I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that categories can't describe?
To be quite specific, I'd be interested any of these:
a.) Objects that can be described by categories that have properties that can't.
b.) Category equivalents of set-theoretic type limits, like how "the set of all sets" causes problems.
c.) Some type of mathematics so pathological it foils, say, associativity. It doesn't need to be a mathematics that's useful in any sense, just one designed specifically to be impossible to describe with categories.