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Would anyone be opposed to giving the definition of pointwise Kan extension as a Kan extension that is preserved by all co-representable functors (i.e. is a pointwise kan extension if where denotes the functor .
I looked this up in Mac Lane's Categories Work, but I'm not sure that I interpreted "preserved by for all objects a" correctly.
Mac Lane also makes the point of assuming that the category has small Hom-sets, but I think that distinction is unnecessary unless we're being cardinality-strict for no reason.
Of course not. Mac Lane's assumption of local smallness is just because a category that is not locally small has no hom-functors landing in Set, so the definition is meaningless. Not everyone believes that proper classes are just sets of some bigger universe in disguise.
I don't see the use of assuming otherwise. Is there a good argument for having hom sets that are actually proper classes? I mean, on questions that are independent of set theory, shouldn't we take the answer that allows us to do the most?
We can systematically excise every occurrence of Set in any mathematical document by replacing Set with U-Set and making the necessary changes to be relative to U.
This gives us the Yoneda embedding for any category, which is no trivial matter.
But we might not want to assume the axiom of universes...
To what end?
centipede mathematics, perhaps. People around here, I've noticed, tend to try and do this sort of thing. You might as well ask, why do without the axiom of associativity? The answer is, of course, why not? If things break, we want to know precisely why and how.
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