Categorical lattice-valued topology Lecture 2: lattice-valued topological systems

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1 Categorcal lattce-valued topology Lecture 2: lattce-valued topologcal systems Sergejs Solovjovs Department o Mathematcs and Statstcs, Faculty o Scence, Masaryk Unversty Kotlarska 2, Brno, Czech Republc Abstract Ths lecture ntroduces lattce-valued topologcal systems as models o topologcal theores, and shows lattcevalued analogues o the system spatalzaton (topologcal spaces rom topologcal systems) and localcaton (locales rom topologcal systems) procedures. As an applcaton o the theory o topologcal systems, we obtan the equvalence between the categores o state property systems o D. Aerts and closure spaces. 1. Bascs on topologcal systems 1.1. Topologcal systems and contnuous maps Remark 1. Topologcal systems were ntroduced by S. Vckers n 1989 n [8] as a common ramework or topologcal spaces and the underlyng algebrac structures o ther topologes locales, n order to provde a convenent way to swtch between spatal (ponted) and localc (pontree) topologcal settngs. Denton 2. A rame s a complete lattce A such that a ( S) = s S (a s) or every a A and every S A. A rame homomorphsm A 1 A2 s a map, whch preserves nte (ncludng the empty) meets and arbtrary jons. Frm s the category (varety) o rames and rame homomorphsms. Loc s the dual category o Frm, whose objects (resp. morphsms) are called locales (resp. localc homomorphsms). Denton 3. A topologcal system s a trple (X, A, ), where X s a set, A s a locale and X A s a bnary relaton (satsacton relaton on (X, A)) such that or every S A and every x X (1) x S x s or some s S; (2) S s nte, then x S x s or every s S. A topologcal system morphsm (X 1, A 1, 1 ) (,) (X 2, A 2, 2 ) (called contnuous map) contans a map X 1 X2 and a localc homomorphsm A 1 A2 such that or every x X 1 and every a A 2, x 1 op (a) (x) 2 a. TopSys s the category o topologcal systems and contnuous maps, whch s concrete over the product category Set Loc. Remark 4. The man examples o topologcal systems are gven by topologcal spaces and locales. Ths lecture course was supported by the ESF Project No. CZ.1.07/2.3.00/ Algebrac methods n Quantum Logc o the Masaryk Unversty n Brno, Czech Republc. Emal address: solovjovs@math.mun.cz (Sergejs Solovjovs) URL: (Sergejs Solovjovs) Preprnt submtted to the Masaryk Unversty n Brno June 27, 2013

2 1.2. Topologcal spaces as topologcal systems Theorem 5. There exsts a ull embeddng Top E TopSys dened by E((X 1, τ 1 ) (X 1, τ 1, 1 ) (,( ) op ) (X 2, τ 2, 2 ), where x U x U. (X 2, τ 2 )) = Proo. To show that the unctor s correct on morphsms, notce that gven x X 1 and V τ 2, t ollows that x (V ) (x) V. To show that the unctor s ull, notce that gven a contnuous map (X 1, τ 1, 1 ) (,) (X 2, τ 2, 2 ), or every V τ 2, t ollows that x op (V ) (x) V x (V ). Theorem 6. The embeddng Top E TopSys has a rght adjont TopSys Spat Top, whch s dened by Spat((X 1, A 1, 1 ) (,) (X 2, A 2, 2 )) = (X 1, {ext(a) a A 1 }) (X 2, {ext(b) b A 2 }), where ext(c) = {x X x c}. Proo. To show that the unctor s correct on objects, notce, e.g., that gven a, b A, t ollows that ext(a) ext(b) = {x X x a and x b} = {x X x (a b)} = ext(a b). To show that the unctor s correct on morphsms, notce that gven a A 2, t ollows that x (ext(a)) (x) ext(a) (x) 2 a x 1 op (a) x ext( op (a)), and