nLab multicategory

Contents

Context

Monoidal categories

Category theory

Idea
Definition

In components
In terms of cartesian monads

Properties

Relation to operads
Relation to monoidal categories
Exponentiability

Examples and special cases
Related concepts
References

Idea

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.

Definition

In components

A multicategory $C$ consists of

A collection of objects, $C_0$ .
A collection of multimorphisms, $C_1$ .
A source map $s: C_1 \to (C_0)*$ to the collection of finite, possibly empty lists of objects (where $(C_0)*$ is the free monoid generated by $C_0$ ), and a target map $t: C_1 \to C_0$ . We write $f: c_1, \ldots, c_n \to c$ to indicate the source and target of a multimorphism $f$ .
Identity and composition laws. The identity law is a map $1_{-}: C_0 \to C_1$ where $1_c: c \to c$ . The composition law assigns, to each $f: c_1, \ldots, c_n \to c$ together with an $n$ -tuple $\langle f_i: \vec{c}_i \to c_i: i = 1, \ldots, n \rangle$ , a composite
$f \circ (f_1, \ldots, f_n): \vec{c}_1, \ldots, \vec{c}_n \to c$

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.

In terms of cartesian monads

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.

First, a $T$ -span from $X$ to $Y$ is a span $p$ from $T X$ to $Y$ , that is, a diagram
$T X \stackrel{p_1}{\leftarrow} P \stackrel{p_2}{\to} Y$

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

$(T X \stackrel{m X}{\leftarrow} T^2 X \stackrel{T e_1}{\leftarrow} T E \stackrel{T e_2}{\to} T Y) \circ (T Y \stackrel{f_1}{\leftarrow} F \stackrel{f_2}{\to} Z)$

where $m: T^2 \to T$ is the monad multiplication. The identity $T$ -span from $X$ to itself is the span

$T X \stackrel{u X}{\leftarrow} X \stackrel{1_X}{\to} X$

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.

Properties

Relation to operads

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.

Relation to monoidal categories

There is a 2-adjunction between the 2-category of multicategories and the 2-category of strict monoidal categories, strict monoidal functors, and monoidal natural transformations. The left adjoint sends a multicategory to its monoidal envelope. The right adjoint sends a strict monoidal category to the multicategory with the same objects, and multimorphisms $A_1, \ldots, A_n \to B$ given by morphisms $A_1 \otimes \cdots \otimes A_n \to B$ (which is a representable multicategory).

The right adjoint is faithful and locally fully faithful?, but not full (in general, the functors of multicategories correspond to lax monoidal functors). Furthermore, this 2-adjunction is 2-monadic and lax-idempotent, and the unit is compontentwise full and faithful, i.e. for each multicategory $M$ , $M \to UFM$ is fully faithful. The pseudoalgebras for the induced 2-monad are precisely the representable multicategories, equivalently the (non-strict) monoidal categories.

The adjunction is also comonadic, and is consequently an example of a nuclear adjunction.

There is also a fully faithful 2-functor $StrMonCat_l \to Multicat$ (abstractly, this is the forgetful functor from the 2-category of strict algebras? and lax morphisms for the induced 2-monad).

For details on some of this material see (Hermida 2000).

Exponentiability

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 $D^C$ .

Examples and special cases

permutative categories with multi-linear functors between them form a multicategory PermCat

Related concepts

References

Multicategories were introduced in:

Joachim Lambek, Deductive systems and categories II. Standard constructions and closed categories, Category Theory, Homology Theory and their Applications I: Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July 19, 1968 Volume One. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.

and developed further in:

Joachim Lambek, Multicategories revisited, Contemp. Math 92 (1989): 217-239.

Hermida constructed a monadic 2-adjunction between the 2-category of nonsymmetric colored operads in $(Set, \times)$ (which he calls multicategories) and the 2-category of strict monoidal categories in Theorem 7.2 of this paper:

Claudio Hermida, Representable multicategories, Advances in Mathematics, Volume 151, Issue 2, 2000, 164-225. (pdf)