ct.category theory - Is there a category of non-well-founded sets?

Starting from any model of a membership-based set theory, be it well-founded or ill-founded, you can construct a category of sets and functions. The basic properties of this category (e.g. it is a well-pointed topos) don't depend on whether the model you started from was well- or ill-founded. In fact, in the presence of the axiom of choice, every ill-founded set is still well-orderable, hence bijective to a well-founded set (a von Neumman ordinal) -- thus the category of sets obtained from a model of ill-founded set theory + choice is equivalent to the subcategory obtained from its submodel of well-founded sets.

You can then ask whether you can reconstruct a model of membership-based set theory from its category of sets; here is where the graphs come in (to model hereditary membership relations), as at nlab:pure set. You can choose to use well-founded graphs or ill-founded ones. If the one you choose matches the type of set theory you started from, then (as long as your set theory is strong enough otherwise) you'll reconstruct the same model. If you choose well-founded graphs starting from an ill-founded set theory, then you'll reconstruct the submodel of well-founded sets, reproducing the proof of relative consistency of the axiom of foundation. And if you choose ill-founded graphs starting from a well-founded set theory, you'll reproduce Aczel's original proof of the relative consistency of the anti-foundation axiom.