Local elements, partial metrics, and diagonals Isar Stubbe - PowerPoint PPT Presentation

Local elements, partial metrics, and diagonals Isar Stubbe Université du Littoral, France Clea Workshop Brussels, 01/12/2013

1. Local elements

( P, ≤ ) is an order (aka ‘preorder’) if: ≤ is a binary relation on a set P such that - if x ≤ y and y ≤ z then x ≤ z , - x ≤ x .

( P, ≤ ) is an order if: [ · ≤ · ]: P × P → 2 is a binary 2 -valued predicate on a set P such that - if x ≤ y and y ≤ z then x ≤ z , - x ≤ x .

( P, ≤ ) is an order if: [ · ≤ · ]: P × P → 2 is a binary 2 -valued predicate on a set P such that - [ x ≤ y ] ∧ [ y ≤ z ] ≤ [ x ≤ z ] , - x ≤ x .

( P, ≤ ) is an order if: [ · ≤ · ]: P × P → 2 is a binary 2 -valued predicate on a set P such that - [ x ≤ y ] ∧ [ y ≤ z ] ≤ [ x ≤ z ] , - 1 ≤ [ x ≤ x ] .

( P, ≤ ) is an order if: [ · ≤ · ]: P × P → 2 is a binary 2 -valued predicate on a set P such that - [ x ≤ y ] ∧ [ y ≤ z ] ≤ [ x ≤ z ] ... in 2 ! - 1 ≤ [ x ≤ x ] ... in 2 !

( P, ≤ ) is an order if: [ · ≤ · ]: P × P → 2 is a binary 2 -valued predicate on a set P such that - [ x ≤ y ] ∧ [ y ≤ z ] ≤ [ x ≤ z ] ... in 2 ! - 1 ≤ [ x ≤ x ] ... in 2 ! We can replace the ‘truth value object’ 2 by ...

( P, ≤ ) is a B -order if: [ · ≤ · ]: P × P → B is a binary B -valued predicate on a set P such that - [ x ≤ y ] ∧ [ y ≤ z ] ≤ [ x ≤ z ] in B , - 1 ≤ [ x ≤ x ] in B . We replaced the ‘truth value object’ 2 by ... any Boolean algebra B .

( P, ≤ ) is an M -order if: [ · ≤ · ]: P × P → M is a binary M -valued predicate on a set P such that - [ x ≤ y ] ∧ [ y ≤ z ] ≤ [ x ≤ z ] in M , - 1 ≤ [ x ≤ x ] in M . We replaced the ‘truth value object’ 2 by ... any meet-semilattice M .

( P, ≤ ) is a V -order if: [ · ≤ · ]: P × P → V is a binary V -valued predicate on a set P such that - [ x ≤ y ] ◦ [ y ≤ z ] ≤ [ x ≤ z ] in V , - 1 ≤ [ x ≤ x ] in V . We replaced the ‘truth value object’ 2 by ... any ordered monoid V . An ordered monoid V = ( V, ≤ , ◦ , 1) is - an ordered set ( V, ≤ ) , - a monoid ( V, ◦ , 1) , - such that x ◦ − and − ◦ y are monotone. In paticular, 1 need not be the top element, and ◦ need not be commutative.

With slight change of notation and terminology, we thus arrive at: An ordered monoid (aka monoidal order ) V is: ◮ an ordered set ( V, ≤ ) , ◮ a monoid ( V, ◦ , 1) , ◮ such that x ◦ − and − ◦ y are monotone. A V -enriched category P = ( P 0 , P 2 ) is ◮ a set P 0 together with ◮ a binary V -valued predicate P 2 ( · , · ): P 0 × P 0 → V such that ◮ P 2 ( x, y ) ◦ P 2 ( y, z ) ≤ P 2 ( x, z ) in V , ◮ 1 ≤ P 2 ( x, x ) in V .

With slight change of notation and terminology, we thus arrive at: An ordered monoid (aka monoidal order ) V is: ◮ an ordered set ( V, ≤ ) , ◮ a monoid ( V, ◦ , 1) , ◮ such that x ◦ − and − ◦ y are monotone. A V -enriched category P = ( P 0 , P 2 ) is ◮ a set P 0 together with ◮ a binary V -valued predicate P 2 ( · , · ): P 0 × P 0 → V such that ◮ P 2 ( x, y ) ◦ P 2 ( y, z ) ≤ P 2 ( x, z ) in V , ◮ 1 ≤ P 2 ( x, x ) in V . The further generalisation from monoidal orders V to a monoidal categories V shall not be our concern today ...

With slight change of notation and terminology, we thus arrive at: An ordered monoid (aka monoidal order ) V is: ◮ an ordered set ( V, ≤ ) , ◮ a monoid ( V, ◦ , 1) , ◮ such that x ◦ − and − ◦ y are monotone. A V -enriched category P = ( P 0 , P 2 ) is ◮ a set P 0 together with ◮ a binary V -valued predicate P 2 ( · , · ): P 0 × P 0 → V such that ◮ P 2 ( x, y ) ◦ P 2 ( y, z ) ≤ P 2 ( x, z ) in V , ◮ 1 ≤ P 2 ( x, x ) in V . The further generalisation from monoidal orders V to a monoidal categories V shall not be our concern today ... but the generalisation from ordered monoids V to ordered categories W will be all the more so!

