Finitary functors: from Set to Preord and Poset - calco 2011
Finitary functors: from Set to Preord and Poset Adriana Balan1 1 University
Alexander Kurz2
Politehnica of Bucharest, Romania
2 University
of Leicester, UK
4th Conference on Algebra and Coalgebra in Computer Science, 2011
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
1 / 27
Motivation
Most of coalgebraic logic is focussed on Set-coalgebras and their associated (Boolean) logics. Investigation of coalgebraic logic over Poset already started – expressivity results [Kurz-Kapulkin-Velebil CMCS2010]. Would deserve a systematic investigation of Poset-functors and their coalgebras. In this talk: we restrict on how to move from (finitary) Set-functors (fairly-well understood) to Preord and Poset-functors with a quick look on their properties and coalgebras.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
2 / 27
Outline
1
Extensions and liftings
2
From Set-functors to Preord-functors Order on variables Order on operations Order both variables and operations
3
Finally, from Preord to Poset
4
Further work
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
3 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary. If Γ : Preord → Preord is a lifting/extension of T , then T = UΓD.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary. If Γ : Preord → Preord is a lifting/extension of T , then T = UΓD. What about the composition Γ = DTU?
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary. If Γ : Preord → Preord is a lifting/extension of T , then T = UΓD. What about the composition Γ = DTU? DTU is not locally monotone. A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Example of a finitary Set-functor having an extension which is not finitary Consider the functor T : Set → Set, TX = {l : N → X | l(n) = l(n + 1) for all but a finite number of n} T is finitary
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
5 / 27
Example of a finitary Set-functor having an extension which is not finitary Consider the functor T : Set → Set, TX = {l : N → X | l(n) = l(n + 1) for all but a finite number of n} T is finitary and has the Preord-extension Γ(X , ≤) = {l : (N, ≤) → (X , ≤) | l(n) ≤ l(n + 1)
for all but a finite number of n}
with the pointwise order. But Γ is not finitary: take the sequence (1, ≤) ⊆ (2, ≤) ⊆ . . . −→ (N, ≤) Then Γ(N, ≤) � colimΓ(n, ≤). A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
5 / 27
More on extensions/liftings
Extensions and liftings are not unique. Examples: Extension T = Id Γ1 = Id Γ2 =(discrete) connected component functor
A. Balan (UPB), A. Kurz (UL)
Lifting TX = 2 × X Γ1 (X , ≤) = 2 × X , product order Γ2 (X , ≤) = 2 � X , lexicographic order
Finitary functors: from Set to Preord
CALCO 2011
6 / 27
About coalgebras Γ extension of T Set �
Coalg(T ) �
D � C
�
˜ D � ˜ C
Preord
�
Coalg(Γ)
Final Γ-coalgebra is the (discrete) final T -coalgebra.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
7 / 27
About coalgebras Γ extension of T Set �
Coalg(T ) �
D � C
�
˜ D � ˜ C
Preord
�
Coalg(Γ)
Final Γ-coalgebra is the (discrete) final T -coalgebra.
A. Balan (UPB), A. Kurz (UL)
Γ lifting of T Set �
Coalg(T ) �
D ⊥ U ˜ D ⊥ ˜ U
�
Preord
�
Coalg(Γ)
Final Γ-coalgebra is the final T -coalgebra with some preorder.
