nLab
endomorphism operad

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 XXX\to X, but also higher-ary “endomorphisms” like XXXXX\otimes X\otimes X\to X.

Definition

The endomorphism operad of an object XX in a monoidal category CC is the full sub-multicategory of the representable multicategory Rep(C)Rep(C) associated to CC on the single object XX. (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 ACA \in C over an operad PP is equivalently a morphism of operads

ρ:PEnd(A) \rho : P \to End(A)

Created on April 7, 2017 02:03:49 by Mike Shulman (76.167.222.204)