Transitivity Question: Can partial evaluations be composed? ((1 + 1) + (1 + 1)) µ TTe T µ (1 + 1) + (1 + 1) (1 + 1 + 1 + 1) (2 + 2) µ µ Te Te µ Te 1 + 1 + 1 + 1 2 + 2 4 9 of 22
Transitivity Question: Can partial evaluations be composed? TTTA µ TTe T µ TTA TTA TTA µ µ Te Te µ Te TA TA TA 9 of 22
Transitivity Question: Can partial evaluations be composed? TTTA µ TTe T µ TTA TTA TTA µ µ Te Te µ Te TA TA TA We are asking for the existence of a “rewriting of rewritings”. 9 of 22
Transitivity Definition: The diagram a b f A B g m C D n c d is called a weak or meek pullback if for every b ∈ B and c ∈ C such that m ( b ) = n ( c ) there exists an a ∈ A such that f ( a ) = b and g ( a ) = c . 10 of 22
Transitivity Definition: The diagram a b f A B g m C D n c d is called a weak or meek pullback if for every b ∈ B and c ∈ C such that m ( b ) = n ( c ) there exists an a ∈ A such that f ( a ) = b and g ( a ) = c . 10 of 22
Transitivity Definition: The diagram a b f A B g m C D n c d is called a weak or meek pullback if for every b ∈ B and c ∈ C such that m ( b ) = n ( c ) there exists an a ∈ A such that f ( a ) = b and g ( a ) = c . 10 of 22
Transitivity Definition: The diagram a b f A B g m C D n c d is called a weak or meek pullback if for every b ∈ B and c ∈ C such that m ( b ) = n ( c ) there exists an a ∈ A such that f ( a ) = b and g ( a ) = c . 10 of 22
Transitivity Definition: A cartesian monad is a monad ( T , η, µ ) such that: • T preserves pullbacks; • All naturality squares of η and µ are pullbacks. 11 of 22
Transitivity Definition [Weber, 2004]: A weakly cartesian monad is a monad ( T , η, µ ) such that: • T preserves weak pullbacks; • All naturality squares of η and µ are weak pullbacks. 11 of 22
Transitivity Definition [Weber, 2004]: A weakly cartesian monad is a monad ( T , η, µ ) such that: • T preserves weak pullbacks; • All naturality squares of η and µ are weak pullbacks. Proposition: • For weakly cartesian monads, composition is always defined; 11 of 22
Transitivity Definition [Weber, 2004]: A weakly cartesian monad is a monad ( T , η, µ ) such that: • T preserves weak pullbacks; • All naturality squares of η and µ are weak pullbacks. Proposition: • For weakly cartesian monads, composition is always defined; • For cartesian monads, composition is always uniquely defined. 11 of 22
Transitivity What is known [Clementino et al., 2014]: 12 of 22
Transitivity What is known [Clementino et al., 2014]: • All monads presented by an operad are cartesian; ◦ Monoid and group action monads; ◦ Free monoid monads; ◦ Maybe monad; 12 of 22
Transitivity What is known [Clementino et al., 2014]: • All monads presented by an operad are cartesian; ◦ Monoid and group action monads; ◦ Free monoid monads; ◦ Maybe monad; • A wide class of monads, including the free commutative monoid monad, are weakly cartesian; 12 of 22
Transitivity What is known [Clementino et al., 2014]: • All monads presented by an operad are cartesian; ◦ Monoid and group action monads; ◦ Free monoid monads; ◦ Maybe monad; • A wide class of monads, including the free commutative monoid monad, are weakly cartesian; • Most monads of classical algebras are not weakly cartesian; 12 of 22
Transitivity What is known [Clementino et al., 2014]: • All monads presented by an operad are cartesian; ◦ Monoid and group action monads; ◦ Free monoid monads; ◦ Maybe monad; • A wide class of monads, including the free commutative monoid monad, are weakly cartesian; • Most monads of classical algebras are not weakly cartesian; • As we prove, the Kantorovich probability monad is weakly cartesian (more on that later). 12 of 22
Examples Factorization Let T be the free commutative monoid monad. Consider ( N , · ) as an algebra. 13 of 22
Examples Factorization Let T be the free commutative monoid monad. Consider ( N , · ) as an algebra. • Given a number n , its partial decompositions are its factorizations. 13 of 22
Examples Factorization Let T be the free commutative monoid monad. Consider ( N , · ) as an algebra. • Given a number n , its partial decompositions are its factorizations. • Each number admits a unique total decomposition , the decomposition into prime factors. 13 of 22
Examples Factorization Let T be the free commutative monoid monad. Consider ( N , · ) as an algebra. • Given a number n , its partial decompositions are its factorizations. • Each number admits a unique total decomposition , the decomposition into prime factors. 6 · 5 · 11 30 · 11 2 · 3 · 5 · 11 2 · 15 · 11 6 · 55 330 2 · 3 · 55 3 · 110 13 of 22
Examples Group or monoid action Let G be an internal monoid (or group) in a (cartesian) monoidal category. 14 of 22
Examples Group or monoid action Let G be an internal monoid (or group) in a (cartesian) monoidal category. • X �→ G × X is a monad; 14 of 22
