representable multicategory

Representable multicategories


The representable multicategory underlying a monoidal category CC is a multicategory whose nn-ary morphisms are the morphisms out of nn-fold tensor products in CC, i.e.

Rep(C) n(c 1,,c n,c):=Hom C(c 1c n,c). Rep(C)_n(c_1, \cdots, c_n,c) := Hom_C(c_1\otimes \cdots \otimes c_n, c) \,.


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

Rep(C)(c 1,,c n;c):=C(c 1c n,c), 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 CC. Moreover, in the symmetric case there is a canonical action of the symmetric group induced.

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

In terms of Cartesian monads

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

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

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

TObj(C)× Obj(C)Mor(C) TObj(C) Mor(C) id s t TObj(C) Obj(C) Obj(C), \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 :TObj(C)C\otimes : T Obj(C) \to C denotes the iterated tensor product in CC, and where the top square is defined to be the pullback, as indicated.



The functor RepRep 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 7.3) 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.


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

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

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 7.3 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

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)

Revised on April 7, 2017 01:59:56 by Mike Shulman (