Category Theory - Morphisms, Uniqueness & Equivalence
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If f:A-->B is an equivalence in a category C and g:B-->A is a morphism such that gf=1_A (the identity on A) and fg=1_B, show that g is unique.
Prove that any two universal (initial) objects in a category C are equivalent.
Prove that any two couniversal (terminal) objects in a category C are equivalent.
In the category of abelian groups, show that the group A_1 x A_2 together with the homomorphisms f:A_1 --> A_1 x A_2, f(x)=(x,e) and g:A_2 --> A_1 x A_2, g(x)=(e,x) is a coproduct for {A_1,A_2}. Why is this not a coproduct in the category of groups?
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Solution Summary
Category equivalence is investigated.
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