nLab representable multicategory

Idea

A multicategory is representable when it admits tensor products of objects that represent its multimorphisms so that, for instance:

$$M(A_1, \ldots, X \otimes Y, \ldots A_n; B) \,\cong\, M(A_1, \ldots, X, Y, \ldots, A_n; B) \,.$$

We may construct from every representable multicategory a monoidal category. Conversely, every monoidal category $C$ gives rise to a representable multicategory $Rep(C)$, by defining

$$Rep(C)_n(c_1, \cdots, c_n;c) \,\coloneqq\, Hom_C(c_1\otimes \cdots \otimes c_n, c) \,.$$

Definition

There is one flavor of representable multicategory for every flavor of generalized multicategory. Here we focus on the two best-known: one for ordinary monoidal categories, giving an ordinary multicategory, and one for symmetric monoidal categories, giving a symmetric multicategory. Mostly the discussion of both cases proceeds in parallel.

We first give the simple pedestrian definition in terms of explicit components, and then a more abstract definition, which is useful for studying general properties.

In terms of components

For $(C,\otimes, I)$ a (symmetric) monoidal category, the representable multicategory $Rep(C)$ is the (symmetric) multicategory whose objects are the objects of $C$, and whose objects of $n$-ary operations are the hom objects

$$Rep(C)(c_1, \cdots, c_n ; c) := C(c_1 \otimes \cdots \otimes c_n,\; c) \,,$$

This comes with the obvious composition operation induced from the composition in $C$. Moreover, in the symmetric case there is a canonical action of the symmetric group induced.

The full sub-multicategory of $Rep(C)$ on an object $c\in C$, being a one-object multicategory, is a (symmetric) operad, called the endomorphism operad of $c\in C$. Note that some authors use the term “endomorphism (colored) operad” for the whole multicategory $Rep(C)$.

In terms of Cartesian monads

Let $T : Set \to Set$ be the free monoid monad. Notice, from the discussion at multicategory, that a multicategory $P$ over Set with object set $C$ is equivalently a monad in the bicategory of $T$-spans

$$\array{ && P \\ & \swarrow && \searrow \\ T C && && C } \,.$$

In this language, for $C$ a (strict) monoidal category, the corresponding representable multicategory is given by the $T$-span

$$\array{ && & & T Obj(C) \times_{Obj(C)} Mor(C) \\ && & \swarrow && \searrow \\ && T Obj(C) && && Mor(C) \\ & {}^{\mathllap{id}}\swarrow && \searrow^{\mathrlap{\otimes}} && {}^{\mathllap{s}}\swarrow && \searrow^{\mathrlap{t}} \\ T Obj(C) &&&& Obj(C) &&&& Obj(C) } \,,$$

where $\otimes : T Obj(C) \to C$ denotes the iterated tensor product in $C$, and where the top square is defined to be the pullback, as indicated.

Properties

Adjunction

The functor $Rep$ is a right adjoint; its left adjoint constructs the free monoidal category on a multicategory, which is also known as the prop associated to the multicategory. See (Hermida, theorem 7.2) for a precise statement in the context of non-symmetric operads and strict monoidal categories. In the case of semicartesian multicategories, this free monoidal category is the category of operators associated to the multicategory.

Related concepts

• structured category

• endomorphism operad

References

The basic definition of representable symmetric multicategories (there called “endomorphism operads”) is for instance in section 1 of

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras Contemp. Math. 431 (2007) 31-58 (arXiv:0512576)

A general account of the definition of representable multicategories is in section 3.3 of

• Tom Leinster, Higher Operads, Higher Categories (arXiv:math/0305049)

The notion of representable multicategory is due to

• Claudio Hermida, Representable multicategories, Adv. Math. 151 (2000), no. 2, 164-225 (pdf)

Discussion of the 2-adjunction with the left adjoint prop-construction is around theorem 7.2 there. Characterization of representable multicategories by fibrations of multicategories is in

• Claudio Hermida, Fibrations for abstract multicategories, Field Institute Communications, Volume 43 (2004) (pdf)

and in section 9 of

• Claudio Hermida, From Coherent Structures to Universal Properties (arXiv:0006161)

Discussion in the context of generalized multicategories is in section 9 of

• G. Cruttwell, Mike Shulman, A unified framework for generalized multicategories Theory and Applications of Categories, Vol. 24, 2010, No. 21, pp 580-655. (TAC)