nLab unitor

Context

2-Category theory

2-category theory

Structures on 2-categories

Higher category theory

higher category theory

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 ther are natural unitor 2-morphisms

$\mathrm{Id}\circ f\stackrel{\simeq }{⇒}f$Id \circ f \stackrel{\simeq}{\Rightarrow} f
$f\circ \mathrm{Id}\stackrel{\simeq }{⇒}f$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, aby monoidal category is equipped with a natural isomorphism

${\ell }_{x}:1\otimes x\to x$\ell_x : 1 \otimes x \to x

called the left unitor, and a natural isomorphism

${r}_{x}:x\otimes 1\to x$r_x : x \otimes 1 \to x

called the right unitor.

Revised on September 17, 2010 06:57:57 by Toby Bartels (173.190.154.103)