Using topological systems to create a framework for institutions

Transcription

1 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 1/34 Using topological systems to create a framework for institutions Jeffrey T. Denniston 1 Austin Melton 1 Stephen E. Rodabaugh 2 Sergejs Solovjovs 3 1 Kent State University, Kent, Ohio, USA 2 Youngstown State University, Youngstown, Ohio, USA 3 Brno University of Technology, Brno, Czech Republic Topology, Algebra, and Categories in Logic 2015 Ischia, Italy June 21 26, 2015

2 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 2/34 Acknowledgements Sergejs Solovjovs acknowledges the support of Czech Science Foundation (GAČR) and Austrian Science Fund (FWF) through bilateral project No. I 1923-N25 New Perspectives on Residuated Posets.

3 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 3/34 Outline 1 Introduction 2 Affine systems 3 Affine theories 4 Affine institutions 5 Conclusion

4 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 4/34 Institutions and topological systems Institutions There is a convenient approach to logical systems in computer science based in institutions of J. A. Goguen and R. M. Burstall. An institution comprises a category of (abstract) signatures, where every signature has its associated sentences, models, and a relation of satisfaction, which is invariant under change of signature, i.e., truth is invariant under change of notation. Institutions include unsorted universal algebra, many-sorted algebra, order-sorted algebra, first-order logic, partial algebra. A number of authors proposed generalizations of institutions in various forms, e.g., using a purely category-theoretic approach.

5 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 4/34 Institutions and topological systems Institutions There is a convenient approach to logical systems in computer science based in institutions of J. A. Goguen and R. M. Burstall. An institution comprises a category of (abstract) signatures, where every signature has its associated sentences, models, and a relation of satisfaction, which is invariant under change of signature, i.e., truth is invariant under change of notation. Institutions include unsorted universal algebra, many-sorted algebra, order-sorted algebra, first-order logic, partial algebra. A number of authors proposed generalizations of institutions in various forms, e.g., using a purely category-theoretic approach.

6 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 4/34 Institutions and topological systems Institutions There is a convenient approach to logical systems in computer science based in institutions of J. A. Goguen and R. M. Burstall. An institution comprises a category of (abstract) signatures, where every signature has its associated sentences, models, and a relation of satisfaction, which is invariant under change of signature, i.e., truth is invariant under change of notation. Institutions include unsorted universal algebra, many-sorted algebra, order-sorted algebra, first-order logic, partial algebra. A number of authors proposed generalizations of institutions in various forms, e.g., using a purely category-theoretic approach.

7 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 4/34 Institutions and topological systems Institutions There is a convenient approach to logical systems in computer science based in institutions of J. A. Goguen and R. M. Burstall. An institution comprises a category of (abstract) signatures, where every signature has its associated sentences, models, and a relation of satisfaction, which is invariant under change of signature, i.e., truth is invariant under change of notation. Institutions include unsorted universal algebra, many-sorted algebra, order-sorted algebra, first-order logic, partial algebra. A number of authors proposed generalizations of institutions in various forms, e.g., using a purely category-theoretic approach.

8 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 5/34 Institutions and topological systems Topological systems Based in the ideas of geometric logic, topological systems of S. Vickers provide a common setting for both topological spaces and their underlying algebraic structures locales. S. Vickers showed system spatialization and localification procedures, i.e., ways to move back and forth between the categories of topological spaces (resp., locales) and topological systems. Recently, topological systems have gained in interest in connection with lattice-valued topology, e.g., one has introduced and studied lattice-valued topological systems; provided lattice-valued system spatialization procedure.

9 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 5/34 Institutions and topological systems Topological systems Based in the ideas of geometric logic, topological systems of S. Vickers provide a common setting for both topological spaces and their underlying algebraic structures locales. S. Vickers showed system spatialization and localification procedures, i.e., ways to move back and forth between the categories of topological spaces (resp., locales) and topological systems. Recently, topological systems have gained in interest in connection with lattice-valued topology, e.g., one has introduced and studied lattice-valued topological systems; provided lattice-valued system spatialization procedure.

