I'm looking for a text that treats category theory from the perspective of MK class theory.


MK is already very well-designed and equipped for the type of abstraction that occurs in category theory, and I suspect that this together with its ability to discuss proper classes with relative ease would make a treatment in that context very elegant and concise.


EDIT: It is also my understanding that many category theorists are less interested in viewing set theory as a foundation for mathematics, and more interested in it as simply a base language with which we can construct collections of objects in an intuitive manner and then manipulate them. It seems to me that a good option in this case is to simply pick an absurdly strong set theory to use as the underpinning 'language', and MK is more powerful than other widely studied set theories without the existence of inaccessible cardinals.



Category theory from MK class theory perspective?
Morse-Kelley set theory doesn't seem adequate for all the things one would like to do in category theory. It provides a nice treatment of proper classes, so it can deal with large categories like the category of sets or the category of groups. It can deal with a functor between two such categories, like the forgetful functor from groups to sets or the "free group" functor from sets to groups. But the category of all functors between two large categories (or even from a large category to a small one) is in general a collection of classes, and that's beyond what MK can handle directly.


For a smooth development of things like functor categories, Kan extensions, and related concepts, especially if you want to handle iterations, like the category of functors between two categories of functors between large categories, you'll want infinitely many levels of the cumulative hierarchy of sets beyond your large categories. In other words,you'll want not only classes but also super-classes (i.e., collections of classes), super-duper classes, etc. That sort of set theory is stronger than MK. It's still considerably weaker than an inaccessible cardinal, but the underlying intuition is pretty close to an inaccessible cardinal. 


Furthermore, once you've introduced all these super-duper-etc.-classes, your original large categories, like the category of sets, don't look so comprehensive any more; you might want a category of all sets, classes, super-classes, etc. Unless I put some serious constraints on my wishful thinking, I'm likely to end up just where Grothendieck did, in a universe full of inaccessible cardinals (while set theorists are looking down at these pitifully "small large cardinals").