thereore, (ext(a)) = ext( op (a)) τ 1. Corollary 7. Top s somorphc to a ull corelectve subcategory o the category TopSys. Remark 8. The unctor o Theorem 6 s called the system spatalzaton procedure Locales as topologcal systems Denton 9. Gven a rame A, P t(a) stands or the set Frm(A, 2) o all rame homomorphsms A 2, whose elements are called the ponts o A and are denoted p. Theorem 10. There exsts a ull embeddng Loc E TopSys, whch s dened by E(A 1 A2 ) = (P t(a 1 ), A 1, 1 ) ((op ) 2,) (P t(a 2 ), A 2, 2 ), where p c p(c) =, and ( op ) 2 (p) = p op. Proo. To show that the unctor s correct on objects, notce that gven p P t(a ) and S A, t ollows that p S p( S) = s S p(s) = p(s) = or some s S p s or some s S. To show that the unctor s correct on morphsms, notce that gven p P t(a 1 ) and a A 2, t ollows that p 1 op (a) p op (a) = (( op ) 2 (p))(a) = ((op ) 2 (p)) 2 a. To show that the unctor s ull, notce that gven a contnuous map (P t(a 1 ), A 1, 1 ) (,) (P t(a 2 ), A 2, 2 ), or every p P t(a 1 ), t ollows that ((p))(a) = (p) 2 a p 1 op (a) p( op (a)) = (( op ) 2 (p))(a) =. Theorem 11. The embeddng Loc E TopSys has a let adjont TopSys Loc Loc, whch s dened by Loc((X 1, A 1, 1 ) (,) (X 2, A 2, 2 )) = A 1 A2. Corollary 12. Loc s somorphc to a ull relectve subcategory o the category TopSys. Remark 13. The unctor o Theorem 11 s called the system localcaton procedure. 2. Lattce-valued topologcal systems Remark 14. In [7], J. T. Dennston and S. E. Rodabaugh consdered relatonshps between topologcal systems and lattce-valued topologcal spaces. It appeared though that the crsp concept o topologcal system s not well-suted or lattce-valued topology. Motvated by ths observaton, J. T. Dennston, A. Melton and S. E. Rodabaugh ntroduced n [4] the noton o lattce-valued topologcal system. 2

3 Denton 15 (Fxed-bass approach). Gven a locale L, an L-topologcal system s a trple (X, A, ), where X s a set, A s a locale, and X A L s a map (L-satsacton relaton on (X, A)) such that A (x, ) L s a rame homomorphsm or every x X. An L-topologcal system morphsm (X 1, A 1, 1 ) (,) (X 2, A 2, 2 ) (called L-contnuous map) contans a map X 1 X2 and a localc homomorphsm A 1 A2 such that 1 (x, op (a)) = 2 ((x), a) or every x X 1 and every a A 2. L-TopSys s the category o L-topologcal systems and L-contnuous maps. Remark 16. The category 2-TopSys s somorphc to the category TopSys o S. Vckers. Denton 17 (Varable-bass approach). Gven a subcategory C o Loc, a C-topologcal system s a tuple (X, L, A, ), whch s an L-topologcal system or some C-object L. A C-topologcal system morphsm (X 1, L 1, A 1, 1 ) (,ψ,) (X 2, L 2, A 2 2 ) (called C-contnuous map) contans a map X 1 X2 and localc homomorphsms A 1 A2, L 1 ψ L2 such that 1 (x, op (b)) = ψ op 2 ((x), a) or every x X 1 and every a A 2. C-TopSys s the category o C-topologcal systems and C-contnuous maps. Remark 18. In [8], S. Vckers provded an alternatve denton o topologcal systems, whch was extended to lattce-valued case by J. T. Dennston, A. Melton and S. E. Rodabaugh n [4]. Proposton 19. Gven sets X, Y, Z, where exsts a bjecton Set(X Y, Z) h Set(Y, Z X ) gven by ((h())(y))(x)=(x, y). Its nverse map Set(Y, Z X ) g Set(X Y, Z) s gven by (g())(x, y) = ((y))(x). Denton 20. Gven a subcategory C o Loc, a C-topologcal system s a tuple (X, L, A, κ), where (X, L, A) s a Set C Loc-object and A κ L X s a rame homomorphsm. A C-topologcal