Homotopy Type Theory unitary isomorphism in a dagger precategory > history (Rev #1)

Definition

A morphism f:hom A(a,b)f:hom_A(a,b) of a dagger precategory AA is a unitary isomorphism or dagger isomorphism if f f=1 af^\dagger \circ f=1_a and ff =1 bf \circ f^\dagger =1_b. We write a ba \cong^\dagger b for the type of unitary isomorphisms.

Properties

For any f:hom A(a,b)f : hom_A(a,b) the type “ff is a unitary isomorphism” is a proposition. Therefore, for any a,b:Aa,b:A the type a ba \cong^\dagger b is a set.

For a,b:Aa,b:A, if p:(a=b)p:(a = b), then idtoiso(p)idtoiso(p) (as defined in precategory) is a unitary isomorphism.

See also

Category theory

References

category: category theory

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