\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \newcommand{\E} {\mathrm{E}} \)
deepdream of a sidewalk
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)
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 [\( (g \, f)^{-1} = \; ? \; \)].