nLab unitor

Contents

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 there are natural unitor 2-morphisms

$Id \circ f \stackrel{\simeq}{\Rightarrow} f$
$f \circ Id \stackrel{\simeq}{\Rightarrow} f$

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, a monoidal category is equipped with a natural isomorphism

$\ell_x : 1 \otimes x \to x$

called the left unitor, and a natural isomorphism

$r_x : x \otimes 1 \to x$

called the right unitor.

Last revised on July 24, 2017 at 15:09:17. See the history of this page for a list of all contributions to it.