On Coalgebras Over Algebras

On coalgebras over algebras

Adriana Balan1 Alexander Kurz2

1University Politehnica of Bucharest, Romania

2University of Leicester, UK

10th International Workshop on Coalgebraic Methods in Computer Science

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 1 / 31 Outline

1 Motivation

2 The ﬁnal coalgebra of a continuous functor

3 Final coalgebra and lifting

4 Commuting pair of endofunctors and their ﬁxed points

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 2 / 31 Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) Locally finitely presentable categories: Hom(B, L) completion of Hom(B, I ) for all finitely presentable objects B [Adamek 2003]

Motivation

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 Locally ﬁnitely presentable categories: Hom(B, L) completion of Hom(B, I ) for all ﬁnitely presentable objects B [Adamek 2003]

Motivation

Starting data: category C, endofunctor H : C −→ C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+)

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 Motivation

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 In this talk

Category: Alg(M) for a Set-monad M Alg(M)-functor: obtained from lifting

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 4 / 31 Outline

1 Motivation

2 The ﬁnal coalgebra of a continuous functor

3 Final coalgebra and lifting

4 Commuting pair of endofunctors and their ﬁxed points

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 5 / 31 The limit of the terminal sequence is the ﬁnal H-coalgebra by cocontinuity ξ = τ −1 : L ' HL

Construction of the ﬁnal coalgebra Assumption 1: functor H : Set −→ Set ωop-continuous Terminal sequence

t Hnt 1 o H1 o ... o Hn1 o ...

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31 The limit of the terminal sequence is the ﬁnal H-coalgebra by cocontinuity ξ = τ −1 : L ' HL

Construction of the ﬁnal coalgebra Assumption 1: functor H : Set −→ Set ωop-continuous Terminal sequence

t Hnt 1 o H1 o ... o Hn1 o ... d O pn L

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31 Construction of the ﬁnal coalgebra Assumption 1: functor H : Set −→ Set ωop-continuous Terminal sequence

t Hnt 1 o H1 o ... o Hn1 o ... ^d O Z pn

L Hpn−1 O τ HL

The limit of the terminal sequence is the ﬁnal H-coalgebra by cocontinuity ξ = τ −1 : L ' HL

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31 Topology: Discrete topology on Hn1. Initial topology on L, HL and C =⇒ L complete ultrametric space. All maps are continuous.

Final coalgebras and anamorphisms

ξ For each coalgebra C −→C HC there is a cone over the terminal sequence

t Hnt 1 o H1 o ... o Hn1 o ... a [ R Hα0

HC f αn

α0 C

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31 Topology: Discrete topology on Hn1. Initial topology on L, HL and C =⇒ L complete ultrametric space. All maps are continuous.

Final coalgebras and anamorphisms

ξ For each coalgebra C −→C HC there is a cone over the terminal sequence

t Hnt 1 o H1 o ... o Hn1 o ... a [ R Z Hα0 pn

HC L f αn O αC α0 C

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31 Final coalgebras and anamorphisms

ξ For each coalgebra C −→C HC there is a cone over the terminal sequence

t Hnt 1 o H1 o ... o Hn1 o ... a [ R Z Hα0 pn

HC L f αn O αC α0 C

Topology: Discrete topology on Hn1. Initial topology on L, HL and C =⇒ L complete ultrametric space. All maps are continuous.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31 Outline

1 Motivation

2 The ﬁnal coalgebra of a continuous functor

3 Final coalgebra and lifting

4 Commuting pair of endofunctors and their ﬁxed points

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 8 / 31 ⇐⇒ Distributive law λ : MH −→ HM

Mλ λM M2H / MHM / HM2

mH Hm λ MH / HM

uH H / MH C CC CC λ Hu CC C! HM

Lifting of H to Alg(M)

H˜ Alg(M) / Alg(M)

UM UM H Set / Set

Lifting functors to algebras over a monad

Monad M = (M, m : M2 −→ M, u : Id −→ M) Adjunction F M a UM : Alg(M) −→ Set Initial object M20 −→ M0, terminal object M1 −→ 1

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 9 / 31 Lifting functors to algebras over a monad

