endomorphism operad



The endomorphism operad of a monoidal category CC – also called the multicategory represented by CC – is an operad whose nn-ary operations are the morphisms out of nn-fold tensor products in CC, i.e.

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


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 (C,,I)(C,\otimes, I) a (symmetric) monoidal category, the endomorphism operad End C(X)End_C(X) of XX in CC is the symmetric operad/ planar operad whose colors are the objects of CC, and whose objects of nn-ary operations are the hom objects

End C(X)(c 1,,c n;c):=C(c 1c n,c), End_C(X)(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.

For SObj(C)S \subset Obj(C) any subset of objects, the SS-colored endomorphism operad of CC is the restriction of the endomorphism operad defined to the set of colors being SS.

In particular, the endomorphism operad of a single object cCc \in C, often denoted End(c)End(c), is the single-colored operad whose nn-ary operations are the morphisms c ncc^{\otimes n}\to c in CC.

In terms of Cartesian monads

Let T:SetSetT : Set \to Set be the free monoid monad. Notice, from the discussion at multicategory, that a planar operad PP over Set with set of colors 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 endomorphism operad 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 structure of an algebra over an operad on an object ACA \in C over PP is equivalently a morphism of operads

ρ:PEnd(A) \rho : P \to End(A)

Relation to categories of operators

To every operad PP is associated its category of operators P 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.


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)

Revised on April 7, 2015 16:31:50 by Noam Zeilberger (