A Sufficient Condition for Liftable Adjunctions between Eilenberg-Moore Categories Koki Nishizawa (Kanagawa Univ.) with Hitoshi Furusawa （Kagoshima Univ. ） RAMiCS 2014/4/30 1 Background Some left adjoints to forgetful functors are given by construction of ideals. Conway's S-algebra Quantale Complete Join Semilattice Closed Semiring *-continuous Idempotent Join Semilattice Kleene algebra Semiring (with ⊥) 2 Background Some left adjoints to forgetful functors are given by construction of ideals. Conway's S-algebra Quantale Complete Join Semilattice Closed Semiring Construction of ideals *-continuous Idempotent Join Semilattice Kleene algebra Semiring (with ⊥) 3 Background Some left adjoints to forgetful functors are given by construction of ideals. Conway's S-algebra Quantale Complete Construction of Join Semilattice countable ideals Construction Closed Semiring of ideals Construction of ideals Construction of *-ideals *-continuous Idempotent Join Semilattice Kleene algebra Semiring (with ⊥) 4 Background Some left adjoints to forgetful functors Why ? are given by construction of ideals. When ? Conway's S-algebra Quantale Complete Construction of Join Semilattice countable ideals Construction Closed Semiring of ideals Construction of ideals Construction of *-ideals *-continuous Idempotent Join Semilattice Kleene algebra Semiring (with ⊥) 5 Background Some left adjoints to forgetful functors Why ? are given by construction of ideals. When ? Conway's S-algebra Quantale Complete Construction of Join Semilattice countable ideals Construction Closed Semiring of ideals Construction of ideals Construction of *-ideals *-continuous Idempotent Join Semilattice Kleene algebra Semiring (with ⊥) 6 Result ： Quantale Complete Join Semilattice Construction of ideals Construction of ideals Idempotent Join Semilattice Semiring (with ⊥) 7 Result ： Complete Join Semilattice Join Semilattice over Monoid over Monoid Quantale Complete Join Semilattice Construction of ideals Construction of ideals Idempotent Join Semilattice Semiring (with ⊥) 8 Result ：from Monoid to T-algebra Complete Join Semilattice Join Semilattice over Monoid over Monoid Complete Quantale Complete Join Semilattice Join Semilattice over T-algebra Construction Construction Construction of ideals of ideals of ideals Join Semilattice Idempotent Join Semilattice over T-algebra Semiring (with ⊥) 9 Contents 1. Construction of left adjoint from join semilattice to complete join semilattice 2. Construction of left adjoint from join semilattice over T-algebra to complete join semilattice over T-algebra 3. Conclusion 10 Contents 1. Construction of left adjoint from join semilattice to complete join semilattice 2. Construction of left adjoint from join semilattice over T-algebra to complete join semilattice over T-algebra 3. Conclusion 11 Ideal in Join Semilattice Def. S is a join semilattice if S is a partially ordered set S has a join ∨X for each finite subset X of S. Def. S is a complete join semilattice if S is a partially ordered set S has a join ∨X for each subset X of S. Def. A subset X of a join semilattice S is an ideal if X is closed under finite joins X is downward-closed 12 Goal Prop. The forgetful functor from CSLat to SLat has a left adjoint, which sends S to the set of all ideals of S. CSLat The category of complete join semilattices and homomorphisms. construction forget of ideals The category of join semilattices SLat and homomorphisms. 