isomorphism class

Let \(\mathcal{C}\) be a category, then the isomorphism class of an object \(A \in \mathrm{Ob}\) is the equivalence class of objects isomorphic to \(A\)

\begin{align*} A \sim B \Leftrightarrow \exists f \in \mathrm{Hom}_{\mathcal{C}}(A,B), \text{ isomorphism} \end{align*}

• reflexive relation follows from the identity map
• symmetric relation follows from the symmetry of the inverses
• transitive relation follows from \((f \circ g) \circ (g^{-1} \circ f^{-1})\)