10 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 5/34 Institutions and topological systems Topological systems Based in the ideas of geometric logic, topological systems of S. Vickers provide a common setting for both topological spaces and their underlying algebraic structures locales. S. Vickers showed system spatialization and localification procedures, i.e., ways to move back and forth between the categories of topological spaces (resp., locales) and topological systems. Recently, topological systems have gained in interest in connection with lattice-valued topology, e.g., one has introduced and studied lattice-valued topological systems; provided lattice-valued system spatialization procedure.

11 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 6/34 Institutions and topological systems Institutions versus systems To find relationships between institutions and topological systems, J. T. Denniston, A. Melton, and S. E. Rodabaugh introduced lattice-valued institutions, and showed that lattice-valued topological systems provide their particular instance. A. Sernadas, C. Sernadas, and J. M. Valença introduced crisp topological institutions based in topological systems, the slogan being that the central concept is the theory, not the formula. The purpose of this talk is to show that a suitably generalized concept of topological system provides a setting for elementary institutions of A. Sernadas, C. Sernadas, and J. M. Valença.

12 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 6/34 Institutions and topological systems Institutions versus systems To find relationships between institutions and topological systems, J. T. Denniston, A. Melton, and S. E. Rodabaugh introduced lattice-valued institutions, and showed that lattice-valued topological systems provide their particular instance. A. Sernadas, C. Sernadas, and J. M. Valença introduced crisp topological institutions based in topological systems, the slogan being that the central concept is the theory, not the formula. The purpose of this talk is to show that a suitably generalized concept of topological system provides a setting for elementary institutions of A. Sernadas, C. Sernadas, and J. M. Valença.

13 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 6/34 Institutions and topological systems Institutions versus systems To find relationships between institutions and topological systems, J. T. Denniston, A. Melton, and S. E. Rodabaugh introduced lattice-valued institutions, and showed that lattice-valued topological systems provide their particular instance. A. Sernadas, C. Sernadas, and J. M. Valença introduced crisp topological institutions based in topological systems, the slogan being that the central concept is the theory, not the formula. The purpose of this talk is to show that a suitably generalized concept of topological system provides a setting for elementary institutions of A. Sernadas, C. Sernadas, and J. M. Valença.

14 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 7/34 Algebraic preliminaries Ω-algebras and Ω-homomorphisms Definition 1 Let Ω = (n λ ) λ Λ be a family of cardinal numbers, which is indexed by a (possibly proper or empty) class Λ. An Ω-algebra is a pair (A, (ω A λ ) λ Λ), comprising a set A and a family of maps A n λ ωa λ A (n λ -ary primitive operations on A). An Ω-homomorphism (A 1, (ω A 1 λ ) λ Λ) ϕ (A 2, (ω A 2 λ ) λ Λ) is a ϕ map A 1 A2 such that ϕ ω A 1 λ = ωa 2 λ ϕn λ for every λ Λ. Alg(Ω) is the construct of Ω-algebras and Ω-homomorphisms.

15 Algebraic preliminaries Varieties and algebras Definition 2 Let M (resp. E) be the class of Ω-homomorphisms with injective (resp. surjective) underlying maps. A variety of Ω-algebras is a full subcategory of Alg(Ω), which is closed under the formation of products, M-subobjects and E-quotients, and whose objects (resp. morphisms) are called algebras (resp. homomorphisms). Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 8/34

16 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 9/34 Algebraic preliminaries Examples of varieties Example 3 1 CSLat( ) is the variety of -semilattices, and CSLat( ) is the variety of -semilattices. 2 Frm is the variety of frames. 3 CBAlg is the variety of complete Boolean algebras. 4 CSL is the variety of closure semilattices, i.e., -semilattices, with the singled out bottom element.