With slight change of notation and terminology, we thus arrive at: An ordered monoid (aka monoidal order ) V is: ◮ an ordered set ( V, ≤ ) , ◮ a monoid ( V, ◦ , 1) , ◮ such that x ◦ − and − ◦ y are monotone. A V -enriched category P = ( P 0 , P 2 ) is ◮ a set P 0 together with ◮ a binary V -valued predicate P 2 ( · , · ): P 0 × P 0 → V such that ◮ P 2 ( x, y ) ◦ P 2 ( y, z ) ≤ P 2 ( x, z ) in V , ◮ 1 ≤ P 2 ( x, x ) in V . The further generalisation from monoidal orders V to a monoidal categories V shall not be our concern today ... but the generalisation from ordered monoids V to ordered categories W will be all the more so! References: Lawvere [1973], Kelly [1982]

Let X be a set and ( P, ≤ ) an order. The set P = R PF ( X, P ) := { f : S → P is a function | S ⊆ X } f g of partial functions from X to P is an archetypical mathematical structure with local elements . X = R

Let X be a set and ( P, ≤ ) an order. The set P = R PF ( X, P ) := { f : S → P is a function | S ⊆ X } f g of partial functions from X to P is an archetypical mathematical structure with local elements . To compare f, g ∈ PF ( X, P ) , it is most natural to compute the “extent to which f is smaller than g ”: X = R { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } ∈ P ( X ) .

Let X be a set and ( P, ≤ ) an order. The set P = R PF ( X, P ) := { f : S → P is a function | S ⊆ X } f g of partial functions from X to P is an archetypical mathematical structure with local elements . To compare f, g ∈ PF ( X, P ) , it is most natural to compute the “extent to which f is smaller than g ”: X = R { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } ∈ P ( X ) . This seems to define a V -enriched category: - V is the Boolean algebra ( P ( X ) , ⊆ , ∩ , X ) , - P 0 = PF ( X, P ) , - P 2 ( f, g ) = { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } ,

Let X be a set and ( P, ≤ ) an order. The set P = R PF ( X, P ) := { f : S → P is a function | S ⊆ X } f g of partial functions from X to P is an archetypical mathematical structure with local elements . To compare f, g ∈ PF ( X, P ) , it is most natural to compute the “extent to which f is smaller than g ”: X = R { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } ∈ P ( X ) . This seems to define a V -enriched category: - V is the Boolean algebra ( P ( X ) , ⊆ , ∩ , X ) , - P 0 = PF ( X, P ) , - P 2 ( f, g ) = { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } , - P 2 ( f, g ) ∩ P 2 ( g, h ) ⊆ P 2 ( f, h ) is okay,

Let X be a set and ( P, ≤ ) an order. The set P = R PF ( X, P ) := { f : S → P is a function | S ⊆ X } f g of partial functions from X to P is an archetypical mathematical structure with local elements . To compare f, g ∈ PF ( X, P ) , it is most natural to compute the “extent to which f is smaller than g ”: X = R { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } ∈ P ( X ) . This seems to define a V -enriched category: - V is the Boolean algebra ( P ( X ) , ⊆ , ∩ , X ) , - P 0 = PF ( X, P ) , - P 2 ( f, g ) = { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } , - P 2 ( f, g ) ∩ P 2 ( g, h ) ⊆ P 2 ( f, h ) is okay, - ... but X ⊆ P 2 ( f, f ) fails , precisely because f is a partial function!

Let X be a set and ( P, ≤ ) an order. The set P = R PF ( X, P ) := { f : S → P is a function | S ⊆ X } f g of partial functions from X to P is an archetypical mathematical structure with local elements . To compare f, g ∈ PF ( X, P ) , it is most natural to compute the “extent to which f is smaller than g ”: X = R { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } ∈ P ( X ) . This seems to define a V -enriched category: - V is the Boolean algebra ( P ( X ) , ⊆ , ∩ , X ) , - P 0 = PF ( X, P ) - P 2 ( f, g ) = { x ∈ dom ( f ) ∩ dom ( g ) | fx ≤ gx in P } , - P 2 ( f, g ) ∩ P 2 ( g, h ) ⊆ P 2 ( f, h ) is okay, - ... but X ⊆ P 2 ( f, f ) fails , precisely because f is a partial function! We must – somehow – keep track of the domains (or types ) of the elements of P 0 ...

An ordered category (more accurately: locally ordered category ) W is: ◮ a category W , ◮ each hom-set W ( X, Y ) is ordered, ◮ composition is monotone in each variable. A W -enriched category P = ( P 0 , P 1 , P 2 ) consists of ◮ a set P 0 , ◮ a unary (“type”) predicate P 1 : P 0 → obj ( W ) and ◮ a binary (“value”) predicate P 2 : P 0 × P 0 → arr ( W ) such that: ◮ P 2 ( x, y ): P 1 ( y ) → P 1 ( x ) , ◮ P 2 ( x, y ) ◦ P 2 ( y, z ) ≤ P 2 ( x, z ) , ◮ 1 P 1 ( x ) ≤ P 2 ( x, x ) .

An ordered category (more accurately: locally ordered category ) W is: ◮ a category W , ◮ each hom-set W ( X, Y ) is ordered, ◮ composition is monotone in each variable. A W -enriched category P = ( P 0 , P 1 , P 2 ) consists of ◮ a set P 0 , ◮ a unary (“type”) predicate P 1 : P 0 → obj ( W ) and ◮ a binary (“value”) predicate P 2 : P 0 × P 0 → arr ( W ) such that: ◮ P 2 ( x, y ): P 1 ( y ) → P 1 ( x ) , ◮ P 2 ( x, y ) ◦ P 2 ( y, z ) ≤ P 2 ( x, z ) , ◮ 1 P 1 ( x ) ≤ P 2 ( x, x ) . An ordered category W with a single object ∗ is the same thing as an ordered monoid V = W ( ∗ , ∗ ) . The above definition agrees with the previous definition of V -enriched category: because the “type” predicate is then obsolete.