Monad M = (M, m : M2 −→ M, u : Id −→ M) Adjunction F M a UM : Alg(M) −→ Set Initial object M20 −→ M0, terminal object M1 −→ 1 Lifting of H to Alg(M) ⇐⇒ Distributive law λ : MH −→ HM

λ H˜ 2 Mλ M 2 Alg(M) / Alg(M) M H / MHM / HM

mH Hm UM UM λ H Set / Set MH / HM

uH H / MH C CC CC λ Hu CC C! HM

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 9 / 31 The ﬁnal coalgebra and the lifting

Assumption 2: there is a lifting He of H to Alg(M) ξ γ Then (L, L−→HL) inherits an algebra structure map ML−→L making it the ﬁnal He-coalgebra. Lemma Mp The cone ML −→n MHn1 −→an Hn1 is induced by the H-coalgebra structure of ML

Mp x n M1 o MH1 o ... o MHn1 o ... ML Mt MHnt

a0 a1 an γ t Hnt 1 o H1 o ... o Hn1 o ... L f pn

Hence the unique coalgebra map γ : ML −→ L is also the anamorphism αML : ML −→ L for the coalgebra ML.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 10 / 31 Topology Discrete topology on both sequences Initial topologies on ML and L

Proposition The ﬁnal H-coalgebra inherits a structure of a topological M-algebra.

The ﬁnal coalgebra and the lifting

Diagram in Alg(M) with limiting lower sequence

Mp x n M1 o MH1 o ... o MHn1 o ... ML Mt MHnt

a0 a1 an γ t Hnt 1 o H1 o ... o Hn1 o ... L f pn

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 11 / 31 Proposition The ﬁnal H-coalgebra inherits a structure of a topological M-algebra.

The ﬁnal coalgebra and the lifting

Diagram in Alg(M) with limiting lower sequence

Mp x n M1 o MH1 o ... o MHn1 o ... ML Mt MHnt

a0 a1 an γ t Hnt 1 o H1 o ... o Hn1 o ... L f pn

Topology Discrete topology on both sequences Initial topologies on ML and L

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 11 / 31 The ﬁnal coalgebra and the lifting

Diagram in Alg(M) with limiting lower sequence

Mp x n M1 o MH1 o ... o MHn1 o ... ML Mt MHnt

a0 a1 an γ t Hnt 1 o H1 o ... o Hn1 o ... L f pn

Topology Discrete topology on both sequences Initial topologies on ML and L

Proposition The ﬁnal H-coalgebra inherits a structure of a topological M-algebra.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 11 / 31 Assumption 3: M0=1 [Adamek 2003] He has also (non empty) initial algebra I built upon this sequence in Alg(M), with unique M-algebra monomorphism f : I −→ L

6 I mmm in mm mmm mmm mmm ... n m ... 1 o / H1 o / o / H 1 o / f t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L

Fixed points of lifted functor

Initial-terminal He-sequences: M0 / HM0 / ... / HnM0 / ...

s Hs Hns ... n ... 1 o t H1 o o H 1 o

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31 [Adamek 2003] He has also (non empty) initial algebra I built upon this sequence in Alg(M), with unique M-algebra monomorphism f : I −→ L

6 I mmm in mm mmm mmm mmm ... n m ... 1 o / H1 o / o / H 1 o / f t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L

Fixed points of lifted functor

Initial-terminal He-sequences: M0 / HM0 / ... / HnM0 / ...

s Hs Hns ... n ... 1 o t H1 o o H 1 o Assumption 3: M0=1

6 I mmm in mm mmm mmm mmm ... n m ... 1 o / H1 o / o / H 1 o / f t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L

Fixed points of lifted functor

Initial-terminal He-sequences: M0 / HM0 / ... / HnM0 / ...

s Hs Hns ... n ... 1 o t H1 o o H 1 o Assumption 3: M0=1

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31 Initial-terminal He-sequences: Assumption 3: M0=1

Fixed points of lifted functor

[Adamek 2003] He has also (non empty) initial algebra I built upon this sequence in Alg(M), with unique M-algebra monomorphism f : I −→ L

6 I mmm in mm mmm mmm mmm ... n m ... 1 o / H1 o / o / H 1 o / f t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31 Main result