system morphsm (X 1, L 1, A 1, κ 1 ) (,ψ,) (X 2, L 2, A 2, κ 2 ) s a Set C Loc-morphsm such that the dagram A 2 op A 1 κ 2 L X2 2 (,ψ) κ 1 L X1 1 commutes. C-TopSys a s the category o C-topologcal systems and C-contnuous maps. Remark 21. Proposton 19 provdes an somorphsm between the categores C-TopSys and C-TopSys a. 3. Categorcal lattce-valued topologcal systems Remark 22. Denton 20 suggests a categorcal approach to lattce-valued topologcal systems, whch s based n powerset operators and topologcal theores. Denton 23. Let X T B op be a t-theory. TopSys(T ) s the comma category (T 1 B op), concrete over the product category X B op, whose objects (T -systems) are trples (X, κ, B), whch are made by B op - morphsms T X κ B (T -satsacton relaton on (X, B)), and whose morphsms (T -contnuous morphsms) (X 1, κ 1, B 1 ) (,) (X 2, κ 2, B 2 ) are X B op -morphsms (X 1, B 1 ) (,) (X 2, B 2 ) such that the dagram T X 1 T T X 2 κ 1 B 1 B 2 κ 2 commutes. 3

4 Example 24. The case o the ground category X = Set S, where S s a subcategory o A op or some varety A, s called varety-based approach. In partcular, TopSys((V A, B)) provdes the category A B -TopSys, whch s the ramework or xed-bass varety-based topologcal systems, whereas TopSys((V, B)) gves the category (S, B)-TopSys (the case A = B s shortened to S-TopSys), whch s the ramework or varablebass varety-based topologcal systems. More specc, (1) TopSys((P, Frm)) s somorphc to the category TopSys o topologcal systems o S. Vckers [8]; (2) TopSys((P, Set)) s somorphc to the category IntSys o nterchange systems o J. T. Dennston, A. Melton and S. E. Rodabaugh [5, 6]; (3) TopSys((R 3, Frm)), where S = Loc, s somorphc to the category Loc-TopSys o lattce-valued topologcal systems o J. T. Dennston et al. [4] Spatalzaton o lattce-valued topologcal systems Remark 25. One o the man results o the theory o lattce-valued topologcal systems s the possblty o representng the category Top(T ) as a ull subcategory (wth convenent propertes) o the category TopSys(T ). The embeddng though s not concrete, snce the ground categores n queston are derent. Theorem 26. (1) There exsts a ull embeddng Top(T ) E TopSys(T ), whch s gven by E((X 1, τ 1 ) (X 2, τ 2 )) = (X 1, e op τ 1, τ 1 ) (,) (X 2, e op τ 2, τ 2 ), where e τ denotes the ncluson τ T X, and op stands or the restrcton τ 2 (T ) op τ 1 τ 2 τ1. (2) There exsts a unctor TopSys(T ) Spat Top(T ) gven by Spat((X 1, κ 1, B 1 ) (X 1, ( 1 ) (B 1 )) (X 2, ( 2 ) (B 2 )). (3) Spat s a rght-adjont-let-nverse to E. (,) (X 2, κ 2, B 2 )) = Proo. Ad (1). To show that the unctor s ull, notce that gven a T -contnuous morphsm (X 1, e op τ 1, τ 1 ) (,) (X 2, e op τ 2, τ 2 ), commutatvty o the dagram τ 2 op τ 1 e τ2 e τ1 T X 2 (T ) op T X 1 mples that ((T ) op ) (τ 2 ) τ 1, and thereore, (X 1, τ 1 ) (X 2, τ 2 ) s T -contnuous wth E = (, op ). Ad (2). To show that Spat(, ) s T -contnuous, notce that gven b B 2, t ollows that (T ) op 2 (b) = 1 op (b) ( 1 ) (B 1 ). Ad (3). Straghtorward computatons show that SpatE = 1 Top(T ). For the rst clam, t wll be enough to show that every system (X, κ, B) has an E-co-unversal arrow,.e., a TopSys(T )-morphsm ESpat(X, κ, B) ε (X, κ, B) such that or every TopSys(T )-morphsm E(X, τ ) (,) (X, κ, B), there exsts a unque Top(T )-morphsm (X, τ ) g Spat(X, κ, B) such that the ollowng trangle commutes E(X, τ ) Eg ESpat(X, κ, B) 4 (,) ε (X, κ, B).

