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\begin{document}

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\section*{tensor product}

\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{in_a_multicategory}{In a multicategory}\dotfill \pageref*{in_a_multicategory} \linebreak
\noindent\hyperlink{in_terms_of_heteromorphisms}{In terms of heteromorphisms}\dotfill \pageref*{in_terms_of_heteromorphisms} \linebreak
\noindent\hyperlink{of_modules_in_a_monoidal_category}{Of modules in a monoidal category}\dotfill \pageref*{of_modules_in_a_monoidal_category} \linebreak
\noindent\hyperlink{of_modules_in_a_bicategory}{Of modules in a bicategory}\dotfill \pageref*{of_modules_in_a_bicategory} \linebreak
\noindent\hyperlink{in_a_virtual_double_category}{In a virtual double category}\dotfill \pageref*{in_a_virtual_double_category} \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}

The term \textbf{tensor product} has many different but closely related meanings.

\begin{itemize}%
\item In its original sense a \emph{tensor product} is a representing object for a suitable sort of \emph{[[bilinear map]]} and \emph{[[multilinear map]]}. The most classical versions are for [[vector spaces]] ([[modules]] over a [[field]]), and more generally [[modules]] over a [[ring]]. In modern language this takes place in a [[multicategory]].


\item Consequently, the [[functor]] $\otimes : C \times C \to C$ which is part of the data of any [[monoidal category]] $C$ is also often called a \emph{tensor product}, since in many examples of monoidal categories it is induced from a tensor product in the above sense (and in fact, \emph{any} monoidal category underlies a multicategory in a canonical way). In parts of the literature (certain) [[abelian category|abelian]] monoidal categories are even addressed as \emph{tensor categories}.


\item Given two objects in a [[monoidal category]] $(C,\otimes)$ with a right and left [[action]], respectively, of some [[monoid]] $A$, their \emph{tensor product over $A$} is the quotient of their tensor product in $C$ by this action. If $A$ is commutative, then this is a special case of the tensor product in a multicategory.


\item This generalizes to modules over [[monad|monads]] in a bicategory, which includes the notion of \emph{tensor product of functors}.


\item Finally, tensor products in a multicategory and tensor products over monads in a bicategory are both special cases of tensor products in an [[virtual double category]].



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

\hypertarget{in_a_multicategory}{}\subsubsection*{{In a multicategory}}\label{in_a_multicategory}

\begin{defn}
\label{}\hypertarget{}{}
For $M$ a [[multicategory]] and $A$ and $B$ [[objects]] in $M$, the \textbf{tensor product} $A \otimes B$ is defined to be an object equipped with a [[universal construction|universal]] [[multimorphism]] $A,B\to A \otimes B$ in that any multimorphism $A,B\to C$ factors uniquely through $A,B\to A \otimes B$ via a (1-ary) morphism $A \otimes B\to C$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
$M$ is the category [[Ab]] of abelian groups, made into a [[multicategory]] using [[multilinear maps]] as the multimorphisms, then we get the usual [[tensor product of abelian groups]]. That is, $A \otimes B$ is equipped with a universal map from $A \times B$ (as a set) to $C$ such that this map is linear (a group homomorphism) in each argument separately. This tensor product can also be constructed explicitly by

\begin{enumerate}%
\item starting with the cartesian product $A\times B$ in sets,
\item generating a \emph{free} abelian group from it, and then
\item quotienting by relations $(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$ and $(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$. (The 0-ary relations $(0,b)\sim 0$ and $(a,0)\sim 0$ follow automatically; you need them explicitly if you generalise to [[abelian monoid]]s.)

\end{enumerate}
\end{example}
Note that in this case, $A\otimes B$ is not a [[subobject]] \emph{or} a [[quotient]] of the [[cartesian product]] $A\times B$. However, in many other cases the tensor product in a multicategory \emph{can} be obtained as a quotient of some other pre-existing product; see \emph{[[tensor product of modules]]} below.

