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\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \)
Math and science::Algebra::Aluffi

Category of Groups

There is a meaningful way to define a category with groups as objects and with morphisms being set functions between the underlying sets of two groups.

Category \( \cat{Grp} \)

The category \( \cat{Grp} \) has groups as objects. Let \( (G, m_G) \) and \( (H, m_H) \) be two groups in \( \cat{Grp} \). Then a set function \( \varphi : G \to H \) is a morphism from \( (G, m_G) \) to \( (H, m_H) \) iff \( \varphi \) preserves the group structure.

The group structure is preserved iff:

[\[ \begin{align} \forall a, b \in G, & \\ & \varphi( \; ? \;) = ?( \; \varphi(a), \varphi(b) \;) \end{align} \]]

\( (G, m_G) \) is the tuple of the underlying set \( G \) and the operation \( m_G \) for group \( G \).

Can you remember a visualization for the preservation of group structure?