Theorem Let H be a Set-endofunctor ωop-continuous and M a monad on Set such that: 1 H admits a lifting H˜ to Alg(M) 2 M0 = 1 in Alg(M) Then the ﬁnal H-coalgebra is the completion of the initial H-algebrae under a suitable (ultra)metric.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 13 / 31 Idea of the proof... Take on I the coarsest topology such that f is continuous

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 14 / 31 Idea of the proof... Take on I the coarsest topology such that f is continuous = initial topology from the cone pn ◦ f

... n ... 1 o / H1 o / o / H 1 o / f t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 14 / 31 Idea of the proof... Take on I the coarsest topology such that f is continuous = initial topology from the cone pn ◦ f

6 I o MI mmm in mm mmm mmm mmm ... n m ... 1 o / H1 o / o / H 1 o / f Mf t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L o ML

Obtain MI −→ I topological algebra.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 14 / 31 Density of initial algebra into the ﬁnal coalgebra Remember L is complete ultrametric space. Then the image of I is dense in L:

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 15 / 31 Density of initial algebra into the ﬁnal coalgebra Remember L is complete ultrametric space. Then the image of I is dense in L:

... n ... o / H 1 o / jUUUU UUUU UUUU pn UUU UUUU UU L

... n n+1 ... o / H 1 o / H 1 o / kWWWWW WWWWW WWWWW pn WWWWW WWWWW WW L

6 I mmm in+1 mmm mmm mmm mmm ... n n+1 ... o / H 1 o / H 1 o / kWWWWW WWWWW WWWWW pn WWWWW WWWWW WW L

6 I mmm in+1 mmm mmm mmm mmm ... n n+1 ... o / H 1 o / H 1 o / f kWWWWW WWWWW WWWWW pn WWWWW WWWWW WW L

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 15 / 31 Particular case: k is a semiring (like B = {0, 1}, N, R≥0). I Consider the monad M = (M, m, u) given by MX = {f : X −→ k|supp(f ) ﬁnite} I Final H-coalgebra: khhAii I Initial He-algebra: khAi

An example

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 16 / 31 I Final H-coalgebra: khhAii I Initial He-algebra: khAi

An example

A Consider HX = k × X . I Coalgebras are Moore automata. A∗ I Final coalgebra is k , initial algebra is empty. I For any monad M, such that k carries an M-algebra structure, a lifting He always exists. A∗ I Hence the theorem applies: k as the completion of initial He-algebra. Particular case: k is a semiring (like B = {0, 1}, N, R≥0). I Consider the monad M = (M, m, u) given by MX = {f : X −→ k|supp(f ) ﬁnite}

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 16 / 31 An example

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 16 / 31 Outline

1 Motivation

2 The ﬁnal coalgebra of a continuous functor

3 Final coalgebra and lifting

4 Commuting pair of endofunctors and their ﬁxed points

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 17 / 31 Lifting of T to Kl(M) ⇐⇒ Distributive law ς : TM −→ MT

ς Mς Tˆ 2 M 2 Kl(M) / Kl(M) TM / MTM / M T O O Tm mT FM FM ς T Set / Set TM / MT

Tu T / TM C CC CC ς uT CC C! MT

Lifting functors to categories of algebras

Lifting of H to Alg(M) ⇐⇒ Distributive law λ : MH −→ HM

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 18 / 31 Lifting functors to categories of algebras

Lifting of H to Alg(M) ⇐⇒ Distributive law λ : MH −→ HM Lifting of T to Kl(M) ⇐⇒ Distributive law ς : TM −→ MT

ς Mς Tˆ 2 M 2 Kl(M) / Kl(M) TM / MTM / M T O O Tm mT FM FM ς T Set / Set TM / MT

Tu T / TM C CC CC ς uT CC C! MT

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 18 / 31 Consider also I : Kl(M) −→ Alg(M)

Construct the left Kan extension T¯ = LanI (ITˆ )

More on Kleisli lift Assume Kleisli lift of T exists

Tˆ Kl(M) / Kl(M) O O FM FM T Set / Set

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 19 / 31 More on Kleisli lift Assume Kleisli lift of T exists Consider also I : Kl(M) −→ Alg(M)

T¯

I Tˆ I ) Alg(M) o Kl(M) / Kl(M) / Alg(M) O O FM FM T Set / Set

Construct the left Kan extension T¯ = LanI (ITˆ )

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 19 / 31 More on Kleisli lift Upper diagram commutes: ITˆ =∼ T¯ I.