17 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 10/34 Affine spaces Affine spaces The following extends the notion of affine set of Y. Diers. Definition 4 Given a functor X T B op, where B is a variety of algebras, AfSpc(T ) is the concrete category over X, whose objects (T -affine spaces or T -spaces) are pairs (X, τ), where X is an X-object and τ is a subalgebra of TX ; morphisms (T -affine morphisms or T -morphisms) (X 1, τ 1 ) (X 2, τ 2 ) are X-morphisms X 1 f X2 with the property that (Tf ) op (α) τ 1 for every α τ 2. The concrete category (AfSpc(T ), ) is topological over X. f

18 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 10/34 Affine spaces Affine spaces The following extends the notion of affine set of Y. Diers. Definition 4 Given a functor X T B op, where B is a variety of algebras, AfSpc(T ) is the concrete category over X, whose objects (T -affine spaces or T -spaces) are pairs (X, τ), where X is an X-object and τ is a subalgebra of TX ; morphisms (T -affine morphisms or T -morphisms) (X 1, τ 1 ) (X 2, τ 2 ) are X-morphisms X 1 f X2 with the property that (Tf ) op (α) τ 1 for every α τ 2. The concrete category (AfSpc(T ), ) is topological over X. f

19 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 10/34 Affine spaces Affine spaces The following extends the notion of affine set of Y. Diers. Definition 4 Given a functor X T B op, where B is a variety of algebras, AfSpc(T ) is the concrete category over X, whose objects (T -affine spaces or T -spaces) are pairs (X, τ), where X is an X-object and τ is a subalgebra of TX ; morphisms (T -affine morphisms or T -morphisms) (X 1, τ 1 ) (X 2, τ 2 ) are X-morphisms X 1 f X2 with the property that (Tf ) op (α) τ 1 for every α τ 2. The concrete category (AfSpc(T ), ) is topological over X. f

20 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 11/34 Affine spaces Examples Example 5 Given a variety B, every subcategory S of B op induces a functor Set S P S B op (f,ϕ), P S ((X 1, B 1 ) (X 2, B 2 )) = B X P 1 S (f,ϕ) 1 B X 2 2, where (P S (f, ϕ)) op (α) = ϕ op α f. Example 6 1 If B = Frm, then AfSpc(P 2 ) = Top (topological spaces). 2 If B = CSL, then AfSpc(P 2 ) = Cls (closure spaces). 3 AfSpc(P B ) is the category AfSet(B) of affine sets of Y. Diers. 4 If B = Frm, then AfSpc(P S ) = S-Top (variable-basis latticevalued topological spaces of S. E. Rodabaugh).

21 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 11/34 Affine spaces Examples Example 5 Given a variety B, every subcategory S of B op induces a functor Set S P S B op (f,ϕ), P S ((X 1, B 1 ) (X 2, B 2 )) = B X P 1 S (f,ϕ) 1 B X 2 2, where (P S (f, ϕ)) op (α) = ϕ op α f. Example 6 1 If B = Frm, then AfSpc(P 2 ) = Top (topological spaces). 2 If B = CSL, then AfSpc(P 2 ) = Cls (closure spaces). 3 AfSpc(P B ) is the category AfSet(B) of affine sets of Y. Diers. 4 If B = Frm, then AfSpc(P S ) = S-Top (variable-basis latticevalued topological spaces of S. E. Rodabaugh).

22 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 12/34 Affine systems Affine systems Definition 7 Given a functor X T B op, AfSys(T ) is the comma category (T 1 B op), concrete over the product category X B op, whose objects (T -affine systems or T -systems) are triples (X, κ, B), made by B op -morphisms TX κ B; morphisms (T -affine morphisms or T -morphisms) (X 1, κ 1, B 1 ) (X 1, B 1 ) (f,ϕ) (f,ϕ) (X 2, κ 2, B 2 ) are X B op -morphisms (X 2, B 2 ), making the next diagram commute TX 1 Tf TX 2 κ 1 B 1 ϕ κ 2 B 2.

