Show Answer
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?