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\begin{document}

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\section*{virtual double category}

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

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\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{higher_categories_of_virtual_double_categories}{Higher categories of virtual double categories}\dotfill \pageref*{higher_categories_of_virtual_double_categories} \linebreak
\noindent\hyperlink{functors_and_transformations}{Functors and transformations}\dotfill \pageref*{functors_and_transformations} \linebreak
\noindent\hyperlink{profunctors}{Profunctors}\dotfill \pageref*{profunctors} \linebreak
\noindent\hyperlink{Monads}{Monads on virtual double categories}\dotfill \pageref*{Monads} \linebreak
\noindent\hyperlink{monoids_and_modules}{Monoids and modules}\dotfill \pageref*{monoids_and_modules} \linebreak
\noindent\hyperlink{generalized_multicategories}{Generalized multicategories}\dotfill \pageref*{generalized_multicategories} \linebreak
\noindent\hyperlink{enriching_categories}{Enriching categories}\dotfill \pageref*{enriching_categories} \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{virtual double category} or \textbf{$fc$-multicategory} is a common generalization of a [[monoidal category]], a [[bicategory]], a [[double category]], and a [[multicategory]]. It contains:

\begin{itemize}%
\item objects
\item vertical arrows, which form a category
\item horizontal arrows, which do not have identities or composites, and
\item 2-cells which have\begin{itemize}%
\item a horizontal source and target, which are vertical arrows,
\item a vertical target, which is a horizontal arrow, and
\item a vertical source, which is a composable string of horizontal arrows.

\end{itemize}


\end{itemize}
2-cells are usually drawn like this:

[[virtual-double-category-cell.png:pic]]

Note that this includes the case when $n=0$, i.e. a cell of ``nullary'' source. In this case, we must have $X_0 = X_n$. Finally, the 2-cells can be composed in a more or less evident way, akin to composition in a multicategory:

[[virtual-double-category-composite.png:pic]]

Virtual double categories are related to double categories precisely as ordinary multicategories are related to monoidal categories (see [[generalized multicategory]] and [[tensor product]]).

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

A virtual double category can be defined in two equivalent ways:

\begin{itemize}%
\item It is a $T$-[[generalized multicategory|multicategory]], in the sense of Leinster, relative to the monad $T$ on [[quiver|directed graphs]] whose algebras are categories. For this reason, Leinster originally called them \textbf{fc-multicategories}, where ``fc'' is a name for this monad $T$ which stands for ``free-category.''


\item It is a generalized multicategory, in the sense of Hermida, Cruttwell-Shulman, and others, relative to the monad $T$ on graphs-internal-to-Cat whose algebras are double categories. This is the origin of the name ``virtual double category,'' in line with the general terminology ``virtual $T$-algebra'' of Cruttwell-Shulman for such generalized multicategories.



\end{itemize}
We can also give an explicit definition, which was more or less already given in the ``Idea'' section: all that is missing are identities and associativity for 2-cell composition.

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

\begin{itemize}%
\item Any double category is an example, and thus also any bicategory viewing the arrows as horizontal.
\item For any [[monoidal category]] $V$, there is a virtual double category of $V$-matrices whose objects are sets, vertical arrows are functions and a horizontal arrow $p : X \to Y$ is a family of objects $p_{y,x}
\in V$ for each $x\in X, y \in Y$, and a 2-cell from $X_0
\overset{p_1}{\to} X_1 \to\dots \to X_n$ to $Y_0 \overset{q}{\to} Y_1$ along $f : X_0 \to Y_0, g : X_n \to Y_n$ is a family of arrows $\alpha_{x_0,...} : p_1(x_1,x_0)\otimes p_2(x_2,x_1)\otimes\dots \to
q(g(x_n),f(x_0))$ in $V$ (using the unit of the monoidal category if the source string is empty). If $V$ has certain colimits that are preserved by $\otimes$ then composites exist and this virtual double category is pseudo.
\item Given a monad on a virtual double category, the [[generalized multicategory| horizontal kleisli double category]] produces a virtual double category that is only pseudo under strong conditions on the monad. In particular, ``free monoid'' monad on the double category of sets and spans does not produce a pseudo double category.

