Nominal Sets, Data Symmetries and OrdNom Hans-Peter Deifel Oberseminar 21.01.2020 1 / 43
Motivation There’s more than one way to skin a category. Abstract syntax with names and name binding “Slightly infjnite” structures Data accessible via a limited interface 2 / 43
Motivation Abstract Syntax: Abstract Syntax Trees Free Variables Name Dependence Binders 3 / 43 α -equivalence
Motivation “Slightly Infjnite” sets: Under consistent bijective renaming: 4 / 43 ( x, y ) ∈ A × A Only two cases: x = y and x � = y “Generated” by: ( a, a ) and a, b
5 / 43 [ G, Set ]
Groups Defjnition A group G is a set equipped with: such that 6 / 43 [ G, Set ] A composition function G · G → G A unit element e ∈ G For each element a ∈ G , an inverse a − 1 ∈ G (unit): e · a = a = a · e for all a ∈ G (assoc): a · ( b · c ) = ( a · b ) · c for all a, b, c ∈ G (invers): a · a − 1 = e for all a ∈ G ⇒ Also: a category with one object!
Example Groups 7 / 43 [ G, Set ] Sym ( X ) : All bijections on X
Group Actions Group Action 8 / 43 [ G, Set ] Action of G on set X is function · : G × X → X , s.t.: a · ( b · x ) = ( a · b ) · x e · x = x G -Sets Sets equipped with action of G called G -Set . G -Sets form a category [ G, Set ] morphisms: Equivariant functions f : A → B , s.t. f ( π · x ) = π · f ( x )
Group Actions Example 9 / 43 [ G, Set ] Σ signature, X variables. Σ -terms over X . Defjne group action of Sym ( X ) recursively on term t : π · x = π ( x ) π · op ( t 1 , . . . , t n ) = op ( π · t 1 , . . . , π · t n )
Constructions Products with action: Coproducts with action: 10 / 43 [ G, Set ] X 1 × · · · × X n Product of G -Sets X 1 , . . . , X n : π · ( x 1 , . . . , x n ) = ( π · x 1 , . . . , π · x n ) Product of G -Sets X 1 , . . . , X n : X 1 + · · · + X n π · inj i x = inj i ( π · x ) ⋔ : Extends to infjnite products/coproducts.
Constructions Exponentials and are equivariant. 11 / 43 [ G, Set ] Given G -sets X , Y : Defjne G -action on set of all functions Y X : ( π · f )( x ) = π · f ( π − 1 · x ) ⋔ app : Y X × X → Y curry ( f : X × Y → Z ): X → Z Y ⋔ : [ G, Set ] is CCC!
Constructions Subobjects Subobject Classifjer 12 / 43 [ G, Set ] A subset S of a G -Set X is an equivariant subset if: π · S = S ∀ π ∈ G Subobjects of G -Set X ⇔ equivariant subsets of X . B = { true , false } = 1 + 1 is subobject classifjer in [ G, Set ] . ⋔ : [ G, Set ] is Boolean topos!
Nominal Sets The Original 13 / 43
Nominal Sets Names Names Abstract “atomic” names Can be compared for equality No further structure 14 / 43 Fix an infjnite set A of names { a, b, c, . . . } . ⋔ Intuition: Variables
Nominal Sets Permutations Names Permutations 15 / 43 Fix an infjnite set A of names { a, b, c, . . . } . Fix the group G = Perm A of fjnite permutations of A . Note: A is itself a Perm ( A ) -Set with action π · a = π ( a ) .
Nominal Sets Name Dependence Support Nominal Set support. 16 / 43 Let X be Perm A -set. A set of names A ⊆ A is a support for x ∈ X if ∀ π ∈ Perm A : ( ∀ a ∈ A. π · a = a ) ⇒ π · x = x A nomial set X is a Perm A -set, s.t. every x ∈ X has fjnite
Nominal Sets Examples Discrete Nominal Set 17 / 43 The set of names A ∀ a ∈ A . supp a = { a } Set X with Perm A -action π · x = x . ∀ x ∈ X. supp x = ∅ Σ -Terms ∀ t ∈ Σ[ A ] . supp t = freevars ( t )
Nominal Sets Least support Least Support 18 / 43 A 1 , A 2 fjnite supports for x ∈ X ⇒ so is A 1 ∩ A 2 X nominal set and x ∈ X . Then: ∩ supp X x = { A ∈ P A | A is fjnite support for x } ⋔ -fact: supp is an equivariant function!
Nominal Sets Nom, Nom, Nom… Nom Nominal sets objects in category: 19 / 43 Morphisms: Equivariant functions, as in [ Perm A , Set ] ⋔ : Full subcategory of [ Perm A , Set ]
Nominal Sets Constructions Coproducts & Finite Products Infjnite Products 20 / 43 Same as in [ Perm A , Set ] supp ( x 1 , . . . , x n ) = supp x 1 ∪ · · · ∪ supp x n supp ( inj i x ) = supp x ∪ i ∈ I supp x i may not be fjnite for inifjnite I But: Nom is co-refmective subcategory of [ Perm A , Set ]
Nominal Sets Products Infjnite Products where 21 / 43 ∪ i ∈ I supp x i may not be fjnite for inifjnite I But: Nom is co-refmective subcategory of [ Perm A , Set ] For each [ Perm A , Set ] set X , there is X fs → X , s.t. Y f ¯ f X fs X X fs = { x ∈ X | x is fjnitely supported in X }
Nominal Sets Topos?! Exponentials Subobjects Subobject Classifjer 22 / 43 Same thing: ( Y X ) fs exponential in Nom Same as in [ Perm A , Set ] Same as in [ Perm A , Set ] ⋔ : Boolean Topos!
