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 = (P0, P2) is

◮ a set P0 together with ◮ a binary V -valued predicate P2(·, ·): P0 × P0 → V

such that

◮ P2(x, y) ◦ P2(y, z) ≤ P2(x, z) in V , ◮ 1 ≤ P2(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 = (P0, P2) is

◮ a set P0 together with ◮ a binary V -valued predicate P2(·, ·): P0 × P0 → V

such that

◮ P2(x, y) ◦ P2(y, z) ≤ P2(x, z) in V , ◮ 1 ≤ P2(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 = (P0, P2) is

◮ a set P0 together with ◮ a binary V -valued predicate P2(·, ·): P0 × P0 → V

such that

◮ P2(x, y) ◦ P2(y, z) ≤ P2(x, z) in V , ◮ 1 ≤ P2(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 = (P0, P2) is

◮ a set P0 together with ◮ a binary V -valued predicate P2(·, ·): P0 × P0 → V

such that

◮ P2(x, y) ◦ P2(y, z) ≤ P2(x, z) in V , ◮ 1 ≤ P2(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]

X = R P = R f g Let X be a set and (P, ≤) an order. The set PF(X, P) := {f : S → P is a function | S ⊆ X}

• f partial functions from X to P is an archetypical

mathematical structure with local elements.

X = R P = R f g Let X be a set and (P, ≤) an order. The set PF(X, P) := {f : S → P is a function | S ⊆ X}

• f 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 ∈ dom(f) ∩ dom(g) | fx ≤ gx in P} ∈ P(X).

X = R P = R f g Let X be a set and (P, ≤) an order. The set PF(X, P) := {f : S → P is a function | S ⊆ X}

• f 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 ∈ 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),
• P0 = PF(X, P),
• P2(f, g) = {x ∈ dom(f) ∩ dom(g) | fx ≤ gx in P},

X = R P = R f g Let X be a set and (P, ≤) an order. The set PF(X, P) := {f : S → P is a function | S ⊆ X}

• f 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 ∈ 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),
• P0 = PF(X, P),
• P2(f, g) = {x ∈ dom(f) ∩ dom(g) | fx ≤ gx in P},
• P2(f, g) ∩ P2(g, h) ⊆ P2(f, h) is okay,

X = R P = R f g Let X be a set and (P, ≤) an order. The set PF(X, P) := {f : S → P is a function | S ⊆ X}

• f 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 ∈ 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),
• P0 = PF(X, P),
• P2(f, g) = {x ∈ dom(f) ∩ dom(g) | fx ≤ gx in P},
• P2(f, g) ∩ P2(g, h) ⊆ P2(f, h) is okay,
• ... but X ⊆ P2(f, f) fails, precisely because f is a partial function!

X = R P = R f g Let X be a set and (P, ≤) an order. The set PF(X, P) := {f : S → P is a function | S ⊆ X}

• f 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 ∈ 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),
• P0 = PF(X, P)
• P2(f, g) = {x ∈ dom(f) ∩ dom(g) | fx ≤ gx in P},
• P2(f, g) ∩ P2(g, h) ⊆ P2(f, h) is okay,
• ... but X ⊆ P2(f, f) fails, precisely because f is a partial function!

We must – somehow – keep track of the domains (or types) of the elements of P0...

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 = (P0, P1, P2) consists of

◮ a set P0, ◮ a unary (“type”) predicate P1 : P0 → obj(W) and ◮ a binary (“value”) predicate P2 : P0 × P0 → arr(W)

such that:

◮ P2(x, y): P1(y) → P1(x), ◮ P2(x, y) ◦ P2(y, z) ≤ P2(x, z), ◮ 1P1(x) ≤ P2(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 = (P0, P1, P2) consists of

◮ a set P0, ◮ a unary (“type”) predicate P1 : P0 → obj(W) and ◮ a binary (“value”) predicate P2 : P0 × P0 → arr(W)

such that:

◮ P2(x, y): P1(y) → P1(x), ◮ P2(x, y) ◦ P2(y, z) ≤ P2(x, z), ◮ 1P1(x) ≤ P2(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.