23 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 13/34 Affine systems Examples Example 8 1 If B = Frm, then AfSys(P 2 ) = TopSys (topological systems of S. Vickers). 2 If B = Set, then AfSys(P B ) = Chu B (Chu spaces over a set B of P.-H. Chu). Definition 9 A T -system (X, κ, B) is called separated provided that TX κ B is an epimorphism in B op. AfSys s (T ) is the full subcategory of AfSys(T ) of separated T -systems. Example 10 For B=CSL, AfSys s (P 2 )=SP (state property systems of D. Aerts).

24 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 13/34 Affine systems Examples Example 8 1 If B = Frm, then AfSys(P 2 ) = TopSys (topological systems of S. Vickers). 2 If B = Set, then AfSys(P B ) = Chu B (Chu spaces over a set B of P.-H. Chu). Definition 9 A T -system (X, κ, B) is called separated provided that TX κ B is an epimorphism in B op. AfSys s (T ) is the full subcategory of AfSys(T ) of separated T -systems. Example 10 For B=CSL, AfSys s (P 2 )=SP (state property systems of D. Aerts).

25 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 13/34 Affine systems Examples Example 8 1 If B = Frm, then AfSys(P 2 ) = TopSys (topological systems of S. Vickers). 2 If B = Set, then AfSys(P B ) = Chu B (Chu spaces over a set B of P.-H. Chu). Definition 9 A T -system (X, κ, B) is called separated provided that TX κ B is an epimorphism in B op. AfSys s (T ) is the full subcategory of AfSys(T ) of separated T -systems. Example 10 For B=CSL, AfSys s (P 2 )=SP (state property systems of D. Aerts).

26 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 14/34 Affine spatialization procedure Affine spatialization procedure Theorem 11 1 AfSpc(T ) E AfSys(T ), E((X 1, τ 1 ) (X 1, e op τ 1, τ 1 ) f (X 2, τ 2 )) = (f,ϕ) (X 2, e op τ 2, τ 2 ) is a full embedding, with e τi the inclusion τ i TX i and ϕ op the restriction τ 2 (Tf ) op τ 1 τ 2 τ1. 2 E has a right-adjoint-left-inverse AfSys(T ) Spat AfSpc(T ), (f,ϕ) Spat((X 1, κ 1, B 1 ) (X 2, κ 2, B 2 )) = (X 1, κ op 1 (B f 1)) (X 2, κ op 2 (B 2)). 3 AfSpc(T ) is isomorphic to a full (regular mono)-coreflective subcategory of AfSys(T ).

27 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 15/34 Affine spatialization procedure Consequences Theorem 12 E and Spat restrict to AfSpc(T ) E AfSys s (T ) and AfSys s (T ) Spat AfSpc(T ), providing an equivalence between the categories AfSpc(T ) and AfSys s (T ) such that Spat E = 1 AfSpc(T ). Corollary 13 AfSpc(T ) is the amnestic modification of AfSys s (T ). Example 14 1 Top is isomorphic to a full (regular mono)-coreflective subcategory of TopSys (system spatialization procedure of S. Vickers). 2 The categories Cls and SP are equivalent.

28 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 15/34 Affine spatialization procedure Consequences Theorem 12 E and Spat restrict to AfSpc(T ) E AfSys s (T ) and AfSys s (T ) Spat AfSpc(T ), providing an equivalence between the categories AfSpc(T ) and AfSys s (T ) such that Spat E = 1 AfSpc(T ). Corollary 13 AfSpc(T ) is the amnestic modification of AfSys s (T ). Example 14 1 Top is isomorphic to a full (regular mono)-coreflective subcategory of TopSys (system spatialization procedure of S. Vickers). 2 The categories Cls and SP are equivalent.