5 There s a T -contnuous morphsm (ESpat(X, κ, B) = (X, e op ( ) (B), (κop ) (B))) ε=(1 X,κ) (X, κ, B). Gven a TopSys(T )-morphsm E(X, τ ) (,) (X, κ, B), the dagram B op τ T X e τ (T ) op T X commutes, and thereore, (X, τ ) (Spat(X, κ, B) = (X, ( ) (B))) s a Top(T )-morphsm, whch makes the above-mentoned trangle commute, and whose unqueness s clear. Corollary 27. Top(T ) s somorphc to a ull (regular mono)-corelectve subcategory o TopSys(T ). Proo. In vew o Theorem 26, t wll be enough to show that gven a T -system (X, κ, B), the map B κop ( ) (B) s a regular epmorphsm n B. Dene C = {(b 1, b 2 ) B B (b 1 ) = (b 2 )} (the kernel o ), and let C π B be gven by π (b 1, b 2 ) = b or {1, 2}. It s easy to see that (, ( ) (B)) s a coequalzer o (π 1, π 2 ), whch proves the clam. Remark 28. The unctor Spat o Theorem 26 extends the system spatalzaton procedure o S. Vckers Localcaton o lattce-valued topologcal systems Remark 29. The second mportant result o the theory o lattce-valued topologcal systems s the possblty o representng the category B op as a ull subcategory (wth convenent propertes) o the category TopSys(T ). Unlke Top(T ), however, the embeddng o B op nto TopSys(T ) s not always possble. Proposton 30. There exsts a unctor TopSys(T ) Loc B op, whch s dened by Loc((X 1, κ 1, B 1 ) (,) (X 2, κ 2, B 2 )) = B 1 B2. Remark 31. The unctor Loc o Proposton 30 extends the system localcaton procedure o S. Vckers. Theorem 32. Gven a t-theory X T B op, the ollowng are equvalent: (1) there exsts an adjont stuaton (η, ε) : T P t : B op X; (2) there exsts a ull embeddng B op E TopSys(T ) such that Loc s a let-adjont-let-nverse to E. B op then s somorphc to a ull (n general, nether mono- nor ep-) relectve subcategory o TopSys(T ). Proo. Ad (1) (2). Dene a unctor B op E TopSys(T ) by E(B1 B2 ) = (P tb 1, ε B1, B 1 ) (P tb 2, ε B2, B 2 ). Correctness o E on morphsms ollows rom commutatvty o the dagram (P t,) T P tb 1 T P t T P tb 2 ε B1 B 1 B 2. ε B2 Moreover, E s clearly an embeddng. To very that E s ull, notce that gven a T -contnuous morphsm (P tb 1, ε B1, B 1 ) (,) (P tb 2, ε B2, B 2 ), commutatvty o the dagram T P t T P tb 1 ε B1 T B 2 T P tb 2 ε B2 B 2 5

6 mples that ε B2 T P t = ε B2 T, and thereore, P t =. Gven a T -system (X, κ, B), straghtorward calculatons show that (X, κ, B) (:=P tκ η X,1 B ) ((P tb, ε B, B) = ELoc(X, κ, B)) provdes an E-unversal arrow or (X, κ, B). It s also easy to see that LocE = 1 B op. Ad (2) (1). Suppose we have an adjuncton Loc E : B op TopSys(T ). There clearly exsts a unctor TopSys(T ) Gr X, whch s dened by Gr((X 1, κ 1, B 1 ) (,) (X 2, κ 2, B 2 )) = X 1 X2 ( Gr s an abbrevaton or ground category ). Moreover, there exsts a ull embeddng X M TopSys(T ), whch (,T ) s gven by M(X 1 X2 ) = (X 1, 1 T X1, T X 1 ) (X 2, 1 T X2, T X 2 ). Straghtorward calculatons show that M s a let-adjont-rght-nverse to Gr, e.g., gven a T -system (X, κ, B), t ollows that (MGr(X, κ, B) = (X, 1 T X, T X)) (1 X,κ) (X, κ, B) provdes an M-co-unversal arrow or (X, κ, B). The