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\section*{compact double category}

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

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

[[!include monoidal categories - contents]]

[[Mike Shulman]]: Original research alert.

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{extraordinary_2cells}{Extraordinary 2-cells}\dotfill \pageref*{extraordinary_2cells} \linebreak
\noindent\hyperlink{structures_in_a_compact_double_category}{Structures in a compact double category}\dotfill \pageref*{structures_in_a_compact_double_category} \linebreak
\noindent\hyperlink{representability}{Representability}\dotfill \pageref*{representability} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{compact double category} is a (usually pseudo) [[double category]] which is [[monoidal double category|symmetric monoidal]], and in which every object has an assigned [[dualizable object|dual]] in the ``proarrow'' direction (as in a [[compact closed category]]). The basic example is the double category $\underline{Cat}$ of categories, functors, and [[profunctors]], in which the dual of a category is its [[opposite category]].

On this page, we will draw the ``functor'' direction horizontally and the ``profunctor'' direction vertically. Thus, in a compact double category, every object has a vertical dual.

Note that if a compact double category is a [[proarrow equipment]], then its [[vertical bicategory]] is a [[symmetric monoidal bicategory]], which is moreover an [[autonomous bicategory]]. However, saying that a double category is compact says more than this: it says that the vertical duals are assigned in a horizontally functorial way. Just as duals in a monoidal category are characterized up to isomorphism, duals in an autonomous bicategory are characterized up to equivalence, which in this case would mean \emph{vertical} equivalence. But equivalence in the bicategory $Prof$ of categories and profunctors is a weaker notion than [[equivalence of categories]], so just saying that $C^{op}$ is a dual of $C$ in $Prof$ does not characterize $C^{op}$ up to equivalence, only up to [[Morita equivalence]] (i.e. equivalence of [[Cauchy completion]]s). The extra structure (not merely extra properties) of the compact double category $\underline{Cat}$ includes the specification of $C^{op}$ up to equivalence (i.e., horizontal equivalence), along with all the attendant structure.

Likewise, a \textbf{compact virtual double category} is a [[virtual double category]] which is compact in a similar sense: it has a monoidal structure and functorially assigned duals. Note that in order to match with our conventions on this page, the role of vertical and horizontal cells in a virtual double category is flipped from the choices made at [[virtual double category]], and the cells must be transposed and have their multi-sources on the left, rather than the top.

By contrast, in a \textbf{virtually compact virtual double category}, the monoidal and compact structure has also been ``virtualized'': rather than coming equipped with a horizontally functorial tensor product, in such a virtual double category there are also horizontal arrows with multi-sources that are finite lists of objects with variance, such as $(A,B^{op},C)$. We can also have a \textbf{virtually compact double category} which is an honest double category, but whose monoidal and compact structure is only virtual.

Finally, if a compact double category or virtual category is a [[proarrow equipment]] or a [[virtual equipment]], we of course call it a \textbf{compact (virtual) proarrow equipment}.

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

Let $T$ be the ``free category'' monad on the [[virtual equipment]] $Span(Quiv)$ of [[spans]] in the category of [[quivers]], and let $T' = Mod(T)$ be the resulting monad on the virtual equipment $Mod(Span(Quiv)) = Cat(Quiv)$ whose objects are internal quivers in $Cat$, or equivalently internal categories in $Quiv$. Then a (pseudo) $T'$-algebra is precisely a (pseudo) double category, while a [[virtual T-algebra]] is precisely a virtual double category. In particular, the virtual equipment $Vdc = nKMod(Cat(Quiv),T')$ consists of virtual double categories, functors, and profunctors between them.

Let $G$ be the [[pseudomonad]] on $Cat(Quiv)$ defined as follows. We may consider an object of $Cat(Quiv)$ as like a double category, but with only horizontal composition: there are no vertical composites or identities. If $C$ is such, then:

\begin{itemize}%
\item The objects of $G(C)$ are finite ordered lists of objects of $C$ with variance, such as $(x,y^{op},z,x^{op})$. Of course, we include the empty list.