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 = (P0, P1, P2) consists of

◮ a set P0, ◮ a unary (“type”) predicate P1 : P0 → obj(W) and ◮ a binary (“value”) predicate P2 : P0 × P0 → arr(W)

such that:

◮ P2(x, y): P1(y) → P1(x), ◮ P2(x, y) ◦ P2(y, z) ≤ P2(x, z), ◮ 1P1(x) ≤ P2(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. References: Walters [1981], Street [1982]

Partial functions done right: Remember, X is a set, (P, ≤) is an order, and PF(X, P) is the set of partially defined functions from X to P. Construct the ordered category W with:

• objects: elements of P(X),
• homs: W(A, B) = {S ∈ P(X) | S ⊆ A ∩ B}, order inherited from (P(X), ⊆),
• composition of A

S B T C is A T ∩S C , identity on A is A A A .

Now setting

• P0 = PF(X, P),
• P1(f) = dom(f),
• P2(f, g) = {x ∈ dom(f) ∩ dom(g) | fx ≤ gx in P}

does produce a W-enriched category. In particular, 1P1(f) ≤ P2(f, f) ⇐ ⇒ dom(f) ⊆ dom(f). (This line of thought extends – vastly – to sheaf theory.)

• 2. Partial metrics

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid.

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff: P(·, ·): P0 × P0 → V is a binary V -valued predicate such that

• P(x, y) ◦ P(y, z) ≤ P(x, z) in V ,
• 1 ≤ P(x, x) in V .

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff: P(·, ·): P0 × P0 → [0, +∞] is a binary [0, +∞]-valued predicate such that

• P(x, y) ◦ P(y, z) ≤ P(x, z) in V ,
• 1 ≤ P(x, x) in V .

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff: P(·, ·): P0 × P0 → [0, +∞] is a binary [0, +∞]-valued predicate such that

• P(x, y) + P(y, z) ≥ P(x, z) in [0, +∞],
• 1 ≤ P(x, x) in V .

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff: P(·, ·): P0 × P0 → [0, +∞] is a binary [0, +∞]-valued predicate such that

• P(x, y) + P(y, z) ≥ P(x, z) in [0, +∞],
• 0 ≥ P(x, x) in [0, +∞].

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), P(·, ·): P0 × P0 → [0, +∞] is a binary [0, +∞]-valued predicate such that

• P(x, y) + P(y, z) ≥ P(x, z) in [0, +∞],
• 0 ≥ P(x, x) in [0, +∞].

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), d: X × X → [0, +∞] is a function such that

• P(x, y) + P(y, z) ≥ P(x, z) in [0, +∞],
• 0 ≥ P(x, x) in [0, +∞].

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), d: X × X → [0, +∞] is a function such that

• d(x, y) + d(y, z) ≥ d(x, z) in [0, +∞],
• 0 ≥ d(x, x) in [0, +∞].

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), d: X × X → [0, +∞] is a function such that

• d(x, y) + d(y, z) ≥ d(x, z) in [0, +∞],
• 0 = d(x, x) in [0, +∞].

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), d: X × X → [0, +∞] is a function such that

• d(x, y) + d(y, z) ≥ d(x, z),
• 0 = d(x, x).

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), d: X × X → [0, +∞] is a function such that

• d(x, y) + d(y, z) ≥ d(x, z),
• 0 = d(x, x).

That is to say, V -enriched categories are precisely generalised metric spaces.

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), d: X × X → [0, +∞] is a function such that

• d(x, y) + d(y, z) ≥ d(x, z),
• 0 = d(x, x).

That is to say, V -enriched categories are precisely generalised metric spaces. Separatedness, symmetry and finiteness are the “missing axioms” for d: X × X → [0, +∞] to be a metric. They can be stated in the full generality of V -enriched categories, but we shall not go into the details here.

Let V = ([0, +∞], ≥, +, 0) be Lawvere’s quantale of positive real numbers:

• take the opposite of the natural order on [0, +∞],
• take the monoidal structure ([0, +∞], +, 0),
• then b ≥ b′ implies a + b ≥ a + b′,

so V is an ordered monoid. Now P is a V -enriched category iff, writing X = P0 and d(x, y) = P(x, y), d: X × X → [0, +∞] is a function such that

• d(x, y) + d(y, z) ≥ d(x, z),
• 0 = d(x, x).

