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Has any work been done on what $MK+CC$ looks like as a foundation for category theory? Is it 'the same' as restricting to inaccessibles in some precise manner?

According to wikipedia, any category with all unrestricted limits is thin. This is typically formulated in $ZFC$ as the statement that any small category with all small limits is thin since we can only quantify over sets, or in $GBC$/$MK$ we can say outright that a large category with all large limits is thin but a large category with small limits may be interesting.

I am interested in the possibility of working in $MK+CC$ as discussed by Joel David Hamkins and Kameryn Williams here, where we have the ability to 'turn $O_n$ into a largest cardinal and keep going', as a foundation for category theory. More specifically, I am interested in any advantages/disadvantages this approach may offer when compared to the standard fare of working with Grothendieck universes and inaccessibles.

In particular, I wonder whether a category which is 'full-sized' as mentioned by Andreas Blass in comments of the above link but only has 'class-sized' limits might be interesting in similar fashion to a large category with small limits.

Mike Shulman has an excellent paper discussing these issues in some depth which I'm currently working through, but has asked a question about class collection axioms since its publication and I would be interested to hear any updates on his opinion.

I am hesitant to restrict to any cardinals irrespective of how large they are because I'm trying to understand the Galois precategory of $[N_0:Frac(\mathfrak{G}(O_n))],$ the surreal numbers over the field of fractions of the Grothendieck ring of the ordinals as a field extension, and both fields will contain a copy of any large cardinals in the universe so restricting to some particular one will cut off part of the field I'm trying to work with. I'm additionally interested in analysis over the surreals and certain analytical statements over the reals are known to have a consistency strength exceeding ZFC, so I intuitively feel that certain analytical statements over the surreals (should they exist) will have greater consistency strength.

It's certainly possible that there is some business about initial chunks of the surreals modeling their full behavior which I'm not seeing; I'm open to an argument in favor of this if one exists.

EDIT: To be precise, the sense in which I mean a 'foundation for category theory' is an axiomatic system capable of modeling all the things a category theorist would typically like to do. For example $MK$ fails to be such an axiomatic system since, despite being able to define large categories, we are left with functor categories between large categories being undefined/empty -- this is no longer the case in $MK+CC$.

Categories are axiomatic objects themselves so this request may seem odd, but in completely precise terms I believe I'm asking if $KM+CC$ is a model of the axioms of category theory, in much the same way that we can axiomatically characterize the reals or construct them in $ZFC$ and say that $ZFC$ models the reals.

Alec Rhea
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  • I believe the question requires formulating more precisely what do you mean by category theory: seems like its most common meaning is different from what you have in mind. – მამუკა ჯიბლაძე Aug 17 '18 at 10:02
  • @მამუკაჯიბლაძე Could you clarify how the standard meaning differs from mine? I mean as an axiomatic system capable of modeling the categorical behavior that category theorists care about -- for example, $MK$ fails to be such a system despite allowing class quantification since the category of functors between large categories is undefined/empty in $MK$, but this is no longer the case in $MK+CC$. (also thank you for catching the typo, it's late over here) – Alec Rhea Aug 17 '18 at 10:10
  • I had in mind just plain elementary theory of categories - an equational first order theory with two sorts (for objects and morphisms); if preferred, there is a version with a single sort for morphisms, but I believe this is not relevant at all. – მამუკა ჯიბლაძე Aug 17 '18 at 10:38
  • Is this what you mean too? I don't see how – მამუკა ჯიბლაძე Aug 17 '18 at 10:42
  • Concerning your last addition - certainly from any model of any version of set theory you can produce a model of the theory I have in mind; in fact, very weak fragments of set theory suffice for that. – მამუკა ჯიბლაძე Aug 17 '18 at 10:46
  • @მამუკაჯიბლაძე I am confused -- how does this mesh with Andreas Blass's answer here: https://mathoverflow.net/questions/273551/category-theory-from-mk-class-theory-perspective? He seems to indicate that $MK$ is insufficient for category theory for the reasons I have outlined above and more. Is it possibly because we're passing to second (or nth) order category theory when we move to classes of functors between large categories etc. and second (or nth) order category theory has larger consistency strength? This stuff is outside my domain of expertise, apologies for any undue ignorance. – Alec Rhea Aug 18 '18 at 19:32
  • I am probably way more ignorant in these questions than you. I only want to say that your question seems to be about comparison of some variants of set theory with some variant of category theory, and while I more or less understand which theories do you have in mind on the set theory side, I have no idea what do you have in mind on the category theory side. – მამუკა ჯიბლაძე Aug 18 '18 at 20:18
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    @მამუკაჯიბლაძე In some sense I'm asking what the category $Sets$ looks like in $MK+CC$; for example in $ZFC$ $Sets$ is well behaved in all the expected ways but we can't define large categories or larger iterations thereof, while in Ackermann set theory we can define large categories and their iterations but $Sets$ is not cartesian closed (see Mike's paper above for a reference). But this is not the whole question since I would also like to know about what categorical constructions can be legitimately undertaken when the background universe we work in is $MK+CC$, and if anything new pops out. – Alec Rhea Aug 19 '18 at 18:20
  • Yes, for me your last comment is in perfect accordance with the content of your question; and I think it shows that the title of the question is actually misleading: what does this have to do with a foundation for category theory? – მამუკა ჯიბლაძე Aug 20 '18 at 04:42
  • @მამუკაჯიბლაძე I am asking about the class theory $MK+CC$ as a foundation for category theory in the same sense that we ask for $ZFC$ to be a foundation for all other parts of mathematics - can we model everything that one would like to do in category theory in the class theory $MK+CC$, in the same way that we can model most of analysis and topology in $ZFC$? There are certain pieces of measure theory that might make a measure theorist want $ZFC$+"there are measurable cardinals" as his background theory, or a topologist might want $ZFC$+continuum hypothesis; do category theorists want $MK+CC$? – Alec Rhea Aug 20 '18 at 19:33
  • @მამუკაჯიბლაძე If you'd like to make an argument in favor of some weak set theory as a foundation for category theory as an answer I'd be interested to read it, but I don't understand how this dovetails with Andreas Blass's answer above which indicates that $MK$ is too weak despite being stronger than $ZFC$. – Alec Rhea Aug 20 '18 at 19:35

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