# nLab associator

### Context

#### 2-Category theory

2-category theory

## Structures on 2-categories

#### Higher category theory

higher category theory

# Contents

## Idea

An associator in category theory and higher category theory is an isomorphism that relaxes the ordinary associativity equality of a binary operation.

### In Bicategories

In a bicategory the composition of 1-morphisms does not satisfy associativity as an equation, but there are natural associator 2-morphisms

$h \circ (g \circ f) \stackrel{\simeq}{\Rightarrow} (h \circ g) \circ f$

that satisfy a coherence law among themselves.

If one thinks of the bicategory as obtained from a geometrically defined 2-category $C$, then the composition opeeration of 1-morphisms is a choise of 2-horn-fillers and the associator is a choice of filler of the spheres $\partial \Delta[3] \to C$ formed by these.

### In monoidal categories

By the periodic table of higher categories a monoidal category is a pointed bicategory with a single object, its objects are the 1-morphisms of the bicategory. Accordingly, here the associator is a natural isomorphism

$a_{x,y,z} : (x \otimes y) \otimes z \to x (\otimes y \otimes z)$

relating the triple tensor products of these objects.

## Examples

Revised on October 4, 2013 01:24:21 by Urs Schreiber (82.169.114.243)