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I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory.

What appears as an obstacle to me is the axiom of regularity, which solves some paradoxical foundational questions and the related well-foundedness concepts: In a category theoretical set theory, e.g. ETCS, the point is to avoid a membership predicate "$\in$" and nestings of sets like $b\in a,\ c\in b,\ d\in c,\dots$. Subobjects are characterized via arrows and the "regular" topoi which are like set theories characterize elements as single injections too.

Does category theory let us formulate any statement which relates to regularity?

Naively it would appear that the theory doesn't leave us any choice but exclude the phenomenon which the classical formulations axiomize away. But then to what extend does the theory leave you a choice and to what extend can you deviate from it?

Appearently non-well-foundedness is a thing now. (And I even read somewhere on MathOverflow that the proof of the abc-conjecture relates to it.) And so I assume you can speak about it in category theory as well.

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mathoverflow.net/a/118009/7206 may interest you (as may the rest of that thread, including the links there) –  Asaf Karagila Jul 4 '13 at 19:35
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Also, note that in $\sf ZFC$ it's perfectly fine to have $a\in b\in\ldots$, but you can't have $\ldots\in b\in a$. –  Asaf Karagila Jul 4 '13 at 19:39
    
@AsafKaragila: Thanks for the link, I'll read now. And I've never though about it that way, but of course, e.g. $a,\{a\},\{\{a\}\},\{\{\{a\}\}\},\dots$ are all legal. The question concernes only the downward chains then. –  NikolajK Jul 4 '13 at 19:50
    
Then you might want to correct your question, which at the moment speaks about upward chains. –  Asaf Karagila Jul 4 '13 at 19:55
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The main question is more important than the comments. –  Asaf Karagila Jul 4 '13 at 20:05

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