symmetric monoidal (∞,1)-category of spectra
The endomorphism operad of a monoidal category $C$ – also called the multicategory represented by $C$ – is an operad whose $n$-ary operations are the morphisms out of $n$-fold tensor products in $C$, i.e.
Endomorphism operads come in two flavors, one being a planar operad, the other a symmetric operad. 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 some general properties of endomorphism operads.
For $(C,\otimes, I)$ a (symmetric) monoidal category, the endomorphism operad $End_C(X)$ of $X$ in $C$ is the symmetric operad/ planar operad whose colors 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.
For $S \subset Obj(C)$ any subset of objects, the $S$-colored endomorphism operad of $C$ is the restriction of the endomorphism operad defined to the set of colors being $S$.
In particular, the endomorphism operad of a single object $c \in C$, often denoted $End(c)$, is the single-colored operad whose $n$-ary operations are the morphisms $c^{\otimes n}\to c$ in $C$.
Let $T : Set \to Set$ be the free monoid monad. Notice, from the discussion at multicategory, that a planar operad $P$ over Set with set of colors $C$ is equivalently a monad in the bicategory of $T$-spans
In this language, for $C$ a (strict) monoidal category, the corresponding endomorphism operad 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 structure of an algebra over an operad on an object $A \in C$ over $P$ is equivalently a morphism of operads
To every operad $P$ is associated its category of operators $P^{\otimes}$, which is a monoidal category.
With that suitably defined, forming endomorphism operads is right 2-adjoint to forming categories of operators. See (Hermida, theorem 7.3) for a precise statement in the context of non-symmetric operads and strict monoidal categories.
endomorphism operad
The basic definition of symmetric 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 category of operators-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