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 $f \in {Hom}_{C} (A, B)$ is an isomorphism iff there exists a [what?] such that both:

[

[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 $f \in {Hom}_{C} (A, B)$ and $g \in {Hom}_{C} (B, C)$ are isomorphisms, then $g f$ is an isomorphism and [ $(g f)^{- 1} = ?$ ].

05.10.2020