29 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 15/34 Affine spatialization procedure Consequences Theorem 12 E and Spat restrict to AfSpc(T ) E AfSys s (T ) and AfSys s (T ) Spat AfSpc(T ), providing an equivalence between the categories AfSpc(T ) and AfSys s (T ) such that Spat E = 1 AfSpc(T ). Corollary 13 AfSpc(T ) is the amnestic modification of AfSys s (T ). Example 14 1 Top is isomorphic to a full (regular mono)-coreflective subcategory of TopSys (system spatialization procedure of S. Vickers). 2 The categories Cls and SP are equivalent.

30 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 16/34 Affine localification procedure Affine localification procedure Proposition 15 AfSys(T ) Loc B op (f,ϕ), Loc((X 1, κ 1, B 1 ) (X 2, κ 2, B 2 )) = B 1 B 2 is a functor. ϕ Theorem 16 Given a functor X T B op, the following are equivalent. 1 There exists an adjoint situation (η, ε) : T Pt : B op X. 2 There exists a full embedding B op E AfSys(T ) such that Loc is a left-adjoint-left-inverse to E. B op is then isomorphic to a full reflective subcategory of AfSys(T ).

31 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 16/34 Affine localification procedure Affine localification procedure Proposition 15 AfSys(T ) Loc B op (f,ϕ), Loc((X 1, κ 1, B 1 ) (X 2, κ 2, B 2 )) = B 1 B 2 is a functor. ϕ Theorem 16 Given a functor X T B op, the following are equivalent. 1 There exists an adjoint situation (η, ε) : T Pt : B op X. 2 There exists a full embedding B op E AfSys(T ) such that Loc is a left-adjoint-left-inverse to E. B op is then isomorphic to a full reflective subcategory of AfSys(T ).

32 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 17/34 Affine localification procedure Examples Remark 17 P B Every functor Set B op has a right adjoint B op Pt B Set, ϕ Pt B (B 1 B2 ) = B(B 1, B) Pt Bϕ B(B 2, B), (Pt B ϕ)(p) = p ϕ op. Example 18 Loc is isomorphic to a full reflective subcategory of TopSys, which gives the system localification procedure of S. Vickers. B op is isomorphic to a full reflective subcategory of AfSys(P B ).

33 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 17/34 Affine localification procedure Examples Remark 17 P B Every functor Set B op has a right adjoint B op Pt B Set, ϕ Pt B (B 1 B2 ) = B(B 1, B) Pt Bϕ B(B 2, B), (Pt B ϕ)(p) = p ϕ op. Example 18 Loc is isomorphic to a full reflective subcategory of TopSys, which gives the system localification procedure of S. Vickers. B op is isomorphic to a full reflective subcategory of AfSys(P B ).

34 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 18/34 Affine theories Affine theories One could like to study the properties of the categories AfSys(T ) and AfSpc(T ) through the properties of the functor X T B op. Definition 19 An affine theory is a functor X T B op with B a variety of algebras.

35 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 18/34 Affine theories Affine theories One could like to study the properties of the categories AfSys(T ) and AfSpc(T ) through the properties of the functor X T B op. Definition 19 An affine theory is a functor X T B op with B a variety of algebras.

36 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 19/34 Affine theories Category of affine theories Definition 20 AfTh is the category given by the following data: objects are affine theories X T B op ; (F,Φ,η) morphisms T 1 T 2 (shortened to η) comprise two functors F Φ X 1 X2, B 1 B2 and a natural transformation T 2 F η Φ op T 1, X 1 T 1 B op 1 F X 2 η Φ op T 2 B op 2 η 1 η 2 composition of two affine theories T 1 T2, T 2 T3 is η 2 η 1 T 3 F 2 F 1 Φ op 2 Φop 1 T η 2F1 1=T 3 F 2 F 1 Φ op 2 T Φ op 2 2F η 1 1 Φ op 2 Φop 1 T 1; identity on a theory T is the identity natural transformation T 1 T T.

