Strong conceptual completeness completeness Applications of for - PowerPoint PPT Presentation

Strong conceptual completeness for Boolean coherent toposes Jesse Han Strong conceptual Strong conceptual completeness completeness Applications of for Boolean coherent toposes strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Jesse Han Exotic functors McMaster University McGill logic, category theory, and computation seminar 5 December 2017

Strong conceptual What is strong conceptual completeness for completeness for Boolean coherent first-order logic? toposes Jesse Han Strong conceptual completeness § A strong conceptual completeness statement for a Applications of logical doctrine is an assertion that a theory in this strong conceptual completeness logical doctrine can be recovered from an appropriate A definability structure formed by the models of the theory. criterion for ℵ 0 -categorical § Makkai proved such a theorem for first-order logic theories Exotic functors showing one could reconstruct a first-order theory T from Mod p T q equipped with structure induced by taking ultraproducts. § Before we dive in, let’s look at a well-known theorem from model theory, with the same flavor, which Makkai’s result generalizes: the Beth definability theorem.

Strong conceptual The Beth theorem completeness for Boolean coherent toposes Theorem. Jesse Han Let L 0 Ď L 1 be an inclusion of languages with no new sorts. Let T 1 be an L 1 -theory. Let F : Mod p T 1 q Ñ Mod pH L 0 q be Strong conceptual completeness the reduct functor. Suppose you know any of the following: Applications of 1. There is a L 0 -theory T 0 and a factorization: strong conceptual completeness F Mod p T 1 q Mod pH L 0 q A definability criterion for ℵ 0 -categorical » theories Mod p T 0 q Exotic functors 2. F is full and faithful. 3. F is injective on objects. 4. F is full and faithful on automorphism groups. 5. F is full and faithful on Hom L 1 p M , M U q for all M P Mod p T 1 q and all ultrafilters U . 6. Every L 0 -elementary map is an L 1 -homomorphism of structures. Then: (*) Every L 1 -formula is T 1 -provably equivalent to an L 0 -formula.

Strong conceptual Useful consequence of Beth’s theorem completeness for Boolean coherent toposes Jesse Han Corollary. Strong conceptual completeness Let T be an L-theory, let S be a finite product of sorts. Let Applications of strong conceptual X : Mod p T q Ñ Set be a subfunctor of M ÞÑ S p M q . completeness A definability Then: if X commutes with ultraproducts on the nose criterion for ℵ 0 -categorical (”satisfies a � Los’ theorem”), then X was definable, i.e. X is theories an evaluation functor for some definable set ϕ P Def p T q . Exotic functors Proof. (Sketch): expand each model M of T by a new sort X p M q . Use commutation with ultraproducts to verify this is an elementary class. Then we are in the situation of 1 ù ñ p˚q from Beth’s theorem.

Strong conceptual How does strong conceptual completeness enter completeness for Boolean coherent this picture? toposes Jesse Han Strong conceptual completeness Applications of strong conceptual § Plain old conceptual completeness (this was one of the completeness key results of Makkai-Reyes) says that if an A definability criterion for interpretation I : T 1 Ñ T 2 induces an equivalence of ℵ 0 -categorical theories I ˚ categories Mod p T 1 q » Mod p T 2 q , then I must have Exotic functors been a bi-interpretation. So, it proves 1 ù ñ p˚q , and therefore the corollary. § Strong conceptual completeness is the following upgrade of the corollary.

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent toposes Jesse Han Theorem. Strong conceptual completeness Let T be an L-theory. Let X be any functor Applications of Mod p T q Ñ Set . Suppose that you have: strong conceptual completeness § for every ultraproduct ś i Ñ U M i a way to identify A definability Φ p Mi q criterion for X p ś i Ñ U M i q » ś i Ñ U X p M i q (”there exists a ℵ 0 -categorical theories transition isomorphism”), such that Exotic functors § p X , Φ q preserves ultraproducts of models/elementary embeddings (”is a pre-ultrafunctor”), and also § preserves all canonical maps between ultraproducts (”preserves ultramorphisms”). Then: there exists a ϕ p x q P T eq such that X » ev ϕ p x q as functors Mod p T q Ñ Set . (We call such X an ultrafunctor.)

