Convexity and the Kalmbach monad Gejza Jena August 10, 2018 Gejza - PowerPoint PPT Presentation

Convexity and the Kalmbach monad Gejza Jenča August 10, 2018 Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 1 / 42

The Plan Monads as generalized varieties Examples of monads Kalmbach monad on bounded posets Effect algebras Convex effect algebras The � product [ 0 , 1 ] -actions Distributive laws The composite monad and its algebras Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 2 / 42

Monads generalize varieties of algebras The ‘free algebra‘ endofunctor Let V be a variety of universal algebras. Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 3 / 42

Monads generalize varieties of algebras The ‘free algebra‘ endofunctor Let V be a variety of universal algebras. Write T V ( X ) for the the underlying set of the free algebra generated by the set X , so elements of T V ( X ) are (equivalence classes) of terms over X . Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 3 / 42

Monads generalize varieties of algebras The ‘free algebra‘ endofunctor Let V be a variety of universal algebras. Write T V ( X ) for the the underlying set of the free algebra generated by the set X , so elements of T V ( X ) are (equivalence classes) of terms over X . Then T V : Set → Set is a functor: f − → Y ): F ( X ) → F ( Y ) T V ( X replaces variable x in terms by the variable f ( x ) . Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 3 / 42

Monads generalize varieties of algebras The unit and the multiplication For every set X there is a natural mapping η X : X → T V ( X ) , given by η X ( x ) = x . Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras The unit and the multiplication For every set X there is a natural mapping η X : X → T V ( X ) , given by η X ( x ) = x . η is the unit of the monad Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras The unit and the multiplication For every set X there is a natural mapping η X : X → T V ( X ) , given by η X ( x ) = x . η is the unit of the monad For every set X there is a natural mapping µ X : T V ( T V ( X )) → T V ( X ) , given by ‘flattening of terms over terms’ or ‘evaluation in the free algebra’. Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras The unit and the multiplication For every set X there is a natural mapping η X : X → T V ( X ) , given by η X ( x ) = x . η is the unit of the monad For every set X there is a natural mapping µ X : T V ( T V ( X )) → T V ( X ) , given by ‘flattening of terms over terms’ or ‘evaluation in the free algebra’. µ is the multiplication of the monad Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 4 / 42

Monads generalize varieties of algebras Example: the free monoid monad T ( X ) is the set of all words over alphabet X : Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras Example: the free monoid monad T ( X ) is the set of all words over alphabet X : X = { a , b , c } [] , [ a ] , [ babbca ] ∈ T ( X ) Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras Example: the free monoid monad T ( X ) is the set of all words over alphabet X : X = { a , b , c } [] , [ a ] , [ babbca ] ∈ T ( X ) For a mapping f : X → Y , T ( f ): T ( X ) → T ( Y ) is given by T ( f )([ x 1 x 2 . . . x n ]) = [ f ( x 1 ) f ( x 2 ) . . . f ( x n )] Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras Example: the free monoid monad T ( X ) is the set of all words over alphabet X : X = { a , b , c } [] , [ a ] , [ babbca ] ∈ T ( X ) For a mapping f : X → Y , T ( f ): T ( X ) → T ( Y ) is given by T ( f )([ x 1 x 2 . . . x n ]) = [ f ( x 1 ) f ( x 2 ) . . . f ( x n )] For a set X , η X : X → T ( X ) is given by η X ( x ) = [ x ] Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

Monads generalize varieties of algebras Example: the free monoid monad T ( X ) is the set of all words over alphabet X : X = { a , b , c } [] , [ a ] , [ babbca ] ∈ T ( X ) For a mapping f : X → Y , T ( f ): T ( X ) → T ( Y ) is given by T ( f )([ x 1 x 2 . . . x n ]) = [ f ( x 1 ) f ( x 2 ) . . . f ( x n )] For a set X , η X : X → T ( X ) is given by η X ( x ) = [ x ] For a set X , µ X : T ( T ( X )) → T ( X ) concatenates the words: µ X ([[ aba ][ acd ][][ da ]]) = [ abaacdda ] Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 5 / 42

