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 [$$(g \, f)^{-1} = \; ? \;$$].