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent toposes Jesse Han Strong conceptual § That is, the specified transition isomorphisms completeness Φ p M i q : X p ś i Ñ U M i q Ñ ś i Ñ U X p M i q make all Applications of strong conceptual diagrams of the form completeness A definability criterion for Φ p Mi q ℵ 0 -categorical X p ś ś i Ñ U M i q i Ñ U X p M i q theories Exotic functors X p i Ñ U f i q ś ś i Ñ U X p f i q X p ś i Ñ U N i q ś i Ñ U X p N i q Φ p Ni q commute (“transition isomorphism/pre-ultrafunctor condition”).

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent toposes What are ultramorphisms? Jesse Han An ultragraph Γ comprises: Strong conceptual § A directed graph whose vertices are partitioned into free completeness nodes Γ f and bound nodes Γ b . Applications of strong conceptual completeness § For any bound node β P Γ b , we assign a triple A definability x I , U , g y df “ x I β , U β , g β y where U is an ultrafilter on I criterion for ℵ 0 -categorical and g is a function g : I Ñ Γ f . theories Exotic functors § An ultradiagram for Γ is a diagram of shape Γ which incorporates the extra data: bound nodes are the ultraproducts of the free nodes given by the functions g . § A morphism of ultradiagrams (for fixed Γ) is just a natural transformation of functors which respects the extra data: the component of the transformation at a bound node is the ultraproduct of the components for the indexing free nodes.

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent toposes Okay, but what are ultramorphisms ? Jesse Han Strong conceptual Definition. completeness Let Hom p Γ , S q be the category of all ultradiagrams of type Γ Applications of strong conceptual inside S with morphisms the ultradiagram morphisms defined completeness above. Any two nodes k , ℓ P Γ define evaluation functors A definability criterion for p k q , p ℓ q : Hom p Γ , S q Ñ S , by ℵ 0 -categorical theories Exotic functors ´ ¯ A Φ Φ k p k q Ñ B “ A p k q Ñ B p k q (resp. ℓ ). An ultramorphism of type x Γ , k , ℓ y in S is a natural transformation δ : p k q Ñ p ℓ q . It’s sufficient to consider the ultramorphisms which come from universal properties of colimits of products in Set .

Strong conceptual Strong conceptual completeness, II completeness for Boolean coherent toposes Now, what’s changed between this statement and that of Jesse Han the useful corollary to Beth’s theorem? Strong conceptual completeness § We dropped the subfunctor assumption! We don’t have Applications of strong conceptual such a nice way of knowing exactly how X p M q is completeness obtained from M . We only have the invariance under A definability criterion for ultra-stuff. We’ve left the placental warmth of the ℵ 0 -categorical theories ambient models and we’re considering some kind of Exotic functors abstract permutation representation of Mod p T q . § Yet, if X respects enough of the structure induced by the ultra-stuff, then X must have been constructible from our models in some first-order way (”is definable”). § (With this new language, the corollary becomes: ”strict sub-pre-ultrafunctors of definable functors are definable.”)

Strong conceptual Strong conceptual completeness, III completeness for Boolean coherent toposes Actually, Makkai proved something more, by doing the Jesse Han following: Strong conceptual completeness § Introduce the notions of ultracategory and ultrafunctors Applications of by requiring all this extra ultra-stuff to be preserved. strong conceptual completeness § Develop a general duality theory between pretoposes A definability (“ Def p T q ”) and ultracategories (“ Mod p T q ”) via a criterion for ℵ 0 -categorical contravariant 2-adjunction (“generalized Stone theories Exotic functors duality”). § In particular, from this adjunction we get Pretop p T 1 , T 2 q » Ult p Mod p T 2 q , Mod p T 1 qq . Therefore, SCC tells us how to recognize a reduct functor in the wild between two categories of models—i.e., if there is some uniformity underlying a functor Mod p T 2 q Ñ Mod p T 1 q due to a purely syntactic assignment T 1 Ñ T 2 . Just check if the ultra-structure is preserved!

Strong conceptual completeness for Boolean coherent toposes Jesse Han Caveat. Of course, one has an infinite list of conditions to Strong conceptual completeness verify here. Applications of strong conceptual § So the only way to actually do this is to recognize some completeness kind of uniformity in the putative reduct functor which A definability criterion for lets you take care of all the ultramorphisms at once. ℵ 0 -categorical theories § But it gives you another way to think about uniformities Exotic functors you need. § It also gives you a way to check that something can never arise from any interpretation!