two adjont stuatons X M Gr TopSys(T ) Loc E B op gve rse to the requred one X T =LocM Theorem 33. Every topologcal theory Set T A B op, whch s nduced by the bp-theory Set V A=( ) A A op, has a rght adjont. Proo. Dene a unctor B op P t A P t A Set by P t A (B 1 B2 ) = P t A B 1 P t A B 2, where P t A B = B(B, A ) and (P t A )(p) = p op. Straghtorward calculatons show that gven a B-algebra B, the map B εop (T A P t A B = A B(B, A ) ), dened by (ε op (b))(p) = p(b), provdes a T A -co-unversal arrow or B. GrE B op. Remark 34. Theorem 33 can not be restated or varable-bass topologcal theores. Proposton 35. Consder the t-theory Set A op ((,) ) op T :=V=( ) A op gven by ((X 1, A 1 ) (,) (X 2, A 2 )) = A X1 1 A X2 2, where (, ) (α) = op α, and, moreover, assume that there exsts an A-algebra A, whose underlyng set s nte, e.g., has the cardnalty n. Then T has no rght adjont. Proo. I T has a rght adjont, then T preserves colmts, and thus, coproducts. Gven a sngleton set 1, T ((1, A) (1, A)) = T ((1 1, A A)) = (A A) (1 1) and T (1, A) T (1, A) = A 1 A 1. I T ((1, A) (1, A)) = T (1, A) T (1, A), then n 4 = Card((A A) (1 1) ) = Card(A 1 A 1 ) = n 2, whch s a contradcton Sobrety-spatalty equvalence revsted Remark 36. Ths subsecton provdes a more general analogue o the sobrety-spatalty equvalence, whch s consdered at the end o Lecture 1. Remark 37. Let X T B op be a t-theory, whch has a rght adjont. One has the adjont stuatons Top(T ) E s Spat Loc TopSys(T ) B op, whch gve rse to the adjuncton Top(T ) more precsely, (η, ε) : O P T : B op Top(T ). E l O:=LocE s P T :=SpatE l Denton 38. Spat s the ull subcategory o B op, whch contans B-algebras B such that OP T B ε B B s an somorphsm. Denton 39. Sob s the ull subcategory o Top(T ), whch contans T -spaces (X, τ) such that (X, τ) η (X,τ) P T O(X, τ) s an somorphsm. B op, or, Proposton 40. The adjuncton Top(T ) O P T B op restrcts to an equvalence Sob O P T Spat. 6

7 Denton 41. A T -space (X, τ) s called separated provded that (X, τ) η (X,τ) P T O(X, τ) s a monomorphsm. Top s (T ) stands or the ull subcategory o Top(T ) o separated T -spaces. Example 42. Separated T -spaces n the category Top((P, Frm)), whch s somorphc to Top (recall Lecture 1), are precsely the crsp T 0 topologcal spaces. Theorem 43. Let T be a t-theory n X. I X s a (Retr, Mono)-category, where Retr (resp. Mono) s the class o retractons (resp. monomorphsms) n X, then Top s (T ) s a relectve subcategory o Top(T ). Example 44. The category Set s a (Retr, Mono)-category, and thereore, Theorem 43 s applcable to every t-theory n Set, whch s generated by, e.g., a bp-theory o the orm Set V A=( ) A A op TopSys(T ) s an essentally algebrac category Remark 45. The category Top(T ) s topologcal over ts ground category X (see Lecture 1). The category TopSys(T ), however, s essentally algebrac over ts ground category X B op. Theorem 46 ([1]). A concrete category (C, ) over X s essentally algebrac the ollowng condtons are satsed: (1) creates somorphsms, (2) has a let adjont, (3) C