nLab
category over an operad

Contents

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 OO in some symmetric monoidal category CC, a category over the operad OO, or OO-category DD is

  • a set/class/whatever D 0D_0, called the set of objects of DD;

  • for each pair x,yD 0x,y \in D_0 an object D(x,y)C 0D(x,y) \in C_0, called the object of morphisms from xx to yy in DD;

  • for each natural number nn and each sequence x 0,x 1,,x nx_0, x_1, \cdots, x_n of objects of D 0D_0 a morphism

    comp (x 0,,x n):(D(x 0,x 1)D(x 1,x 2)D(x n1,x n))O(n)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 D(x_{n-1},x_n) \right) \otimes O(n) \to D(x_0, x_n)

    called the nn-ary composition operation;

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

Examples

Revised on July 21, 2014 10:36:01 by Nate Bottman? (18.111.7.72)