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The endomorphism operad of a monoidal category – also called the multicategory represented by – is an operad whose -ary operations are the morphisms out of -fold tensor products in , 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.
In terms of components
For a (symmetric) monoidal category, the endomorphism operad of in is the symmetric operad/ planar operad whose colors are the objects of , and whose objects of -ary operations are the hom objects
This comes with the obvious composition operation induced from the composition in . Moreover, in the symmetric case there is a canonical action of the symmetric group induced.
For any subset of objects, the -colored endomorphism operad of is the restriction of the endomorphism operad defined to the set of colors being .
In particular, the endomorphism operad of a single object , often denoted , is the single-colored operad whose -ary operations are the morphism in .
In terms of Cartesian monads
Let be the free monoid monad. Notice, from the discussion at multicategory, that a planar operad over Set with set of colors is equivalently a monad in the bicategory of -spans
In this language, for a (strict) monoidal category, the corresponding endomorphism operad is given by the -span
where denotes the iterated tensor product in , and where the top square is defined to be the pullback, as indicated.
The structure of an algebra over an operad on an object over is equivalently a morphism of operads
Relation to categories of operators
To every operad is associated its category of operators , 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.
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
- Claudio Hermida, Representable multicategories, Adv. Math. 151 (2000), no. 2, 164-225 (pdf)
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
- 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)