Explore BrainMass
Share

Explore BrainMass

    Category Theory - Morphisms, Uniqueness & Equivalence

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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?

    © BrainMass Inc. brainmass.com October 9, 2019, 10:44 pm ad1c9bdddf
    https://brainmass.com/math/linear-transformation/category-theory-morphisms-uniqueness-equivalence-230525

    Solution Summary

    Category equivalence is investigated.

    $2.19