Strong conceptual completeness for Boolean Applications of strong - PowerPoint PPT Presentation

Strong conceptual Useful consequence of Beth’s theorem completeness for Boolean coherent classifying toposes Jesse Han Corollary. Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Useful consequence of Beth’s theorem completeness for Boolean coherent classifying 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 criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Useful consequence of Beth’s theorem completeness for Boolean coherent classifying 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

Strong conceptual Useful consequence of Beth’s theorem completeness for Boolean coherent classifying 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? classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual How does strong conceptual completeness enter completeness for Boolean coherent this picture? classifying 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.

Strong conceptual How does strong conceptual completeness enter completeness for Boolean coherent this picture? classifying 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 How does strong conceptual completeness enter completeness for Boolean coherent this picture? classifying 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 classifying toposes Jesse Han Theorem. Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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”).

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 .

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying toposes What are ultramorphisms? Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying toposes What are ultramorphisms? Jesse Han An ultragraph Γ comprises: Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 .

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying toposes Okay, but what are ultramorphisms ? Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 .

Strong conceptual Strong conceptual completeness, I completeness for Boolean coherent classifying 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 classifying toposes Now, what’s changed between this statement and that of Jesse Han the useful corollary to Beth’s theorem? Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, II completeness for Boolean coherent classifying 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 .

Strong conceptual Strong conceptual completeness, II completeness for Boolean coherent classifying 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”).

Strong conceptual Strong conceptual completeness, II completeness for Boolean coherent classifying 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 classifying toposes Actually, Makkai proved something more, by doing the Jesse Han following: Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, III completeness for Boolean coherent classifying 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 A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Strong conceptual completeness, III completeness for Boolean coherent classifying 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”).

Strong conceptual Strong conceptual completeness, III completeness for Boolean coherent classifying 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 .

Strong conceptual Strong conceptual completeness, III completeness for Boolean coherent classifying 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 classifying toposes Jesse Han Caveat. Of course, one has an infinite list of conditions to Strong conceptual completeness verify here. Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual completeness for Boolean coherent classifying 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 Exotic functors

Strong conceptual completeness for Boolean coherent classifying 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.

Strong conceptual completeness for Boolean coherent classifying 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!

Strong conceptual Ultramorphisms, I completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Ultramorphisms, I completeness for Boolean coherent classifying toposes § Part of the critera for p X , Φ q (a functor Jesse Han X : Mod p T q Ñ Set plus a choice of transition Strong conceptual isomorphism Φ) to be definable was “preserving completeness ultramorphisms.” Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Ultramorphisms, I completeness for Boolean coherent classifying toposes § Part of the critera for p X , Φ q (a functor Jesse Han X : Mod p T q Ñ Set plus a choice of transition Strong conceptual isomorphism Φ) to be definable was “preserving completeness ultramorphisms.” Applications of strong conceptual § What are ultramorphisms? Loosely speaking, completeness A definability ultraproducts are a kind of universal construction in criterion for Set , and so there are certain canonical comparison ℵ 0 -categorical theories maps between them induced by their universal Exotic functors properties. (By the Los theorem, these things are “absolute” in the sense that no matter what first-order structure you put on a set, these maps will always be elementary embeddings.)

Strong conceptual Ultramorphisms, I completeness for Boolean coherent classifying toposes § Part of the critera for p X , Φ q (a functor Jesse Han X : Mod p T q Ñ Set plus a choice of transition Strong conceptual isomorphism Φ) to be definable was “preserving completeness ultramorphisms.” Applications of strong conceptual § What are ultramorphisms? Loosely speaking, completeness A definability ultraproducts are a kind of universal construction in criterion for Set , and so there are certain canonical comparison ℵ 0 -categorical theories maps between them induced by their universal Exotic functors properties. (By the Los theorem, these things are “absolute” in the sense that no matter what first-order structure you put on a set, these maps will always be elementary embeddings.) § Out of mercy, I will spare you the formal definition (because then I’d have to define ultragraphs, ultradiagrams, and ultratransformations...)