37 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 20/34 Affine theories Models of affine theories Definition 21 AfStm is the category, whose objects are categories of the form AfSys(T ) and whose morphisms are functors between them. Theorem 22 AfTh AfSys AfStm, AfSys(T 1 η T2 ) = AfSys(T 1 ) AfSysη (f,ϕ) AfSys(T 2 ), AfSysη((X, κ, B) (X, κ, B )) = (FX, Φ op κ η X, Φ op B) (Ff,Φop ϕ) (FX, Φ op κ η X, Φ op B ) is a functor. The respective functor for affine spaces requires more effort.

38 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 20/34 Affine theories Models of affine theories Definition 21 AfStm is the category, whose objects are categories of the form AfSys(T ) and whose morphisms are functors between them. Theorem 22 AfTh AfSys AfStm, AfSys(T 1 η T2 ) = AfSys(T 1 ) AfSysη (f,ϕ) AfSys(T 2 ), AfSysη((X, κ, B) (X, κ, B )) = (FX, Φ op κ η X, Φ op B) (Ff,Φop ϕ) (FX, Φ op κ η X, Φ op B ) is a functor. The respective functor for affine spaces requires more effort.

39 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 20/34 Affine theories Models of affine theories Definition 21 AfStm is the category, whose objects are categories of the form AfSys(T ) and whose morphisms are functors between them. Theorem 22 AfTh AfSys AfStm, AfSys(T 1 η T2 ) = AfSys(T 1 ) AfSysη (f,ϕ) AfSys(T 2 ), AfSysη((X, κ, B) (X, κ, B )) = (FX, Φ op κ η X, Φ op B) (Ff,Φop ϕ) (FX, Φ op κ η X, Φ op B ) is a functor. The respective functor for affine spaces requires more effort.

40 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 21/34 Institutions and their morphisms Institutions Definition 23 An institution I consists of: a category Sign of signatures, Σ denoting an arbitrary object, a functor Sign Mod Cat op giving Σ-models and Σ-morphisms, a functor Sign Sen Cat giving Σ-sentences and Σ-proofs, a satisfaction relation = Σ Ob(ModΣ) Ob(SenΣ) for every Σ Ob(Sign) such that satisfaction: m = Σ Senφ(s) iff Modφ(m ) = Σ s for every m Ob(ModΣ ), s Ob(SenΣ), Σ φ Σ in Sign, soundness: m = Σ s and s s in SenΣ imply m = Σ s for m Ob(ModΣ).

41 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 22/34 Institutions and their morphisms Institution morphisms Definition 24 An institution morphism I (Φ,α,β) I comprises a functor Sign Φ Sign, natural transformations Sen Φ α Sen and Mod β Mod Φ, Sign Sen Cat α Φ Sign Cat Sen Sign Mod Cat op such that the following satisfaction condition holds m = Σ α Σ (s ) iff β Σ (m) = ΦΣ s Φ Sign β Mod Cat op for every Σ-model m from I and every ΦΣ-sentence s from I.

42 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 23/34 Institutions and their morphisms Elementary institutions Definition 25 An institution is called elementary provided that the category Cat is replaced with the category Set. Inst (resp. ElInst) is the category of (resp. elementary) institutions and their morphisms.

43 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 24/34 Topological institutions and their morphisms Topological institutions and their morphisms Definition 26 A topological institution is a functor Sign T TopSys op, where Sign is a category of (abstract) signatures. A topological institution morphism (Sign, T ) (Φ,α) (Sign, T ) consists of a functor Sign Φ Sign and a natural transformation T α T Φ. TpInst is the category of topological institutions and their morphisms.

44 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 25/34 Affine institutions and their morphisms Affine institutions and their morphisms Definition 27 An affine institution is a functor S I AfSys(T ), where S is a category of (abstract) signatures. An affine institution morphism (S 1, I 1, T 1 ) (Φ,α,η) (S 2, I 2, T 2 ) comprises a functor S 1 Φ S2, an affine theory morphism T 1 η T 2, and a natural transformation AfSysηI 1 α I2 Φ, S 1 I 1 AfSys(T 1 ) Φ S 2 α AfSysη I 2 AfSys(T 2 ). AfInst is the category of affine institutions and their morphisms.

