Relations are hard to model in category theory


WARNING: This is a somewhat rambling post about category theory.  If half-baked mathematical philosophy is not your thing, please consider navigating away right now.

Anyway, the thing that I want to write about today is the difference between category theory and what I shall call for lack of a better term the older “set theoretic” approach.  My goal is to try to articulate what I see as the main difference is between these two structures, and why I think that while category theory offers many insights and new perspectives it is probably hopeless to try to shoe horn all of mathematics into that symbolism.

Relation Theory

If you’ve ever taken an advanced math course, you probably already know that set theory is default “programming language” of modern mathematics.  Modern set theory is built upon two basic components:

A set is any well defined unordered collection of objects, the…

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