nLab unitor



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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 with tensor product \otimes and tensor unit11” is equipped with a natural isomorphism of the form

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.

Last revised on April 5, 2023 at 15:41:30. See the history of this page for a list of all contributions to it.