Group Theory/Groups with structure

Definition (group with structure):

Let be a concrete category. Then the category of groups with structure is the subcategory of which is defined as follows:

  • Its objects are the objects of such that the categorical product exists (and, on a set level, equals the set-theoretic product of with itself, projections included) and whose underlying sets bear a group structure such that the group law is a morphism in and inversion is a morphism in .
  • Its morphisms are morphisms in that, on the set level, are also group homomorphisms.

Proposition (multipication by an element is an automorphism in the underlying concrete category for every group with structure in a category admitting enough constant morphisms):

Let be a group with structure whose additional structure is given by the concrete category , such that every constant morphism is a morphism of . Further, suppose that . Then the function

is an automorphism of in the category .

Proof: If we show that the given function is a morphism, we've completed the proof, since an inverse is given by left multiplication by .

Thus, consider the morphism whose first component is given by the constant function associated to and whose second component is given by the identity on : Postcomposing it with the group law yields the morphism in the theorem statement, which is hence a morphism of .