\item There can exist a horizontal arrow of $G(C)$ from one list $\vec{x}$ to another $\vec{y}$ only if the two lists have the same length, say $n$. In this case, such an arrow is given by a permutation $\sigma \in S_n$ such that the variance of $x_i$ matches the variance of $y_{\sigma (i)}$, together with for each $i$ a horizontal arrow $x_i \to y_{\sigma(i)}$ in $C$. Composition of these is defined in the evident way using composition in $C$ and multiplying permutations.


\item A vertical arrow from $\vec{x}$ to $\vec{y}$ in $G(C)$ is given by a ``graph'' from $\vec{x}$ to $\vec{y}$ labeled by vertical arrows of $C$, together with an ordered list of endo-vertical-arrows of $C$ (called ``loops''). To be precise, by such a ``graph'' we mean a fixed-point-free involution of $\vec{x} \sqcup \vec{y}^{op}$ which reverses variance. Here $\vec{y}^{op}$ means reverse the variance all through $\vec{y}$; thus if $x_i$ is matched with $y_j$, they must have the \emph{same} variance, while if $x_i$ is matched with $x_j$, or $y_i$ with $y_j$, they must have opposite variances.

To define a vertical arrow in $G(C)$, together with such a graph we also require, for each matched pair, a vertical arrow in $C$, according to the following rules. If $x_i$ is matched with $y_j$ and neither is ``opped,'' then we require a vertical arrow $x_i\to y_j$, while if both are ``opped,'' we require instead a vertical arrow $y_j\to x_i$. And if $x_i$ is matched with $x_j^{op}$, we require a vertical arrow $x_i\to x_j$, while if $y_i$ is matched with $y_j^{op}$, we require a vertical arrow $y_j\to y_i$. (And, in addition to all this, we also require an ordered list of loops.)


\item The squares in $G(C)$ are defined in a straightforward way, incorporating two graphs which are related by a pair of permutations, and a collection of labeling squares from $C$ with appropriately chosen boundaries.



\end{itemize}
With this definition, $G$ is evidently an endofunctor of the category of $Cat$-quivers. We can extend it to the virtual equipment $Cat(Quiv)$ in a straightforward way, mimicking the definition above for horizontal arrows and squares to define it on the proarrows (which are like [[double profunctors]] but, again, without vertical composites).

Note that the action of $G$ on vertical arrows is very much like that of the monad on $Cat$ whose algebras are [[compact closed categories]]. The main difference is that in the latter, rather than an \emph{ordered} list of loops (an element of the [[free monoid]] on endomorphims), we have an element of the free \emph{commutative} monoid on the endomorphisms. This change is what will make $G$ be only a pseudomonad, rather than a monad, but it is also crucial for the applications.

We next extend $G$ to a pseudomonad in the [[Gray-category]] of [[virtual equipments]]. The unit $C\to G(C)$ is easy to define and strictly natural: an object $x$ goes to the unary list $(x)$, horizontal arrows are labeled with the unique permutation of one element, and vertical arrows are labeled with the unique graph between two such unary lists.

The multiplication is somewhat trickier\ldots{}.

The multiplication defined in this way is still strictly natural, and it satisfies the laws relating it to the unit transformation strictly, but its ``associativity'' law is only satisfied up to isomorphism, making $G$ into a fairly strict sort of pseudomonad.

We now claim that there is a [[distributive law]] relating $T'$ to $G$, and therefore $G$ has the structure of a pseudomonad on the object $(Cat(Quiv),T')$ in the category of monads-on-virtual-equipments\ldots{}.

