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Exercise 30 - Chapter 3

edited June 2018 in Exercises

Show that in any category \(C\), for any object \( c \in C \), the identity \( id_c \) is an isomorphism.

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  • 1.

    Per the definition of isomorphism, it is sufficient to find a morphism \(g\in\mathcal{C}(c,c)\)) such that \(g\circ\text{id}_c=\text{id}_c\) and \(\text{id}_c\circ g=\text{id}_c\). For any object in any category, the identity morphism is guaranteed to exist and \(\text{id}_c\circ\text{id}_c=\text{id}_c\). So, \(\text{id}_c\) is an isomorphism with inverse given by \(\text{id}_c\).

    Comment Source:Per the definition of isomorphism, it is sufficient to find a morphism \\(g\in\mathcal{C}(c,c)\\)) such that \\(g\circ\text{id}_c=\text{id}_c\\) and \\(\text{id}_c\circ g=\text{id}_c\\). For any object in any category, the identity morphism is guaranteed to exist and \\(\text{id}_c\circ\text{id}_c=\text{id}_c\\). So, \\(\text{id}_c\\) is an isomorphism with inverse given by \\(\text{id}_c\\).
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