s (Ep, Mono-Source)-actorzable. Proposton 47. The unctor TopSys(T ) X B op creates somorphsms. Proo. Gven an X B op -somorphsm (X 1, B 1 ) (,) (X 2, κ 2, B 2 ), the unque T -satsacton relaton on (X 1, B 1 ), whch makes (, ) an somorphsm n TopSys(T ), can be dened by κ 1 = 1 κ 2 T. Proposton 48. The unctor TopSys(T ) X B op has a let adjont. Proo. It s enough to show that every X B op -object (X, A) has a -unversal arrow,.e., an X B op - morphsm (X 1, B 1 ) η (X 2, κ 2, B 2 ) such that or every X B op -morphsm (X 1, B 1 ) (,) (X 3, κ 3, B 3 ), there exsts a unque T -contnuous morphsm (X 2, κ 2, B 2 ) (g,ψ) (X 3, κ 3, B 3 ) such that the trangle (X 1, B 1 ) η (,) (X 2, κ 2, B 2 ) (g,ψ) (X 3, κ 3, B 3 ) commutes. There exsts an X B op -morphsm (X 1, B 1 ) η=(1 X 1,π op B ) 1 (X 1, π op T X 1, T X 1 B 1 ), where T X 1 π B1 π T X1 T X 1 B 1 B1 s the product o T X 1 and B 1 (recall that B s a varety). Gven an X B op -morphsm (X 1, B 1 ) (,) ψ (X 3, κ 3, B 3 ), there exsts a unque B-homomorphsm B 3 T X1 B 1, whch s dened by commutatvty o the ollowng dagram: T X 3 (T ) op 3 B 3 T X 1 T X 1 B 1 π T X1 7 ψ op π B1 B 1.

8 The let-hand sde o the dagram mples that (X 1, π op T X 1, T X 1 B 1 ) (,ψ) (X 3, κ 3, B 3 ) s a TopSys(T )- morphsm, and the rght-hand sde o the dagram gves commutatvty o the above-mentoned trangle. Gven another TopSys(T )-morphsm (X 1, π op T X 1, T X 1 B 1 ) (,ψ ) (X 3, κ 3, B 3 ) wth the same property, t ollows that (, ) = (, ψ ) η = (, ψ ) (1 X1, π op B 1 ) = (, ψ π op B 1 ), and thus, = and π B1 ψ op = op. Moreover, (T ) op 3 = (T ) op 3 = π T X1 ψ op by the condton o T -contnuous morphsms, and thereore, ψ = ψ op by the unversal property o products. Proposton 49. I every source S n X has an (Ep, Mono-Source)-actorzaton S = M e such that (T e) op s an njectve map, then the category TopSys(T ) s (Ep, Mono-Source)-actorzable. Proo. Let S = ((X, κ, B) (,) (X, κ, B )) I be a source n TopSys(T ). By the assumpton, there exsts an (Ep, Mono-Source)-actorzaton X X = X e X m X o the source K = (X X ) I wth the above-mentoned property. Dene B = ) (B ) (recall that S s the smallest B-subalgebra I contanng S), let B ψ ψ B be the ncluson, and let B B be dened by ψ (b) = (b). op ψ ( op It ollows ψ that B B = B B B s an (Ep-Snk, Mono)-actorzaton o the snk T = (B B) I n B. Moreover, easy consderatons show that B = {ω B ( c j ) c j K} wth K = ) (B ), where ω B ( c j ) s a word (n the algebrac sense), consstng o operatons on B and takng elements o K as arguments (notce that we delberately omt the set, j ranges over, snce, n general, t conssts o the unon o some n λ ; also bear n mnd that the word may contan no operatons at all, beng just an element o K). In vew o ths characterzaton, dene a map B κop T X by (ω B ( c j )) = ω T X ( (T m j ) op (c j) ) wth c j = j (b j ). To very correctness o the denton, suppose that ω B ( c j ) = ω B ( c j ), and get then (T e) op (ω T X ( (T m j ) op j (b j) )) = ω T X ( (T e) op (T m j ) op j (b j) ) = ω T X ( (T j ) op j (b j) ) = ω