Canonical extension of coherent categories

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Canonical extension of coherent categories Dion Coumans Radboud University Nijmegen Topology, Algebra and Categories in Logic (TACL) Marseilles, July 2011 1 / 33

Outline 1 Canonical extension of distributive lattices (,,, ) 2 Algebraic semantics for coherent logic (,,,, ): Polyadic distributive lattices (pdl s) Coherent categories 3 Canonical extension of pdl s and coherent categories 4 Relation to other constructions (Makkai s topos of types) 5 Future work 2 / 33

Canonical extension of distributive lattices DL + U distributive lattices J Dwn P rf l ClDwn posets U Priestley spaces DL + = completely distributive algebraic lattices Priestley spaces = totally order-disconnected compact Hausdorff spaces 3 / 33

Canonical extension of distributive lattices ( ) δ DL + U distributive lattices J Dwn P rf l ClDwn posets U Priestley spaces DL + = completely distributive algebraic lattices Priestley spaces = totally order-disconnected compact Hausdorff spaces 4 / 33

Canonical extension of distributive lattices DL + = completely distributive algebraic lattices. Canonical extension is left adjoint to DL + DL. Universal characterization of canonical extension: L e L δ f f K where L DL and K, L δ DL +. 5 / 33

Algebraic semantics for coherent logic We start from Signature: Σ = (f 0,..., f k 1, R 0,..., R l 1, c 0,..., c m 1 ) Set of var s / sorts: X = {x 0, x 1,...} / {A, B,...} Equality: = Connectives:,,,, Derivability notion: (given by axioms and rules) Question: What properties does the logic over Σ have? 6 / 33

Algebraic semantics for coherent logic We start from Signature: Σ = (f 0,..., f k 1, R 0,..., R l 1, c 0,..., c m 1 ) Set of var s / sorts: X = {x 0, x 1,...} / {A, B,...} Equality: = Connectives:,,,, Derivability notion: (given by axioms and rules) Question: What properties does the logic over Σ have? First observation: For each n N, (F m(x 0,..., x n 1 )/, ) is a distributive lattice. 7 / 33

Algebraic semantics for coherent logic [] [x 0 ] [x 0, x 1 ]... 8 / 33

Algebraic semantics for coherent logic c φ(c) φ(x 0 ) [] [x 0 ] [x 0, x 1 ]... Substitutions: x 0 c φ(x 0 ) φ(c) 9 / 33

Algebraic semantics for coherent logic c φ(c) φ(x 0 ) [] [x 0 ] [x 0, x 1 ]... Substitutions: x 0 c φ(x 0 ) φ(c) 10 / 33

Algebraic semantics for coherent logic Contexts and substitutions form a category B: Objects: natural numbers (contexts) / sorts Morphism n m: m-tuple t 0,..., t m 1 s.t. F V (t i ) {x 0,..., x n 1 } x 0 [] c [x 0 ] x 0, f(x 0 ) [x 0, x 1 ]... c, f(c) 11 / 33

Algebraic semantics for coherent logic Contexts and substitutions form a category B: Objects: natural numbers (contexts) / sorts Morphism n m: m-tuple t 0,..., t m 1 s.t. F V (t i ) {x 0,..., x n 1 } This category has finite products: x 2 x 0, x 1 [x 0, x 1 ] [x 0, x 1, x 2 ] [x 0 ] t 0, t 1, s 0 t 0, t 1 s 0 [...] 12 / 33

Algebraic semantics for coherent logic Formulas and substitutions: functor B op DL n F m(x 0,..., x n 1 ) n t 0,...,t m 1 m F m(x 0,..., x m 1 ) F m(x 0,..., x n 1 ) φ(x 0,..., x m 1 ) φ(t 0,..., t m 1 ) φ(c, f(c)) φ(x0, f(x0)) φ(x0, x1) [] c [x 0 ] x 0, f(x 0) [x 0, x 1 ]... c, f(c) 13 / 33

Algebraic semantics for coherent logic Existential quantification: related to the inclusion map i φ(x 0 ) x1 ψ(x 0, x 1 ) x 0 [x 0 ] [x 0, x 1 ] x1 (ψ(x 0, x 1 )) φ(x 0 ) ψ(x 0, x 1 ) φ(x 0 ) 14 / 33

Algebraic semantics for coherent logic Existential quantification: related to the inclusion map i φ(x 0 ) x1 ψ(x 0, x 1 ) x 0 [x 0 ] [x 0, x 1 ] x1 (ψ(x 0, x 1 )) x0 φ(x 0 ) ψ(x 0, x 1 ) x0,x 1 i(φ(x 0 )) 15 / 33

Algebraic semantics for coherent logic Existential quantification: interaction with substitutions f(x 0 ) x1 ψ(x 0, x 1 ) f(x 0 ), x 1 [x 0 ] [x 0, x 1 ] x1 (ψ(x 0, x 1 ))[f(x 0 )/x 0 ] = x1 (ψ(f(x 0 ), x 1 )) (Beck-Chevalley) 16 / 33

Algebraic semantics for coherent logic Existential quantification: interaction with substitutions i φ(x 0 ) x1 ψ(x 0, x 1 ) x 0 [x 0 ] [x 0, x 1 ] x1 [i(φ(x 0 ) ψ(x 0, x 1 )] = φ(x 0 ) x1 [ψ(x 0, x 1 )] (Frobenius) 17 / 33