That is to say, V -enriched categories are precisely generalised metric spaces. Separatedness, symmetry and finiteness are the “missing axioms” for d: X × X → [0, +∞] to be a metric. They can be stated in the full generality of V -enriched categories, but we shall not go into the details here. Reference: Lawvere [1973]

Consider the set of (infinite) sequences in some finite alphabet, say X = AN. It is a metric space if we set, for x = (xi)i and y = (yi)i in X, d(x, y) := 2−k for k the smallest index such that xk = yk.

Consider the set of (infinite) sequences in some finite alphabet, say X = AN. It is a metric space if we set, for x = (xi)i and y = (yi)i in X, d(x, y) := 2−k for k the smallest index such that xk = yk. If we “add” the set A∗ of (finite) words in the alphabet A to this space (think of words as approximations of sequences, e.g. for reasons of computability), then some properties

• f the metric d are destroyed, notably

d(w, w) = 0 whenever w is a word.

Consider the set of (infinite) sequences in some finite alphabet, say X = AN. It is a metric space if we set, for x = (xi)i and y = (yi)i in X, d(x, y) := 2−k for k the smallest index such that xk = yk. If we “add” the set A∗ of (finite) words in the alphabet A to this space (think of words as approximations of sequences, e.g. for reasons of computability), then some properties

• f the metric d are destroyed, notably

d(w, w) = 0 whenever w is a word. The relevant mathematical structure in this example (and others) is: A partial metric space is a set X together with a function p: X × X → R such that

◮ 0 ≤ p(x, x) ≤ p(x, y), ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z).

Consider the set of (infinite) sequences in some finite alphabet, say X = AN. It is a metric space if we set, for x = (xi)i and y = (yi)i in X, d(x, y) := 2−k for k the smallest index such that xk = yk. If we “add” the set A∗ of (finite) words in the alphabet A to this space (think of words as approximations of sequences, e.g. for reasons of computability), then some properties

• f the metric d are destroyed, notably

d(w, w) = 0 whenever w is a word. The relevant mathematical structure in this example (and others) is: A partial metric space is a set X together with a function p: X × X → R such that

◮ 0 ≤ p(x, x) ≤ p(x, y), ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z).

References: Matthews [1994], Bukatin et al [2009]

Consider the set of (infinite) sequences in some finite alphabet, say X = AN. It is a metric space if we set, for x = (xi)i and y = (yi)i in X, d(x, y) := 2−k for k the smallest index such that xk = yk. If we “add” the set A∗ of (finite) words in the alphabet A to this space (think of words as approximations of sequences, e.g. for reasons of computability), then some properties

• f the metric d are destroyed, notably

d(w, w) = 0 whenever w is a word. The relevant mathematical structure in this example (and others) is: A partial metric space is a set X together with a function p: X × X → R such that

◮ 0 ≤ p(x, x) ≤ p(x, y), ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z).

References: Matthews [1994], Bukatin et al [2009] So what is the categorical content of this definition?

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → R such that

◮ 0 ≤ p(x, x) ≤ p(x, y), ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z).

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → [0, ∞] such that

◮ 0 ≤ p(x, x) ≤ p(x, y), ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z).

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → [0, ∞] such that

◮ p(x, x) ≤ p(x, y) = +∞, ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z).

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → [0, ∞] such that

◮ p(x, x) ≤ p(x, y), ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z), ◮ p(x, y) = +∞.

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → [0, ∞] such that

◮ p(x, x) ≤ p(x, y), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z), ◮ if p(x, x) = p(x, y) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) = +∞.

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → [0, ∞] such that

◮ p(x, x) ≤ p(x, y), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z), ◮ if p(x, x) = p(x, y) = p(y, x) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) = +∞.

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → [0, ∞] such that

◮ p(x, x) ≤ p(x, y), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z), ◮ if p(x, x) = p(x, y) = p(y, x) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) = +∞.