Strong conceptual Ultramorphisms, I completeness for Boolean coherent classifying toposes § Part of the critera for p X , Φ q (a functor Jesse Han X : Mod p T q Ñ Set plus a choice of transition Strong conceptual isomorphism Φ) to be definable was “preserving completeness ultramorphisms.” Applications of strong conceptual § What are ultramorphisms? Loosely speaking, completeness A definability ultraproducts are a kind of universal construction in criterion for Set , and so there are certain canonical comparison ℵ 0 -categorical theories maps between them induced by their universal Exotic functors properties. (By the Los theorem, these things are “absolute” in the sense that no matter what first-order structure you put on a set, these maps will always be elementary embeddings.) § Out of mercy, I will spare you the formal definition (because then I’d have to define ultragraphs, ultradiagrams, and ultratransformations...) § Keep in mind these two examples:

Strong conceptual Ultramorphisms, II completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Examples. Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Ultramorphisms, II completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Examples. Applications of strong conceptual completeness § The diagonal embedding into an ultrapower. A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Ultramorphisms, II completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Examples. Applications of strong conceptual completeness § The diagonal embedding into an ultrapower. A definability criterion for § Generalized diagonal embeddings. More generally, let ℵ 0 -categorical theories f : I Ñ J be a function, let U be an ultrafilter on I and Exotic functors let V be the pushforward ultrafilter on J.

Strong conceptual Ultramorphisms, II completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Examples. Applications of strong conceptual completeness § The diagonal embedding into an ultrapower. A definability criterion for § Generalized diagonal embeddings. More generally, let ℵ 0 -categorical theories f : I Ñ J be a function, let U be an ultrafilter on I and Exotic functors let V be the pushforward ultrafilter on J. Then for any I-indexed sequence of structures p M i q i P I , there is a canonical map δ f : ś j Ñ V M f p i q Ñ ś i Ñ U M i given by taking the diagonal embedding along each fiber of f .

Strong conceptual ∆-functors induce continuous maps on completeness for Boolean coherent automorphism groups classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual ∆-functors induce continuous maps on completeness for Boolean coherent automorphism groups classifying toposes Jesse Han § Why should we expect ultramorphisms to help us Strong conceptual identify evaluation functors in the wild? completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual ∆-functors induce continuous maps on completeness for Boolean coherent automorphism groups classifying toposes Jesse Han § Why should we expect ultramorphisms to help us Strong conceptual identify evaluation functors in the wild? completeness Applications of § Here’s an result which might indicate that knowing that strong conceptual completeness they’re preserved tells us something nontrivial. A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual ∆-functors induce continuous maps on completeness for Boolean coherent automorphism groups classifying toposes Jesse Han § Why should we expect ultramorphisms to help us Strong conceptual identify evaluation functors in the wild? completeness Applications of § Here’s an result which might indicate that knowing that strong conceptual completeness they’re preserved tells us something nontrivial. A definability criterion for ℵ 0 -categorical Definition. theories Exotic functors Say that X : Mod p T q Ñ Mod p T 1 q is a ∆ -functor if it preserves ultraproducts and diagonal maps into ultrapowers.

Strong conceptual ∆-functors induce continuous maps on completeness for Boolean coherent automorphism groups classifying toposes Jesse Han § Why should we expect ultramorphisms to help us Strong conceptual identify evaluation functors in the wild? completeness Applications of § Here’s an result which might indicate that knowing that strong conceptual completeness they’re preserved tells us something nontrivial. A definability criterion for ℵ 0 -categorical Definition. theories Exotic functors Say that X : Mod p T q Ñ Mod p T 1 q is a ∆ -functor if it preserves ultraproducts and diagonal maps into ultrapowers. Equip automorphism groups with the topology of pointwise convergence.

Strong conceptual ∆-functors induce continuous maps on completeness for Boolean coherent automorphism groups classifying toposes Jesse Han § Why should we expect ultramorphisms to help us Strong conceptual identify evaluation functors in the wild? completeness Applications of § Here’s an result which might indicate that knowing that strong conceptual completeness they’re preserved tells us something nontrivial. A definability criterion for ℵ 0 -categorical Definition. theories Exotic functors Say that X : Mod p T q Ñ Mod p T 1 q is a ∆ -functor if it preserves ultraproducts and diagonal maps into ultrapowers. Equip automorphism groups with the topology of pointwise convergence. Theorem. If X is a ∆ -functor from Mod p T q to Mod p T 1 q , then X restricts to a continuous map Aut p M q Ñ Aut p X p M qq for every M P Mod p T q .

