I had a slightly ironic experience on Math Overflow. A couple of months ago, I started wondering to what extent you could develop category theory “below a cardinal”. When you consider the category of groups (for example), you’re probably not literally interested in groups of arbitrarily large sizes — you just want enough space so that you can perform any operation you need to. I started writing this post here arguing that for concrete categories, sets smaller than a limit cardinal were big enough. Limit cardinals are not usually large cardinals in the sense of set theory, but they’re pretty big — the category of sets smaller than a limit cardinal is closed under the power set operation, for example.
Before I finished the post, I thought I should check the claim and look over some proofs in a category theory book. I realized that, under the usual definition of a diagram in the literature, my proposed restriction would make the category of sets fail to be either complete or cocomplete — even countable diagrams could have to have limits or colimits. You could finesse the issue by changing the definition of diagram, but I thought “No one will stand for that”. Under the standard definition, the construction of limits or colimits requires the Axiom of Replacement, which means that the right condition is inaccessibility, or equivalently you need Grothendieck universes.
So now I thought I understood the big picture. Completeness required replacement, which leads naturally to Groethendieck universes, which explains why the main competitor in textbooks to either Goedel-Bernays or Morse-Kelley set theory is to postulate one or more Grothendieck universes. The only thing that puzzled me was that while people using category theory seemingly made casual use of replacement, people would also argue that replacement is never used in ordinary mathematics. I thought that maybe I was confused on some issue, so I asked on Math Overflow.
It turns out that at least some people really don’t want to use replacement. They would rather change the definition of what it means to be a small diagram so as to be able to avoid the axiom. Avoiding replacement has lots of little consequences. For example, you have to require that the image of a small diagram is a set. Even with the corrected definition, the General Adjoint Functor Theorem becomes false as stated, and you have to strengthen the solution set condition. It means lots of fiddly little details have to be changed. You also no longer have as clean of a distinction between large and small. (You can have categories that are locally small, and have only countably many objects, and yet are not small categories, for example.)
But I could have stuck with my original idea for this post.