Examples Group or monoid action Let G be an internal monoid (or group) in a (cartesian) monoidal category. • X �→ G × X is a monad; • The algebras e : G × A → A are G -spaces. 14 of 22
Examples Group or monoid action Let G be an internal monoid (or group) in a (cartesian) monoidal category. • X �→ G × X is a monad; • The algebras e : G × A → A are G -spaces. Let ( g , x ) , ( h , y ) ∈ G × A . 14 of 22
Examples Group or monoid action Let G be an internal monoid (or group) in a (cartesian) monoidal category. • X �→ G × X is a monad; • The algebras e : G × A → A are G -spaces. Let ( g , x ) , ( h , y ) ∈ G × A . A partial evaluation from ( g , x ) to ( h , y ) is an element ( h , ℓ, x ) ∈ G × G × A such that h ℓ = g and ℓ · x = y . 14 of 22
Examples Group or monoid action Let G be an internal monoid (or group) in a (cartesian) monoidal category. • X �→ G × X is a monad; • The algebras e : G × A → A are G -spaces. Let ( g , x ) , ( h , y ) ∈ G × A . A partial evaluation from ( g , x ) to ( h , y ) is an element ( h , ℓ, x ) ∈ G × G × A such that h ℓ = g and ℓ · x = y . g x ℓ · x g · x ℓ h 14 of 22
Examples Group or monoid action Let G be an internal monoid (or group) in a (cartesian) monoidal category. • X �→ G × X is a monad; • The algebras e : G × A → A are G -spaces. Let ( g , x ) , ( h , y ) ∈ G × A . A partial evaluation from ( g , x ) to ( h , y ) is an element ( h , ℓ, x ) ∈ G × G × A such that h ℓ = g and ℓ · x = y . g x ℓ · x g · x ℓ h 14 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C X 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX PX X 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX • Unit δ : X → PX PX X 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. ? 1/2 1/2 • Base category C • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. ? ? 1/2 1/2 • Base category C 3/4 1/4 • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. ? ? 1/2 1/2 • Base category C 3/4 1/4 • Functor X �→ PX • Unit δ : X → PX 1/2 1/2 1/2 1/2 • Composition E : PPX → PX PX PPX 15 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a e : PA → A are “convex spaces” b A 16 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a a b e : PA → A are “convex spaces” a b b a b A PA 16 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a a b e : PA → A are “convex spaces” a b b a b A PA 16 of 22
Probability monads Idea [Giry, 1982]: Spaces of random elements as formal convex combinations. • Algebras a a b e : PA → A are λ a + (1 − λ ) b “convex spaces” a b • Formal averages are b mapped to actual a b averages A PA 16 of 22
Probability monads Kantorovich monad [van Breugel, 2005, Fritz and Perrone, 2017]: • Given a complete metric space X , PX is the set of Radon probability measures of finite first moment, equipped with the Wasserstein distance , or Kantorovich-Rubinstein distance , or earth mover’s distance : � � � � � d PX ( p , q ) = sup f ( x ) d ( p − q )( x ) � � � � f : X → R X 17 of 22
Probability monads Kantorovich monad [van Breugel, 2005, Fritz and Perrone, 2017]: • Given a complete metric space X , PX is the set of Radon probability measures of finite first moment, equipped with the Wasserstein distance , or Kantorovich-Rubinstein distance , or earth mover’s distance : � � � � � d PX ( p , q ) = sup f ( x ) d ( p − q )( x ) � � � � f : X → R X • The assignment X �→ PX is part of a monad on the category of complete metric spaces and short maps. 17 of 22
Probability monads Kantorovich monad [van Breugel, 2005, Fritz and Perrone, 2017]: • Given a complete metric space X , PX is the set of Radon probability measures of finite first moment, equipped with the Wasserstein distance , or Kantorovich-Rubinstein distance , or earth mover’s distance : � � � � � d PX ( p , q ) = sup f ( x ) d ( p − q )( x ) � � � � f : X → R X • The assignment X �→ PX is part of a monad on the category of complete metric spaces and short maps. • Algebras of P are closed convex subsets of Banach spaces. 17 of 22
Probability monads Idea: Partial evaluations for P are “partial expectations”. 18 of 22
Probability monads Idea: Partial evaluations for P are “partial expectations”. 18 of 22
Probability monads Idea: Partial evaluations for P are “partial expectations”. 18 of 22
Probability monads Idea: Partial evaluations for P are “partial expectations”. 18 of 22
Probability monads Idea: Partial evaluations for P are “partial expectations”. 18 of 22
Probability monads Idea: Partial evaluations for P are “partial expectations”. Properties: 1. A partial expectation makes a distribution “more concentrated”, or “less random” (closer to its center of mass); 18 of 22
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