A Sufficient Condition for Liftable Adjunctions Between Eilenberg�Moore Categories

A Sufficient Condition for Liftable Adjunctions between EilenbergMoore 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 Salgebra 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 Salgebra 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 Salgebra 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 Salgebra 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

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 Talgebra

Complete Join Semilattice Join Semilattice over Monoid over Monoid

Complete Quantale Complete Join Semilattice Join Semilattice over Talgebra Construction Construction Construction of ideals of ideals of ideals

Join Semilattice Idempotent Join Semilattice over Talgebra 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 Talgebra to complete join semilattice over Talgebra

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 Talgebra to complete join semilattice over Talgebra

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 downwardclosed 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'

14 A sufficient condition of adjunction [Toposes, Triples and Theories]

G:D' →D has a left adjoint if D'

G C

15 A sufficient condition of adjunction [Toposes, Triples and Theories]

G:D' →D has a left adjoint if D'

G C

16 A sufficient condition of adjunction [Toposes, Triples and Theories]

G:D' →D has a left adjoint if D'

G C

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

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

24 When G is the functor which precomposes a monad map

C has absolute G:D' →D has a left adjoint if coequalizers of D' certain diagrams P'Alg

G C

PAlg 25 When G is the functor which precomposes 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

PAlg 26 C has an absolute coequalizer of Theorem 1 P'(b) P'Pc P'c

G:D' →D has a left adjoint ' ・P'i c if D' for each Palgebra P'Alg b : Pc →c

G(a : P'c →c) G C = a ・ ic : Pc →c

by monad map i : P →P' D

PAlg 27 Theorem 1

G : P'Alg →PAlg 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 Palgebra (b:Pc →c) ' ・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 Talgebra to complete join semilattice over Talgebra

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

PAlg

39 Theorem 2

P' TAlg

forget TAlg

PTAlg

40 Theorem 2 C

P'TAlg

forget TAlg

PTAlg

41 Theorem 2 P'TAlg has coequalizers of C certain diagrams P'TAlg

forget TAlg

PTAlg

42 C has an absolute coequalizer of Theorem 2 P'TAlg has P'(b) P'Pc coequalizers of P'c C ・ certain diagrams ' P'i c P'TAlg for each PTalgebra (b : Pc →c, t:Tc →c)

forget TAlg

PTAlg

43 C has an absolute coequalizer of

Theorem 2 P'(b) P'Pc P'c C ' ・P'i c

P'TAlg for each PTalgebra (b : Pc →c, t:Tc →c)

forget TAlg

PTAlg

44 Theorem 2

G : P'TAlg →PTAlg 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 ' ・P'i c for each PTalgebra (b:Pc →c, t:Tc →c)

45 Theorem 2

G : P'TAlg →PTAlg 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 ' ・P'i c for each PTalgebra (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 ℘(℘ ()) ℘() ∪

℘TAlg for each idempotent semiring (S, ∨,・,e) by construction of ideals forget TAlg by embedding i : ℘ →℘

℘ TAlg

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. Construction of left adjoint from join semilattice to complete join semilattice

2. Construction of left adjoint from join semilattice over Talgebra to complete join semilattice over Talgebra

3. Conclusion

49 Main Theorem and Corollary

To give a sufficient condition for monad P,P',T such that

P' Talgebra

Absolute coequalizers

PTalgebra

50 Main Theorem and Corollary

To give a sufficient condition for monad P,P',T such that Complete Join Semilattice P' Talgebra over Talgebra

Absolute Construction coequalizers of ideals

PTalgebra Join Semilattice over Talgebra 51 Nonideal Example

complete ℘Alg join semilattices CSLat

⊆ | ∈ ,∨ over Talgebra

, , ⊥

Set * pointed sets 1 + − Alg 52 Future Work

Conway's Salgebra 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 ⊥) 53 Thank you