> You'll see why it's useful to have a formal justification for treating isomorphic objects as equal (this is called "univalence"), which is something you may have even been doing informally/subconsciously all along.
Would you be able to discuss how this compares to quotienting? It sounds similar, but I assume has very important differences. I have, very formally, been doing quotienting. I identify an equivalence relation, quotient it, prove that the properties and operations of interest are well-defined when lifted to the equivalence classes, and then work exclusively at the level of the equivalence classes and the lifted operations. Say, when I work exclusively with reals, which are equivalence classes of Cauchy sequences, or with cosets in group theory.
Would you be able to discuss how this compares to quotienting? It sounds similar, but I assume has very important differences. I have, very formally, been doing quotienting. I identify an equivalence relation, quotient it, prove that the properties and operations of interest are well-defined when lifted to the equivalence classes, and then work exclusively at the level of the equivalence classes and the lifted operations. Say, when I work exclusively with reals, which are equivalence classes of Cauchy sequences, or with cosets in group theory.