nLab
endomorphism operad

Idea

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.

$$\mathrm{End}(C{)}_n(c_1,\cdots ,c_n,c):={\mathrm{Hom}}_C(c_1\otimes \cdots \otimes c_n,c).$$End(C)_n(c_1, \cdots, c_n,c) := Hom_C(c_1\otimes \cdots \otimes c_n, c) \,.

Definition

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 $(C,\otimes ,I)$ a (symmetric) monoidal category, the endomorphism operad ${\mathrm{End}}_C(X)$ 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

$${\mathrm{End}}_C(X)(c_1,\cdots ,c_n;c):=C(c_1\otimes \cdots \otimes c_n,c),$$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 $C$. Moreover, in the symmetric case there is a canonical action of the symmetric group induced.

For $S\subset \mathrm{Obj}(C)$ 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 $\mathrm{End}(c)$, is the single-colored operad whose $n$-ary operations are the morphism $c^{\otimes n}\to c$ in $C$.

In terms of Cartesian monads

Let $T:\mathrm{Set}\to \mathrm{Set}$ 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

$$\begin{array}{ccc}& & P\\ & \swarrow & & \searrow \\ TC& & & & C\end{array}.$$\array{ && P \\ & \swarrow && \searrow \\ T C && && C } \,.

In this language, for $C$ a (strict) monoidal category, the corresponding endomorphism operad is given by the $T$-span

$$\begin{array}{ccccc}& & & & T\mathrm{Obj}(C){\times }_{\mathrm{Obj}(C)}\mathrm{Mor}(C)\\ & & & \swarrow & & \searrow \\ & & T\mathrm{Obj}(C)& & & & \mathrm{Mor}(C)\\ & {}^{\mathrm{id}}\swarrow & & {\searrow }^{\otimes }& & {}^s\swarrow & & {\searrow }^t\\ T\mathrm{Obj}(C)& & & & \mathrm{Obj}(C)& & & & \mathrm{Obj}(C)\end{array},$$\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 $\otimes :T\mathrm{Obj}(C)\to C$ denotes the iterated tensor product in $C$, and where the top square is defined to be the pullback, as indicated.

Properties

Algebras

The structure of an algebra over an operad on an object $A\in C$ over $P$ is equivalently a morphism of operads

$$\rho :P\to \mathrm{End}(A)$$\rho : P \to End(A)

Relation to categories of operators

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.

Related concepts

• endomorphism

• endomorphism monoid, endomorphism ring

• endomorphism operad

References

The basic definition of symmetric endomorphism operads is for instance in section 1 of

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras Contemp. Math. 431 (2007) 31-58 (arXiv:0512576)

A general account of the definition of representable multicategories is in section 3.3 of

• Tom Leinster, Higher Operads, Higher Categories (arXiv:math/0305049)

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

• Claudio Hermida, From Coherent Structures to Universal Properties (arXiv:0006161)

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)