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.