Categorical semantics of metric spaces and continuous logic Simon - PowerPoint PPT Presentation

Categorical semantics of metric spaces and continuous logic Simon Cho CT 2019, University of Edinburgh July 12, 2019 Simon Cho (UMich) Continuous semantics July 12, 2019 1 / 25

Outline Motivation & perspective Brief review of continuous logic An examination of the category of metric spaces Reverse engineering a “continuous semantics” A continuous subobject classifier in the style of Barr Categories of presheaves of metric spaces Simon Cho (UMich) Continuous semantics July 12, 2019 2 / 25

Motivation & perspective Lawvere noticed that metric spaces are categories enriched over R ≥ 0 = (0 ← · · · ← r ← · · · ← ∞ ) with tensor given by addition. X ( a , b ) is “the distance from a to b ”, and the condition X ( a , b ) + X ( b , c ) − → X ( a , c ) is the triangle inequality. X ( a , b ) expresses degree of truth of the equality predicate on a and b 0 is “true” and ∞ is “false” The triangle inequality is the transitivity of equality We work with I = (0 ← · · · ← r ← · · · ← 1) with truncated addition. Simon Cho (UMich) Continuous semantics July 12, 2019 3 / 25

Motivation & perspective On the one hand, continuous logic is a relatively new [0 , 1]-valued first order logic important to the model theory community. One reason for this is that surprisingly many of the (non-continuous) model theoretic notions have sensible continuous analogues. On the other hand, categorical semantics has been extremely successful at analyzing the logical structure of categories. Simon Cho (UMich) Continuous semantics July 12, 2019 4 / 25

Brief review of continuous logic Continuous logic is the same as the usual first order logic, except: Sorts are interpreted as metric spaces (as opposed to as sets) with diameter ≤ 1. Function & predicate symbols come with a specified modulus of uniform continuity; their interpretations must obey the modulus Predicates are interpreted as uniformly continuous maps X → [0 , 1] (as opposed to as set functions X → { 0 , 1 } ) The distance function on a space X plays the role of the equality predicate Universal/existential quantification is sup/inf So the interpretation of the syntax of continuous logic takes place in the category Met whose objects are metric spaces of diameter ≤ 1, and whose morphisms are uniformly continuous maps. We work in the category pMet , which allows pseudo metric spaces. Simon Cho (UMich) Continuous semantics July 12, 2019 5 / 25

The category of metric spaces Categorical semantics informs us what structures are required of a given category in order to support various fragments of logic. To get all of first order logic, sufficient to require the category be geometric has finite limits has images which are stable under pullback for each object X , Sub m X is small-complete lattice with structure preserved by pullback interpret predicates on X as subobjects of X Example: Set Simon Cho (UMich) Continuous semantics July 12, 2019 6 / 25

The category of metric spaces There is a variant using regular monos instead of monos: we require that the category has finite limits has regular images which are stable under pullback composing two regular monos yields a regular mono for each X , the lattice Sub X of regular subobjects is small-complete, with structure preserved by pullback interpret predicates on X as regular subobjects of X Example: pMet Simon Cho (UMich) Continuous semantics July 12, 2019 7 / 25

Barr’s equivalence Barr established the following equivalence of categories, given a locale L : Fuz( L ) ≃ Mon( L + ) Fuz( L ) is the category whose objects are set functions ξ : X → | L | and morphisms are (noncommutative) triangles f X Y η ξ | L | for which f ◦ η ≤ L op ξ Mon( L + ) is the category of sheaves of monos ∗ on L + , where L + = L ∪ { i } and the topology is the logic topology Simon Cho (UMich) Continuous semantics July 12, 2019 8 / 25

Barr’s equivalence Fuz( L ) ≃ Mon( L + ) The actual maps are given by �→ ( r �→ { x | ξ ( x ) ≤ L op r } ) ξ inf { r ∈ L op | x ∈ R ( r ) } ) ( x �→ ← � R R ∈ Mon( L + ) is (up to iso) a meet-preserving functor R : L op → Sub( R tot ) � where R tot = r ∈ L op R ( r ). The slogan is: A function on X valued in | L | is equivalent to the data of meet-preserving ( L + ) op -indexed sublevelsets of X . Simon Cho (UMich) Continuous semantics July 12, 2019 9 / 25

Indexed subobjects As a trivial special case of Barr’s equivalence. Write ✷ = (0 ← 1) and | ✷ | = { 0 , 1 } . In Set , a predicate R on X is: Function χ R : X − → | ✷ | Functor R : ✷ op − → Sub X where R (0) = { x ∈ X | χ R ( x ) ≤ ✷ op 0 } R (1) = { x ∈ X | χ R ( x ) ≤ ✷ op 1 } = X , i.e. a meet-preserving functor R : ✷ op → Sub X So the subobject classifier in Set is just a Barr-style equivalence between functions into classical truth values and functors of subobjects on classical truth values. Simon Cho (UMich) Continuous semantics July 12, 2019 10 / 25