( Set , η, µ ) . So we have data of the following type: Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

( Set , η, µ ) . So we have data of the following type: a category Set , Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

( Set , η, µ ) . So we have data of the following type: a category Set , a functor T : Set → Set , Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

( Set , η, µ ) . So we have data of the following type: a category Set , a functor T : Set → Set , a natural transformation η : id Set → T , Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

( Set , η, µ ) . So we have data of the following type: a category Set , a functor T : Set → Set , a natural transformation η : id Set → T , a natural transformation µ : T 2 → T . Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 6 / 42

The monad laws Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 7 / 42

� � Axioms of a monad Right unit axiom T ( η X ) � T 2 ( X ) T ( X ) µ X id T ( X ) T ( X ) Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 8 / 42

� � � � � Axioms of a monad Right unit axiom T ( η X ) � T 2 ( X ) [ abac ] ✤ T ( X ) [[ a ][ b ][ a ][ c ]] ❴ ✌ µ X id T ( X ) T ( X ) [ abac ] Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 8 / 42

� � � Axioms Left unit axiom η T ( X ) T 2 ( X ) T ( X ) µ X id T ( X ) T ( X ) Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 9 / 42

✤ � � � � � � Axioms Left unit axiom η T ( X ) T 2 ( X ) T ( X ) [[ abac ]] [ abac ] ✹ ❴ µ X id T ( X ) [ abac ] T ( X ) Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 9 / 42

� � Axioms of a monad Associativity axiom T ( µ X ) � T 3 ( X ) T 2 ( X ) µ T ( X ) µ X µ X � T ( X ) T 2 ( X ) Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 10 / 42

� � � � � Axioms of a monad Associativity axiom T ( µ X ) � T 3 ( X ) T 2 ( X ) µ T ( X ) µ X µ X � T ( X ) T 2 ( X ) �� ✤ �� �� � � [ ab ][ bc ] [ ca ] [ abbc ][ ca ] ❴ ❴ � ✤ � [ abbcca ] � [ ab ][ bc ][ ca ] Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 10 / 42

The definition Definition Let C be a category. A monad over C is a triple ( T , η, µ ) such that T : C → C , η : id C → T , µ : T 2 → T such that the unit and associativity axioms hold. Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 11 / 42

Algebras for a monad Definition Let ( T , η, µ ) be a monad over C . Then an algebra for T (or T -algebra) is a pair ( X , α ) , where X is an object of C and Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 12 / 42

Algebras for a monad Definition Let ( T , η, µ ) be a monad over C . Then an algebra for T (or T -algebra) is a pair ( X , α ) , where X is an object of C and α : T ( X ) → X Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 12 / 42

� � � � � Algebras for a monad Definition Let ( T , η, µ ) be a monad over C . Then an algebra for T (or T -algebra) is a pair ( X , α ) , where X is an object of C and α : T ( X ) → X such that the following diagrams commute η X � µ X T 2 ( X ) X T ( X ) T ( X ) α T ( α ) α id X X � X T ( X ) α Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 12 / 42

Intuition/meaning of all this An algebra α : T ( X ) → X equips the set X with evaluation of terms over X . Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 13 / 42

Morphisms of algebras Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 14 / 42

� � Morphisms of algebras Definition If ( A , α ) , ( B , β ) are algebras for a monad T , then a morphism of algebras f : ( A , α ) → ( B , β ) is a morphism f : A → B in the underlying category such that the square T ( f ) � T ( A ) T ( B ) α β � B A f commutes. Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 14 / 42

The Eilenberg-Moore category Definition Let ( T , η, µ ) be a monad on a category C . The category C T of T -algebras and their morphisms is called the Eilenberg-Moore category of the monad T . Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 15 / 42

Algebras are algebras If V is a variety of algebras and T V is the monad associated with V , then Set T V ≃ V . Gejza Jenča Convexity and the Kalmbach monad August 10, 2018 16 / 42