The n-Category Café

We did indeed construct the quotient of a set by an equivalence relation, and prove its universal property and the first isomorphism theorem for sets (which is the core of its more famous algebraic cousins). And we did indeed construct coproducts, using a technique that looks much more like the ZFC trick of “$\{0\} \times X \cup \{1\} \times Y$” than it looks like Paré’s sublime construction of colimits in a topos.

But that’s not all! We also did an isomorphism-invariant version of “there is no set of all sets”, proving that for any set-indexed family of sets $(X_i)_{i \in I}$, there is some set not isomorphic to any of its members $X_i$. And, most excitingly, we used coproducts to prove the Cantor–Bernstein theorem: if you’ve got sets $X$ and $Y$ and know that there exist injections $X \leftrightarrows Y$, then $X$ and $Y$ must, in fact, be isomorphic.