Stands for the Elementary Theory of the Category of Sets.

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### Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...

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### Can we define geometric morphisms (between ETCS categories) elementarily?

The ETCS axioms give conditions on a category for it to be a category of sets. These axioms can be written out in first order language, resulting in a finite axiomatisation of the category of sets. ...

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### How do we compare models of ETCS?

The elementary theory of the category of sets (nLab) gives axioms on a category such that it is a category of sets. In answering this MO question I realised that we might have trouble comparing ...