The three last axioms express separatedness, symmetry and finiteness of p: X × X → [0, ∞]. We leave these aside for today—but rest assured: they can be expressed in a purely categorical manner.

Slightly rewrite the definition of partial metric: A partial metric space is a set X together with a function p: X × X → [0, ∞] such that

◮ p(x, x) ≤ p(x, y), ◮ p(x, y) + p(y, z) − p(y, y) ≥ p(x, z), ◮ if p(x, x) = p(x, y) = p(y, x) = p(y, y) then x = y, ◮ p(x, y) = p(y, x), ◮ p(x, y) = +∞.

The three last axioms express separatedness, symmetry and finiteness of p: X × X → [0, ∞]. We leave these aside for today—but rest assured: they can be expressed in a purely categorical manner. But what is the categorical content of the first two axioms, i.e. of generalised partial metric spaces?

Partial metric spaces done right: Construct the ordered category W with:

• objects: elements of [0, +∞],
• arrows: W(a, b) = {s ∈ [0, +∞] | s ≥ a ∨ b}, order inherited from ([0, +∞], ≥),
• composition of a

s b t c is a t+s−b c , identity on a is a a a .

Now setting

• P0 = X,
• P1(x) = p(x, x),
• P2(x, y) = p(x, y)

produces a W-enriched category from a generalised partial metric; and conversely, each such W-enriched category is exactly a partial metric. That is, W-enriched categories are precisely generalised partial metric spaces. Reference: Höhle and Kubiak [2011], Stubbe [2013]

• 3. Diagonals

For partial functions, from V = (P(X), ⊆, ∩, X) we built W as:

• objects: elements of P(X),
• homs: W(A, B) = {S ∈ P(X) | S ⊆ A ∩ B}, order inherited from (P(X), ⊆),
• composition of A

S B T C is A T ∩S C , identity on A is A A A .

For partial metrics, from V = ([0, +∞], ≥, +, 0) we built W as:

• objects: elements of [0, +∞],
• arrows: W(a, b) = {s ∈ [0, +∞] | s ≥ a ∨ b}, order inherited from ([0, +∞], ≥),
• composition of a

s b t c is a t+s−b c , identity on a is a a a .

The formal resemblance is not a coincidence: in both cases, the appropriate W is the universal “splitting of everything” in V , i.e. the “best possible way to turn every value into a type”...

From now on, we make some extra assumptions on the ordered monoid V : A quantale V = (V, , ◦, 1) is:

◮ a join-lattice (V, ), ◮ a monoid (V, ◦, 1), ◮ such that x ◦ − and − ◦ y distribute over arbitrary joins.

From now on, we make some extra assumptions on the ordered monoid V : A quantale V = (V, , ◦, 1) is:

◮ a join-lattice (V, ), ◮ a monoid (V, ◦, 1), ◮ such that x ◦ − and − ◦ y distribute over arbitrary joins.

Because x ◦ −: V → V preserves joins, it has a (meet-preserving) right adjoint x ց −: V → V . The adjointness is precisely expressed by: x ◦ y ≤ z ⇐ ⇒ y ≤ (x ց z).

From now on, we make some extra assumptions on the ordered monoid V : A quantale V = (V, , ◦, 1) is:

◮ a join-lattice (V, ), ◮ a monoid (V, ◦, 1), ◮ such that x ◦ − and − ◦ y distribute over arbitrary joins.

Because x ◦ −: V → V preserves joins, it has a (meet-preserving) right adjoint x ց −: V → V . The adjointness is precisely expressed by: x ◦ y ≤ z ⇐ ⇒ y ≤ (x ց z). Similarly, − ◦ y : V → V has a right adjoint − ւ y : V → V , characterised by: x ◦ y ≤ z ⇐ ⇒ x ≤ (z ւ y).

From now on, we make some extra assumptions on the ordered monoid V : A quantale V = (V, , ◦, 1) is:

◮ a join-lattice (V, ), ◮ a monoid (V, ◦, 1), ◮ such that x ◦ − and − ◦ y distribute over arbitrary joins.

