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Math and science::Algebra::Aluffi

Isomorphism

For [...] between sets, we defined the notion of injective, surjective and bijective/isomorphic functions. We do the same for morphisms between objects of a category. This card covers isomorphisms; other cards cover monomorphisms and epimorphisms.

An isomorphism is defined as follows:

Isomorphism. Definition.

Let \( C \) be a category. A morphism \( f \in \cathom{C}(A, B) \) is an isomorphism iff there exists a [what?] such that both:

[\[  \text{[statement], [statement]} \]]

\( g \) is said to be a (two sided) inverse of \( f \).

Uniqueness

Can \( f \) have multiple inverses? Uniqueness of \( g \) is not built into the definition above; however, it is indeed true that \( g \) is unique:

Theorem

An inverse of an isomorphism is unique.

Proof on the other side.

Since an inverse is unique, there is no ambiguity in denoting it as \( f^{-1} \).

Auxiliary propositions

Here are three useful propositions related to isomorphisms:

  • [Every something] is an isomorphism and is its own inverse.
  • If \( f \) is an isomorphism then [something] is an isomorphism and [\( \, ? \; = f \) ].
  • If \( f \in \cathom{C}(A, B) \) and \( g \in \cathom{C}(B, C) \) are isomorphisms, then \( g\, f \) is an isomorphism and [\( (g \, f)^{-1} = \; ? \; \)].