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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 Grp

The category Grp has groups as objects. Let (G,mG) and (H,mH) be two groups in Grp. Then a set function φ:GH is a morphism from (G,mG) to (H,mH) iff φ preserves the group structure.

The group structure is preserved iff:

[(1)a,bG,(2)φ(?)=?(φ(a),φ(b))]

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

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