T¯

Tˆ ' Alg(M) o Kl(M) / Kl(M) / Alg(M) I I eKK O O M 9 KK F sss KKK FM FM ss F M KK sss K T ss Set / Set

It follows that T¯ Alg(M) / Alg(M) O O F M =∼ F M T Set / Set

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 20 / 31 Then

MT = UMF MT =∼ UMTF¯ M =∼ UMHFe M = HUMF M = HM

Hence MT =∼ HM

Commuting pair of Set-endofunctors

Take two functors T , H on Set such that:

I H has a lift He to Alg(M) I T has a lift Tˆ to Kl(M), hence an extension T¯ to Alg(M) ∼ ¯ I He = T

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 21 / 31 Commuting pair of Set-endofunctors

Take two functors T , H on Set such that:

I H has a lift He to Alg(M) I T has a lift Tˆ to Kl(M), hence an extension T¯ to Alg(M) ∼ ¯ I He = T Then

MT = UMF MT =∼ UMTF¯ M =∼ UMHFe M = HUMF M = HM

Hence MT =∼ HM

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 21 / 31 Trivial examples: T = H = Id or T = H = M, M any monad T = H = A + (−), M = B + (−) T = H = A × (−), M = B × (−) M idempotent monad, H = M, T = Id or H = Id, T = M

Commuting pair of Set-endofunctors

Deﬁnition Let M = (M, m, u) be a monad on Set. A pair of Set-endofunctors (T , H) such that MT =∼ HM is called an M-commuting pair.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 22 / 31 Commuting pair of Set-endofunctors

Deﬁnition Let M = (M, m, u) be a monad on Set. A pair of Set-endofunctors (T , H) such that MT =∼ HM is called an M-commuting pair.

Trivial examples: T = H = Id or T = H = M, M any monad T = H = A + (−), M = B + (−) T = H = A × (−), M = B × (−) M idempotent monad, H = M, T = Id or H = Id, T = M

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 22 / 31 If the algebra lift of H is isomorphic to the algebra extension of T , then H and T form a commuting pair by an algebra isomorphism HM =∼ MT .

Commuting pair of Set-endofunctors

He =∼ T¯ implies not only natural isomorphism MT =∼ HM, but also isomorphism of algebras

λMX HmX MHMX / HM2X / HMX

=∼ =∼ 2 / MTX M TX mTX

because of HFe M =∼ TF¯ M =∼ F MT

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 23 / 31 Commuting pair of Set-endofunctors

He =∼ T¯ implies not only natural isomorphism MT =∼ HM, but also isomorphism of algebras

λMX HmX MHMX / HM2X / HMX

=∼ =∼ 2 / MTX M TX mTX

because of HFe M =∼ TF¯ M =∼ F MT If the algebra lift of H is isomorphic to the algebra extension of T , then H and T form a commuting pair by an algebra isomorphism HM =∼ MT .

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 23 / 31 Assume M, T , H ﬁnitary. Then T¯ is determined by its action on ﬁnitely free algebras, and so is He (because it preserves sifted colimits) Obtain H˜ =∼ T¯

Commuting pair of Set-endofunctors

Conversely, assume a commuting pair (T , H) such that corresponding lifts exists, and HMX =∼ MTX as algebras. This implies H˜ =∼ T¯ on free algebras.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 24 / 31 Obtain H˜ =∼ T¯

Commuting pair of Set-endofunctors

Conversely, assume a commuting pair (T , H) such that corresponding lifts exists, and HMX =∼ MTX as algebras. This implies H˜ =∼ T¯ on free algebras. Assume M, T , H ﬁnitary. Then T¯ is determined by its action on ﬁnitely free algebras, and so is He (because it preserves sifted colimits)

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 24 / 31 Commuting pair of Set-endofunctors

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 24 / 31 Commuting pair and algebra lift-extension isomorphism

