Abstract: Lawvere's Elementary Theory of the Category of Sets (ETCS) characterises when a category E behaves 'similarly' to the category of sets. More formally, it is an axiomitization of categorical models of ZFC except for the axiom schema of replacement. As such, it provides the language for a category-theoretic foundation of mathematics. In this talk, I will define relevant category-theoretic notions and give an idea of how these relate to the ZFC axioms. I will also exemplify how one can do logic internal to a category satisfying ETCS (or more generally, in an elementary topos). No prior category theoretic knowledge is required. This talk is a precursor to some work that I have been doing with Arian on developing a 2-dimensional version of this; the elementary theory of the 2-category of small categories.