Indexed subobjects Recall ■ = 0 ← · · · ← r ← · · · ← 1, and write | ■ | = [0 , 1]. In continuous logic, predicates on X ∈ pMet are uniformly continuous maps X → | I | . These should correspond to appropriate functors I op → Sub X . Given f : X → | I | , should look at functor R f : I op → Sub X defined by R f ( r ) = { x ∈ X | f ( x ) ≤ I op r } Continuity of f should translate into some property of R f ... Simon Cho (UMich) Continuous semantics July 12, 2019 11 / 25

Formalizing the metric Given X ∈ pMet with metric d X , have D X : I op → Sub( X × X ) defined by D X ( r ) = { ( x , y ) ∈ X × X | d X ( x , y ) ≤ I op r } Proposition There is a choice of distinguished D X : I op → Sub( X × X ) for each X ∈ pMet , as well as a choice of product X × Y for each X , Y ∈ pMet , such that the following hold: Simon Cho (UMich) Continuous semantics July 12, 2019 12 / 25

Formalizing the metric D X (0) contains the diagonal; ∼ = The symmetry iso X × X − → X × X takes D X to itself; Letting π i , j : ( X × X × X ) → ( X × X ) denote the projection onto i th and j th factors respectively, π ∗ i , j D X ( r ) ∧ π ∗ j , k D X ( s ) ≤ π ∗ i , k D X ( r + s ) i r i for r , r i ∈ I op , then D X ( r ) = � Letting r = inf D X ( r i ). i Let π X × X : ( X × Y × X × Y ) → ( X × X ) and π Y × Y : ( X × Y × X × Y ) → ( Y × Y ) denote the projections preserving the ordering of the factors. Then D X × Y ( r ) = ( π X × X ) ∗ D X ( r ) ∧ ( π Y × Y ) ∗ D Y ( r ) Simon Cho (UMich) Continuous semantics July 12, 2019 13 / 25

Formalizing continuity An inf- and 0-preserving increasing function ǫ : [0 , 1] → [0 , 1] is a modulus of continuity for f : X → Y when for all r ∈ [0 , 1] d X ( a , b ) ≤ r = ⇒ d Y ( f ( a ) , f ( b )) ≤ ǫ ( r ) . Translating into our setting, we say an inf- and 0-preserving functor ǫ : I op → I op is a modulus of continuity for f : X → Y when for all r ∈ I op D X ( r ) ≤ ( f × f ) ∗ D Y ( ǫ ( r )) . Let E ⊆ End( I op ) be the submonoid (under composition) of all such ǫ . Can vary E to allow only Lipschitz or 1-Lipschitz maps Important real analysis properties of pMet follow categorically from our formulation. Simon Cho (UMich) Continuous semantics July 12, 2019 14 / 25

Maps into [0 , 1] Consider | I | = [0 , 1] with the obvious metric. Define T I : I op → Sub | I | by T I ( r ) = [0 , r ] . Lemma A map f : X → | I | is continuous w.r.t. ǫ ∈ E iff for all r , s ∈ I op , we have ( π 1 ) ∗ f ∗ T I ( r ) ∧ D X ( s ) ≤ ( π 2 ) ∗ f ∗ T I ( r + ǫ ( s )) Simon Cho (UMich) Continuous semantics July 12, 2019 15 / 25

Continuous predicates Definition Given X ∈ pMet and ǫ ∈ E , call R : I op → Sub X an ǫ -predicate on X when i r i in I op , R ( r ) = � For r = inf R ( r i ) i For all r , s ∈ I op , ( π 1 ) ∗ R ( r ) ∧ D X ( s ) ≤ ( π 2 ) ∗ R ( r + ǫ ( s )) Write Sub ǫ X ⊆ [ I op , Sub X ] for the full subcategory on ǫ -predicates on X . Proposition For f : X → Y with modulus ǫ f , and R ∈ Sub ǫ Y , we have that f ∗ R ∈ [ I op , Sub X ] is an ( ǫ ◦ ǫ f )-predicate. Simon Cho (UMich) Continuous semantics July 12, 2019 16 / 25

Continuous predicate classifier Recall T I : I op → Sub | I | is defined as T I ( r ) = [0 , r ] so clearly T I ∈ Sub 1 I op | I | . Theorem Given f : X → | I | with modulus ǫ ∈ E , R f := f ∗ T I is an ǫ -predicate on X . Given R ∈ Sub ǫ X , the function f R : X → | I | defined by f R ( x ) = inf { r ∈ I op | x ∈ R ( r ) } is a uniformly continuous map with modulus ǫ ∈ E . These operations are inverse to each other, and natural in X . Simon Cho (UMich) Continuous semantics July 12, 2019 17 / 25

Sanity check For any geometric category C (e.g. Set ), we have ✷ -valued “metrics”: given any X ∈ C , set D X (0) = diagonal D X (1) = X × X Also let { 1 ✷ op } = E ⊆ End( ✷ op ) Then a “continuous predicate” on X is exactly just a subobject of X . If we have some Ω ∈ C and some given T ✷ ∈ Sub Ω, the analogous statement of the previous theorem (with Ω in place of | I | ) precisely means that Ω is a subobject classifier. Simon Cho (UMich) Continuous semantics July 12, 2019 18 / 25