Theorem Let H, T two endofunctors and M a monad on Set, such that H and T have algebra lift H,e respectively Kleisli lift with respect to the monad M, with T¯ the corresponding left Kan extension to algebras.Then: If He =∼ T¯ , then (T , H) form an M-commuting pair and HMX =∼ MTX as algebras for any X . If M, H, T are ﬁnitary and MT =∼ HM as algebras, then He =∼ T.¯

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 25 / 31 Commuting pair and algebra lift-extension isomorphism

Corollary Let H, T two endofunctors and M a monad on Set, such that: M, H, T are ﬁnitary H is ωop-continuous H has algebra lift, T has Kleisli lift MT =∼ HM as algebras M0 = 1 as algebras Then the ﬁnal H-coalgebra is the completion of the free M-algebra built on the initial T -algebra.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 26 / 31 Assume Alg(M) has biproducts. Then T¯ is the lifting to Alg(M) of the Set-endofunctor HX = M1 × X A. Hence (T , H) form a commuting pair.

An example

Consider TX = 1 + A · X and M any monad. Kleisli lift exists Algebra extension TX¯ = F M1 + A · X

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 27 / 31 Hence (T , H) form a commuting pair.

An example

Consider TX = 1 + A · X and M any monad. Kleisli lift exists Algebra extension TX¯ = F M1 + A · X Assume Alg(M) has biproducts. Then T¯ is the lifting to Alg(M) of the Set-endofunctor HX = M1 × X A.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 27 / 31 An example

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 27 / 31 Given monad M and (T , H) commuting pair, ﬁnd both distributive laws.

More on commuting pairs

Given T and H, ﬁnd the linking monad such that (T , H) form a commuting pair.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 28 / 31 More on commuting pairs

Given T and H, ﬁnd the linking monad such that (T , H) form a commuting pair. Given monad M and (T , H) commuting pair, ﬁnd both distributive laws.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 28 / 31 Particular case: T contains products, as in T1X = A × X or T2X = X × X M M Then T¯1X = F A ⊗ X , respectively T¯2X = X ⊗ X (as F is monoidal) If M is ﬁnitary, then UM sends (X , x) ⊗ (Y , y) to the reﬂexive Set-coequalizer of

M(x×y) M(MX × MY ) ⇒ M(X × Y ) mX ×Y ◦Mϕ2

Hence for any such T and M, a corresponding commuting pair (T , H) can be constructed.

More on commuting pairs

The Kleisli lift M commutative monad, T analytic functor =⇒ distributive law exists

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 29 / 31 If M is ﬁnitary, then UM sends (X , x) ⊗ (Y , y) to the reﬂexive Set-coequalizer of

M(x×y) M(MX × MY ) ⇒ M(X × Y ) mX ×Y ◦Mϕ2

Hence for any such T and M, a corresponding commuting pair (T , H) can be constructed.

More on commuting pairs

The Kleisli lift M commutative monad, T analytic functor =⇒ distributive law exists Particular case: T contains products, as in T1X = A × X or T2X = X × X M M Then T¯1X = F A ⊗ X , respectively T¯2X = X ⊗ X (as F is monoidal)

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 29 / 31 Hence for any such T and M, a corresponding commuting pair (T , H) can be constructed.

More on commuting pairs

M(x×y) M(MX × MY ) ⇒ M(X × Y ) mX ×Y ◦Mϕ2

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 29 / 31 More on commuting pairs

M(x×y) M(MX × MY ) ⇒ M(X × Y ) mX ×Y ◦Mϕ2

Hence for any such T and M, a corresponding commuting pair (T , H) can be constructed.

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 29 / 31 H constant functor, then the image of H must be carrier of an M-algebra HX = A × X n, then ∃ lifting =⇒ A is the carrier of an M-algebra HX = A + X or HX = X + X , there is no obvious distributive law MH −→λ HM

More on commuting pairs

The algebra lift More complicated, even for simplest cases of polynomial functors:

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 30 / 31 More on commuting pairs

The algebra lift More complicated, even for simplest cases of polynomial functors: H constant functor, then the image of H must be carrier of an M-algebra HX = A × X n, then ∃ lifting =⇒ A is the carrier of an M-algebra HX = A + X or HX = X + X , there is no obvious distributive law MH −→λ HM

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 30 / 31 Thank you!

A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 31 / 31