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\begin{document}

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\section*{premonoidal category}

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

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

[[!include monoidal categories - contents]]

\hypertarget{premonoidal_categories}{}\section*{{Premonoidal categories}}\label{premonoidal_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{slick_version}{Slick version}\dotfill \pageref*{slick_version} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A premonoidal category is a generalisation of a [[monoidal category]], applied by [[John Power]] and his collaborators to [[denotational semantics]] in [[computer science]]. There, the [[Kleisli category]] of a [[strong monad]] provides a model of [[call-by-value]] programming languages. In general, if the original category is monoidal, the Kleisli category will only be premonoidal.

Recall that a [[bifunctor]] from $C$ and $D$ to $E$ (for $C,D,E$ [[categories]]) is simply a [[functor]] to $E$ from the [[product category]] $C \times D$. We can think of this as an operation which is `jointly functorial'. But just as a [[function]] to $X$ from $Y$ and $Z$ (for $X,Y,Z$ [[topological spaces]]) may be [[continuous map|continuous]] in each variable yet not [[jointly continuous function|jointly continuous]] (continuous from the [[Tychonoff product]] $Y \times Z$), so an operation between categories can be functorial in each variable separately yet not jointly functorial.

Recall that a [[monoidal category]] is a [[category]] $C$ equipped with a bifunctor $C \times C \to C$ (equipped with [[extra structure]] such as the [[associator]]). Similarly, a premonoidal category is a category equipped with an operation $C \times C \to C$, which is (at least) a [[function]] on [[objects]] as shown, but one which is functorial only in each variable separately.

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

A \textbf{binoidal category} is a [[category]] $C$ equipped with

\begin{itemize}%
\item for each pair $x,y$ of [[objects]] of $C$, an object $x \otimes y$;
\item for each object $x$ a [[functor]] $x \rtimes -$ whose action on objects sends $y$ to $x \otimes y$
\item for each object $x$ a [[functor]] $- \ltimes x$ whose action on objects sends $y$ to $y \otimes x$

\end{itemize}
A [[morphism]] $f\colon x \to y$ in a binoidal category is \textbf{central} if, for every morphism $f'\colon x' \to y'$, the diagrams

\begin{displaymath}
\array {
   x \otimes x'                      & \overset{x \rtimes f'}\to  & x \otimes y' \\
   \mathllap{f \ltimes x'}\downarrow &                            & \downarrow\mathrlap{f \ltimes y'} \\
   y \otimes x'                      & \underset{y \rtimes f'}\to & y \otimes y' \\
   }
\end{displaymath}
and

\begin{displaymath}
\array {
   x' \otimes x                      & \overset{x' \rtimes f}\to  & x' \otimes y \\
   \mathllap{f' \ltimes x}\downarrow &                            & \downarrow\mathrlap{f' \ltimes y} \\
   y' \otimes x                      & \underset{y' \rtimes f}\to & y' \otimes y \\
   }
\end{displaymath}
commute. In this case, we denote the common composites $f \otimes f'\colon x \otimes x' \to y \otimes y'$ and $f' \otimes f\colon x' \otimes x \to y' \otimes y$.

A \textbf{premonoidal category} is a binoidal category equipped with:

\begin{itemize}%
\item an object $I$;
\item for each triple $x,y,z$ of objects, a central [[isomorphism]] $\alpha_{x,y,z}\colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$; and
\item for each object $x$, central isomorphisms $\lambda_x\colon x \otimes I \to x$ and $\rho_x\colon I \otimes x \to x$;

\end{itemize}
such that the following conditions hold.

\begin{itemize}%
\item all possible [[natural transformation|naturality squares]] for $\alpha$, $\lambda$, and $\rho$ (which make sense since we have central morphisms) commute. Note that when written out explicitly in terms of the functors $x\rtimes -$ and $-\ltimes x$, we need three different naturality squares for $\alpha$. (But it is possible to rephrase $\alpha$ as a single natural transformation using the slick version below.)
\item the pentagon law holds for $\alpha$, as in a [[monoidal category]].
\item the triangle law holds for $\alpha$, $\lambda$, and $\rho$, as in a monoidal category.

