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Question 1.3 - Is it an isomorphism?

Is it an isomorphism?

Suppose that someone tells you that their category \(C \) has two objects \(c,d \) and two non-identity morphisms, \(f : c \to d \) and \(g : d \to c \), but no other morphisms. Does \(f \) have to be the inverse of \(g \), i.e. is it forced by the category axioms that \( g◦f = id_c \) and \(f◦g= id_d \)?

Comments

  • 1.

    My argument is as follows :

    (i) \(f◦g\) is a morphism from \(d \to d \). Since the only morphism in \(C \) from \(d \to d \) is \(id_d \) we must have \(f◦g= id_d \).

    (ii) \(g◦f\) is a morphism from \(c \to c \). Since the only morphism in \(C \) from \(c \to c \) is \(id_c \) we must have \(g◦f= id_c \).

    Comment Source:My argument is as follows : (i) \\(f◦g\\) is a morphism from \\(d \to d \\). Since the only morphism in \\(C \\) from \\(d \to d \\) is \\(id_d \\) we must have \\(f◦g= id_d \\). (ii) \\(g◦f\\) is a morphism from \\(c \to c \\). Since the only morphism in \\(C \\) from \\(c \to c \\) is \\(id_c \\) we must have \\(g◦f= id_c \\).
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