Symmetric Containers

Transcription

1 Symmetric Containers Representing functors on groupoids Håkon Robbestad Gylterud May 2013 Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 1 / 14

2 Context Based on the work by Thorsten Altenkirch and others on containers and differentiation of containers. Generalised notions of containers. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 2 / 14

3 The groupoid interpretation of type theory Martin Hofmann and Thomas Streicher (1996) We may give the following interpretation to Martin-Löf type theory: Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 3 / 14

4 The groupoid interpretation of type theory Martin Hofmann and Thomas Streicher (1996) We may give the following interpretation to Martin-Löf type theory: Types is interpreted as groupoids. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 3 / 14

5 The groupoid interpretation of type theory Martin Hofmann and Thomas Streicher (1996) We may give the following interpretation to Martin-Löf type theory: Types is interpreted as groupoids. Terms are objects of the groupoids. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 3 / 14

6 The groupoid interpretation of type theory Martin Hofmann and Thomas Streicher (1996) We may give the following interpretation to Martin-Löf type theory: Types is interpreted as groupoids. Terms are objects of the groupoids. More generally depentendent types are modelled by families of groupoids. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 3 / 14

7 The groupoid interpretation of type theory Martin Hofmann and Thomas Streicher (1996) We may give the following interpretation to Martin-Löf type theory: Types is interpreted as groupoids. Terms are objects of the groupoids. More generally depentendent types are modelled by families of groupoids. A family of groupoids is a pair (G, F), where G : Grpd and F : G Grpd is a functor. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 3 / 14

8 Examples Examples Since our types now may be groupoids (or families of groupoids) let of give some examples what these might be: Sets are discrete groupoids. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 4 / 14

9 Examples Examples Since our types now may be groupoids (or families of groupoids) let of give some examples what these might be: Sets are discrete groupoids. Setoids are groupoids. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 4 / 14

10 Examples Examples Since our types now may be groupoids (or families of groupoids) let of give some examples what these might be: Sets are discrete groupoids. Setoids are groupoids. Groups are groupoids. In particular finite groups such as The cyclic groups Z n for n N +. The permutation groups S n for n N. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 4 / 14

11 Examples Examples Since our types now may be groupoids (or families of groupoids) let of give some examples what these might be: Sets are discrete groupoids. Setoids are groupoids. Groups are groupoids. In particular finite groups such as The cyclic groups Z n for n N +. The permutation groups S n for n N. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 4 / 14

12 Examples Examples (cont.) We may take disjoint unions of groups. For instance Z = n N Z n S = n N S n Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 5 / 14

13 Examples Examples (cont.) We may take disjoint unions of groups. For instance Z = n N Z n S = n N S n We are especially interested in families of groupoids where the fibres are (decidable) sets (discrete groupoids). Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 5 / 14

14 Examples Examples (cont.) We may take disjoint unions of groups. For instance Z = n N Z n S = n N S n We are especially interested in families of groupoids where the fibres are (decidable) sets (discrete groupoids). Examples of such are C : Z Grpd, where each object n : Ob(Z ) is mapped to the set Fin(n) = {0, 1,, n 1}, and the arrows (i.e. the elements of the group Z n ) act on Fin(n) by additon. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 5 / 14

15 Examples Examples (cont.) We may take disjoint unions of groups. For instance Z = n N Z n S = n N S n We are especially interested in families of groupoids where the fibres are (decidable) sets (discrete groupoids). Examples of such are C : Z Grpd, where each object n : Ob(Z ) is mapped to the set Fin(n) = {0, 1,, n 1}, and the arrows (i.e. the elements of the group Z n ) act on Fin(n) by additon. P : S Grpd, where each object n : Ob(S ) is mapped to the set Fin(n) = {0, 1,, n 1}, and the arrows (i.e. permutations) act on Fin(n) by permutation. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 5 / 14

16 Definition of symmetric containers Definition A symmetric container, (S P), is a groupoid S together with a functor P : S Set. I.e. a family of groupoid where the fibres are discrete. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 6 / 14

17 Definition of symmetric containers Definition A symmetric container, (S P), is a groupoid S together with a functor P : S Set. I.e. a family of groupoid where the fibres are discrete. We call S the groupoid of shapes, and for each shape s : Ob(S) we call P(s) the set of positions in s. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 6 / 14

18 Definition of symmetric containers Definition A symmetric container, (S P), is a groupoid S together with a functor P : S Set. I.e. a family of groupoid where the fibres are discrete. We call S the groupoid of shapes, and for each shape s : Ob(S) we call P(s) the set of positions in s. Definition A morphism of symmetric containers, (f, σ) : (S P) (T Q), is a functor f : S T together with a natural transformation σ : Q f P. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 6 / 14

