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    $\begingroup$ What do you mean by a core which uses category theory? Like first order logic with ETCS (or ETCC) instead of ZFC? If not, then one could argue that type theory is just formal category theory. (Although I guess one could take a category-theoretic-first approach to designing the type system making sure it is exactly the semantics of a particular family of categories, like a topos. Maybe homotopy type theory fits that bill.) $\endgroup$
    – Jason Rute
    Commented May 7 at 20:36
  • $\begingroup$ Or is the point that “category” should be the fundamental notion instead of say “set”, “type”, or “function”? I think this is what ETCC does, but I don’t think it is in any proof assistant. I guess I have also heard of people trying to come up with a type theory where the types represent (higher?) categories (instead of sets, groupoids, etc). I don’t know if there has been progress. $\endgroup$
    – Jason Rute
    Commented May 7 at 20:43
  • $\begingroup$ I mean your second comment, that "category" (maybe along with functors, natural transformations, etc.) is the fundamental notion. I ran across this paper A Formal Logic for Formal Category Theory that seems to be somewhat in that direction, embedded in Agda. I have not seen any standalone proof assistants. $\endgroup$
    – Chris Henson
    Commented May 8 at 0:05
  • $\begingroup$ Does something like Metamath + first-order theory of an abstract category count? $\endgroup$
    – siegment
    Commented May 8 at 4:24