symmetric monoidal (∞,1)-category of spectra
The representable multicategory underlying a monoidal category $C$ is a multicategory whose $n$-ary morphisms are the morphisms out of $n$-fold tensor products in $C$, i.e.
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.
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
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 $Rep(C)$.
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
In this language, for $C$ a (strict) monoidal category, the corresponding representable multicategory is given by the $T$-span
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.
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 7.3) 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
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
and in section 9 of
Discussion in the context of generalized multicategories is in section 9 of