[PPT] - On coalgebras over algebras Adriana Balan 1 Alexander Kurz 2 1 PowerPoint Presentation

SLIDE 1

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

SLIDE 2

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

SLIDE 3

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]

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 3 / 31

SLIDE 4

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

f 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

SLIDE 5

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

f 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]

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 3 / 31

SLIDE 6

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

SLIDE 7

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

SLIDE 8

Construction of the final coalgebra

Assumption 1: functor H : Set − → Set ωop-continuous Terminal sequence 1 H1

t

. . .
Hn1
. . .

Hnt

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 6 / 31

SLIDE 9

Construction of the final coalgebra

Assumption 1: functor H : Set − → Set ωop-continuous Terminal sequence 1 H1

t

. . .
Hn1
. . .

Hnt

L
pn
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 6 / 31

SLIDE 10

Construction of the final coalgebra

Assumption 1: functor H : Set − → Set ωop-continuous Terminal sequence 1 H1

t

. . .
Hn1
. . .

Hnt

L
pn
HL

τ

Hpn−1
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

SLIDE 11

Final coalgebras and anamorphisms

For each coalgebra C

ξC

− → HC there is a cone over the terminal sequence 1 H1

t

. . .
Hn1
. . .

Hnt

HC

Hα0

C
α0
αn
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 7 / 31

SLIDE 12

Final coalgebras and anamorphisms

For each coalgebra C

ξC

− → HC there is a cone over the terminal sequence 1 H1

t

. . .
Hn1
. . .

Hnt

HC

Hα0

L

pn

C
αC
α0
αn
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 7 / 31

SLIDE 13

Final coalgebras and anamorphisms

For each coalgebra C

ξC

− → HC there is a cone over the terminal sequence 1 H1

t

. . .
Hn1
. . .

Hnt

HC

Hα0

L

pn

C
αC
α0
αn
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

SLIDE 14

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

SLIDE 15

Lifting functors to algebras over a monad

Monad M = (M, m : M2 − → M, u : Id − → M) Adjunction F M ⊣ 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

SLIDE 16

Lifting functors to algebras over a monad

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

˜ H

UM
Alg(M)

UM

Set

H

Set

⇐ ⇒ Distributive law λ : MH − → HM M2H

Mλ mH

MHM

λM

HM2

Hm

MH

λ

HM

H

uH Hu

MH

λ

HM
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 9 / 31

SLIDE 17

The final coalgebra and the lifting

Assumption 2: there is a lifting H of H to Alg(M) Then (L, L

ξ

− →HL) inherits an algebra structure map ML

γ

− →L making it the final H-coalgebra.

Lemma

The cone ML

Mpn

− → MHn1

an

− → Hn1 is induced by the H-coalgebra structure

f ML

M1

a0

MH1

Mt

a1
. . .
MHn1
an
. . .

MHnt

ML

Mpn

γ
1

H1

t

. . .
Hn1
. . .

Hnt

L

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

SLIDE 18

The final coalgebra and the lifting

Diagram in Alg(M) with limiting lower sequence M1

a0

MH1

Mt

a1
. . .
MHn1
an
. . .

MHnt

ML

Mpn

γ
1

H1

t

. . .
Hn1
. . .

Hnt

L

pn

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 11 / 31

SLIDE 19

The final coalgebra and the lifting

Diagram in Alg(M) with limiting lower sequence M1

a0

MH1

Mt

a1
. . .
MHn1
an
. . .

MHnt

ML

Mpn

γ
1

H1

t

. . .
Hn1
. . .

Hnt

L

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

SLIDE 20

The final coalgebra and the lifting

Diagram in Alg(M) with limiting lower sequence M1

a0

MH1

Mt

a1
. . .
MHn1
an
. . .

MHnt

ML

Mpn

γ
1

H1

t

. . .
Hn1
. . .

Hnt

L

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

SLIDE 21

Fixed points of lifted functor

Initial-terminal H-sequences: M0

s

HM0
Hs
. . .

HnM0

Hns

. . .

1 H1

t

. . .
Hn1
. . .
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 12 / 31

SLIDE 22

Fixed points of lifted functor

Initial-terminal H-sequences: M0

s

HM0
Hs
. . .

