List Objects with Algebraic Structure Marcelo Fiore and Philip - PowerPoint PPT Presentation

List objects are initial algebras Definition An algebra for a functor F : C → C is a pair ( A , α : FA → A ). 12 / 26

List objects are initial algebras Definition An algebra for a functor F : C → C is a pair ( A , α : FA → A ). Lemma If ( C , ⊗ , I ) is a monoidal category with finite coproducts (0 , +) and ω -colimits, both preserved by all ( − ) ⊗ P for P ∈ C , then the initial � � I + X ⊗ ( − ) algebra of the functor is a list object on X. 12 / 26

List objects are initial algebras Definition An algebra for a functor F : C → C is a pair ( A , α : FA → A ). Lemma If ( C , ⊗ , I ) is a monoidal category with finite coproducts (0 , +) and ω -colimits, both preserved by all ( − ) ⊗ P for P ∈ C , then the initial � � I + X ⊗ ( − ) algebra of the functor is a list object on X. Remark This result relies on a general theory of parametrised initial algebras . 12 / 26

The story so far 13 / 26

The story so far list objects 13 / 26

The story so far list objects ◮ well-understood datatype 13 / 26

The story so far list objects ◮ well-understood datatype ◮ are free monoids 13 / 26

The story so far list objects ◮ well-understood datatype ◮ are free monoids ◮ described by µ A . ( I + X ⊗ A ). 13 / 26

Rest of this talk list objects T -list objects � (new work) ◮ well-understood datatype ◮ are free monoids ◮ extends datatype of lists ◮ described by ◮ are free T -monoids ◮ described by µ A . ( I + X ⊗ A ). µ A . T ( I + X ⊗ A ). 13 / 26

Rest of this talk list objects T -list objects � (new work) ◮ well-understood datatype ◮ are free monoids ◮ extends datatype of lists ◮ described by ◮ are free T -monoids ◮ described by µ A . ( I + X ⊗ A ). µ A . T ( I + X ⊗ A ). ...and instantiate this for applications 13 / 26

Compatible algebraic structure 14 / 26

Compatible algebraic structure Definition A monad on a category C is a functor T : C → C equipped with a multiplication µ : T 2 → T and a unit η : Id C → T satisfying associativity and unit laws. 14 / 26

Compatible algebraic structure Definition A monad on a category C is a functor T : C → C equipped with a multiplication µ : T 2 → T and a unit η : Id C → T satisfying associativity and unit laws. Definition An algebra for a monad ( T , µ, η ) is a pair ( A , α : TA → A ) satisfying unit and associativity laws. 14 / 26

Compatible algebraic structure Definition A monad on a category C is a functor T : C → C equipped with a multiplication µ : T 2 → T and a unit η : Id C → T satisfying associativity and unit laws. Definition An algebra for a monad ( T , µ, η ) is a pair ( A , α : TA → A ) satisfying unit and associativity laws. Definition A strong monad T is a monad on a monoidal category ( ⊗ , I ) that is equipped with a natural transformation st A , B : T ( A ) ⊗ B → T ( A ⊗ B ) satisfying coherence laws. 14 / 26

List objects with algebraic structure 15 / 26

T -list objects 15 / 26

T -list objects Let ( T , st ) be a strong monad on a monoidal category ( ⊗ , I ). A T-list object M ( X ) on X consists of I M ( X ) X ⊗ M ( X ) cons nil 15 / 26

T -list objects Let ( T , st ) be a strong monad on a monoidal category ( ⊗ , I ). A T-list object M ( X ) on X consists of T ( M ( X )) τ I M ( X ) X ⊗ M ( X ) cons nil 15 / 26

T -list objects Let ( T , st ) be a strong monad on a monoidal category ( ⊗ , I ). A T-list object M ( X ) on X consists of T ( M ( X )) τ I M ( X ) X ⊗ M ( X ) cons nil such that for every structure TA α P A X ⊗ A n c 15 / 26

T -list objects Let ( T , st ) be a strong monad on a monoidal category ( ⊗ , I ). A T-list object M ( X ) on X consists of T ( M ( X )) τ I M ( X ) X ⊗ M ( X ) cons nil such that for every structure TA α P A X ⊗ A n c there exists a unique mediating map it ( n , c , α ) : M ( X ) ⊗ P → A 15 / 26

T -list objects such that nil ⊗ P cons ⊗ P I ⊗ P M ( X ) ⊗ P X ⊗ M ( X ) ⊗ P ∼ = it ( n , c ,α ) X ⊗ it ( n , c ,α ) P A X ⊗ A n c and st M ( X ) , P T ( it ( n , c ,α )) � � T M ( X ) ⊗ P T ( M ( X ) ⊗ P ) TA τ ⊗ P α M ( X ) ⊗ P A it ( n , c ,α ) 16 / 26

T -list objects Remark Every list object is a T -list object. If every ( − ) ⊗ P has a right adjoint, the iterator it ( n , c , α ) is a T -algebra homomorphism. 16 / 26

