∞,1)-category]]


\item [[symmetric monoidal category]], [[symmetric monoidal (∞,1)-category]]


\item \textbf{closed monoidal category} , [[closed monoidal (∞,1)-category]]

\begin{itemize}%
\item [[closed monoidal functor]]


\item [[indexed closed monoidal category]]



\end{itemize}

\item [[string diagram]], [[Kelly-Mac Lane diagram]], [[linguistics|natural language syntax]]



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

Textbook account for symmetric closed categories:

\begin{itemize}%
\item [[Francis Borceux]], vol 2 section 6.1 of \emph{[[Handbook of Categorical Algebra]]}, Cambridge University Press (1994)

\end{itemize}
Original articles studying monoidal biclosed categories are

\begin{itemize}%
\item [[Joachim Lambek]], \emph{Deductive systems and categories}, Mathematical Systems Theory 2 (1968), 287-318.


\item [[Joachim Lambek]], \emph{Deductive systems and categories II}, Lecture Notes in Math. 86, Springer-Verlag (1969), 76-122.



\end{itemize}
For more historical development see at \emph{\href{linear+type+theory#HistoryCategoricalSemantics}{linear type theory -- History of linear categorical semantics}}.

In [[enriched category theory]] the enriching category is taken to be closed monoidal. Accordingly the standard textbook on enriched category theory

\begin{itemize}%
\item [[Max Kelly]], \emph{Basic concepts of enriched category theory}, section 1.5, (\href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html}{tac})

\end{itemize}
has a chapter on just closed monoidal categories.

See also the article

\begin{itemize}%
\item [[Samuel Eilenberg]] and [[Max Kelly]], \emph{Closed categories}. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965).

\end{itemize}
on the concept of [[closed categories]].

[[!redirects closed monoidal category]] [[!redirects closed monoidal categories]] [[!redirects monoidal closed category]] [[!redirects monoidal closed categories]] [[!redirects closed monoidal structure]] [[!redirects closed monoidal structures]]

[[!redirects closed symmetric monoidal category]] [[!redirects closed symmetric monoidal categories]] [[!redirects symmetric monoidal closed category]] [[!redirects symmetric monoidal closed categories]] [[!redirects closed symmetric monoidal structure]] [[!redirects closed symmetric monoidal structures]]

[[!redirects biclosed monoidal category]] [[!redirects biclosed monoidal categories]] [[!redirects monoidal biclosed category]] [[!redirects monoidal biclosed categories]] [[!redirects left closed monoidal category]] [[!redirects left closed monoidal categories]] [[!redirects right closed monoidal category]] [[!redirects right closed monoidal categories]]



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