On coalgebras over algebras Adriana Balan 1 Alexander Kurz 2 1 University Politehnica of Bucharest, Romania 2 University 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 Motivation 1 The final coalgebra of a continuous functor 2 Final coalgebra and lifting 3 Commuting pair of endofunctors and their fixed points 4 A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 2 / 31
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
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 H 0 = 0, important cases not covered (as A × ( − ) n , D , P κ + ) A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31
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 H 0 = 0, important cases not covered (as A × ( − ) n , D , P κ + ) 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
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 Motivation 1 The final coalgebra of a continuous functor 2 Final coalgebra and lifting 3 Commuting pair of endofunctors and their fixed points 4 A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 5 / 31
� � � � Construction of the final coalgebra → Set ω op -continuous Assumption 1: functor H : Set − Terminal sequence H n t t . . . . . . H n 1 1 H 1 A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31
� � � � � � Construction of the final coalgebra → Set ω op -continuous Assumption 1: functor H : Set − Terminal sequence H n t t . . . . . . H n 1 1 H 1 p n L A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31
� � � � � � � � � Construction of the final coalgebra → Set ω op -continuous Assumption 1: functor H : Set − Terminal sequence H n t t . . . . . . H n 1 1 H 1 p n L Hp n − 1 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
� � � � � � � � Final coalgebras and anamorphisms ξ C For each coalgebra C − → HC there is a cone over the terminal sequence H n t t . . . . . . H n 1 1 H 1 H α 0 HC α n α 0 C A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31
� � � � � � � � � � Final coalgebras and anamorphisms ξ C For each coalgebra C − → HC there is a cone over the terminal sequence H n t t . . . . . . H n 1 1 H 1 H α 0 p n HC L α n α C α 0 C A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31
� � � � � � � � � � Final coalgebras and anamorphisms ξ C For each coalgebra C − → HC there is a cone over the terminal sequence H n t t . . . . . . H n 1 1 H 1 H α 0 p n HC L α n α C α 0 C Topology: Discrete topology on H n 1. 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 Motivation 1 The final coalgebra of a continuous functor 2 Final coalgebra and lifting 3 Commuting pair of endofunctors and their fixed points 4 A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 8 / 31
Lifting functors to algebras over a monad Monad M = ( M , m : M 2 − → M , u : Id − → M ) Adjunction F M ⊣ U M : Alg ( M ) − → Set Initial object M 2 0 − → M 0, terminal object M 1 − → 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 : M 2 − → M , u : Id − → M ) Adjunction F M ⊣ U M : Alg ( M ) − → Set Initial object M 2 0 − → M 0, terminal object M 1 − → 1 Lifting of H to Alg ( M ) ⇐ ⇒ Distributive law λ : MH − → HM λ M M λ � � HM 2 ˜ H M 2 H MHM Alg ( M ) Alg ( M ) m H Hm U M U M λ � HM H � Set MH Set u H � H MH � � � � � λ � Hu � � 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 � H of H to Alg ( M ) ξ γ Then ( L , L − → HL ) inherits an algebra structure map ML − → L making it the final � H -coalgebra. Lemma Mp n a n → MH n 1 → H n 1 is induced by the H-coalgebra structure The cone ML − − of ML Mp n . . . . . . MH n 1 M 1 MH 1 ML MH n t Mt γ a 0 a 1 a n H n t t . . . . . . H n 1 1 H 1 L p n Hence the unique coalgebra map γ : ML − → L is also the anamorphism α ML : ML − → L for the coalgebra ML . A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 10 / 31
� � � � � � � � � � � � � � The final coalgebra and the lifting Diagram in Alg ( M ) with limiting lower sequence Mp n . . . . . . MH n 1 M 1 MH 1 ML MH n t Mt a 0 a 1 a n γ H n t t . . . . . . H n 1 1 H 1 L p n 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 n . . . . . . MH n 1 M 1 MH 1 ML MH n t Mt a 0 a 1 a n γ H n t t . . . . . . H n 1 1 H 1 L p n 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 n . . . . . . MH n 1 M 1 MH 1 ML MH n t Mt a 0 a 1 a n γ H n t t . . . . . . H n 1 1 H 1 L p n 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
� � � � � � � � Fixed points of lifted functor Initial-terminal � H -sequences: � HM 0 � H n M 0 . . . � . . . M 0 s H n s Hs . . . . . . H n 1 1 H 1 t A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31
� � � � � � � � Fixed points of lifted functor Initial-terminal � H -sequences: � HM 0 � H n M 0 . . . � . . . M 0 s H n s Hs . . . . . . H n 1 1 H 1 t Assumption 3: M0=1 A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31
� � � � � Fixed points of lifted functor Initial-terminal � H -sequences: � HM 0 � H n M 0 . . . � . . . M 0 s H n s Hs . . . . . . H n 1 1 H 1 t Assumption 3: M0=1 A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31
� � � � � � 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 � � � � i n � � � � � � � � � � H 1 � H n 1 � . . . � � . . . � 1 f � ��������������� t H n t Ht p n L A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31
Main result Theorem Let H be a Set-endofunctor ω op -continuous and M a monad on Set such that: 1 H admits a lifting ˜ H to Alg ( M ) 2 M 0 = 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
� 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
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