Proof. Strong conceptual completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Proof. Strong conceptual completeness for Boolean coherent classifying toposes § The topology of pointwise convergence is sequential, so Jesse Han to check continuity it suffices to check convergent sequences of automorphisms are preserved. Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Proof. Strong conceptual completeness for Boolean coherent classifying toposes § The topology of pointwise convergence is sequential, so Jesse Han to check continuity it suffices to check convergent sequences of automorphisms are preserved. Strong conceptual completeness § If f i Ñ f in Aut p M q , then since the cofinite filter is Applications of strong conceptual contained in any ultrafilter, ś i Ñ U f i agrees with completeness i Ñ U f over the diagonal copy of M in M U . ś A definability criterion for ℵ 0 -categorical theories Exotic functors

Proof. Strong conceptual completeness for Boolean coherent classifying toposes § The topology of pointwise convergence is sequential, so Jesse Han to check continuity it suffices to check convergent sequences of automorphisms are preserved. Strong conceptual completeness § If f i Ñ f in Aut p M q , then since the cofinite filter is Applications of strong conceptual contained in any ultrafilter, ś i Ñ U f i agrees with completeness i Ñ U f over the diagonal copy of M in M U . That is, ś A definability criterion for p ś i Ñ U f i q ˝ ∆ M “ p ś i Ñ U f q ˝ ∆ M . ℵ 0 -categorical theories Exotic functors

Proof. Strong conceptual completeness for Boolean coherent classifying toposes § The topology of pointwise convergence is sequential, so Jesse Han to check continuity it suffices to check convergent sequences of automorphisms are preserved. Strong conceptual completeness § If f i Ñ f in Aut p M q , then since the cofinite filter is Applications of strong conceptual contained in any ultrafilter, ś i Ñ U f i agrees with completeness i Ñ U f over the diagonal copy of M in M U . That is, ś A definability criterion for p ś i Ñ U f i q ˝ ∆ M “ p ś i Ñ U f q ˝ ∆ M . ℵ 0 -categorical theories § Applying X and using that X is a ∆-functor, conclude Exotic functors that ś i Ñ U X p f i q agrees with ś i Ñ U X p f q over the diagonal copy of X p M q inside X p M q U .

Proof. Strong conceptual completeness for Boolean coherent classifying toposes § The topology of pointwise convergence is sequential, so Jesse Han to check continuity it suffices to check convergent sequences of automorphisms are preserved. Strong conceptual completeness § If f i Ñ f in Aut p M q , then since the cofinite filter is Applications of strong conceptual contained in any ultrafilter, ś i Ñ U f i agrees with completeness i Ñ U f over the diagonal copy of M in M U . That is, ś A definability criterion for p ś i Ñ U f i q ˝ ∆ M “ p ś i Ñ U f q ˝ ∆ M . ℵ 0 -categorical theories § Applying X and using that X is a ∆-functor, conclude Exotic functors that ś i Ñ U X p f i q agrees with ś i Ñ U X p f q over the diagonal copy of X p M q inside X p M q U . § For any point a P X p M q , the above says the sequence p X p f i qp a qq i P I “ U p X p f qp a qq i P I .

Proof. Strong conceptual completeness for Boolean coherent classifying toposes § The topology of pointwise convergence is sequential, so Jesse Han to check continuity it suffices to check convergent sequences of automorphisms are preserved. Strong conceptual completeness § If f i Ñ f in Aut p M q , then since the cofinite filter is Applications of strong conceptual contained in any ultrafilter, ś i Ñ U f i agrees with completeness i Ñ U f over the diagonal copy of M in M U . That is, ś A definability criterion for p ś i Ñ U f i q ˝ ∆ M “ p ś i Ñ U f q ˝ ∆ M . ℵ 0 -categorical theories § Applying X and using that X is a ∆-functor, conclude Exotic functors that ś i Ñ U X p f i q agrees with ś i Ñ U X p f q over the diagonal copy of X p M q inside X p M q U . § For any point a P X p M q , the above says the sequence p X p f i qp a qq i P I “ U p X p f qp a qq i P I . § Since U was arbitrary and the cofinite filter on I is the intersection of all non-principal ultrafilters on I , we conclude that the above equation holds cofinitely. Hence, X p f i q Ñ X p f q .

