Interpreting ETCS in ZFC I am currently doing a project on foundation of mathematics. Set theory and category theory are two different approaches to describe the core principles. ZFC is an axiomatic system which is inspired by set theory that is commonly accepted as the foundation of maths. Then there's also ETCS axiomatisation which has categorical origins.
I am looking into category theory as of now to have a better undertstanding
of the context and motivation behing ETCS and looking to interpret ETCS in ZFC. However, i am very overwhelmed and not sure where to start my research into category theory and struggling to find any good reasources that shows the connection between the two axiomatic systems. So i would greatly appreciate if you guys could let me know some good introductory resources.
Thank you in advance.
 A: The most introductory material related to ETCS is Trimble’s informal notes starting here: https://ncatlab.org/nlab/show/Trimble+on+ETCS+I
However, this may not be introductory enough, depending on your background, since you say you’re quite new to category theory. Leinster’s book Basic Category Theory is where’d I’d usually send an undergraduate (my best guess at your level, do let me know if I’ve misjudged) wanting to learn some category theory proper, which is likely to be necessary to understand much about ETCS. You would also likely find Lawvere and Rosebrugh’s book Sets for Mathematics a useful and initially gentle introduction to both category theory and how it thinks about set theory.
It might be worth noting a warning here that ETCS is not explicitly used as a logical foundation by any (as far as I know) working mathematicians in the way that ZFC is. Rather it purports to better explicate how such mathematicians actually assume sets to behave. Thus the purposes of the two systems are not perfectly comparable. Penelope Maddy’s papers on “believing the axioms” might be a good starting point if you hope to clarify your thoughts about what exactly a foundation is for.