\end{itemize}
\hypertarget{higher_categories_of_virtual_double_categories}{}\subsection*{{Higher categories of virtual double categories}}\label{higher_categories_of_virtual_double_categories}

There are notions of \textbf{functor}, \textbf{transformation}, and \textbf{profunctor} between virtual double categories. The neatest way to define all of these notions at once is to use the general framework of [[generalized multicategories]]: from the monad $fc$ on the [[virtual equipment]] $Span = Span(Set)$ we can construct a new virtual equipment $vDblProf = KMod(Span,fc)$ whose objects are virtual double categories, whose arrows are functors between them, whose proarrows are profunctors between them, and whose cells are transformations. But we can also give explicit definitions of all of these notions.

\hypertarget{functors_and_transformations}{}\subsubsection*{{Functors and transformations}}\label{functors_and_transformations}

A \textbf{functor} of virtual double categories is fairly obvious; it takes each kind of morphism/cell to the same kind, preserving sources, targets, composition, and identities.

The relevant \textbf{transformations} are a ``virtual'' version of [[vertical transformations]] between ordinary double categories. Specifically, a transformation $\alpha\colon F\to G$ has a vertical arrow component $\alpha_X\colon F X\to G X$ for each object $X$ of the domain, and a cell component

\begin{displaymath}
\itexarray{F X & \overset{F p}{\to} & F Y\\
  ^{\alpha_X}\downarrow & \Downarrow ^{\alpha_p}& \downarrow^{\alpha_Y}\\
  G X& \underset{G p}{\to} & G Y}
\end{displaymath}
for each horizontal arrow $p\colon X\to Y$ in the domain. These must be natural with respect to vertical composition of arrows and of 2-cells, where we must of course allow composites with arbitrary arities in the latter case.

Virtual double categories, functors, and transformations form a [[strict 2-category]], and thus we can apply all notions of 2-category theory to it. In particular, we have a notion of a [[monad]] \emph{on} a virtual double category, which is the starting point for one theory of [[generalized multicategories]].

\hypertarget{profunctors}{}\subsubsection*{{Profunctors}}\label{profunctors}

The \textbf{profunctors} between virtual double categories are a similar ``virtualization'' of the notion of [[double profunctor]] between double categories. Explicitly, a profunctor $H\colon C &#8696; D$ consists of:

\begin{itemize}%
\item An ordinary [[profunctor]] $H_0\colon C_0 &#8696; D_0$ between the categories of objects and vertical arrows.


\item For each string of horizontal arrows $X_0 \overset{p_1}{\to} X_1 \to\dots \to X_n$ in $D$, each horizontal arrow $Y_0 \overset{q}{\to} Y_1$ in $C$, and each pair of elements $f \in H_0 (X_0,Y_0)$ and $g\in H_0(X_n,Y_1)$, a set of ``hetero-cells'' of shape

[[virtual-double-category-cell.png:pic]]


\item The hetero-cells are acted on by the 2-cells of $D$ on the top, and by the 2-cells of $C$ on the bottom, in an evident way, respecting the given action of vertical arrows of $D$ and $C$ on the elements of $H_0$.



\end{itemize}
Every [[double profunctor]] induces such a profunctor in an evident way, but even if $C$ and $D$ are (non-virtual) double categories, not every ``virtual double profunctor'' from $C$ to $D$ need be a double functor; only those for which the ``hetero-cells'' also factor uniquely through the opcartesian cells in $D$ which make it ``representable.''

As mentioned above in the context of the abstract definition, virtual double categories, functors, transformations, and profunctors form another virtual double category, which is in fact a [[virtual equipment]].

\hypertarget{Monads}{}\subsubsection*{{Monads on virtual double categories}}\label{Monads}

\begin{udefn}
A \emph{monad on a virtual double category} is a [[monad]] in the [[2-category]] [[vDbl]].