Finitary functors: from Set to Preord
CALCO 2011
7 / 27
Outline
1
Extensions and liftings
2
From Set-functors to Preord-functors Order on variables Order on operations Order both variables and operations
3
Finally, from Preord to Poset
4
Further work
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
8 / 27
Outline
1
Extensions and liftings
2
From Set-functors to Preord-functors Order on variables Order on operations Order both variables and operations
3
Finally, from Preord to Poset
4
Further work
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
9 / 27
First construction: order on variables T finitary Set-functor ⇐⇒ quotient of a polynomial functor. �
n
Alexander Kurz2
Politehnica of Bucharest, Romania
2 University
of Leicester, UK
4th Conference on Algebra and Coalgebra in Computer Science, 2011
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
1 / 27
Motivation
Most of coalgebraic logic is focussed on Set-coalgebras and their associated (Boolean) logics. Investigation of coalgebraic logic over Poset already started – expressivity results [Kurz-Kapulkin-Velebil CMCS2010]. Would deserve a systematic investigation of Poset-functors and their coalgebras. In this talk: we restrict on how to move from (finitary) Set-functors (fairly-well understood) to Preord and Poset-functors with a quick look on their properties and coalgebras.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
2 / 27
Outline
1
Extensions and liftings
2
From Set-functors to Preord-functors Order on variables Order on operations Order both variables and operations
3
Finally, from Preord to Poset
4
Further work
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
3 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary. If Γ : Preord → Preord is a lifting/extension of T , then T = UΓD.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary. If Γ : Preord → Preord is a lifting/extension of T , then T = UΓD. What about the composition Γ = DTU?
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Extensions and liftings We fix a Set-functor T D ⊥ U
Recall the adjunction Set � Extension: Preord �
Γ
D
Set
� Preord �
�
Preord Lifting: Preord
D T
� Set
Γ
� Preord
U
�
Set
U T
�
� Set
Similarly we can define extensions/liftings to Poset. We require for Γ (lifting or extension) to be locally monotone and also finitary if T is finitary. If Γ : Preord → Preord is a lifting/extension of T , then T = UΓD. What about the composition Γ = DTU? DTU is not locally monotone. A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
4 / 27
Example of a finitary Set-functor having an extension which is not finitary Consider the functor T : Set → Set, TX = {l : N → X | l(n) = l(n + 1) for all but a finite number of n} T is finitary
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
5 / 27
Example of a finitary Set-functor having an extension which is not finitary Consider the functor T : Set → Set, TX = {l : N → X | l(n) = l(n + 1) for all but a finite number of n} T is finitary and has the Preord-extension Γ(X , ≤) = {l : (N, ≤) → (X , ≤) | l(n) ≤ l(n + 1)
for all but a finite number of n}
with the pointwise order. But Γ is not finitary: take the sequence (1, ≤) ⊆ (2, ≤) ⊆ . . . −→ (N, ≤) Then Γ(N, ≤) � colimΓ(n, ≤). A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
5 / 27
More on extensions/liftings
Extensions and liftings are not unique. Examples: Extension T = Id Γ1 = Id Γ2 =(discrete) connected component functor
A. Balan (UPB), A. Kurz (UL)
Lifting TX = 2 × X Γ1 (X , ≤) = 2 × X , product order Γ2 (X , ≤) = 2 � X , lexicographic order
Finitary functors: from Set to Preord
CALCO 2011
6 / 27
About coalgebras Γ extension of T Set �
Coalg(T ) �
D � C
�
˜ D � ˜ C
Preord
�
Coalg(Γ)
Final Γ-coalgebra is the (discrete) final T -coalgebra.
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
7 / 27
About coalgebras Γ extension of T Set �
Coalg(T ) �
D � C
�
˜ D � ˜ C
Preord
�
Coalg(Γ)
Final Γ-coalgebra is the (discrete) final T -coalgebra.
A. Balan (UPB), A. Kurz (UL)
Γ lifting of T Set �
Coalg(T ) �
D ⊥ U ˜ D ⊥ ˜ U
�
Preord
�
Coalg(Γ)
Final Γ-coalgebra is the final T -coalgebra with some preorder.
Finitary functors: from Set to Preord
CALCO 2011
7 / 27
Outline
1
Extensions and liftings
2
From Set-functors to Preord-functors Order on variables Order on operations Order both variables and operations
3
Finally, from Preord to Poset
4
Further work
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
8 / 27
Outline
1
Extensions and liftings
2
From Set-functors to Preord-functors Order on variables Order on operations Order both variables and operations
3
Finally, from Preord to Poset
4
Further work
A. Balan (UPB), A. Kurz (UL)
Finitary functors: from Set to Preord
CALCO 2011
9 / 27
First construction: order on variables T finitary Set-functor ⇐⇒ quotient of a polynomial functor. �
n
