A multicategory (or coloured operad, if you like) is a category where instead of (only) morphisms between objects we have multimorphisms from finite lists of objects to (single) objects. There may be further bells and whistles, e.g. a symmetric group action. Morally, they are monoidal categories where we know what the morphisms out of monoidal products are supposed to be but do not know (or do not care) what morphisms into monoidal products are. (Famously, the tensor product of modules is defined in this way: multilinear maps are conceptually prior!)
Question. What are some textbook-style references for basic multicategory theory? Is there an equivalent of Mac Lane's Categories for the working mathematician? For example, I would like to see precise definitions, statements, and proofs for the following:
Given a monoidal category, we can define the "underlying" multicategory to have as its multimorphisms $[X_1, \ldots, X_n] \to Y$ the morphisms $( \cdots (X_1 \otimes X_2) \otimes \cdots ) \otimes X_n \to Y$; lax monoidal functors then give rise to multicategory functors in the obvious way, and so on.
Given objects $A_1, \ldots, A_n$ in a multicategory, a monoidal representation is an object $A_1 \otimes \cdots \otimes A_n$ and a multimorphism $\eta : [A_1, \ldots, A_n] \to A_1 \otimes \cdots \otimes A_n$ such that, for every multimorphism $f : [X, \ldots, A_1, \ldots, A_n, Y, \ldots] \to Z$, there is a unique multimorphism $[X, \ldots, A_1 \otimes \cdots \otimes A_n, Y, \ldots] \to Z$ such that precomposing $\eta$ yields $f$. (This what Hermida calls strongly universal.)
If every pair of objects in a multicategory has a monoidal representation, then every finite list of positive length has a monoidal representation.
If every finite list of objects in a multicategory has a monoidal representation – i.e. if the multicategory is representable – then it is equivalent to the "underlying" multicategory of a monoidal category.
Likewise for symmetric (resp. braided) monoidal categories and symmetric (resp. braided) multicategories.
Given a closed category, we can define the "underlying" multicategory to have as its multimorphisms $[X_1, \ldots, X_n] \to Y$ the morphisms $X_1 \to X_2 \multimap (\cdots \multimap (X_n \multimap Y) \cdots)$; closed functors then give rise to multicategory functors, and so on.
Given objects $A_1, \ldots, A_n$ and $B$ in a multicategory, an internal hom representation is an object $[A_1, \ldots, A_n] \multimap B$ and a multimorphism $\epsilon : [A_1, \ldots, A_n, [A_1, \ldots, A_n] \multimap B] \to B$ such that, for every multimorphism $f : [A_1, \ldots, A_n, Y, \ldots] \to B$, there is a unique multimorphism $[Y, \ldots] \to [A_1, \ldots, A_n] \multimap B$ such that postcomposing $\epsilon$ yields $f$. (We could define a two-sided version of this, but since it does not appear in Manzyuk's paper, I assume it is not necessary.)
If every $A$ and $B$ has an internal hom representation, then every $A_1, \ldots, A_n$ and $B$ has an internal hom representation.
If, in addition, the empty list has a monoidal representation, then the multicategory is equivalent to the "underlying" multicategory of a closed category.
Possible definitions of colimits in a multicategory, and more generally universal objects characterised by outgoing morphisms. (I presume there is no controversy about the correct definition of universal objects characterised by incoming multimorphisms, but unfortunately multicategory theory does not have arrow-reversing duality...)
The definitions of adjunction between multicategories, and some translation of the fact that in a monoidal adjunction the left adjoint is strong monoidal.
Given a monad on a multicategory, we can form Kleisli and Eilenberg–Moore multicategories of algebras. The Kleisli multicategory is representable if the base is, and the Eilenberg–Moore multicategory is representable under some additional hypotheses regarding reflexive coequalisers.
