54A05-Isomorphism.tex

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\usepackage{pmmeta}
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A morphism $f: A \longrightarrow B$ in a category is an \emph{isomorphism} if there exists a morphism $f^{-1}: B \longrightarrow A$ which is its inverse. The objects $A$ and $B$ are \emph{isomorphic} if there is an isomorphism between them.

A morphism which is both an isomorphism and an endomorphism is called an \emph{automorphism}. The set of automorphisms of an object $A$ is denoted $\operatorname{Aut}(A)$.

Examples:
\begin{itemize}
\item In the category of sets and functions, a function $f: A \longrightarrow B$ is an isomorphism if and only if it is bijective.
\item In the category of groups and group homomorphisms (or rings and ring homomorphisms), a homomorphism $\phi: G \longrightarrow H$ is an isomorphism if it has an inverse map $\phi^{-1}: H \longrightarrow G$ which is also a homomorphism.
\item In the category of vector spaces and linear transformations, a linear transformation is an isomorphism if and only if it is an invertible linear transformation.
\item In the category of topological spaces and continuous maps, a continuous map is an isomorphism if and only if it is a homeomorphism.
\end{itemize}
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