nLab
associator

Context

2-Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations


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(gf)(hg)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 CC, 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 Δ[3]C\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:(xy)zx(yz) 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)