Natural numbers in Cpo , revisited Flat natural numbers , Lazy natural numbers , Strict natural µ A . (1 + A ): µ A . (1 + A ) ⊥ : numbers , µ A . A ⊥ : · · · · · · · · · 0 1 2 3 · · · s 2 ( ⊥ ) 1 1 ⊥ s ( ⊥ ) 0 0 ⊥ ⊥ 17 / 26

Natural numbers in Cpo as T -list objects on the unit Flat natural numbers , Lazy natural numbers , Strict natural µ A . (1 + A ): µ A . (1 + A ) ⊥ : numbers , µ A . A ⊥ : · · · · · · · · · 0 1 2 3 · · · s 2 ( ⊥ ) 1 1 ⊥ s ( ⊥ ) 0 0 ⊥ ⊥ 17 / 26

Natural numbers in Cpo as T -list objects on the unit Flat natural numbers , Lazy natural numbers , Strict natural µ A . (1 + A ): µ A . (1 + A ) ⊥ : numbers , µ A . A ⊥ : · · · · · · · · · 0 1 2 3 · · · s 2 ( ⊥ ) 1 1 ⊥ s ( ⊥ ) 0 0 ⊥ ⊥ T -list object with ( × , 1) structure and monad T = Id 17 / 26

Natural numbers in Cpo as T -list objects on the unit Flat natural numbers , Lazy natural numbers , Strict natural µ A . (1 + A ): µ A . (1 + A ) ⊥ : numbers , µ A . A ⊥ : · · · · · · · · · 0 1 2 3 · · · s 2 ( ⊥ ) 1 1 ⊥ s ( ⊥ ) 0 0 ⊥ ⊥ T -list object with T -list object with ( × , 1) structure ( × , 1) structure and monad T = Id and T := ( − ) ⊥ the lifting monad 17 / 26

Natural numbers in Cpo as T -list objects on the unit Flat natural numbers , Lazy natural numbers , Strict natural µ A . (1 + A ): µ A . (1 + A ) ⊥ : numbers , µ A . A ⊥ : · · · · · · · · · 0 1 2 3 · · · s 2 ( ⊥ ) 1 1 ⊥ s ( ⊥ ) 0 0 ⊥ ⊥ T -list object with T -list object with T -list object with ( × , 1) structure ( × , 1) structure (+ , 0) structure and monad T = Id and T := ( − ) ⊥ the and T := ( − ) ⊥ the lifting monad lifting monad 17 / 26

Monoids with compatible algebraic structure 18 / 26

T -monoids 18 / 26

T -monoids Let ( T , st ) be a strong monad on on a monoidal category ( ⊗ , I ). A � � T-monoid EM-monoid (Pir´ og) is a monoid I M M ⊗ M 18 / 26

T -monoids Let ( T , st ) be a strong monad on on a monoidal category ( ⊗ , I ). A � � T-monoid EM-monoid (Pir´ og) is a monoid equipped with a T -algebra TM τ I M M ⊗ M 18 / 26

T -monoids Let ( T , st ) be a strong monad on on a monoidal category ( ⊗ , I ). A � � T-monoid EM-monoid (Pir´ og) is a monoid equipped with a T -algebra TM τ I M M ⊗ M compatible in the sense that st C , C Tm T ( C ) ⊗ C T ( C ⊗ C ) TC c c ⊗ C C ⊗ C C m 18 / 26

T -monoids Let ( T , st ) be a strong monad on on a monoidal category ( ⊗ , I ). A � � T-monoid EM-monoid (Pir´ og) is a monoid equipped with a T -algebra TM τ I M M ⊗ M compatible in the sense that st C , C Tm T ( C ) ⊗ C T ( C ⊗ C ) TC c c ⊗ C C ⊗ C C m Remark T -monoids generalise both monoids and T -algebras. 18 / 26

T -monoids Remark In the context of abstract syntax, T is freely generated from some theory, and T -monoids are models of this theory. 19 / 26

T -monoids Remark In the context of abstract syntax, T is freely generated from some theory, and T -monoids are models of this theory. Lemma For every monoid M the endofunctor T := M ⊗ ( − ) is a monad, � � and T-Mon ( C ) ≃ M / Mon ( C ) . 19 / 26

T -monoids Remark In the context of abstract syntax, T is freely generated from some theory, and T -monoids are models of this theory. Lemma For every monoid M the endofunctor T := M ⊗ ( − ) is a monad, � � and T-Mon ( C ) ≃ M / Mon ( C ) . Example In particular, a T -monoid for the endofunctor T := S ⊗ ( − ) is precisely an algebraic operation with signature S in the sense of η Jaskelioff, and can be identified with a map S − → L ( S ) → M interpreting S inside M . 19 / 26

