TITLE: Is there a category of non-well-founded sets?
QUESTION [12 upvotes]: 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.

REPLY [2 votes]: According to this doctoral thesis http://www.andrew.cmu.edu/~awodey/students/hughes.pdf by Jesse Hughes (supervised by Steve Awodey), cogebras = coalgebras are the appropriate category to study non-wellfounded sets.