My understanding is that it depends deeply on the version of set theory you're using. If you're using something like ZFC without large cardinal axioms, categories and morphisms are proper classes. If you're using something like Tarski-Grothendeick set theory, then most (but not all) categories and morphisms can be put into sets. The exception seems to be the category Set, which would appear to be a class in any set theory you're working with. The workaround most people use is "the category of all sets smaller than some strongly inaccessable cardinal," which gives you a model of ZFC, but not of all the sets possible in your particular set theory. -Scott On Jul 13, 2009 9:23pm, Fred lunnon <fred.lunnon@gmail.com> wrote:
Having for the moment run out of steam as far as geometric algebra is
concerned,
I thought I might for a while resume banging my head against an
ancient brick wall.
It hasn't taken me long to stub my toe (or to mix my metaphors).
Upon earnestly scanning the following introductory documents,
each equipped with apparently impeccable credentials, I find that none
of them can
agree about whether the objects of a category belong to a set
(Stanford), a class
(Wikipedia), a "collection" (Baez & Stay), or an "aggregate" (anon ---
mislaid this one).
Furthermore, they are similarly contradictory (and in one case
confused) about whether
the morphisms thereof comprise a set or a class.
I'm not going to insist that authors commit to some particular flavour
of set theory,
appreciating that category theory may well provide it with an
alternative foundation.
However, I feel entitled to expect that --- after nearly half a
century --- the pedagogy
of this discipline might have managed to reach agreement concerning fundamental
definitions!
Would anybody care to cast some light on this disturbing
inconsistency? Fred Lunnon
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