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

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

\( (G, m_G) \) 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?


Notation

In the more usual notation, the preservation of group structure is expressed as:

\[ \forall a, b \in G, \quad \phi( a \cdot_G b) = \varphi(a) \cdot_H \varphi(b) \]

Or even more tersely, with the operations being implicit:

\[ \forall a, b \in G, \quad \quad \varphi( a \cdot b) = \varphi(a) \cdot \varphi(b) \]

Preservation Condition

Below is a figure expressing the group preservation condition for a set function \( f : G \to H \).

In words:

If \( f : G \to H \) is a set function, then for any \( a, b \in G \) and any \( x \in H \), \( f \) applied to \( a \) and \( b \) individually before \( \cdot_H \) must map to the same element as \( f \) applied to the result of \( \cdot_G \).

TODO: add the commutative diagram. TODO: talk about the morphism preserving identity and inverse. TODO: talk about composition and associativity of the group morphism.