T X ( op j (b j) ) = ω B ( op j (b j) ) = ω B ( c j ) = ω B ( c j ) = (T e)op (ω T X ( (T m j ) op j (c j ) )), and thereore, ωt X ( (T m j ) op j (b j) ) = ω T X ( (T m j ) op j (b j ) ) by the assumpton o the proposton. To prove that s a homomorphsm, take λ Λ and d j B or j n λ, gettng (ωλ B( d j nλ )) = (ωλ B( ωj B ( c js ) nλ )) = ωλ T X( ωj T X ( (T m js ) op j s (b js ) ) nλ ) = ωλ T X( κop (d j ) nλ ), whch proves that (X, κ, B) s a T -system and, at the same tme, obtanng the ollowng dagram I ( op op j B op B ψ B ψ (T m ) op T X T X (T e) op (T ) op T X. It remans to show commutatvty o both ts let and rght rectangles. For the ormer, notce that gven b B, (T e) op ψ (b ) = (T e) op op (b ) = (T e) op (T m ) op (b ) by the denton o, and thereore, (T e) op ψ = (T e) op (T m ) op k op, whch mples ψ = (T m ) op k op. For the latter, use the act that (T e) op ψ = (T e) op (T m ) op = (T ) op = op = ψ ψ or every I, mples (T e) op = ψ. As a result, (X, κ, B) (,) (X, κ, B ) = (X, κ, B) (e,ψop ) (X, κ, B) (m,ψop ) (X, κ, B ) s the requred (Ep, Mono-Source)-actorzaton o S. 8

9 Theorem 50. Gven a t-theory X T B op, wth the property that every source S n X has an (Ep, Mono- Source)-actorzaton S = M e such that (T e) op has an njectve underlyng map, the category TopSys(T ) s essentally algebrac over X B op. Proo. Follows rom Propostons 47, 48, 49. Remark 51. By Theorem 50, all the categores o the orm TopSys((V, B)) (e.g., the category TopSys o S. Vckers) are essentally algebrac. 4. An applcaton o the theory o lattce-valued topologcal systems 4.1. State property systems o D. Aerts and closure spaces Remark 52. State property systems were ntroduced by D. Aerts n 1999 n [2] as the basc mathematcal structure n the Geneva-Brussels approach to the oundaton o physcs. Moreover, the category o state property systems s equvalent to the category o closure spaces [3]. Denton 53. A closure space s a par (X, F), where X s a set, and F s a amly o subsets o X, whch satses the ollowng propertes: (1) F; (2) F F or I, then I F F. A map X 1 X2 between closure spaces (X 1, F 1 ) and (X 2, F 2 ) s sad to be contnuous provded that ( ) (F 2 ) F 1. Cls s the category o closure spaces and contnuous maps. Denton 54. CSL s the varety o closure semlattces (c-semlattces),.e., -semlattces, wth the sngled out bottom element. Remark 55. Top((P, CSL)) s somorphc to the category Cls. Denton 56. A state property system s a trple (X, A, κ), where X s a set, A s a c-semlattce, and A κ PX s an njectve c-semlattce homomorphsm (Cartan map). A state property system morphsm (X 1, A 1, κ 1 ) (,) (X 2, A 2, κ 2 ) s a Set CSL op -morphsm (X 1, A 1 ) (,) (X 2, A 2 ) such that the dagram A 2 op A 1 κ 2 κ 1 PX 2 PX 1 commutes. SP s the category o state property systems and state property system morphsms. Theorem 57 ([3]). The categores SP and Cls are equvalent Lattce-valued state property systems and closure spaces Denton 58. A T -system (X, κ, B) s called separated provded that the map B κop T X s njectve. TopSys s (T ) stands or the ull subcategory o TopSys(T ) o separated T -systems. Example 59. TopSys s ((P, CSL)) s somorphc to the category SP o D. Aerts. Proposton 60. There exst the restrctons Top(T ) E TopSys s (T ) and TopSys s (T ) Spat Top(T ) o the unctors E and Spat o Theorem 26, respectvely. 9