45 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 26/34 Affine institutions and their morphisms Examples of affine institutions Definition 28 Given an affine theory T, AfInst(T ) stands for the subcategory of AfInst consisting of affine institutions (S, I, T ) (shortened to (S, I )) and their respective morphisms (Φ, α, 1 T ) (shortened to (Φ, α)). Example 29 1 For B = Frm, AfInst(P 2 ) is a modification of TpInst. 2 Set P 2 Set op := Set P 2 CBAlg op op Set op gives the category AfInst( P 2 ), which is a modification of ElInst.

46 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 26/34 Affine institutions and their morphisms Examples of affine institutions Definition 28 Given an affine theory T, AfInst(T ) stands for the subcategory of AfInst consisting of affine institutions (S, I, T ) (shortened to (S, I )) and their respective morphisms (Φ, α, 1 T ) (shortened to (Φ, α)). Example 29 1 For B = Frm, AfInst(P 2 ) is a modification of TpInst. 2 Set P 2 Set op := Set P 2 CBAlg op op Set op gives the category AfInst( P 2 ), which is a modification of ElInst.

47 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 27/34 Affine institution spatialization procedure Spatial affine institutions Definition 30 Let T be an affine theory. A spatial affine T -institution is a functor S I AfSpc(T ), where S is a category of (abstract) signatures. A spatial affine T -institution morphism (S 1, I 1 ) (Φ,α) (S 2, I 2 ) Φ comprises a functor S 1 S2 and a natural transformation α I 1 I2 Φ. SAfInst(T ) is the category of spatial affine T -institutions and their morphisms.

48 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 28/34 Affine institution spatialization procedure Affine institution spatialization procedure Theorem 31 1 SAfInst(T ) IE AfInst(T ), IE((S 1, I 1 ) (Φ,α) (S 2, I 2 )) = (S 1, EI 1 ) (Φ,Eα) (S 2, EI 2 ) is a full embedding. 2 AfInst(T ) ISpat SAfInst(T ), ISpat((S 1, I 1 ) (Φ,α) (S 2, I 2 )) = (Φ,Spatα) (S 2, SpatI 2 ) is a right-adjoint-left- (S 1, SpatI 1 ) inverse to IE. 3 SAfInst(T ) is isomorphic to a full coreflective subcategory of AfInst(T ). This answers the question on spatialization construction for topological institutions of A. Sernadas, C. Sernadas, and J. M. Valença.

49 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 28/34 Affine institution spatialization procedure Affine institution spatialization procedure Theorem 31 1 SAfInst(T ) IE AfInst(T ), IE((S 1, I 1 ) (Φ,α) (S 2, I 2 )) = (S 1, EI 1 ) (Φ,Eα) (S 2, EI 2 ) is a full embedding. 2 AfInst(T ) ISpat SAfInst(T ), ISpat((S 1, I 1 ) (Φ,α) (S 2, I 2 )) = (Φ,Spatα) (S 2, SpatI 2 ) is a right-adjoint-left- (S 1, SpatI 1 ) inverse to IE. 3 SAfInst(T ) is isomorphic to a full coreflective subcategory of AfInst(T ). This answers the question on spatialization construction for topological institutions of A. Sernadas, C. Sernadas, and J. M. Valença.

50 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 29/34 Affine institution localification procedure Localic affine institutions Definition 32 Let X T B op be an affine theory. A localic affine T -institution is a functor S I B op, where S is a category of (abstract) signatures. A localic affine T -institution morphism (S 1, I 1 ) (Φ,α) (S 2, I 2 ) Φ comprises a functor S 1 S2 and a natural transformation α I 1 I2 Φ. LAfInst(T ) is the category of localic affine T -institutions and their morphisms.

