Is there a category (in the category theory sense) of non-well-founded sets (something analogous to Set, the category of sets), and has it been (well-)studied? Any references are appreciated.



Is there a category of non-well-founded sets?
Yes, there is. See Peter Aczel's 1988 notes on models of non-well-founded set theory, [here](http://standish.stanford.edu/pdf/00000056.pdf). The basic idea is that you can model sets as graphs (ie, as a collection of element sets together with a binary relation intuitively suggesting membership), and then the foundation axiom corresponds exactly to the assertion that the membership relation is well-founded. Allowing non-well-founded sets corresponds to allowing arbitrary relations. 


These graphs are perfectly ordinary mathematical objects, and so we can collect them into a category exactly the way we do with "normal" sets. The nLab has some notes on this on [their page on pure sets](http://ncatlab.org/nlab/show/pure+set), and also on their page on [material set theory](http://ncatlab.org/nlab/show/set+theory#material_set_theory_4).


