## 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

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

Created on April 7, 2017 at 02:03:49. See the history of this page for a list of all contributions to it.