Morley derivative as non-classical Cantor-Bendixson derivative?

Morley derivative as non-classical Cantor-Bendixson derivative?

[Mamuka Jibladze]

Mention of the Morley derivative by Vaughan Pratt reminded me of my recent
suspicion that I think would be natural to share here.

Is there a chance to obtain the Morley derivative by performing construction
of the Cantor-Bendixson derivative internally on an appropriately chosen
internal locale in the topos of set-valued functors on an appropriate
subcategory of spaces and continuous maps?

Mamuka Jibladze


----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: <categories@mta.ca>
Sent: Sunday, September 03, 2006 10:59 PM
Subject: categories: Re: no membership-respecting morphisms


> My request for precursors to Mike Barr's deprecation of set membership
> seems to have set loose a thread that has veered off from set theory
> into model theory.  The following extract from the introduction to
> Gerald Sacks' "Saturated Model Theory" (335 pages, W.A. Benjamin, 1972)
> serendipitously ties up this loose thread while at the same time
> promising that category theory offers deeper insight into categoricity,
> a central notion of model theory, than the alternatives.
>
> "It is true that model theory bears a disheartening resemblance to set
> theory, a fascinating branch of mathematics with little to say about
> fundamental logical questions, and in particular to the arithmetic of
> cardinals and ordinals.  But the resemblance is more of manners than of
> ideas, because the central notions of model theory are absolute, and
> absoluteness, unlike cardinality, is a logical concept.  That is why
> model theory does not founder on that rock of undecidability, the
> generalized continuum hypothesis, and why the Los conjecture is
> decidable: A theory T is k-categorical if all models of T of cardinality
> k are isomorphic.  Los conjectured and Morley proved (Theorem 37.4) that
> if a countable theory is k-categorical for some uncountable k, then it
> is k-categorical for every uncountable k.  The property 'T is
> k-categorical for every uncountable k' is of course an absolute property
> of T.
>
> The notion of rank of 1-types was invented by Morley to prove Los's
> conjecture.  There are proofs of it that make no mention of rank, but
> they leave one ill-prepared to prove Shelah's uniqueness theorem
> (Section 36).  I have made rank a central idea of the book, because it
> is the central idea of current model theory.  ...  Morley's notion of
> rank was inspired by the Bendixson differentiation of a closed subset of
> a compact Hausdorff space; however, the Morley derivative differs from
> the Cantor-Bendixson derivative in that the former commutes with the
> inverse limit operation.  The Morley derivative is expounded in section
> 29 as a transformation which acts on functors of a class common in model
> theory.  One advantage of a category theoretic treatment of Morley rank
> is that it applies equally well to other notions [Shelah] of rank of
> 1-types.  Section 25 reviews the apparatus of category theory needed in
> section 29."
>
> The difference between this recommendation of category theory for model
> theory and (for example) the literature on accessible categories is that
> Sacks was not a card-carrying category theorist but a recursion
> theorist.  While category theory has no bias towards Goedel's notion of
> absoluteness (that I'm aware of), it seems reasonable to infer from
> Sacks' acceptance of CT that neither is CT biased away from absoluteness
> but rather is a neutral general-purpose tool.
>
> Vaughan Pratt
>
>
> V. Schmitt wrote:
>> Dear Paul, as far as i remember model theorists have
>> an extremely elegant way of comparing structures:
>> no morphisms there but "games" of finite isomorphism
>> extensions (see Fraisse' and Ehrenfeucht for the game
>> aspect or B.Poizat's book)
>> All the first order syntax is subsumed by those games.
>> I quite like categories and for sure I do not know everything,
>> but I have not seen so far a convincing categorical counterpart
>> for these games.
>> (And model theory is good stuff!)
>>
>> Best,
>> Vincent.
>
>
>