nLab
unitor

• associator , unitor, Jacobiator

• pentagonator, hexagonator

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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

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

$$f\circ \mathrm{Id}\stackrel{\simeq }{\Rightarrow }f$$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, 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.