\end{udefn}
So a monad on a $X \in vDbl$ consists of a functor

\begin{displaymath}
T : \mathbb{X} \to \mathbb{X}
\end{displaymath}
and transformations $\eta : Id  \to T$ and $\mu : T T \to T$ satisfying [[associativity]] and [[unitality]].

\hypertarget{monoids_and_modules}{}\paragraph*{{Monoids and modules}}\label{monoids_and_modules}

\begin{udefn}
For $T$ a monad on $\mathbb{X} \in$ [[vDbl]], a \textbf{$T$-monoid} is

\begin{itemize}%
\item an object $X_0 \in \mathbb{X}$;


\item a horizontal morphism $X_0 \stackrel{X}{&#8696;} T X_0$


\item an [[action]] [[2-morphism]]

\begin{displaymath}
\itexarray{
     X_0 &\stackrel{X}{&#8696;} & T X_0 
      & \stackrel{T X}{&#8696;} T^2 &  X_0
     \\
     {}^{\mathllap{=}}\downarrow && \Downarrow^{\bar x}
     &&  \downarrow^{\mathrlap{\mu}}
     \\
     X_0 && \underset{X}{&#8696;}  && T X_0
  }
\end{displaymath}
and a [[unit]] 2-morphism

\begin{displaymath}
\itexarray{
     && X_0
     \\
     & {}^{\mathllap{=}}\nearrow &\Downarrow^{\bar x}& \searrow^{\mathrlap{\eta}}
     \\
     X_0 &&\underset{X}{&#8696;}&& T X_0
  }
\end{displaymath}


\end{itemize}
satisfying the evident compatibility conditions.

\end{udefn}
This is (\hyperlink{CruttwellShulman}{CruttwellShulman, def. 4.2}).

\begin{udefn}
A $T$-monoid $X_0 \stackrel{X}{&#8696;} T X_0$ is called \textbf{normalized} if its unit 2-morphism

\begin{displaymath}
\itexarray{
    X_0 &\stackrel{U_{X_0}}{\to}& X_0
    \\
    {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{\eta}}
    \\
    X_0 &\underset{X}{&#8696;}& T X_0
  }
\end{displaymath}
is a \hyperlink{...}{cartesian 2-morphism}.

\end{udefn}
\hypertarget{generalized_multicategories}{}\paragraph*{{Generalized multicategories}}\label{generalized_multicategories}

\begin{udefn}
A \textbf{[[generalized multicategory]]} is a normalized $T$-monoid for some monad $T$ on a [[virtual equipment]] $\mathbb{X} \in$ [[vDbl]].

\end{udefn}
This is (\hyperlink{CruttwellShulman}{CruttwellShulman, page 7}).

\hypertarget{enriching_categories}{}\subsection*{{Enriching categories}}\label{enriching_categories}

Virtual double categories can be viewed as ``the natural place in which to enrich categories.'' Specifically, for any set $A$, there is a virtual double category $A_{ch}$ which has $A$ as its objects, only identity vertical arrows, exactly one horizontal arrow from every object to every other object, and exactly one 2-cell in every possible niche. For any other virtual double category $W$, a functor $A_{ch}\to W$ of virtual double categories is the same as a $W$-enriched category with object set $A$.

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

\begin{itemize}%
\item [[double category]]
\item [[virtual equipment]]
\item [[hypervirtual double category]]

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

\begin{itemize}%
\item [[Tom Leinster]], \emph{Higher Operads, Higher Categories}, \href{http://www.maths.gla.ac.uk/~tl/book.html}{link}, \href{http://arxiv.org/abs/math/0305049}{arXiv:0305049}


\item [[Geoff Cruttwell]] and [[Mike Shulman]], \emph{A unified framework for generalized multicategories}, \href{http://arxiv.org/abs/0907.2460}{arXiv:0907.2460}



\end{itemize}
[[!redirects virtual double categories]] [[!redirects fc-multicategory]] [[!redirects fc-multicategories]]



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