51 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 30/34 Affine institution localification procedure Affine institution localification procedure Theorem 33 Let T be an affine theory such that there exists an adjoint situation (η, ε) : T Pt : B op X. 1 LAfInst(T ) IE AfInst(T ), IE((S 1, I 1 ) (Φ,α) (S 2, I 2 )) = (S 1, EI 1 ) (Φ,Eα) (S 2, EI 2 ) is a full embedding. 2 IE has a left-adjoint-left-inverse AfInst(T ) ILoc LAfInst(T ), ILoc((S 1, I 1 ) (S (Φ,α) 2, I 2 )) = (S 1, LocI 1 ) (S (Φ,Locα) 2, LocI 2 ). 3 LAfInst(T ) is isomorphic to a full reflective subcategory of AfInst(T ).

52 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 31/34 Final remarks Conclusion Following the concept of topological institution, we introduced the notion of affine institution and showed its respective spatialization and localification procedures. Affine institutions seem to provide a good framework for elementary institutions and topological institutions, since they do not require the employed algebraic structures to be frames. While A. Sernadas, C. Sernadas, and J. M. Valença. impose the frame structure on the set of theories (certain closed subsets of the set of sentences) of a given signature, which results in technical difficulties, we suggest the use of an arbitrary algebraic structure, which could be determined in each concrete case.

53 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 31/34 Final remarks Conclusion Following the concept of topological institution, we introduced the notion of affine institution and showed its respective spatialization and localification procedures. Affine institutions seem to provide a good framework for elementary institutions and topological institutions, since they do not require the employed algebraic structures to be frames. While A. Sernadas, C. Sernadas, and J. M. Valença. impose the frame structure on the set of theories (certain closed subsets of the set of sentences) of a given signature, which results in technical difficulties, we suggest the use of an arbitrary algebraic structure, which could be determined in each concrete case.

54 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 31/34 Final remarks Conclusion Following the concept of topological institution, we introduced the notion of affine institution and showed its respective spatialization and localification procedures. Affine institutions seem to provide a good framework for elementary institutions and topological institutions, since they do not require the employed algebraic structures to be frames. While A. Sernadas, C. Sernadas, and J. M. Valença. impose the frame structure on the set of theories (certain closed subsets of the set of sentences) of a given signature, which results in technical difficulties, we suggest the use of an arbitrary algebraic structure, which could be determined in each concrete case.

55 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 32/34 References References I J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats, Dover Publications, D. Aerts, E. Colebunders, A. van der Voorde, and B. van Steirteghem, State property systems and closure spaces: a study of categorical equivalence, Int. J. Theor. Phys. 38 (1999), no. 1, J. T. Denniston, A. Melton, and S. E. Rodabaugh, Lattice-valued institutions, Abstracts of the 35th Linz Seminar on Fuzzy Set Theory (T. Flaminio, L. Godo, S. Gottwald, and E. P. Klement, eds.), Johannes Kepler Universität, Linz, 2014, pp Y. Diers, Categories of algebraic sets, Appl. Categ. Struct. 4 (1996), no. 2-3,

56 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 33/34 References References II J. A. Goguen and R. M. Burstall, Introducing institutions, Logics of Programs (Pittsburgh, Pa., 1983), Lecture Notes in Comput. Sci., vol. 164, Springer, Berlin, 1984, pp A. Sernadas, C. Sernadas, and J. M. Valença, A topological view on institutions, Tech. report, CLC, Department of Mathematics, Instituto Superior Técnico, Lisboa, Portugal, A. Sernadas, C. Sernadas, and J. M. Valença, A theory-based topological notion of institution, Recent Trends in Data Type Specification (E. Astesiano, G. Reggio, and A. Tarlecki, eds.), Springer Berlin Heidelberg, 1995, pp S. Vickers, Topology via Logic, Cambridge University Press, 1989.

57 Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 34/34 Thank you for your attention!