\end{itemize}
A \textbf{strict premonoidal category} is a premonoidal category in which $(x \otimes y) \otimes z = x \otimes (y \otimes z)$, $x \otimes I = x$, and $I \otimes x = x$, and in which $\alpha_{x,y,z}$, $\lambda_x$, and $\rho_x$ are all [[identity morphisms]]. (We need the underlying category $C$ to be a [[strict category]] for this to make sense.)

Similarly, a \textbf{symmetric premonoidal category} is a premonoidal category equipped with a central natural isomorphism $x\otimes y \cong y\otimes x$ (as for $\alpha$, there are two naturality squares unless we use the slick approach), satisfying the usual axioms of a symmetry.

\hypertarget{slick_version}{}\subsection*{{Slick version}}\label{slick_version}

As a [[strict monoidal category]] is a [[monoid]] in the [[cartesian monoidal category]] [[Cat]], so a strict premonoidal category is a monoid in the [[symmetric monoidal category]] $(Cat,\Box)$, where $\Box$ is the [[funny tensor product]].

From this point of view, a binoidal category is just a category $C$ with a functor $C \Box C \to C$

It may be possible to make $(Cat,\Box)$ a [[symmetric monoidal 2-category]], in which a [[pseudomonoid]] object is precisely a non-strict premonoidal category, but if so, nobody seems to have written this up yet. It is possible, however, to describe part of the structure of a non-strict premonoidal category in terms of $(Cat,\Box)$. For instance, a binoidal structure on $C$ is precisely a functor $C\Box C \to C$, and the naturality of the associator $\alpha$ can be expressed by saying that it is a natural transformation (with central components) between functors $C\Box C\Box C \to C$.

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

\begin{itemize}%
\item Every [[monoidal category]] is a premonoidal category.


\item If $T$ is a [[strong monad]] on a monoidal category $C$, then the [[Kleisli category]] $C_T$ of $T$ inherits a premonoidal structure, such that the functor $C\to C_T$ is a strict premonoidal functor. This premonoidal structure is only a monoidal structure if $T$ is a [[commutative monad]].


\item A strict premonoidal category is the same as a [[sesquicategory]] with one object, so any object of a sesquicategory has a corresponding premonoidal category whose objects are endomorphisms and arrows are 2-cells.



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

The central morphisms of a premonoidal category $C$ form a [[subcategory]] $Z(C)$, called the \textbf{centre} of $C$, which is a [[monoidal category]]. This defines a [[right adjoint functor]] to the inclusion $MonCat \hookrightarrow PreMonCat$ using the definition of functor of premonoidal categories in \hyperlink{PR97}{Power-Robinson 97}.

In the same way that a (strict) monoidal category can be identified with a (strict) [[2-category]] with one object, a strict premonoidal category can be identified with a [[sesquicategory]] with one object. In fact, a sesquicategory is precisely a category [[enriched category|enriched]] over the monoidal category $(Cat,\otimes)$ described above.

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

\begin{itemize}%
\item [[John Power]] and [[Edmund Robinson]], \emph{Premonoidal categories and notions of computation}, Math. Structures Comput. Sci., 7(5):453--468, 1997. Logic, domains, and programming languages (Darmstadt, 1995). \href{http://www.eecs.qmul.ac.uk/~edmundr/pubs/mscs97/premoncat.ps}{PostScript}


\item Alan Jeffrey, \emph{Premonoidal categories and a graphical view of programs,} \href{http://fpl.cs.depaul.edu/ajeffrey/papers/premonA.pdf}{pdf file}



\end{itemize}
[[!redirects premonoidal category]] [[!redirects premonoidal categories]] [[!redirects binoidal category]] [[!redirects binoidal categories]]



\end{document}