HnM0

Hns

. . .

1 H1

t

. . .
Hn1
. . .
Assumption 3: M0=1
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 12 / 31

SLIDE 23

Fixed points of lifted functor

Initial-terminal H-sequences: M0

s

HM0

Hs

. . .

HnM0

Hns

. . .

1 H1

t

. . .
Hn1
. . .
Assumption 3: M0=1
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 12 / 31

SLIDE 24

Fixed points of lifted functor

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

f

1

H1

t

. . .

Ht

Hn1

in

Hnt
. . .
L

pn

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 12 / 31

SLIDE 25

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-algebra under a suitable (ultra)metric.

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 13 / 31

SLIDE 26

Idea of the proof...

Take on I the coarsest topology such that f is continuous I

f

L
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 14 / 31

SLIDE 27

Idea of the proof...

Take on I the coarsest topology such that f is continuous = initial topology from the cone pn ◦ f I

f

1

H1

t

. . .

Ht

Hn1

Hnt

. . .
L

pn

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 14 / 31

SLIDE 28

Idea of the proof...

Take on I the coarsest topology such that f is continuous = initial topology from the cone pn ◦ f I

f

MI
Mf
1

H1

t

. . .

Ht

Hn1

in

Hnt
. . .
L

pn

ML
Obtain MI −

→ I topological algebra.

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 14 / 31

SLIDE 29

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

SLIDE 30

Density of initial algebra into the final coalgebra

Remember L is complete ultrametric space. Then the image of I is dense in L: L

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 15 / 31

SLIDE 31

Density of initial algebra into the final coalgebra

Remember L is complete ultrametric space. Then the image of I is dense in L: . . .

Hn1

. . .
L

pn

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 15 / 31

SLIDE 32

Density of initial algebra into the final coalgebra

Remember L is complete ultrametric space. Then the image of I is dense in L: . . .

Hn1

Hn+11
. . .
L

pn

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 15 / 31

SLIDE 33

Density of initial algebra into the final coalgebra

Remember L is complete ultrametric space. Then the image of I is dense in L: I . . .

Hn1

Hn+11

in+1

. . .
L

pn

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 15 / 31

SLIDE 34

Density of initial algebra into the final coalgebra

Remember L is complete ultrametric space. Then the image of I is dense in L: I

f

. . .

Hn1

Hn+11

in+1

. . .
L

pn

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 15 / 31

SLIDE 35

An example

Consider HX = k × X A.

◮ Coalgebras are Moore automata. ◮ Final coalgebra is kA∗, initial algebra is empty. ◮ For any monad M, such that k carries an M-algebra structure, a lifting

H always exists.

◮ Hence the theorem applies: kA∗ as the completion of initial

H-algebra.

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 16 / 31

SLIDE 36

An example

Consider HX = k × X A.

◮ Coalgebras are Moore automata. ◮ Final coalgebra is kA∗, initial algebra is empty. ◮ For any monad M, such that k carries an M-algebra structure, a lifting

H always exists.

◮ Hence the theorem applies: kA∗ as the completion of initial

H-algebra.

Particular case: k is a semiring (like B = {0, 1}, N, R≥0).

◮ 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

SLIDE 37

An example

Consider HX = k × X A.

◮ Coalgebras are Moore automata. ◮ Final coalgebra is kA∗, initial algebra is empty. ◮ For any monad M, such that k carries an M-algebra structure, a lifting

H always exists.

◮ Hence the theorem applies: kA∗ as the completion of initial

H-algebra.

Particular case: k is a semiring (like B = {0, 1}, N, R≥0).