Because x ◦ −: V → V preserves joins, it has a (meet-preserving) right adjoint x ց −: V → V . The adjointness is precisely expressed by: x ◦ y ≤ z ⇐ ⇒ y ≤ (x ց z). Similarly, − ◦ y : V → V has a right adjoint − ւ y : V → V , characterised by: x ◦ y ≤ z ⇐ ⇒ x ≤ (z ւ y). Reading x ◦ y as a “conjunction”, and x ց z and z ւ y as “implications”, we have in particular a “modus ponens” for each implication: x ◦ (x ց z) ≤ z and (z ւ y) ◦ y ≤ z.

From now on, we make some extra assumptions on the ordered monoid V : A quantale V = (V, , ◦, 1) is:

◮ a join-lattice (V, ), ◮ a monoid (V, ◦, 1), ◮ such that x ◦ − and − ◦ y distribute over arbitrary joins.

Because x ◦ −: V → V preserves joins, it has a (meet-preserving) right adjoint x ց −: V → V . The adjointness is precisely expressed by: x ◦ y ≤ z ⇐ ⇒ y ≤ (x ց z). Similarly, − ◦ y : V → V has a right adjoint − ւ y : V → V , characterised by: x ◦ y ≤ z ⇐ ⇒ x ≤ (z ւ y). Reading x ◦ y as a “conjunction”, and x ց z and z ւ y as “implications”, we have in particular a “modus ponens” for each implication: x ◦ (x ց z) ≤ z and (z ւ y) ◦ y ≤ z. (So the values in V are structured by a logic.)

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if ∃fa, fb ∈ V : fa ◦ a = f = b ◦ fb.

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if ∃fa, fb ∈ V : fa ◦ a = f = b ◦ fb.

a f b

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if ∃fa, fb ∈ V : fa ◦ a = f = b ◦ fb.

a f fb b fa

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f).

a f fb b fa

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f).

a f bցf b fւa

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f). Diagonals compose: given f : a b and g : b c,

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f). Diagonals compose: given f : a b and g : b c,

a f bցf b g cցg c fւa gւb

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f). Diagonals compose: given f : a b and g : b c, let g ◦b f : a c be

a f bցf b g cցg c fւa gւb

any of the (equal) paths “from top left to bottom right”: e.g. g ◦b f := g ◦ (b ց f).

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f). Diagonals compose: given f : a b and g : b c, let g ◦b f : a c be g ◦b f := g ◦ (b ց f). For this composition, the identity on a is a: a a itself:

a aցa a a aւa

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f). Diagonals compose: given f : a b and g : b c, let g ◦b f : a c be g ◦b f := g ◦ (b ց f). For this composition, the identity on a is a: a a itself. Ordering diagonals “as in V ”, this defines the ordered category D(V ) whose objects are the elements of V and whose arrows are the diagonals in V .

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f). Diagonals compose: given f : a b and g : b c, let g ◦b f : a c be g ◦b f := g ◦ (b ց f). For this composition, the identity on a is a: a a itself. Ordering diagonals “as in V ”, this defines the ordered category D(V ) whose objects are the elements of V and whose arrows are the diagonals in V . The construction V → D(V ) is much more widely applicable and has many interesting features—but we shall not go into details today.

For a, b, f in a quantale V , f is a diagonal from a to b, written f : a b, if (f ւ a) ◦ a = f = b ◦ (b ց f). Diagonals compose: given f : a b and g : b c, let g ◦b f : a c be g ◦b f := g ◦ (b ց f). For this composition, the identity on a is a: a a itself. Ordering diagonals “as in V ”, this defines the ordered category D(V ) whose objects are the elements of V and whose arrows are the diagonals in V . The construction V → D(V ) is much more widely applicable and has many interesting features—but we shall not go into details today. Note how every element of V becomes an object of D(V ): “every value becomes a type”.

