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バーミンガム大学のSteve Vickers先生の講演のお知らせです。 どうぞふるってご参加ください。
問合せ先： 石原 哉 北陸先端科学技術大学院大学 情報科学系 e-mail: [email protected] -----------------------------------------------
* JAIST Logic Seminar Series *
* The seminar below is held as a part of JSPS Core-to-Core Program, A. Advanced Research Networks, EU FP7 Marie Curie Actions IRSES project CORCON. (http://www.jaist.ac.jp/logic/ja/core2core, https://corcon.net/), and EU Horizon 2020 Marie Skłodowska-Curie actions RISE project CID.
Date: Monday 17, April, 2017, 15:20-17:00
Place: JAIST, Collaboration room 6 (I-57g) (Access: http://www.jaist.ac.jp/english/location/access.html)
Speaker: Steve Vickers (University of Birmingham)
Title: Arithmetic universes as generalized point-free spaces
Abstract: Point-free topology in all its guises (e.g. locales, formal topology) can be understood as presenting a space as a _logical theory_, for which the points are the models and the opens are the formulae. The logic in question is geometric logic, its connectives being finite conjunctions and arbitrary disjunctions, and then the Lindenbaum algebra (formulae modulo equivalence) for a theory T is a frame O[T], a complete lattice with binary meet distributing over all joins. Locales are frames but with the morphisms reversed.
Grothendieck proposed Grothendieck toposes as the generalized point-free spaces got when one moves to the first-order form of geometric logic. Then the opens (giving truth values for each point) are not enough, and one must move to sheaves (giving sets for each point). The Lindenbaum algebra now becomes a Grothendieck topos Set[T], the classifying topos for T, constructed using presheaves with a pasting condition, and closed under finite limits and arbitrary colimits in accordance with Giraud's theorem. The topos Set[T] canonically represents the generalized space of models of T.
Grothendieck used the category Set of classical sets, but we now know that it can be replaced by any elementary topos S. This base will determine the infinities available for "arbitrary" disjunctions, as well as governing the construction of the classifier S[T]. However, for theories in which all the disjunctions are countable (such as the formal space of reals) it doesn't matter which S is used, as long as it has a natural numbers object (nno). Thus the generalized space of models of T is not absolutely fixed as a mathematical object.
In my talk I shall present the idea of using Joyal's _arithmetic universes_ (AUs), pretoposes with parameterized list objects, as a base-independent substitute for Grothendieck toposes in which countable disjunctions are intrinsic to the logic rather than being supplied extrinsically by a base S. In [1] I have defined a 2-category Con whose objects ("contexts") serve as geometric theories that are sufficiently countable in nature, and whose morphisms are the maps of models. In [2] I showed how to use Con to prove results for Grothendieck toposes, fibred over choice of base topos. Thus we start to see AUs providing a free-standing foundations for a significant fragment of geometric logic and Grothendieck toposes, independent of base S.
My two papers - [1] "Sketches for arithmetic universes" (arXiv:1608.01559) [2] "Arithmetic universes and classifying toposes" (arXiv:1701.04611)