Algebraic semantics for coherent logic A polyadic distributive lattice is a functor P : B op DL s.t. 1 (Contexts & substitutions) B is a category with finite products; 2 (Existential quantification) for all I, J B, P (π): P (I) P (I J) has a left adjoint π satisfying Beck-Chevalley and Frobenius; 3 (Equality) for all I, J B, P (δ): P (I I J) P (I J) has a left adjoint δ satisfying Beck-Chevalley and Frobenius, (where δ = π 1, π 1, π 2 : I J I I J). 18 / 33

Algebraic semantics for coherent logic Examples of polyadic distributive lattices (pdl s): Syntactic pdl B = contexts and substitutions F : B op DL n F m(x 0,..., x n 1 )/ Powerset pdl B = Set P : B op DL A P(A) A f B P(B) f 1 P(A). 19 / 33

Polyadic distr. lattices and coherent categories Polyadic distr. lattices Functor P : B op DL s.t. B has finite products; P (π) and P (δ) have left adjoints satisfying BC and Frobenius. Coherent categories Category C s.t. C has finite limits; C has stable finite unions; C has stable images. 20 / 33

Polyadic distr. lattices and coherent categories Polyadic distr. lattices Functor P : B op DL s.t. B has finite products; P (π) and P (δ) have left adjoints satisfying BC and Frobenius. Coherent categories Category C s.t. C has finite limits; C has stable finite unions; C has stable images. Proposition There is an adjunction A: pdl Coh: S, A S. For C Coh, S(C) = S C : C op DL A Sub C (A) and A(S(C)) C. 21 / 33

Canonical extension of pdl s Recall: canonical extension for DL s is a functor DL ( )δ DL +. Definition For a pdl P : B op DL we define: P δ : B op P DL ( )δ DL. Proposition For a pdl P, P δ is again a pdl. Proof: check that P δ (π) and P δ (δ) have left adjoints satisfying BC and Frobenius. 22 / 33

Canonical extension of coherent categories We have: adjunction A: pdl Coh: S, C A(S C ) for a pdl P, P δ : B op P DL ( )δ DL Definition For a coherent category C we define: C δ = A(S δ C ) Proposition For a distributive lattice L, A(S δ L ) Lδ. 23 / 33

Canonical extension of coherent categories Properties of C δ = A(SC δ ): 1 subobject lattices are in DL + 2 pullback morphisms are complete lattice homomorphisms Coh + = coherent categories satisfying (1) and (2). 24 / 33

Canonical extension of coherent categories Properties of C δ = A(SC δ ): 1 subobject lattices are in DL + 2 pullback morphisms are complete lattice homomorphisms Coh + = coherent categories satisfying (1) and (2). Universal characterization: C M 0 C δ M M E where C Coh, E, C δ Coh +, M a coherent functor satisfying: 25 / 33

Canonical extension of coherent categories Properties of C δ = A(SC δ ): 1 subobject lattices are in DL + 2 pullback morphisms are complete lattice homomorphisms Coh + = coherent categories satisfying (1) and (2). Universal characterization: C M 0 C δ M M E where C Coh, E, C δ Coh +, M a coherent functor satisfying: for all A α B in C, ρ (prime) filter in S C (A), M(α) ( {M(U) U ρ}) = { M(α) (M(U)) U ρ}. 26 / 33

Topos of types Note: S δ C : Cop DL + is an internal frame in Set Cop = Ĉ. Then ShĈ(SC δ ) T (C) = topos of types of C. 27 / 33

Topos of types Note: S δ C : Cop DL + is an internal frame in Set Cop = Ĉ. Then ShĈ(SC δ ) T (C) = topos of types of C. Topos of types was introduced by Makkai in 1979 as: a reasonable codification of the discrete (non topological) syntactical structure of types of the theory a tool to prove representation theorems conceptual tool meant to enable us to formulate precisely certain natural intuitive questions Some later work by: Magnan & Reyes and Butz. 28 / 33

Topos of types and the class of models For a distributive lattice L, prime filters of L = lattice homomorphisms L 2 = models of L. L δ = D(Mod(L)) Categorical analogue: Mod(C) = coherent functors M : C Set. Study: Set Mod(C). We have to restrict to an appropriate subcategory K of Mod(C). Question: How does Set K relate to T (C) = ShĈ(S δ C )? 29 / 33

Topos of types and the class of models K appropriate subcategory of M od(c). Question: How does Set K relate to T (C) = ShĈ(S δ C )? Evaluation functor ev : C Set K A ev(a): K Set M M(A) Gives a geometric morphism φ ev : Set K Set Cop T (C) Set K φ ev Set Cop 30 / 33

Topos of types and the class of models K appropriate subcategory of M od(c). Question: How does Set K relate to T (C) = ShĈ(S δ C )? Evaluation functor ev : C Set K A ev(a): K Set M M(A) Gives a geometric morphism φ ev : Set K Sh(C, J coh ) T (C) Set K Sh(C, J coh ) φ ev 31 / 33

Topos of types and the class of models K appropriate subcategory of M od(c). Question: How does Set K relate to T (C) = ShĈ(S δ C )? Evaluation functor ev : C Set K A ev(a): K Set M M(A) Gives a geometric morphism φ ev : Set K Sh(C, J coh ) T (C) Set K Sh(C, J coh ) φ ev Makkai: T (C) functors in Set K with finite support property 32 / 33

Future work We have: notion of canonical extension for coherent categories We would like to: Study the following diagram (where K Mod(C)): T (C) Set K Sh(C, J coh ) φ ev Generalize to Heyting categories and study addition of axioms Apply the developed theory in the study of first order logics In particular: study interpolation problems for first order logics, e.g. for IPL + (φ ψ) (ψ φ) 33 / 33

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