[math-fun] category theory --- hmm...
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, http://en.wikipedia.org/wiki/Category_theory http://plato.stanford.edu/entries/category-theory/ http://math.ucr.edu/home/baez/rosetta.pdf 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
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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On 7/14/09, sctfen@gmail.com <sctfen@gmail.com> wrote:
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.
Ouch! --- when am I ever going to learn _not_ to go looking beneath stones ... Perhaps the approach outlined below might circumvent these complications? WFL << An alternative, suggested by Lawvere in the early sixties, is to develop an adequate language and background framework for a category of categories ... the basic idea is to define what are called weak n-categories (and weak ω-categories), and what had been called categories would then be called weak 1-categories (and sets would be weak 0-categories) >> [from the Stanford article, by Jean-Pierre Marquis].
I am as skilled a category therist as I am a dentist. That having been said, I shall walk out on a limb here. Category Theory attempts to capture the notion of thingies and functions that map thingies to thingies, in the greatest possible generality. (Yes, I know that the thingies in the range might be of a different sort than the thingies in the domain.) If Category Theory could not capture the category of "sets and functions", it would fail at the outset. So a category can't be restricted to anything as paltry as a set. Any approach that starts out by saying that a category is a kind of set, is pretty much doomed. On Mon, Jul 13, 2009 at 11:22 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 7/14/09, sctfen@gmail.com <sctfen@gmail.com> wrote:
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.
Ouch! --- when am I ever going to learn _not_ to go looking beneath stones ... Perhaps the approach outlined below might circumvent these complications? WFL
<< An alternative, suggested by Lawvere in the early sixties, is to develop an adequate language and background framework for a category of categories ... the basic idea is to define what are called weak n-categories (and weak ω-categories), and what had been called categories would then be called weak 1-categories (and sets would be weak 0-categories) >> [from the Stanford article, by Jean-Pierre Marquis].
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On 7/14/09, Allan Wechsler <acwacw@gmail.com> wrote:
I am as skilled a category therist as I am a dentist. That having been said, I shall walk out on a limb here. Category Theory attempts to capture the notion of thingies and functions that map thingies to thingies, in the greatest possible generality. (Yes, I know that the thingies in the range might be of a different sort than the thingies in the domain.)
If Category Theory could not capture the category of "sets and functions", it would fail at the outset.
So a category can't be restricted to anything as paltry as a set. Any approach that starts out by saying that a category is a kind of set, is pretty much doomed.
But in fact "objects" are only in there for historical reasons, and to pander to our regressive preference for having (relatively) concrete "things" around for "relations" to hold between. The category definitions could instead be rephrased to start from "morphisms" as fundamental, omitting "objects" entirely. It's not at this stage clear to me whether we then have to consider the entire class of all morphisms, or might instead restrict ourselves to discussing sets of morphisms between given pairs of objects. This approach would chime with my growing conviction that ontological problems concerning "existence" of this or that "thing" are a linguistic artifact. Does one ever hear of a passionate discussion about whether such-and-such a relation exists (in general), rather than whether a particular instance of that relation happens to hold in some particular situation? So --- let's hear it for 0-ary relations! Fred Lunnon
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