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\begin{document}

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\section*{promonoidal category}

\hypertarget{promonoidal_categories}{}\section*{{Promonoidal categories}}\label{promonoidal_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{versus_monoidal_categories}{Versus monoidal categories}\dotfill \pageref*{versus_monoidal_categories} \linebreak
\noindent\hyperlink{day_convolution}{Day convolution}\dotfill \pageref*{day_convolution} \linebreak
\noindent\hyperlink{versus_multicategories}{Versus multicategories}\dotfill \pageref*{versus_multicategories} \linebreak
\noindent\hyperlink{notes}{Notes}\dotfill \pageref*{notes} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{promonoidal category} is like a [[monoidal category]] in whose structure (namely, tensor product and unit object) we have replaced [[functors]] by [[profunctors]]. It is a categorification of the idea of a [[boolean algebra]].

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

A \textbf{promonoidal category} is a [[pseudomonoid]] in the [[monoidal bicategory]] [[Prof]]. This means that it is a [[category]] $A$ together with

\begin{itemize}%
\item A profunctor $P \colon A\times A &#8696; A$.
\item A profunctor $J\colon 1$  $A$.
\item Associativity and unit isomorphisms $P \odot (P\times 1) \cong P\odot (1\times P)$, $P\odot (J\times 1) \cong 1$, and $P\odot (1\times J) \cong 1$.
\item The usual pentagon and unit conditions hold, as in a [[monoidal category]].

\end{itemize}
Recalling that a profunctor $A$  $B$ is defined to be a functor $B^{op}\times A \to Set$, we can make this more explicit. We can also generalize it by replacing $Set$ by a [[Benabou cosmos]] $V$ and $A$ by a $V$-[[enriched category]]; then a profunctor is a $V$-functor $B^{op}\times A \to V$.

Thus, we obtain the following as an explicit definition of \textbf{promonoidal $V$-category}: we have the following data

\begin{enumerate}%
\item A $V$-category $A$.


\item A $3$-ary functor $P:A^\op \otimes A \otimes A\to V$. For notational clarity, we may write $P(a,b,c)$ as $P(a,b \diamond c)$.


\item A $V$-functor $J:A^{op}\to V$.



\end{enumerate}
and natural isomorphisms

\begin{enumerate}%
\item $\lambda_{ab}:\int^x (J(x) \otimes P(b,a \diamond x))\to A(b,a)$


\item $\rho_{ab}: \int^x ( J(x)\otimes P(b,x \diamond a))\to A(b,a)$


\item $\alpha_{abcd}: \int^x (P(x,a\diamond b)\otimes P(d,x\diamond c)) \to \int^x(P(x,b\diamond c)\otimes P(d,a\diamond x))$



\end{enumerate}
satisfying the pentagon and unit axioms.

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

\hypertarget{versus_monoidal_categories}{}\subsubsection*{{Versus monoidal categories}}\label{versus_monoidal_categories}

Since any functor induces a representable profunctor, any monoidal category can be regarded as a promonoidal category. A given promonoidal category arises in this way if and only if the profunctors $P$ and $J$ are representable.

\hypertarget{day_convolution}{}\subsubsection*{{Day convolution}}\label{day_convolution}

A promonoidal structure on $A$ suffices to induce a monoidal structure on $V^{A^{op}}$ by [[Day convolution]]. In fact, given a small $V$-category $A$, there is an equivalence of categories between

\begin{enumerate}%
\item the category of pro-monoidal structures on $A$, with strong pro-monoidal functors between them, and


\item the category of biclosed monoidal structures on $V^{A^{op}}$, with strong monoidal functors between them.



\end{enumerate}
\hypertarget{versus_multicategories}{}\subsubsection*{{Versus multicategories}}\label{versus_multicategories}

A promonoidal structure on $A$ can be identified with a particular sort of [[multicategory]] structure on $A^{op}$, i.e. with a co-multicategory structure on $A$. The set $P(x, y, z)$ is regarded as the set of co-multimorphisms $x \to (y,z)$.

More generally, we define a co-multicategory $\bar A$ as follows. The objects of $\bar A$ are the objects of $A$. The co-multimorphisms $b\to a_1\dots a_n$ in $\bar A$ are defined by induction on $n$ as follows: $\bar A(b;)=Jb$, and $\bar A(b;a_1,\dots,a_{n+1})=\int^x\bar A(x;a_1,\dots,a_n)\otimes P(b,x\diamond a_{n+1})$.

Not every co-multicategory arises from a promonoidal one in this way. Roughly, a promonoidal category is a co-multicategory whose $n$-ary co-multimorphisms are determined by the binary, unary, and nullary morphisms. In general, co-multicategories can be identified with a certain sort of ``lax promonidal category''.

\hypertarget{notes}{}\subsection*{{Notes}}\label{notes}

[[Brian Day]] introduced the notion of a ``premonoidal'' category in \hyperlink{Day70}{(Day 1970)}, and later renamed this to a ``promonoidal'' category in \hyperlink{Day74}{(Day 1974)} while reformulating the identity and associativity isomorphisms $\lambda,\rho,\alpha$ explicitly in terms of profunctor composition. However, note that his definition is op'd from the definition used in this article, in the sense that a Day-promonoidal structure on a category $C$ corresponds to a pseudomonoid structure on $C^{op}$ in [[Prof]]. In particular, one example Day considers is that of a [[closed category]], which is actually a co-promonoidal category in the sense used here (analogous to the co-promonoidal structure on a multicategory described above).

Regarding monoidal categories as promonoidal is useful in order to express extra structure on them, such as [[closed monoidal category|closedness]], [[star-autonomous category|$\ast$-autonomy]], or [[compact closed category|compact closedness]], in abstract bicategorical terms: these notions can be defined by adding extra structure to a [[pseudomonoid]] in the [[monoidal bicategory]] [[Prof]] (i.e. a promonoidal category), but the extra structure does not lie inside the sub-monoidal bicategory [[Cat]].

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[monoidal category]], [[multicategory]]
\item [[Day convolution]]
\item A (co-)promonoidal [[poset]] is called a [[ternary frame]], when its Day convolution monoidal category is used to model [[substructural logic]].

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

\begin{itemize}%
\item [[Brian Day]], On closed categories of functors, \emph{Lecture Notes in Mathematics} 137 (1970), 1-38.


\item [[Brian Day]], An embedding theorem for closed categories, \emph{Lecture Notes in Mathematics} 420 (1974), 55-64.


\item Day, Panchadcharam and Street, \emph{On centres and lax centres for promonoidal categories}.



\end{itemize}
[[!redirects promonoidal categories]] [[!redirects pro-monoidal category]] [[!redirects pro-monoidal categories]] [[!redirects promonoidal structure]] [[!redirects pro-monoidal structure]] [[!redirects promonoidal structures]] [[!redirects pro-monoidal structures]]



\end{document}