# Is there a category of non-well-founded sets?

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.

-
Thank you Neel and Mike for your answers. I found the nLab notes helpful, as well as <a href="plato.stanford.edu/entries/nonwellfounded-set-theory/… section</a> of the SEP article. Sorry, though, I think I meant to ask a more open-ended question, perhaps unsuitable for mathoverflow. I am interested in references where a category of non-well-founded sets is used as an alternative to the category of sets. –  Noam Zeilberger Feb 4 '10 at 22:41

Yes, there is. See Peter Aczel's 1988 notes on models of non-well-founded set theory, here. 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, and also on their page on material set theory.

-
I was aware of Aczel's notes, but the nLab links were helpful, and this answers the title question. My question was meant a bit more open-endedly, though, especially whether a category of non-well-founded sets has been studied/used systematically, like say in categorical logic. –  Noam Zeilberger Feb 4 '10 at 22:48