19 Definition of symmetric containers Definition A symmetric container, (S P), is a groupoid S together with a functor P : S Set. I.e. a family of groupoid where the fibres are discrete. We call S the groupoid of shapes, and for each shape s : Ob(S) we call P(s) the set of positions in s. Definition A morphism of symmetric containers, (f, σ) : (S P) (T Q), is a functor f : S T together with a natural transformation σ : Q f P. Definition A 2-morphism of symmetric containers, ɛ : (f, σ) (g, τ), is a natural transformation ɛ : f g such that τ Qɛ = σ. We assign the name SCon to the 2-category of symmetric containers, their morphism and 2-morphisms. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 6 / 14

20 Representation theorem Representing functors We think about symmetric containers as data structures with some symmetry actions which has to be respected. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 7 / 14

21 Representation theorem Representing functors We think about symmetric containers as data structures with some symmetry actions which has to be respected. Let (S P) : SCon. Given a groupoid (or a set) X, we can construct a groupoid, S P (X ), of ways to select a shape s S, and fill the positions P(s) with objects from X. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 7 / 14

22 Representation theorem Representing functors We think about symmetric containers as data structures with some symmetry actions which has to be respected. Let (S P) : SCon. Given a groupoid (or a set) X, we can construct a groupoid, S P (X ), of ways to select a shape s S, and fill the positions P(s) with objects from X.That is: Ob( S P (X )) = X P(s) (1) s S Mor S P (X ) ((s, φ), (t, ψ)) = (φ ψp(m)) (2) m:s t Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 7 / 14

23 Represenation theorem Theorem Let Grpd Grpd be the 2-category of 2-functors from Grpd to itself. There is a full and faithful strict 2-functor : SCon Grpd Grpd. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 8 / 14

24 Properties of symmetric containers The 2-category of symmetric containers is closed under (and respects): Finite products and co-products. Composition. That is, given (S P) and (T Q) there is a symmetric container (S P) (T Q) such that (S P) (T Q) = S P T Q Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 9 / 14

25 Differentiation On containers (in the sense of Altenkirch et al) there is a differentiation operation, satisfying leibniz rule, the chain rule and distribution over sums. This defintion extends to a differentiation operation on symmetric containers. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 10 / 14

26 Differentiation On containers (in the sense of Altenkirch et al) there is a differentiation operation, satisfying leibniz rule, the chain rule and distribution over sums. This defintion extends to a differentiation operation on symmetric containers. This structure allow to use techniques such as Taylor-series to analyse containers, and similar techniques can be used on symmetric containers. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 10 / 14

27 Differentiation, definition Definition Given a symmetric container (S P) where the sets of positions are all decidable, we may define the derivative symmetric container (S P) = (S P ) as follows: Ob(S ) = P(s) s S Mor S ((s, p), (t, q)) = P(m)(p) = q m:s t Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 11 / 14

28 Differentiation, definition Definition Given a symmetric container (S P) where the sets of positions are all decidable, we may define the derivative symmetric container (S P) = (S P ) as follows: Ob(S ) = P(s) s S Mor S ((s, p), (t, q)) = P(m)(p) = q m:s t P(s, p) = P(s) p P(m) = P(m) p Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 11 / 14

29 Properties of differentiation Let F, G : SCon be symmetric containers, then (F ) = 0 whenever F represents a constant functor. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 12 / 14

30 Properties of differentiation Let F, G : SCon be symmetric containers, then (F ) = 0 whenever F represents a constant functor. (F + G) = F + G. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 12 / 14

31 Properties of differentiation Let F, G : SCon be symmetric containers, then (F ) = 0 whenever F represents a constant functor. (F + G) = F + G. (F G) = F G + F G. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 12 / 14

32 Properties of differentiation Let F, G : SCon be symmetric containers, then (F ) = 0 whenever F represents a constant functor. (F + G) = F + G. (F G) = F G + F G. (F G) = ( F G) G. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 12 / 14

33 Properties of differentiation Let F, G : SCon be symmetric containers, then (F ) = 0 whenever F represents a constant functor. (F + G) = F + G. (F G) = F G + F G. (F G) = ( F G) G. Theorem All containers with discrete set of shapes and finite, decidable sets of positions, have an anti-derivative. That is, if F : SCon satisfy the hypothesis, then there is some G such that G F. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 12 / 14

34 Current project Formalise definitions in Agda using E-categories and E-bicategories. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 13 / 14

35 Current project Formalise definitions in Agda using E-categories and E-bicategories. Investigate fixed points of functors represented by symmetric containers. Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 13 / 14

36 Thank you all for listening! Håkon Robbestad Gylterud () Symmetric Containers Stockholm University 14 / 14