It follows by the functoriality of the construction of [[generalized multicategories]] that $G$ induces a pseudomonad $G' = nKMod(G,T')$ on the virtual equipment $Vdc = nKMod(Cat(Quiv),T')$ of virtual double categories. Moreover, we can verify that $G'$ preserves pseudo double categories, and induces a monad $G''$ on the virtual equipment of double categories and [[double profunctors]]. Finally, we can define:

\begin{itemize}%
\item A \textbf{compact double category} is a pseudo $G''$-algebra. By general nonsense about distributive laws, this should be the same as a pseudo $G T'$-algebra, where $G T'$ is the composite pseudomonad on $Cat(Quiv)$ resulting from the distributive law.


\item A \textbf{compact virtual double category} is a pseudo $G'$-algebra.


\item A \textbf{virtually compact virtual double category} is a virtual $G T'$-algebra (in $Cat(Quiv)$). I \emph{don't} think this is quite the same as a virtual $G'$-algebra in $Vdc$.


\item A \textbf{virtually compact double category} is a virtual $G''$-algebra.



\end{itemize}
In each of the above cases, we can replace ``double category'' by [[proarrow equipment]] if the (virtual) double category in question is additional a ([[virtual equipment|virtual]]) equipment.

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

The basic example is $\underline{Cat}$, in which the objects are categories, the horizontal arrows are functors, and the vertical arrows are profunctors. This is a compact proarrow equipment. There are similar examples $V \underline{Prof}$ for any Benabou [[cosmos]] $V$. In fact, as long as $V$ is any symmetric [[multicategory]], we can define a virtually compact virtual equipment $V \underline{Prof}$.

\hypertarget{extraordinary_2cells}{}\subsection*{{Extraordinary 2-cells}}\label{extraordinary_2cells}

Let $C$ be a virtually compact virtual double category with units (it could be an equipment). Note that the source of a general 2-cell in $C$ is a graph whose edges are labeled by composable strings of vertical arrows in $C$. Suppose also that $f\colon \vec{x} \to z$ and $g\colon \vec{y}\to z$ are horizontal arrows in $C$. An \textbf{extraordinary 2-cell} in $C$ is defined to be a 2-cell whose target is the unit/identity $U_z$, and whose source is a loop-free graph whose edges are all labeled by empty strings (or, equivalently, by identities). One can verify that in $\underline{Cat}$, this reproduces the usual notion of [[extraordinary natural transformation]].

\hypertarget{structures_in_a_compact_double_category}{}\subsection*{{Structures in a compact double category}}\label{structures_in_a_compact_double_category}

In a (possibly virtual) compact double category, we can define internal notions of ``closed category,'' ``closed monoidal category,'' and so on.

\ldots{}

\hypertarget{representability}{}\subsection*{{Representability}}\label{representability}

\ldots{}

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

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/category/2010/03/extraordinary_2multicategories.html}{Blog post} about extraordinary 2-multicategories and their ilk.

\end{itemize}
[[!redirects compact virtual double category]] [[!redirects virtually compact double category]] [[!redirects virtually compact virtual double category]] [[!redirects compact double categories]] [[!redirects compact virtual double categories]] [[!redirects virtually compact double categories]] [[!redirects virtually compact virtual double categories]] [[!redirects compact closed double category]] [[!redirects compact closed double categories]] [[!redirects compact equipment]] [[!redirects compact proarrow equipment]] [[!redirects compact virtual equipment]] [[!redirects virtually compact equipment]] [[!redirects virtually compact proarrow equipment]] [[!redirects virtually compact virtual equipment]]

[[!redirects autonomous double category]] [[!redirects autonomous virtual double category]] [[!redirects virtually autonomous double category]] [[!redirects virtually autonomous virtual double category]] [[!redirects autonomous double categories]] [[!redirects autonomous virtual double categories]] [[!redirects virtually autonomous double categories]] [[!redirects virtually autonomous virtual double categories]] [[!redirects compact closed double category]] [[!redirects compact closed double categories]] [[!redirects autonomous equipment]] [[!redirects autonomous proarrow equipment]] [[!redirects autonomous virtual equipment]] [[!redirects virtually autonomous equipment]] [[!redirects virtually autonomous proarrow equipment]] [[!redirects virtually autonomous virtual equipment]]



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