Other examples of tensor products in multicategories:

\begin{example}
\label{}\hypertarget{}{}
The [[Gray tensor product]] of [[strict 2-category|strict 2-categories]] is a tensor product in the multicategory of 2-categories and [[cubical functor]]s. Likewise for Sjoerd Crans' tensor product of Gray-categories.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
In particular, any [[closed category]] (even if not monoidal) has an underlying multicategory. Tensor products in this multicategory are characterized by the adjointness relation

\begin{displaymath}
\hom(A\otimes B, C) \cong \hom(A, \hom(B,C)).
\end{displaymath}
This may be the oldest notion of tensor product, since the definition of the internal-hom of abelian groups and vector spaces, unlike that of their tensor product, is intuitively obvious.

\end{example}
While the universal property referred to above (every bilinear map $A,B\to C$ factors uniquely through $A,B\to A\otimes B$ via a map $A\otimes B \to C$) suffices to define the tensor product, it does not suffice to prove that it is associative and unital. For this we need the stronger property that any multilinear map $D_1,\dots,D_m,A,B,E_1,\dots, E_n \to C$ factors uniquely through $A,B\to A\otimes B$ via a multilinear map $D_1,\dots,D_m, A\otimes B ,E_1,\dots, E_n \to C$.

\hypertarget{in_terms_of_heteromorphisms}{}\subsubsection*{{In terms of heteromorphisms}}\label{in_terms_of_heteromorphisms}

An alternative approach is to define the tensor product via an inter-categorical universal property involving [[heteromorphism|heteromorphisms]]. Tensor products do not always arise via an adjunction, but we can observe that $hom (a \otimes b, c) \simeq het (\langle a, b \rangle, c)$ in general. That is to say, any morphism from $a \otimes b$ to $c$ in some category $C$ corresponds to a heteromorphism from $\langle a, b \rangle$ in $C \times C$ to $c$ in $C$. In other words, the tensor product is a left representation of $het (\langle a, b \rangle, c)$.

When tensor products exist, we have a canonical het $\eta_{\langle a, b \rangle} \colon \langle a, b \rangle \to a \otimes b$ from $id_{a \otimes b} \in hom (a \otimes b, a \otimes b)$. Given another het $\phi \colon \langle a, b \rangle \to c$, we get the following commutative diagram.

\begin{displaymath}
\begin{array}{cccC}
                                                 & {\langle a, b \rangle} &  &   & \\
          \eta_{\langle a, b \rangle} & \downarrow           & \overset{\phi}\searrow       &     & \\
                                                 & a \otimes b            & \underset{f}\dashrightarrow & c &  \\ \end{array}
\end{displaymath}
This represents an example of a more general method for translating universal properties in multicategories into ones involving heteromorphisms.

\hypertarget{of_modules_in_a_monoidal_category}{}\subsubsection*{{Of modules in a monoidal category}}\label{of_modules_in_a_monoidal_category}

Let $R$ be a [[commutative ring]] and consider the multicategory $R$[[Mod]] of $R$-[[modules]] and $R$-[[multilinear maps]]. In this case the [[tensor product of modules]] $A\otimes_R B$ of $R$-modules $A$ and $B$ can be constructed as the [[quotient]] of the [[tensor product of abelian groups]] $A\otimes B$ underlying them by the [[action]] of $R$; that is,

\begin{displaymath}
A\otimes_R B = A\otimes B / (a,r\cdot b) \sim (a\cdot r,b).
\end{displaymath}
More category-theoretically, this can be constructed as the [[coequalizer]] of the two maps

\begin{displaymath}
A\otimes R \otimes B \;\rightrightarrows\; A\otimes B
\end{displaymath}
given by the action of $R$ on $A$ and on $B$.

If $R$ is a [[field]], then $R$-modules are vector spaces; this gives probably the most familiar case of a tensor product spaces, which is also probably the situation where the concept was first conceived.

This tensor product can be generalized to the case when $R$ is not commutative, as long as $A$ is a right $R$-module and $B$ is a left $R$-module. More generally yet, if $R$ is a [[monoid]] in any [[monoidal category]] (a ring being a monoid in [[Ab]] with its tensor product), we can define the tensor product of a left and a right $R$-module in an analogous way. If $R$ is a commutative monoid in a [[symmetric monoidal category]], so that left and right $R$-modules coincide, then $A\otimes_R B$ is again an $R$-module, while if $R$ is not commutative then $A\otimes_R B$ will no longer be an $R$-module of any sort.

\begin{itemize}%
\item Not all tensor products in multicategories are instances of this construction. In particular, the tensor product in [[Ab]] is not the tensor product of modules over any monoid in the cartesian monoidal category [[Set]]. Abelian groups \emph{can} be considered as ``sets with an action by something,'' but that something is more complicated than a monoid: it is a special sort of [[monad]] called a [[commutative theory]].


