Talk:PlanetPhysics/Isomorphism

Original TeX Content from PlanetPhysics Archive

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\begin{document}

\textbf{Definition 0.1} \bigbreak A \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $f: A \to B$ in a \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} \textbf{$C$} is an \emph{isomorphism} when there exists an \emph{inverse morphism} of $f$ in \textbf{$C$}, denoted by $\inv f: B \to A$, such that $f \circ \inv f =id_A = 1_A: A \to A$.

One also writes: $A \cong B$, expressing the fact that the \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} A is isomorphic with object B under the isomorphism $f$.

Note also that an isomorphism is both a \htmladdnormallink{monomorphism}{http://planetphysics.us/encyclopedia/InjectiveMap.html} and an epimorphism; moreover, an isomorphism is both a section and a retraction. However, an isomorphism is not the same as an \emph{\htmladdnormallink{equivalence relation}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}}.

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