nLab
unitor

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

A unitor in category theory and higher category theory is an isomorphism that relaxes the ordinary uniticity equality of a binary operation.

In bicategories

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

Idff Id \circ f \stackrel{\simeq}{\Rightarrow} f
fIdf f \circ Id \stackrel{\simeq}{\Rightarrow} f

that satisfy a coherence law among themselves.

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, a monoidal category is equipped with a natural isomorphism

x:1xx \ell_x : 1 \otimes x \to x

called the left unitor, and a natural isomorphism

r x:x1x r_x : x \otimes 1 \to x

called the right unitor.

Revised on July 24, 2017 15:09:17 by Peter Heinig (2003:58:aa1f:f900:8285:32a9:47b1:f47c)