T -monoids Remark In the context of abstract syntax, T is freely generated from some theory, and T -monoids are models of this theory. Lemma For every monoid M the endofunctor T := M ⊗ ( − ) is a monad, � � and T-Mon ( C ) ≃ M / Mon ( C ) . Example Set F , y (1) , • � � Thinking of a Lawvere theory as a monoid L M in , we can identify Lawvere theories extending L M with T -monoids for T := M • ( − ). 19 / 26

T -list objects are free T -monoids 20 / 26

T -list objects are free T -monoids For a strong monad ( T , st ) on a monoidal category ( ⊗ , I ), 20 / 26

T -list objects are free T -monoids For a strong monad ( T , st ) on a monoidal category ( ⊗ , I ), Lemma 1. Every T-list object M ( X ) is a T-monoid. 20 / 26

T -list objects are free T -monoids For a strong monad ( T , st ) on a monoidal category ( ⊗ , I ), Lemma 1. Every T-list object M ( X ) is a T-monoid. 2. This T -monoid is the free T -monoid on X , with universal map ∼ = X ⊗ nil → X ⊗ M ( X ) cons X − → X ⊗ I − − − − − → M ( X ) 20 / 26

T -list objects are free T -monoids For a strong monad ( T , st ) on a monoidal category ( ⊗ , I ), Lemma 1. Every T-list object M ( X ) is a T-monoid. 2. This T -monoid is the free T -monoid on X , with universal map ∼ = X ⊗ nil → X ⊗ M ( X ) cons X − → X ⊗ I − − − − − → M ( X ) We can reason concretely about free T -monoids by reasoning about T -lists. 20 / 26

T -list objects are initial algebras 21 / 26

T -list objects are initial algebras For a strong monad ( T , st ) on a monoidal category ( ⊗ , I ), Lemma If every ( − ) ⊗ P preserves binary coproducts, and the initial algebra exists, then µ A . T ( I + X ⊗ A ) is a T-list object on X. 21 / 26

Theorem Let T be a strong monad on a monoidal category ( C , I , ⊗ ) with binary coproducts (+) . If 1. for every P ∈ C , the endofunctor ( − ) ⊗ P preserves binary coproducts, and 2. for every X ∈ C , the initial algebra of T ( I + X ⊗ − ) exists Then C has all T-list objects and, thereby, the free T-monoid monad M T . 22 / 26

Theorem Let T be a strong monad on a monoidal category ( C , I , ⊗ ) with binary coproducts (+) . If 1. for every P ∈ C , the endofunctor ( − ) ⊗ P preserves binary coproducts, and 2. for every X ∈ C , the initial algebra of T ( I + X ⊗ − ) exists Then C has all T-list objects and, thereby, the free T-monoid monad M T . Remark Thinking in terms of T -list objects makes the proof straightforward! 22 / 26

Technical contribution 23 / 26

Technical contribution µ A . ( I + X ⊗ A ) � list object � free monoid 23 / 26

Technical contribution µ A . ( I + X ⊗ A ) � list object � free monoid T -list object 23 / 26

Technical contribution µ A . ( I + X ⊗ A ) � list object � free monoid T -list object � free T -monoid 23 / 26

Technical contribution µ A . ( I + X ⊗ A ) � list object � free monoid µ A . T ( I + X ⊗ A ) � T -list object � free T -monoid 23 / 26

Technical contribution µ A . ( I + X ⊗ A ) � list object � free monoid µ A . T ( I + X ⊗ A ) � T -list object � free T -monoid Remark A natural extension: algebraic structure encapsulated by Lawvere theories or operads . This gives rise to a notion of near-semiring category , which underlies many of the applications. 23 / 26

Applications 24 / 26

Applications T -NNOs In a a monoidal category ( ⊗ , I ): NNO = list object on I T -NNO = T -list object on I In Cpo : gives rise to the flat- , lazy- and strict natural numbers. 24 / 26

Applications Functional programming ◮ In the bicartesian closed setting: Jaskelioff’s monadic list transformer Lt ( T ) X := µ A . T (1 + X × A ) is just the free T -monoid monad. 24 / 26

Applications Functional programming ◮ In the bicartesian closed setting: Jaskelioff’s monadic list transformer Lt ( T ) X := µ A . T (1 + X × A ) is just the free T -monoid monad. ◮ In the category of endofunctors over a cartesian category: the MonadPlus type class Mp ( F ) X := µ A . List ( X + FA ) of Rivas et al. is a List -list object. 24 / 26

Applications Functional programming ◮ In the bicartesian closed setting: Jaskelioff’s monadic list transformer Lt ( T ) X := µ A . T (1 + X × A ) is just the free T -monoid monad. ◮ In the category of endofunctors over a cartesian category: the MonadPlus type class Mp ( F ) X := µ A . List ( X + FA ) of Rivas et al. is a List -list object. ◮ In the category of endofunctors over a cartesian category: the datatype Bun ( F ) X := µ A . (1 + X × A + F ( A ) × A + A × A ) is an instance of Spivey’s Bunch type class that is a T-list object for T the extension of the theory of monoids with a unary operator. 24 / 26