Strong conceptual ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han § A first-order theory T is ℵ 0 -categorical if it has one countable model up to isomorphism. Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han § A first-order theory T is ℵ 0 -categorical if it has one countable model up to isomorphism. Strong conceptual completeness § ℵ 0 -categorical theories have only finitely many types in Applications of strong conceptual each sort. (Caveat: when I say “type”, I mean an atom completeness in E p T q .) A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han § A first-order theory T is ℵ 0 -categorical if it has one countable model up to isomorphism. Strong conceptual completeness § ℵ 0 -categorical theories have only finitely many types in Applications of strong conceptual each sort. (Caveat: when I say “type”, I mean an atom completeness in E p T q .) A definability criterion for § A theorem of Coquand, Ahlbrandt and Ziegler says ℵ 0 -categorical theories that, given two ℵ 0 -categorical theories T and T 1 with Exotic functors countable models M and M 1 , a topological isomorphism Aut p M q » Aut p M 1 q induces a bi-interpretation M » M 1 .

Strong conceptual ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han § A first-order theory T is ℵ 0 -categorical if it has one countable model up to isomorphism. Strong conceptual completeness § ℵ 0 -categorical theories have only finitely many types in Applications of strong conceptual each sort. (Caveat: when I say “type”, I mean an atom completeness in E p T q .) A definability criterion for § A theorem of Coquand, Ahlbrandt and Ziegler says ℵ 0 -categorical theories that, given two ℵ 0 -categorical theories T and T 1 with Exotic functors countable models M and M 1 , a topological isomorphism Aut p M q » Aut p M 1 q induces a bi-interpretation M » M 1 . § Since we know ∆-functors induce continuous maps on automorphism groups, they’re a good candidate for definable functors.

Strong conceptual ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han § A first-order theory T is ℵ 0 -categorical if it has one countable model up to isomorphism. Strong conceptual completeness § ℵ 0 -categorical theories have only finitely many types in Applications of strong conceptual each sort. (Caveat: when I say “type”, I mean an atom completeness in E p T q .) A definability criterion for § A theorem of Coquand, Ahlbrandt and Ziegler says ℵ 0 -categorical theories that, given two ℵ 0 -categorical theories T and T 1 with Exotic functors countable models M and M 1 , a topological isomorphism Aut p M q » Aut p M 1 q induces a bi-interpretation M » M 1 . § Since we know ∆-functors induce continuous maps on automorphism groups, they’re a good candidate for definable functors. § Boolean coherent toposes split into a finite coproduct of E p T i q , where each T i is ℵ 0 -categorical.

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of Theorem. strong conceptual completeness Let X : Mod p T q Ñ Set . If T is ℵ 0 -categorical, the A definability criterion for following are equivalent: ℵ 0 -categorical theories Exotic functors

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of Theorem. strong conceptual completeness Let X : Mod p T q Ñ Set . If T is ℵ 0 -categorical, the A definability criterion for following are equivalent: ℵ 0 -categorical theories 1. For some transition isomorphism, p X , Φ q is a ∆ -functor Exotic functors (preserves ultraproducts and diagonal maps).

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of Theorem. strong conceptual completeness Let X : Mod p T q Ñ Set . If T is ℵ 0 -categorical, the A definability criterion for following are equivalent: ℵ 0 -categorical theories 1. For some transition isomorphism, p X , Φ q is a ∆ -functor Exotic functors (preserves ultraproducts and diagonal maps). 2. For some transition isomorphism, p X , Φ q is definable.

