category with duals (list of them)
dualizable object (what they have)
Recall that a category consists of a collection of morphisms each having a single object as source or input, and a single object as target or output, together with laws for composition and identity obeying associativity and identity axioms. A multicategory is like a category, except that one allows multiple inputs and a single output.
Another term for multicategory is coloured operad.
A multicategory consists of
where the source is obtained by concatenating lists in the evident way.
Many people (especially non-category theorists) use the word multicategory or the word colored operad to mean what we would call a symmetric multicategory / symmetric operad. These are multicategories equipped with an action of the symmetric group on the multimorphisms such that the composition is equivariant with respect to these actions.
An efficient abstract method for defining multicategories and related structures is through the formalism of cartesian monads. For ordinary categories, one uses the identity monad on Set; for ordinary multicategories, one uses the free monoid monad . This is a special case of the yet more general notion of generalized multicategory.
A -span is often written as .
When is the free monoid monad on , a -span from to itself is called a multigraph on .
-spans are the 1-cells of a bicategory. A 2-cell between -spans is a 2-cell between ordinary spans from to . To horizontally compose -spans and , take the ordinary span composite of
where is the monad multiplication. The identity -span from to itself is the span
where is the monad unit. The verification of the bicategory axioms uses the cartesianness of in concert with the corresponding axioms on the bicategory of spans.
A -multicategory is defined to be a monad in the bicategory of -spans.
When is the free monoid monad on sets, then a -multicategory is a multicategory as defined above. For more examples and generalizations, see generalized multicategory.
A nonpermutative (or Stasheff-) operad in Set may be defined as an ordinary multicategory with exactly one object. Likewise, a symmetric operad in any symmetric monoidal category is equivalent to a -enriched multicategory with one object.
Fully generally, for each cartesian monad , there is a corresponding notion of -operad, namely a -multicategory whose underlying -span has the form .
Ordinary (permutative/symmetric) operads, and their generalization to symmetric multicategories, can also be treated in the framework of generalized multicategories, but they require a framework more general than that of cartesian monads.
There is a faithful functor from monoidal categories to multicategories, given by forming represented multicategories. Conversely, to any multicategory there is an associated (strict) monoidal category , whose objects (respectively, arrows) are lists of objects (respectively, arrows) of , and where the tensor product in is given by concatenation.
See also the examples at operad.