# 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

$h\circ \left(g\circ f\right)\stackrel{\simeq }{⇒}\left(h\circ g\right)\circ f$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 $\partial \Delta \left[3\right]\to C$ 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

${a}_{x,y,z}:\left(x\otimes y\right)\otimes z\to x\left(\otimes y\otimes z\right)$a_{x,y,z} : (x \otimes y) \otimes z \to x (\otimes y \otimes z)

relating the triple tensor products of these objects.

Revised on May 26, 2011 10:38:14 by Anonymous Coward (145.116.22.47)