[ ] A Basic Nominal Techniques Bartek Klin University of Warsaw - PowerPoint PPT Presentation

FoPSS 2019 [ ] A Basic Nominal Techniques Bartek Klin University of Warsaw Warsaw, 10-11 September, 2019

[ ] A Alternative Formulations

Finite vs. arbitrary atom renamings Let be the group of finite bijections on . Perm( A ) A (i.e. such that for all but finitely many ) π ( a ) = a a Perm( A ) canonically acts on the universe , U and the definition of support may be repeated. Fact: whether we use or , Aut( A ) Perm( A ) the same sets are legal and they have the same finite supports. NB. Not so easy to prove! Essentially a topological argument. FoPSS, Warsaw, 10-11/09/19 3

Categories Legal nominal sets and finitely supported functions form a category. A category : C - a collection of objects |C| - for each , a set of morphisms C ( X, Y ) X, Y ∈ |C| - composition operations: : C ( Y, Z ) ⇥ C ( X, Y ) ! C ( X, Z ) � - identity morphisms: id X ∈ C ( X, X ) + axioms Another category: equivariant sets and functions. Nom FoPSS, Warsaw, 10-11/09/19 4

Continuous -sets G For an equivariant set , atom renaming acts on : X X : X × Aut( A ) → X · Fact: for each there is a finite x ∈ X S ⊆ A s.t. for every π , σ ∈ Aut( A ) we know this! if then . π | S = σ | S x · π = x · σ In other words: is a continuous group action · ( discrete, with product topology) Aut( A ) X Nom ≈ continuous -sets Aut( A ) with equivariant functions between them FoPSS, Warsaw, 10-11/09/19 5

Sheaves Fix an equivariant set . X For a finite , define: S ⊆ A ˆ X ( S ) = { x ∈ X | supp( x ) ⊆ S } ⊆ X For an injective function : f : S → T ⊆ A - pick any that extends π ∈ Aut( A ) f X ( f ) : ˆ ˆ X ( S ) → ˆ - define by: X ( T ) ˆ X ( f )( x ) = x · π we know this! X ( f )( x ) ∈ ˆ ˆ Fact: X ( T ) ˆ Fact: does not depend on the choice of X ( f ) π FoPSS, Warsaw, 10-11/09/19 6

Sheaves We have just shown that is a functor: ˆ X ˆ X : I → Set I : the category of finite subsets of A sets and injective functions This extends to a correspondence between equivariant functions and natural transformations! But: not all functors from to arise in this way. Set I Sheaves do. sheaves on and natural transformations Nom ≈ I FoPSS, Warsaw, 10-11/09/19 7

[ ] A Orbit Finite Sets

An example problem revisited - nodes: a 6 = b ab - edges: a 6 = c ab bc ab ad bc be ca cd db de ea ec Is 3-colorability decidable? FoPSS, Warsaw, 10-11/09/19 9

Orbits The orbit of is the set { x · π | π ∈ Aut( A ) } x Every equivariant set is a disjoint union of orbits. Orbit-finite set if the union is finite. More generally: the -orbit of is S x { x · π | π ∈ Aut S ( A ) } Fact: An orbit-finite set is -orbit-finite S for every finite . S FoPSS, Warsaw, 10-11/09/19 10

Examples Orbit-finite sets: ✓ A ◆ A A n n A / = {{ ( a, b, c ) , ( b, c, a ) , ( c, a, b ) } | a, b, c ∈ A } - closed under finite union, intersection difference, finite Cartesian product - but not under (even finite) powerset! Not orbit-finite: P fin ( A ) A ∗ FoPSS, Warsaw, 10-11/09/19 11

Group representation ✓ A ◆ Some single-orbit sets: A ( n ) A / n More generally, for and G ≤ Sym( n ) n ∈ N A ( n ) define an equiv. relation on : ∼ G ( a 1 , . . . , a n ) ∼ G ( a σ (1) , . . . , a σ ( n ) ) for . σ ∈ G A ( n ) / ∼ G Fact: is an equivariant, single-orbit set. Theorem: Every equivariant, single-orbit set is in equivariant bijection with one of this form. FoPSS, Warsaw, 10-11/09/19 12

Example Remember the graph? - nodes: { | a 6 = b 2 A } ab n = 2 G = 1 - edges: { | a 6 = c 2 A } ab bc n = 3 G = 1 Problems: - not well suited for modular representation - inefficient: has exponentially many orbits A n FoPSS, Warsaw, 10-11/09/19 13

Example ctd. Still the same puzzle: - nodes: { | a 6 = b 2 A } ab - edges: { | a 6 = c 2 A } ab bc This is a reasonable finite presentation already! We keep writing down finite descriptions of infinite sets all the time. Let’s make that formal. FoPSS, Warsaw, 10-11/09/19 14

Logical presentation A set-builder expression: { e | a 1 , . . . , a n ∈ A , φ [ a 1 , . . . , a n , b 1 , . . . , b m ] } expression bound variables free variables FO( )-formula = Add also and . ∅ ∪ Fact: s.-b. e. + interpretation of free vars. as atoms = a hereditarily orbit-finite set with atoms Fact: Every h. o.-f. set is of this form. FoPSS, Warsaw, 10-11/09/19 15

Examples The graph: G = ( V, E ) V = { ( a, b ) | a, b 2 A , a 6 = b } E = {{ ( a, b ) , ( b, c ) } | a, b, c 2 A , a 6 = b 6 = c } (encode pairs with standard set-theoretic trickery) Descriptions like this can be input to algorithms, for example: Is 3-colorability of orbit-finite graphs decidable? FoPSS, Warsaw, 10-11/09/19 16

Exercises. Prove that: 1. An equivariant orbit-finite set has only finitely many equivariant subsets. 2. For equivariant, orbit-finite sets and , X Y there are finitely many equivariant functions from to . X Y 3. If is orbit-finite then every S ⊆ fin A X supports only finitely many elements of . X 4. The converse implication to 3. does not hold. FoPSS, Warsaw, 10-15/09/19 17

Set theory with atoms Nominal sets form a topos A lot of mathematics can be done with atoms set nominal set finite orbit-finite function equivariant function EXCEPT: - axiom of choice fails, even orbit-finite choice - powerset does not preserve orbit-finiteness FoPSS, Warsaw, 10-11/09/19 18

Slogans X = set, function, relation, automaton, Turing machine, grammar, graph, system of equations... Nominal X Infinite but with lots of symmetries orbit-finite Infinite but symbolically finitely presentable We can compute on them FoPSS, Warsaw, 10-11/09/19 19