Introduction to nonwell-founded set theory I'm interested in playing with nonwell-founded variants of set theory and weaker/different axioms of induction/extensionality.
I have a hunch coalgebraic methods could better handle weirdness like modelling homotopy type theory.
I also have just been interested in the idea of coinduction as primitive as opposed to induction.
Weak limits like weak function spaces are also useful in Computer Science for higher order abstract syntax. So set-theory minus induction/extensionality could lead to cleaner encodings of stuff like the lambda calculus.
But I really don't know where to start with nonwell-founded set theory in the first place. Unfortunately nonwell-founded set theory seems a bit esoteric.

One version of non-well founded set theory arises when we replace the Axiom of Foundation with the Anti-Foundation Axiom (AFA), explored in an influential book written by Peter Aczel in 1988.
The obvious place to start finding out more about this is the Stanford Encyclopaedia article https://plato.stanford.edu/entries/nonwellfounded-set-theory/