13 A sufficient condition of adjunction [Toposes, Triples and Theories] G:D' →D has a left adjoint if D' G D 14 A sufficient condition of adjunction [Toposes, Triples and Theories] G:D' →D has a left adjoint if D' G C D 15 A sufficient condition of adjunction [Toposes, Triples and Theories] G:D' →D has a left adjoint if D' G C D 16 A sufficient condition of adjunction [Toposes, Triples and Theories] G:D' →D has a left adjoint if D' G C D 17 A sufficient condition of adjunction [Toposes, Triples and Theories] D' has G:D' →D has a left adjoint if coequalizers of D' certain diagrams G C D 18 D' has coequalizers of A sufficient condition of adjunction [Toposes, TriplesF'UFUd and Theories]F'UFUd F'UFη' Ud F'UFU'F'Ud G:D' →D has a left adjoint if ≅ D' F'UFUGF'Ud F'Uε d F'Uε GF'Ud F'UGF'Ud C G ≅ F'U'F'Ud ε' F'Ud D F'Ud F'Ud for each d ∈D 19 + Beck's Theorem D' has G:D' →D has a left adjoint if coequalizers of D' certain diagrams G C Beck's Theorem: D A monadic functor U creates a coequalizer of parallel arrows f,g if Uf and Ug has an absolute coequalizer. 20 + Beck's Theorem D' has G:D' →D has a left adjoint if coequalizers of D' certain diagrams G C absolute coequalizer Beck's Theorem: = coequalizerD preserved by all functors A monadic functor U creates a coequalizer of parallel arrows f,g if Uf and Ug has an absolute coequalizer . 21 + Beck's Theorem D' has G:D' →D has a left adjoint if coequalizers of D' certain diagrams G C Beck's Theorem: D A monadic functor U creates a coequalizer of parallel arrows f,g if Uf and Ug has an absolute coequalizer . 22 + Beck's Theorem C has absolute G:D' →D has a left adjoint if coequalizers of D' certain diagrams G C Beck's Theorem: D A monadic functor U creates a coequalizer of parallel arrows f,g if Uf and Ug has an absolute coequalizer. 23 + Beck's Theorem C has absolute G:D' →D has a left adjoint if coequalizers of D' certain diagrams G C D 24 When G is the functor which pre-composes a monad map C has absolute G:D' →D has a left adjoint if coequalizers of D' certain diagrams P'-Alg G C D P-Alg 25 When G is the functor which pre-composes a monad map C has absolute G:D' →D has a left adjoint if coequalizers of D' certain diagrams P'-Alg G(a : P'c →c) G C = a ・ ic : Pc →c by monad map i : P →P' D P-Alg 26 C has an absolute coequalizer of Theorem 1 P'(b) P'Pc P'c G:D' →D has a left adjoint µ' ・P'i c if D' for each P-algebra P'-Alg b : Pc →c G(a : P'c →c) G C = a ・ ic : Pc →c by monad map i : P →P' D P-Alg 27 Theorem 1 G : P'-Alg →P-Alg has a left adjoint if P,P' are monads on C i : P →P' is a monad map P'(b) C has an absolute coequalizer of P'Pc P'c for each P-algebra (b:Pc →c) µ' ・P'i c G(a : P'c →c)=(a ・ic : Pc →c) 28 Corollary 1 power set complete monad ℘ join semilattices CSLat forget Set SLat finite power set join semilattices monad ℘!$) 29 Corollary 1 power set complete monad ℘ join semilattices CSLat forget Set by embedding i : ℘!$) →℘ SLat finite power set join semilattices monad ℘!$) 30 Ideals give an absolute Corollary 1 coequalizer of ℘ ∨ power set ℘(℘!$) (͍)) ℘(͍) complete monad ℘ ∪ join semilattices CSLat for each join semilattice (S, ∨) by construction of ideals forget Set by embedding i : ℘!$) →℘ SLat finite power set join semilattices monad ℘!