nLab
generalized multicategory

Idea

An ordinary category consists of a set of objects and a set of arrows, each with a single input or domain object and a single output or codomain object. Arrows are composed by plugging outputs into inputs, as when one composes unary operations.

An ordinary multicategory consists of a set of objects and a set of “multi-arrows”, each with a finite list of inputs or domain objects and a single output or codomain object. These multi-arrows are composed by means of multiple plug-ins or substitutions, as when one substitutes a list of $m$ operations of varying arities into an $m$-ary operation.

Both categories and multicategories can be seen as monads in an appropriate bicategory of span-like objects. Categories are monads in the category of ordinary spans (of sets). Multicategories are monads in a bicategory of spans of shape

$$X^{*}\to Y$$X^* \to Y

where $X^{*}$ is the set of finite lists of elements of $X$.

In order to compose spans of this type, one takes advantage of certain properties of the list monad or free monoid monad $(-{)}^{*}$, namely that it is a cartesian monad $T$. This suggests the following wide extrapolation.

Definition

Let $V$ be a finitely complete category, and let $T$ be a cartesian monad on $V$. Then there is a bicategory of $T$-spans in $V$, whose objects $X$ are objects of $V$, whose morphisms $X\to Y$ are spans from $TX$ to $Y$, and whose 2-cells are morphisms of such spans. The identity $X\to X$ is the span

$$TX\stackrel{uX}{\leftarrow }X\stackrel{\mathrm{id}}{\to }X$$T X \overset{u X}{\leftarrow} X \overset{id}{\to} X

where $u:1_V\to T$ is the unit of $T$, and morphisms $X\to Y$, $Y\to Z$ are composed by taking pullbacks of shape

$$\begin{array}{ccccccccccc}& & & & & & R\circ S& & & & \\ & & & & & \swarrow & & \searrow & & & \\ & & & & TS& & & & R& & \\ & & & Tf\swarrow & & \searrow Tg& & h\swarrow & & \searrow k& \\ TX& \stackrel{mX}{\leftarrow }& TTX& & & & TY& & & & Z\end{array}$$\array{& & & & & & R \circ S & & & &\\ & & & & & \swarrow & & \searrow & & & \\ & & & & T S & & & & R & &\\ & & & T f \swarrow & & \searrow T g & & h \swarrow & & \searrow k & \\ T X & \overset{m X}{\leftarrow} & T T X & & & & T Y & & & & Z}

A $T$-multicategory in $V$ is by definition a monad in the bicategory described above.

References

• G.S.H. Cruttwell and Mike Shulman; A unified framework for generalized multicategories