nLab
category over an operad

Idea

A category over an operad is the horizontal categorification of an algebra over an operad. It is like an enriched category in which the composition operation is not necessarily binary, but parameterized by the operad.

Definition

Given an operad $O$ in some symmetric monoidal category $C$, a category over the operad $O$, or $O$-category $D$ is

• a set/class/whatever $D_0$, called the set of objects of $D$;

• for each pair $x,y\in D_0$ an object $D(x,y)\in C_0$, called the object of morphisms from $x$ to $y$ in $D$;

• for each natural number $n$ and each sequence $x_0,x_1,\cdots ,x_n$ of objects of $D_0$ a morphism

  $${\mathrm{comp}}_{(x_0,\cdots ,x_n}:(D(x_0,x_1)\otimes D(x_1,x_2)\otimes \cdots \otimes C(x_{n-1},x_n))\otimes O(n)\to D(x_0,x_n)$$comp_{(x_0, \cdots, x_n} : \left(D(x_0,x_1) \otimes D(x_1,x_2) \otimes \cdots \otimes C(x_{n-1},x_n) \right) \otimes O(n) \to D(x_0, x_n)

  called the $n$-ary composition operation;

• such that the composition operations satisfy the obvious compatibility conditions with the operad composition operation, directly analogous to those for $O$-algebras.

Examples

• Let $\mathrm{Associative}$ be the ordinary associtive operad in Set. An $\mathrm{Associative}$-category is an ordinary Set-enriched category i.e. a locally small category.

• Let $A$ be the ordinary associative operad in vector spaces. An $A$-category is a Vect-enriched category.

• We may regard the operad $A$ as a dg-operad i.e. an operad in the category of cochain complexes. As such, $A$ has a resolution, the A-infinity-operad operad $A_{\mathrm{\infty }}$. An $A_{\mathrm{\infty }}$-category is an A-infinity-category (see there).

• A class of definitions of infinity-categories is operadic in this sense, or in a generalization thereof. See

  • Trimble n-category

  • Eugenia Cheng, Comparing operadic definitions of $n$-category (arXiv)