\item Conversely, if $R$ is a \emph{commutative} monoid in a symmetric monoidal category, there is a multicategory of $R$-modules whose tensor product agrees with the coequalizer defined above, but if $R$ is not commutative this is impossible. However, see the section on tensor products in virtual double categories, below.



\end{itemize}
\hypertarget{of_modules_in_a_bicategory}{}\subsubsection*{{Of modules in a bicategory}}\label{of_modules_in_a_bicategory}

The tensor product of left and right modules over a noncommutative monoid in a monoidal category is a special case of the tensor product of modules for a [[monad]] in a [[bicategory]]. If $R: x\to x$ is a monad in a bicategory $B$, a right $R$-module is a 1-cell $A: y\to x$ with an action by $R$, a left $R$-module is a 1-cell $B: x\to z$ with an action by $R$, and their tensor product, if it exists, is a 1-cell $y\to z$ given by a similar coequalizer. Regarding a monoidal category as a 1-object bicategory, this recovers the above definition.

For example, consider the bicategory $V-Mat$ of $V$-valued [[matrix|matrices]] for some monoidal category $V$. A monad in $V-Mat$ is a $V$-[[enriched category]] $A$, an $(A,I)$-bimodule is a functor $A\to V$, an $(I,A)$-bimodule is a functor $A^{op}\to V$, and their tensor product in $V-Mat$ is a classical construction called the \textbf{tensor product of functors}. It can also be defined as a [[end|coend]].

\hypertarget{in_a_virtual_double_category}{}\subsubsection*{{In a virtual double category}}\label{in_a_virtual_double_category}

A [[virtual double category]] is a common generalization of a multicategory and a bicategory (and actually of a [[double category]]). Among other things, it has objects, 1-cells, and ``multi-2-cells.'' We leave it to the reader to define a notion of tensor product of 1-cells in such a context, analogous to the tensor product of objects in a multicategory. A multicategory can be regarded as a 1-object virtual double category, so this generalizes the notion of tensor product in a multicategory.

On the other hand, in any bicategory (in fact, any double category) there is a virtual double category whose objects are monads and whose 1-cells are bimodules, and the tensor product in this virtual double category is the tensor product of modules in a bicategory defined above. Thus, tensor products in a virtual double category include all notions of tensor product discussed above.

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

\begin{itemize}%
\item [[tensor product of abelian groups]]


\item [[tensor product of modules]]


\item [[tensor product of vector spaces]]

\begin{itemize}%
\item [[inductive tensor product]]

\end{itemize}

\item [[tensor product of representations]]


\item [[tensor product of vector bundles]]


\item [[tensor product of algebras]]


\item [[tensor product of chain complexes]]


\item [[tensor product of Banach spaces]]


\item [[tensor product of functors]]


\item [[Deligne tensor product of abelian categories]]


\item [[tensor product of presentable (infinity,1)-categories]]


\item in terms of [[linear type theory]] the tensor product is the [[categorical semantics]] of the [[multiplicative conjunction]].



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

\begin{itemize}%
\item [[tensor calculus]]


\item [[cartesian product]]


\item [[external tensor product]]


\item [[internal hom]]


\item [[composite system]]


\item [[cotensor product]]



\end{itemize}
[[!include homotopy-homology-cohomology]]

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

Exposition of the tensor product of $R$-[[modules]] is for instance in

\begin{itemize}%
\item Keith Conrad, \emph{Tensor products} (\href{http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf}{pdf})

\end{itemize}
A characterization for tensor products of $R$-modules in terms of heteromorphisms appears in section 9 of.

\begin{itemize}%
\item David Ellerman, \emph{Mac Lane, Bourbaki, and Adjoints: A Heteromorphic Retrospective} (\href{http://www.ellerman.org/wp-content/uploads/2015/06/Maclane-Bourbaki-Redux.pdf}{pdf})

\end{itemize}
[[!redirects tensor products]]



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