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 *YX^* \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 XY are spans from TX to Y, and whose 2-cells are morphisms of such spans. The identity XX is the span

TXuXXidXT X \overset{u X}{\leftarrow} X \overset{id}{\to} X

where u:1 VT is the unit of T, and morphisms XY, YZ are composed by taking pullbacks of shape

RS TS R Tf Tg h k TX mX TTX TY Z\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