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.