◮ Consider the monad M = (M, m, u) given by

MX = {f : X − → k|supp(f ) finite}

◮ Final H-coalgebra: kA ◮ Initial

H-algebra: kA

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 16 / 31

SLIDE 38

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

SLIDE 39

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

SLIDE 40

Lifting functors to categories of algebras

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

ˆ T

Kl(M)

Set

FM

T

Set

FM

⇐

⇒ Distributive law ς : TM − → MT TM2

ςM Tm

MTM

Mς M2T mT

TM

ς

MT

T

Tu uT

TM

ς

MT
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 18 / 31

SLIDE 41

More on Kleisli lift

Assume Kleisli lift of T exists Kl(M)

ˆ T

Kl(M)

Set

FM

T

Set

FM

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 19 / 31

SLIDE 42

More on Kleisli lift

Assume Kleisli lift of T exists Consider also I : Kl(M) − → Alg(M) Alg(M)

¯ T

Kl(M)

I

ˆ

T

Kl(M)

I

Alg(M)

Set

FM

T

Set

FM

Construct the left Kan extension ¯

T = LanI(I ˆ T)

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 19 / 31

SLIDE 43

More on Kleisli lift

Upper diagram commutes: I ˆ T ∼ = ¯ TI. Alg(M)

¯ T

Kl(M)

I

ˆ

T

Kl(M)

I

Alg(M)

Set

F M

FM
T

Set

F M

FM
It follows that

Alg(M)

¯ T

∼

=

Alg(M) Set

F M

T

Set

F M

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 20 / 31

SLIDE 44

Commuting pair of Set-endofunctors

Take two functors T, H on Set such that:

◮ H has a lift

H to Alg(M)

◮ T has a lift ˆ

T to Kl(M), hence an extension ¯ T to Alg(M)

◮

H ∼ = ¯ T

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 21 / 31

SLIDE 45

Commuting pair of Set-endofunctors

Take two functors T, H on Set such that:

◮ H has a lift

H to Alg(M)

◮ T has a lift ˆ

T to Kl(M), hence an extension ¯ T to Alg(M)

◮

H ∼ = ¯ T

Then MT = UMF MT ∼ = UM ¯ TF M ∼ = UM HF M = HUMF M = HM Hence MT ∼ = HM

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 21 / 31

SLIDE 46

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

SLIDE 47

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

SLIDE 48

Commuting pair of Set-endofunctors

H ∼

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

∼ =

λMX HM2X

HmX HMX ∼ =

M2TX

mTX

MTX

because of HF M ∼ = ¯ TF M ∼ = F MT

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 23 / 31

SLIDE 49

Commuting pair of Set-endofunctors

H ∼

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

∼ =

λMX HM2X

HmX HMX ∼ =

M2TX

mTX

MTX

because of HF 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

SLIDE 50

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

SLIDE 51

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 H (because it preserves sifted colimits)

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 24 / 31

SLIDE 52

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 H (because it preserves sifted colimits) Obtain ˜ H ∼ = ¯ T

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 24 / 31

SLIDE 53

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, respectively Kleisli lift with respect to the monad M, with ¯ T the corresponding left Kan extension to algebras.Then: If H ∼ = ¯ 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 H ∼ = ¯ T.

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 25 / 31

SLIDE 54

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

n the initial T-algebra.
A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 26 / 31

SLIDE 55

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

SLIDE 56

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

SLIDE 57

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. Hence (T, H) form a commuting pair.

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 27 / 31

SLIDE 58

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

SLIDE 59

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

SLIDE 60

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

SLIDE 61

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 Then ¯ T1X = F MA ⊗ X, respectively ¯ T2X = X ⊗ X (as F M is monoidal)

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 29 / 31

SLIDE 62

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 Then ¯ T1X = F MA ⊗ X, respectively ¯ T2X = X ⊗ X (as F M is monoidal) If M is finitary, then UM sends (X, x) ⊗ (Y , y) to the reflexive Set-coequalizer of M(MX × MY )

M(x×y)

⇒

mX×Y ◦Mϕ2

M(X × Y )

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 29 / 31

SLIDE 63

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 Then ¯ T1X = F MA ⊗ X, respectively ¯ T2X = X ⊗ X (as F M is monoidal) If M is finitary, then UM sends (X, x) ⊗ (Y , y) to the reflexive Set-coequalizer of M(MX × MY )

M(x×y)

⇒

mX×Y ◦Mϕ2

M(X × Y ) 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

SLIDE 64

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

SLIDE 65

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

SLIDE 66

Thank you!

A. Balan (UPB), A. Kurz (UL)

On coalgebras over algebras CMCS 2010 31 / 31