$) 31 Ideals give an absolute Corollary 1 coequalizer of ℘ ∨ power set ℘(℘!$) (͍)) ℘(͍) complete monad ℘ ∪ join semilattices CSLat for each join semilattice (S, ∨) by construction of ideals forget Set by embedding i : ℘!$) →℘ SLat finite power set join semilattices monad ℘!$) 32 Lemma For each join semilattice S, ideal completion of S gives an absolute coequalizer of ℘(∨) and ∪ in Set. ℘ ∨ ℘(℘!$) (͍)) ℘(͍) Ideal(S) ∪ Ideal Completion 33 Lemma For each join semilattice S, ideal completion of S gives an absolute coequalizer of ℘(∨) and ∪ in Set. ℘ ∨ ℘(℘!$) (͍)) ℘(͍) Ideal(S) ∪ Ideal Completion 34 embedding Ideal(S) ℘(͍) Proof of Ideal the Lemma Identity Completion Ideal(S) ℘!$) ℘(͍) ℘(℘!$) (͍)) down ℘(͍) ℘(℘!$) (͍)) ∪ ℘ ∨ Identity ℘(͍) Identity ℘(͍) ℘ ∨ ℘(℘!$) (͍)) ℘(͍) Ideal(S) ∪ Ideal Completion ℘ !$) ℘ ∨ down ℘(͍) ℘(℘!$) (͍)) ℘(͍) ℘(℘!$) (͍)) Ideal Completion ∪ embedding Ideal(S) ℘(͍) 35 Contents 1. Construction of left adjoint from join semilattice to complete join semilattice 2. Construction of left adjoint from join semilattice over T-algebra to complete join semilattice over T-algebra 3. Conclusion 36 Definitions Def. (S, ∨,・,e) is an idempotent semiring if (S, ∨) is a join semilattice (S, ・,e) is a monoid a・(∨X)= ∨{a ・x | x ∈X} and ( ∨X) ・a= ∨{x ・a | x ∈X} for each a ∈S and each finite subset X of S Def. (S, ∨,・,e) is a quantale (i.e., complete idempotent semiring) if (S, ∨) is a complete join semilattice (S, ・,e) is a monoid a・(∨X)= ∨{a ・x | x ∈X} and ( ∨X) ・a= ∨{x ・a | x ∈X} for each a ∈S and each subset X of S 37 Goal Prop. The forgetful functor from Qt to ISRng has a left adjoint, which sends S to the set of all ideals of S. Qt The category of quantales and homomorphisms. construction forget of ideals The category of idempotent semirings ISRng and homomorphisms. 38 Theorem 1 P'-Alg forget C P-Alg 39 Theorem 2 P' T-Alg forget T-Alg PT-Alg 40 Theorem 2 C P'T-Alg forget T-Alg PT-Alg 41 Theorem 2 P'T-Alg has coequalizers of C certain diagrams P'T-Alg forget T-Alg PT-Alg 42 C has an absolute coequalizer of Theorem 2 P'T-Alg has P'(b) P'Pc coequalizers of P'c C ・ certain diagrams µ' P'i c P'T-Alg for each PT-algebra (b : Pc →c, t:Tc →c) forget T-Alg PT-Alg 43 C has an absolute coequalizer of Theorem 2 P'(b) P'Pc P'c C µ' ・P'i c P'T-Alg for each PT-algebra (b : Pc →c, t:Tc →c) forget T-Alg PT-Alg 44 Theorem 2 G : P'T-Alg →PT-Alg has a left adjoint if P,P' are monads on C Assumption i : P →P' is a monad map P'(b) of C has an absolute coequalizer of P'Pc P'c Theorem 1 µ' ・P'i c for each PT-algebra (b:Pc →c, t:Tc →c) 45 Theorem 2 G : P'T-Alg →PT-Alg has a left adjoint if P,P' are monads on C Assumption i : P →P' is a monad map P'(b) of C has an absolute coequalizer of P'Pc P'c Theorem 1 µ' ・P'i c for each PT-algebra (b:Pc →c, t:Tc →c) T is a monad on C θ : TP →PT is a distributive law θ' : TP' →P'T is a distributive law (id, i) : θ →θ' is a map between distributive laws G(a : P'c →c, t : Tc →c)=(a ・i : Pc →c, t : Tc →c) c 46 Ideals give an absolute coequalizer of Corollary 2 ℘ ∨ Set ℘(℘!$) (͍)) ℘(͍) ∪ ℘T-Alg for each idempotent semiring (S, ∨,・,e) by construction of ideals forget T-Alg by embedding i : ℘!$) →℘ ℘!$) T-Alg 47 Ideals give an absolute coequalizer of Corollary 3 ℘ ∨ Set ℘(℘!$) (͍)) ℘(͍) ∪ Qt for each idempotent semiring (S, ∨,・,e) by construction of ideals forget Monoid by embedding i : ℘!$) →℘ ISRng 48 Contents 1.