With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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 $C$ consists of
where the source is obtained by concatenating lists in the evident way.
These operations are subject to associativity and identity axioms which the reader can probably figure out, but see for example (Leinster, page 35 ff.), for details.
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 $S_n$ on the multimorphisms $c_1, \ldots, c_n \to c$ 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 $(-)*: Set \to Set$. This is a special case of the yet more general notion of generalized multicategory.
We summarize here how the theory applies to the case of a cartesian monad $T$ on a category with pullbacks; see generalized multicategory for the fully general context.
A $T$-span is often written as $p: X ⇸ Y$.
When $T$ is the free monoid monad on $Set$, a $T$-span from $X$ to itself is called a multigraph on $X$.
$T$-spans are the 1-cells of a bicategory. A 2-cell between $T$-spans $e, f: X ⇸ Y$ is a 2-cell between ordinary spans from $T X$ to $Y$. To horizontally compose $T$-spans $e: X ⇸ Y$ and $f: Y ⇸ Z$, take the ordinary span composite of
where $m: T^2 \to T$ is the monad multiplication. The identity $T$-span from $X$ to itself is the span
where $u: I \to T$ is the monad unit. The verification of the bicategory axioms uses the cartesianness of $T$ in concert with the corresponding axioms on the bicategory of spans.
A $T$-multicategory is defined to be a monad in the bicategory of $T$-spans.
When $T$ is the free monoid monad on sets, then a $T$-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 $V$ is equivalent to a $V$-enriched multicategory with one object.
More generally, the notion of multi-colored planar operad is equivalent to that of multicategory, and the notion of multi-colored symmetric operad is equivalent to that of symmetric multicategory.
Fully generally, for each cartesian monad $T$, there is a corresponding notion of $T$-operad, namely a $T$-multicategory whose underlying $T$-span has the form $1 ⇸ 1$.
For example, in Batanin’s approach to (weak) $\infty$-categories, a globular operad is a $T$-operad, where $T$ is the free (strict) $\omega$-category monad on the category of globular sets.
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 $C$ there is an associated (strict) monoidal category $F(C)$, whose objects (respectively, arrows) are lists of objects (respectively, arrows) of $C$, and where the tensor product in $F(C)$ is given by concatenation.
A multicategory is exponentiable if and only if it is promonoidal (Proposition 2.8 of Pisani 2014). In particular, representable multicategories and sequential multicategories are exponentiable. This gives an abstract construction of the Day convolution tensor product on $[C, D]$ for any promonoidal category $C$ and any monoidal category $D$: it is precisely the (representable) multicategory structure on the functor multicategory $D^C$.
See also the examples at operad.
Last revised on January 1, 2024 at 11:31:12. See the history of this page for a list of all contributions to it.