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Proof. Jesse Han (Sketch.) Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Proof. Jesse Han (Sketch.) Strong conceptual § One direction is immediate by SCC: definable functors completeness are ultrafunctors are at least ∆-functors. Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Proof. Jesse Han (Sketch.) Strong conceptual § One direction is immediate by SCC: definable functors completeness are ultrafunctors are at least ∆-functors. Applications of strong conceptual completeness § Let M be the countable model. Use the lemma about A definability ∆-functors p X , Φ q inducing continuous maps on the criterion for ℵ 0 -categorical automorphism groups (equivalently, p X , Φ q has the theories finite support property) to cover each Aut p M q -orbit of Exotic functors X p M q by a projection from an Aut p M q -orbit of M . By ω -categoricity, the kernel relation of this projection is definable, so we know that X p M q looks like an ( a priori , possibly infinite) disjoint union of types.

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Proof. Jesse Han (Sketch.) Strong conceptual § One direction is immediate by SCC: definable functors completeness are ultrafunctors are at least ∆-functors. Applications of strong conceptual completeness § Let M be the countable model. Use the lemma about A definability ∆-functors p X , Φ q inducing continuous maps on the criterion for ℵ 0 -categorical automorphism groups (equivalently, p X , Φ q has the theories finite support property) to cover each Aut p M q -orbit of Exotic functors X p M q by a projection from an Aut p M q -orbit of M . By ω -categoricity, the kernel relation of this projection is definable, so we know that X p M q looks like an ( a priori , possibly infinite) disjoint union of types. § By Aut p M q U orbit-counting, there are actually only finitely many types.

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Proof. Jesse Han (Sketch.) Strong conceptual § One direction is immediate by SCC: definable functors completeness are ultrafunctors are at least ∆-functors. Applications of strong conceptual completeness § Let M be the countable model. Use the lemma about A definability ∆-functors p X , Φ q inducing continuous maps on the criterion for ℵ 0 -categorical automorphism groups (equivalently, p X , Φ q has the theories finite support property) to cover each Aut p M q -orbit of Exotic functors X p M q by a projection from an Aut p M q -orbit of M . By ω -categoricity, the kernel relation of this projection is definable, so we know that X p M q looks like an ( a priori , possibly infinite) disjoint union of types. § By Aut p M q U orbit-counting, there are actually only finitely many types. § Invoke the Keisler-Shelah theorem to transfer to all N | ù T .

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han Corollary. Strong conceptual Let T and T 1 be ℵ 0 -categorical. Let X be an equivalence of completeness categories Applications of strong conceptual X completeness Mod p T 1 q » Mod p T 2 q . A definability criterion for Then X was induced by a bi-interpretation T 1 » T 2 if and ℵ 0 -categorical theories only if X was a ∆ -functor. Exotic functors

Strong conceptual A definability criterion for ℵ 0 -categorical theories completeness for Boolean coherent classifying toposes Jesse Han Corollary. Strong conceptual Let T and T 1 be ℵ 0 -categorical. Let X be an equivalence of completeness categories Applications of strong conceptual X completeness Mod p T 1 q » Mod p T 2 q . A definability criterion for Then X was induced by a bi-interpretation T 1 » T 2 if and ℵ 0 -categorical theories only if X was a ∆ -functor. Exotic functors In particular, Bodirsky, Evans, Kompatscher and Pinkser gave an example of two ℵ 0 -categorical theories T , T 1 with abstractly isomorphic but not topologically isomorphic automorphism groups of the countable model. This abstract isomorphism induces an equivalence Mod p T q » Mod p T 1 q and since it can’t come from an interpretation, from the corollary we conclude that it fails to preserve an ultraproduct or a diagonal map was not preserved.

Strong conceptual Exotic pre-ultrafunctors completeness for Boolean coherent classifying toposes Jesse Han Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Exotic pre-ultrafunctors completeness for Boolean coherent classifying toposes Jesse Han In light of the previous result, a natural question to ask is: Strong conceptual completeness Applications of strong conceptual completeness A definability criterion for ℵ 0 -categorical theories Exotic functors

Strong conceptual Exotic pre-ultrafunctors completeness for Boolean coherent classifying toposes Jesse Han In light of the previous result, a natural question to ask is: Strong conceptual completeness Question. Applications of strong conceptual Is being a ∆ -functor enough for SCC? That is, do completeness non-definable ∆ -functors exist? A definability criterion for ℵ 0 -categorical theories Exotic functors