10 Proo. It wll be enough to show that gven a T -space (X, τ), E(X, τ) s a separated T -system, whch ollows mmedately rom Theorem 26 (1). Theorem 61. The unctors Top(T ) E TopSys s (T ) and TopSys s (T ) Spat Top(T ) provde an equvalence between the categores Top(T ) and TopSys s (T ). Proo. By Theorem 26 (3), Spat s a rght-adjont-let-nverse to E. To prove the theorem, t s enough to show that or every separated T -system (X, κ, B), the E-co-unversal arrow ESpat(X, κ, B) ε=(1 X,κ) (X, κ, B) rom the proo o Theorem 26 (3) s an somorphsm. The clam ollows rom the denton o ε, snce B κop ( ) (B) s always surjectve, and t s njectve by the property o separated T -systems. Corollary 62. The category TopSys s (T ) s a ull (regular mono)-corelectve subcategory o TopSys(T ). Proo. Combne Theorem 61 and Corollary 27. Remark 63. The relevance o Theorem 61 s slghtly undermned by the act that Top(T ) and TopSys s (T ) have derent ground categores,.e., X and X B op, respectvely. In partcular, notwthstandng the result that Top(T ) s topologcal over ts ground category, TopSys s (T ) never needs to have the same property. On the other hand, beng topologcal, Top(T ) s (co)complete provded that ts ground category s (co)complete, and that can be transerred mmedately to the category TopSys s (T ). Reerences [1] J. Adámek, H. Herrlch, and G. E. Strecker, Abstract and Concrete Categores: The Joy o Cats, Dover Publcatons (Mneola, New York), [2] D. Aerts, Foundatons o quantum physcs: a general realstc and operatonal approach, Int. J. Theor. Phys. 38 (1999), no. 1, [3] D. Aerts, E. Colebunders, A. van der Voorde, and B. van Sterteghem, State property systems and closure spaces: a study o categorcal equvalence, Int. J. Theor. Phys. 38 (1999), no. 1, [4] J. T. Dennston, A. Melton, and S. E. Rodabaugh, Lattce-valued topologcal systems, Abstracts o the 30th Lnz Semnar on Fuzzy Set Theory (U. Bodenhoer, B. De Baets, E. P. Klement, and S. Samnger-Platz, eds.), Johannes Kepler Unverstät, Lnz, 2009, pp [5] J. T. Dennston, A. Melton, and S. E. Rodabaugh, Lattce-valued predcate transormers and nterchange systems, Abstracts o the 31st Lnz Semnar on Fuzzy Set Theory (P. Cntula, E. P. Klement, and L. N. Stout, eds.), Johannes Kepler Unverstät, Lnz, 2010, pp [6] J. T. Dennston, A. Melton, and S. E. Rodabaugh, Formal concept analyss and lattce-valued nterchange systems, Abstracts o the 32nd Lnz Semnar on Fuzzy Set Theory (D. Dubos, M. Grabsch, R. Mesar, and E. P. Klement, eds.), Johannes Kepler Unverstät, Lnz, 2011, pp [7] J. T. Dennston and S. E. Rodabaugh, Functoral relatonshps between lattce-valued topology and topologcal systems, Quaest. Math. 32 (2009), no. 2, [8] S. Vckers, Topology va Logc, Cambrdge Unversty Press,