For the (commutative) quantale V = (P(X), , ∩, X),

• the (unique) “implication” is S ⇒ T = Sc ∪ T,
• therefore A

S

B iff (S ⇐ A) ∩ A = S = B ∩ (B ⇒ S) iff S ⊆ A ∩ B,

• and for A

S

B

T

C, T ◦B S = T ∩ (B ⇒ S) = T ∩ (Bc ∪ S) = T ∩ S. For Lawvere’s (commutative) quantale of positive real numbers V = ([0, +∞], , +, 0),

• the (unique) “implication” is a ⇒ c = (c − a) ∨ 0,
• therefore a

s

b iff (s ⇐ a) + a = s = b + (b ⇒ s) iff s ∨ a = s = b ∨ s iff s ≥ a ∨ b,

• and for a

s

b

t

c, t ◦b s = t + (b ⇒ s) = t + ((s − b) ∨ 0) = t + s − b.

Further abstraction: A quantale V = (V, ≤, ◦, 1) is divisible if one (and thus all) of the following equivalent conditions holds:

• 1. 1 = ⊤ and ∀a, b ∈ V : a ◦ (a ց b) = a ∧ b = (b ւ a) ◦ a,
• 2. 1 = ⊤ and ∀b ≤ a ∈ V : a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 3. ∀a, b ∈ V we have b ≤ a iff a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 4. ∀a ∈ V : D(V )(a, a) = ↓ a,
• 5. ∀a, b ∈ V : D(V )(a, b) = ↓ (a ∧ b).

(The first four conditions make sense in the more general context of quantaloids.)

Further abstraction: A quantale V = (V, ≤, ◦, 1) is divisible if one (and thus all) of the following equivalent conditions holds:

• 1. 1 = ⊤ and ∀a, b ∈ V : a ◦ (a ց b) = a ∧ b = (b ւ a) ◦ a,
• 2. 1 = ⊤ and ∀b ≤ a ∈ V : a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 3. ∀a, b ∈ V we have b ≤ a iff a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 4. ∀a ∈ V : D(V )(a, a) = ↓ a,
• 5. ∀a, b ∈ V : D(V )(a, b) = ↓ (a ∧ b).

(The first four conditions make sense in the more general context of quantaloids.) Other examples of divisible quantales in multi-valued logic are:

• BL-algebras, and in particular BL-chains,
• continuous t-norms ([0, 1], ⋆, 1) (continuous and commutative multiplication).

Further abstraction: A quantale V = (V, ≤, ◦, 1) is divisible if one (and thus all) of the following equivalent conditions holds:

• 1. 1 = ⊤ and ∀a, b ∈ V : a ◦ (a ց b) = a ∧ b = (b ւ a) ◦ a,
• 2. 1 = ⊤ and ∀b ≤ a ∈ V : a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 3. ∀a, b ∈ V we have b ≤ a iff a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 4. ∀a ∈ V : D(V )(a, a) = ↓ a,
• 5. ∀a, b ∈ V : D(V )(a, b) = ↓ (a ∧ b).

(The first four conditions make sense in the more general context of quantaloids.) Other examples of divisible quantales in multi-valued logic are:

• BL-algebras, and in particular BL-chains,
• continuous t-norms ([0, 1], ⋆, 1) (continuous and commutative multiplication).

Divisible quantales (and quantaloids) have strong properties, which seem to be related to some form of continuity of the multiplication; this needs further study.

Further abstraction: A quantale V = (V, ≤, ◦, 1) is divisible if one (and thus all) of the following equivalent conditions holds:

• 1. 1 = ⊤ and ∀a, b ∈ V : a ◦ (a ց b) = a ∧ b = (b ւ a) ◦ a,
• 2. 1 = ⊤ and ∀b ≤ a ∈ V : a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 3. ∀a, b ∈ V we have b ≤ a iff a ◦ (a ց b) = b = (b ւ a) ◦ a,
• 4. ∀a ∈ V : D(V )(a, a) = ↓ a,
• 5. ∀a, b ∈ V : D(V )(a, b) = ↓ (a ∧ b).

(The first four conditions make sense in the more general context of quantaloids.) Other examples of divisible quantales in multi-valued logic are:

• BL-algebras, and in particular BL-chains,
• continuous t-norms ([0, 1], ⋆, 1) (continuous and commutative multiplication).

Divisible quantales (and quantaloids) have strong properties, which seem to be related to some form of continuity of the multiplication; this needs further study. References: Stubbe [2013]