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?