nLab endomorphism operad

Idea

Just as every object in a category has a monoid of endomorphisms, every object of a monoidal category, or more generally a multicategory, has an operad of endomorphisms. This includes not just ordinary endomorphisms $X\to X$, but also higher-ary “endomorphisms” like $X\otimes X\otimes X\to X$.

Definition

The endomorphism operad of an object $X$ in a monoidal category $C$ is the full sub-multicategory of the representable multicategory $Rep(C)$ associated to $C$ on the single object $X$. (Note that an non-symmetric operad is, essentially by definition, a one-object multicategory.) More generally, we can consider any one-object full subcategory of a multicategory to be an “endomorphism operad”. This can also be generalized to symmetric monoidal categories and symmetric operads, and also to other kinds of generalized multicategories.

Properties

Algebras

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

$$\rho : P \to End(A)$$

Related concepts

• endomorphism

• endomorphism monoid, endomorphism ring, endomorphism monoid object

• representable multicategory

• operad of set operations