Quotients, of course, also make perfect sense structurally. You can even define the quotient by an equivalence relation as its set of equivalence classes; this involves only "local" membership (elements of a set belonging to subsets of that set) so it makes sense in ETCS etc. On Fri, Dec 8, 2017 at 5:20 PM, Neil Barton <bartonna@gmail.com> wrote:
Dear All,
Thanks once again for the responses---I appreciate it!
@Cory. Thanks! This was helpful to see that the two perspectives respond to different `needs' within mathematics, and that for many applications the algebraic perspective suffices.
I did just want to pick up on the point about large cardinals. I suspect that those sorts of `arithmetical definability' characterisations will only work for cardinals you can characterise from below (i.e. by stating that there's a fixed point for some sort of describability property) and thus are likely to be consistent with V=L. However, there are really interesting categorial characterisations of *very* strong large cardinals, some of them dating back to the 1960s (see Brooke-Taylor and Bagaria, `On Colimits and Elementary Embeddings'). But I'm not really on top of this material, so don't have a feel for how the proof works yet (other than that the embeddability in the category-theoretic setting somehow transfers to the existence of large cardinal embeddings in set theory).
@Mike. Thanks for the points about the meta-theory---all clear to me now. Similarly for ordinals.
I was talking about the etale space. My claim wasn't meant to be that it *couldn't* be done structurally, just that sometimes the set-theoretic perspective is useful for seeing what's going on in a particular construction (but maybe this is just a holdover of how I first came across sheaves---then there was a lot of quotienting in material set theory).
You're right of course about the large cardinals (as I mentioned to Cory). This strikes me as very interesting stuff that I'm just not on top of yet. The example of L is also super-nice...thanks! I will have a think about this.
@Patrik: I'm also unsure how category theory is not a `theory'. Obviously were interested in many and varied categories but the core axiomatisation of what a category is is still first-order. Similarly in material set theory, even though ZFC is the `core' set theory (for many people anyway) we still consider lots of different theories. So I don't see how the phrase `is category theory a theory? I think not.' wouldn't apply mutatis mutandis to material set theory.
I *do* think however there's an important philosophical difference: Set theory aims at an intended interpretation (the cumulative hierarchy), whereas category theory doesn't (the whole *point* of it is to apply across diverse contexts). But maybe things are more subtle than I realise [plus I'm maybe straying into off-list philosophical territory with this claim].
Best Wishes,
Neil
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