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Note: This question takes a pluralistic / "multiverse" view of sets (including "classes" and "collections") and set theories.

My understanding (from reading many nLab articles) is that there are at least two category-theoretic notions$^\dagger$ of "set":

  1. ("intramural" / "within-category" / "internal") a "set" is any discrete category${^\dagger} {^\dagger}$.

  2. ("extramural" / "between-categories" / "external") a "category of sets" is a pretopos, possibly with additional regularity conditions${^\dagger} {^\dagger} {^\dagger}$. A "set" is then just an object of a "category of sets".

Question: To what extent are these two definitions equivalent?

Pointers to references would be appreciated.

For example, given any category all of whose objects are discrete categories and whose morphisms are functors, is there always an embedding of categories ("inclusion" / faithful functor injective on objects) into (some) pretopos?

And conversely, given any pretopos is it always equivalent (even isomorphic) to some category all of whose objects are discrete categories and whose morphisms are functors?

Further notes:
The "intramural" vs. "extramural" terminology comes from here.

$^\dagger$: I.e. "definitional set theories" (in the sense of this article) that define sets / set theories in terms of category theory, models (or rather interpretations?) of sets / set theories in categorical model theory, studying sets / set theories as object theories with (the first-order theory of) category theory as the metatheory. Cf. (1), (2), (3), (4), (5), (6), (7), etc. This is in contrast to the common approach of studying category theory as the object theory with a fixed set theory (ZFC) as the metatheory.

${^\dagger} {^\dagger}$: I.e. any $0$-category, any category equivalent to its $0$-truncation.

Seemingly defining whether a category is "small" requires studying category theory as the object theory with a fixed set theory as the metatheory, whereas this question is about defining non-fixed set theories as object theories with category theory as the metatheory. Seemingly whether the "skeletal" requirement is acceptable depends on how one feels about first order logic with equality vs. without equality, but because it is not difficult to translate between the two flavors of logic, and because any category is equivalent to any of its skeletons, I don't care whether the "skeletal" requirement is included or not.

${^\dagger} {^\dagger} {^\dagger}$: In ETCS the definition is a cocomplete well-pointed topos with a natural numbers object and satisfying the axiom of choice, others say more generally just a cocomplete well-pointed topos, others would just say an (elementary) topos, some constructive or predicativist set theories correspond to just a pretopos, and "categories of classes" and/or "categories with class structure" are also pretoposes / pretopoi (with additional properties / regularity conditions). So "pretopos" seems to be the most general / pluralistic definition. (Again I want to ignore size issues because that seems to presuppose a fixed set theory as metatheory, rather than considering non-fixed set theories as object theories, so the definition should work for any category of "classes" / "collections", not just for categories of "proper" / "small" sets.)

Related questions: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

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  • $\begingroup$ @DmitriP. I'm not sure I understand the point being made, but one thing that came to mind for me that seems related: the same way a category of "all" categories can't be defined consistently due to size and self-referential related issues, probably a category of "all" discrete categories can't be defined consistently either. In other words the "extramural" definition inherently has more restrictions than the "intramural" one. That's fine though, but how do we know the intramural definition is consistent with any choice of extramural definition (the latter doesn't look at internal structure)? $\endgroup$ Mar 16 at 13:40
  • $\begingroup$ In particular though, regarding the claim "the answer to the main question is tautological" using "the traditional definition of a (small) category", I don't understand how that is true at all. Why is it "tautological" that a category of all small discrete categories (a definition that makes zero reference to extramural constructions) is necessarily a pretopos (a definition that makes zero reference to any internal / "intramural" properties of the objects of the corresponding category)? And why is a pretopos necessarily equivalent to a category of (small) discrete categories "tautologically"? $\endgroup$ Mar 16 at 13:43
  • $\begingroup$ @DmitriP. " The traditional definition of a category (and therefore of a discrete category) makes use of an ambient category of sets or classes, necessarily extramural" this makes it sound as if you did not read the question, because the question is not asking about category theory interpreted using the first order theory of ZFC (or ZF or ZC or something else), but about ZF/ZFC/ZC interpreted using the first order theory of categories, which does not require an ambient category of sets or classes to be defined. Possibly I am misunderstanding the purpose of the comment however. $\endgroup$ Mar 19 at 23:20
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    $\begingroup$ The first-order theory of categories (as defined in the cited reference ncatlab.org/nlab/show/theory+of+categories) does not say how to define the category of categories or the category of discrete categories. If you take the category of models of a first-order theory in the usual sense, it depends on the ambient set theory, making the answer to the original question tautological. If not, then a precise definition of the “category of discrete categories” should be given first, otherwise the question is far too vague. $\endgroup$
    – Dmitri P.
    Mar 20 at 2:19
  • $\begingroup$ I don't disagree. In light of that, the question could be rephrased: 'How to give a precise definition of the "category of discrete categories" (without using an ambient set theory), and why make the choices in that precise definition?'. It seems the choices for precise definitions of categories of sets / classes usually involve a pretopos. (Where "classes", even if 'small' in the 'universe' of 'all categories' definable only in the meta-meta-theory, could still be 'large' in the metatheory, e.g. basically first-order formulas of the metatheory math.stackexchange.com/a/1896385/606791 $\endgroup$ Mar 20 at 14:43

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