Data Symmetries More Polish! 23 / 43
Data Symmetries Data Symmetry Intuition: only accessible by limited interface 24 / 43 ( D , G ) D a set of data G ≤ Sym ( D ) D data with some limited structure or
Data Symmetries Examples i.e. the thing from the last few slides. i.e. the thing from the few last slides. 25 / 43 Equality Symmetry ( A , Perm A ) In polish notation: A = N Total Order Symmetry ( Q , Mob ( Q )) Mob ( Q ) = { π ∈ Sym ( Q ) | x < y ⇒ π · x < π · y } Integer Symmetry ( Z , Trans ( Z )) Trans ( Z ) group of translations i �→ i + c
Data Symmetries Orbits Orbit 26 / 43 For x in G -set X , G · x = { π · x | x ∈ G } ⊆ X is called orbit of x . ⋔ -fact: Any G -Set partitionend into its orbits.
Data Symmetries Orbits Orbit Example Equality Symmetry: Two orbits: Total Order Symmetry: Three orbits: 27 / 43 G · x = { π · x | x ∈ G } ⊆ X Consider D 2 : { ( a, a ) | a ∈ A } { ( a, b ) | a � = b ∈ A } { ( x, y ) | x < y ∈ Q } { ( x, x ) | x ∈ Q } { ( x, y ) | x > y ∈ Q }
Data Symmetries Support Support, nominal set, equivariant function defjned as in Nom. Least Support Equality Symmetry: Yes (see previous slides) Total Order Symmetry: Yes 28 / 43 Does a fjnitely supported x ∈ X have a least support? ( N × N , { π × π | π ∈ Sym ( N ) } ) : Nope! See next slide.
Data Symmetries Support 29 / 43 Data symmetry: ( N × N , { π × π | π ∈ Sym ( N ) } ) . The element (0 , 1) ∈ N × N has three minimal supports: { (0 , 1) } { (1 , 0) } { (0 , 0) , (1 , 1) } ⋔ : But no least support
Representations 30 / 43
Representations Single Orbits Coset 31 / 43 Given a subgroup H ≤ G , π ∈ G : πH = { πσ | σ ∈ H } ⊆ G ⋔ -fact: G partitioned into cosets for H .
Representations Single Orbits Coset Space 32 / 43 G H = { πH | π ∈ G } ⧸ G -action on coset space: σ · ( πH ) = ( σπ ) H ⋔ -fact: G ⧸ H is single-orbit G -Set
Representations Single Orbits Stabilizer 33 / 43 Given G -set X , x ∈ X : G x = { π ∈ G | π · x = x } ≤ G X ∼ = G ⧸ G x Any single orbit G -set X is isomorphic to G G x for any x ∈ X . ⧸
Representations Single Orbits Bijective Well defjned Iso 34 / 43 X ∼ = G ⧸ G x Any single orbit G -set X is isomorphic to G G x for any x ∈ X . ⧸ f : X → G ⧸ G x f ( π · x ) = π · G x
Representations Single Orbits Win? 34 / 43 X ∼ = G ⧸ G x Any single orbit G -set X is isomorphic to G G x for any x ∈ X . ⧸ Every G ⧸ H single orbit G -set Every G -set equivalent to a G ⧸ H Every G -set partition of its orbits.
Representations Single Orbits Win? Nope, there are too many of them! 34 / 43 X ∼ = G ⧸ G x Any single orbit G -set X is isomorphic to G G x for any x ∈ X . ⧸ Every G ⧸ H single orbit G -set Every G -set equivalent to a G ⧸ H Every G -set partition of its orbits.
Representations Single Orbits What are we going to do? 35 / 43
Representations Support Stabilizer again Open Subgroup def Support 36 / 43 Idea: Restrict to nominal G -sets: G C = { π ∈ G | π · c = c ∀ c ∈ C } H ≤ G open ⇔ G C ≤ H for fjnite C ∈ D H ≤ G open with G C ≤ H ⇒ C supports H
Representations Support 37 / 43 (1-orbit) Nominal Sets ⇔ Open Subgroups
Representations Finite Subgroups Problem: Open subgroups are unwieldy Extension 38 / 43 Solution: Assume ( D , G ) admits least support and is fungible ( 🍅 ) ext G ( S ) = { π ∈ G | π | C ∈ S } ≤ G for S ≤ G | C C supports ext G ( S ) ⇒ ext G ( S ) is open subgroup For open subgroup H ≤ G with support C : H = ext G ( H | C )
Representations Support 39 / 43 (1-orbit) Nominal Sets ⇔ ( C, S ) where S ≤ Sym ( C )
Total Order! 40 / 43
Total Order Fun Facts Finite Monotone Permutations The only monotone permutation on a fjnite set is Id Homogeneity 41 / 43 Reminder: ( Q , Mob ( Q )) For fjnite C ⊆ Q , D ⊆ Q : | C | = | D | ⇒ ∃ π. π · C = D
Total Order Representation Representation 1 Representation 2 42 / 43 1-orbit nominal set X with x ∈ X supported by C : X ∼ = P | C | ( Q ) 1-orbit nominal set X with x ∈ X supported by C : X fully described by | C |
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