Towards a directed HoTT based on 4 kinds of variance
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1 Towards a directed HoTT based on 4 kinds of variance Promoter: Frank Piessens Supervisors: Jesper Cockx, Dominique Devriese Dept. Computer Science, KU Leuven June 29, 2015
2 Table of Contents 1 Introduction 2 Foundations 3 Inductive type families 4 The mystery of invariance, zigzags and relations
3 Directed HoTT: What and why? HoTT -groupoids Homotopy Theory type A -groupoid A space A element a : A object a obj(a) point a A p : a = A b isomorphism p : a b path from a to b function A B functor A B continuous map A B Directed HoTT (ω, ω)-categories Directed Homot. Theory ϕ : a A b morphism ϕ : a b directed path from a to b Why? Math: Constructive theory for (ω, ω)-categories, Math: Domain-specific morphism is type-theoretic morphism, Progr: Automatic implementation of fmap for functions (but: canonicity problem), Bonus: Constructive (co)limits.
4 The approach Knowledge from (higher) category theory was not central in the development. Made sense of variance for dependent functions, Investigated the variance of important dependencies in HoTT, Adapted foundations of HoTT so as to incorporate functoriality from the start, Generalized univalence to give meaning to paths & morphisms, Inductive definition of morphisms with induction principle turns out to be possible. Work in progress: Still a bit shabby. Semantics/consistency? Mysterious extra structure on types (zigzags?)
5 2D Directed Type Theory (Licata & Harper, 2011) Directed: For all a,b : A, we can speak of morphisms a b. 2-dimensional: Distinct elements a,b : A, Distinct morphisms ϕ, χ : a b, Either ϕ χ or not. New: Co/Contravariance of assumptions (x : ± A): Γ,x : A B[x]type Γ x:a B[x]type Opportunities for improvement: 2DTT excludes identity type. E.g. let a = be covariant. ϕ : a b gives ϕ : (a = a) (a = b), so ϕ (refla) : a = b. a = should be invariant. Decouple variance in types and in elements. 2DTT: a:a C(a) contravariant in A. Therefore: C(a) and f(a). Remarkably: + that is covariant in domain.
6 Meaning of structure: univalence In HoTT: 2 structures: paths and functions. Univalence: path in universe is invertible function: (A = U B) (A B). In directed HoTT 3 structures: paths, morphisms and covariant functions. Directed univalence: morphism in universe is covariant function: (A U B) (A + B). Categorical univalence: path in any type is invertible morphism: (a = A b) (a = A b). (Cfr. precategory vs. category)
7 Variance At least 4 kinds of variance appear in HoTT: Covariant: f : A + C a A b implies f(a) C f(b), Contravariant: f : A C a A b implies f(a) C f(b), Invariant: f : A C a A b implies (almost) nothing, Isovariant: f : A = C a A b implies f(a) = C f(b). All functions preserve equality ( isomorphism), as in HoTT.
8 Meaning of variance for dept. functions Tool: inductive type families for heterogenious equality/morphisms. Let C : A v U. Any f yields f(a) = C f(b) p : a = A b, Isovariant f yields f(a) = C f(b) Covariant f yields f(a) C f(b) Contravariant f yields f(a) C f(b) ϕ : a A b. Whenever p : a = A b, ( c =C d p : a = A b p : C(a) C(b), ) ( p (c) = C(b) d ). Meaning depends on variance of C.
9 Meaning of variance for dept. functions Tool: inductive type families for heterogenious equality/morphisms. Let C : A v U. Any f yields f(a) = C f(b) p : a = A b, Isovariant f yields f(a) = C f(b) Covariant f yields f(a) C f(b) Contravariant f yields f(a) C f(b) ϕ : a A b. Meaning depends on variance of C. Assume C(x) covariant in x. Whenever ϕ : a A b, ( c =C d ϕ : a A b ϕ : C(a) + C(b), ) ( ϕ (c) = C(b) d ).
10 Meaning of variance for dept. functions Tool: inductive type families for heterogenious equality/morphisms. Let C : A v U. Any f yields f(a) = C f(b) p : a = A b, Isovariant f yields f(a) = C f(b) Covariant f yields f(a) C f(b) Contravariant f yields f(a) C f(b) ϕ : a A b. Assume C(x) contravariant in x. Whenever ϕ : a A b, ϕ : C(b) + C(a), ( ) c =C d ( c = ϕ : a C(a) ϕ (d) ). A b Meaning depends on variance of C.
11 Meaning of variance for dept. functions Tool: inductive type families for heterogenious equality/morphisms. Let C : A v U. Any f yields f(a) = C f(b) p : a = A b, Isovariant f yields f(a) = C f(b) Covariant f yields f(a) C f(b) Contravariant f yields f(a) C f(b) ϕ : a A b. Assume C(x) isovariant in x. Whenever ϕ : a A b, ϕ : C(a) C(b), ( ) c =C d ( ϕ (c) = ) ϕ : a C(b) d. A b Meaning depends on variance of C.
12 Meaning of variance for dept. functions Tool: inductive type families for heterogenious equality/morphisms. Let C : A v U. Any f yields f(a) = C f(b) p : a = A b, Isovariant f yields f(a) = C f(b) Covariant f yields f(a) C f(b) Contravariant f yields f(a) C f(b) ϕ : a A b. Meaning depends on variance of C. When C(x) invariant in x: Heterogeneous types are still defined and often have a meaning. Invariant type families map morphisms to relations? Invariant functions map morphisms to spans: a cospans: a zigzags: a b?... b? b?
13 Meaning of variance for dept. functions Tool: inductive type families for heterogenious equality/morphisms. Let C : A v U. Any f yields f(a) = C f(b) p : a = A b, Isovariant f yields f(a) = C f(b) Covariant f yields f(a) C f(b) Contravariant f yields f(a) C f(b) ϕ : a A b. Meaning depends on variance of C. Example: ( ) C : (X : U) X + X. Given morphism h : A + B, ( ) ( ) ( ) A + h A A + h B B + B ( ) f = C g (h f = g h). dua(h) : A B Commutative diagram = path along morphism Commutative diagram for every morphism? Isovariance. E.g. id X.
14 Example: a cospan type Idea: (a b) c:a (a c) (b c) Inductive type family Cospan type family a A b has constructors triv : + a:a a a, Contravariance in a: (a a) + (a b) + (a b), Contravariance in b: (b b) + (a b) + (a b ). To define f :? a,b:a + w:a b f triv : + a:a C(a,a,triva), + C(a,b,w), you need: C(a,b,w) contravariant in a and b. Computation: f(a,a,triva) f triv (a), f is isovariant in a and b.
15 The morphism type Inductive type family Morphism type family a A b has constructors id : = a:a a a, Contravariance in a: (a a) + (a b) + (a b), Covariance in b: (b b ) + (a b) + (a b ). To define f :? a,b:a + + ϕ:a b C(a,b,ϕ), you need: f id : = a:a C(a,a,ida), C(a,b,ϕ) contravariant in a and covariant in b. Computation: f(a,a,ida) f id (a), f is isovariant in a and b.
16 The identity type Inductive type family Identity type family a = A b has constructors refl : = a:a a = a, Invariance in a and b. Although invariance cannot be written as a constructor, the scheme seems to generalize: To define f :? a,b:a + + p:a=b C(a,b,p), you need: f refl : = a:a C(a,a,refla), C(a,b,p) invariant in a and b (this condition is void). Computation: f(a,a,refla) f refl (a), f is isovariant in a and b.
17 Isovariance of refl and id ida is isovariant in a, because ida=idb ϕ:a b commutativity of: ϕ a b is equivalent to ida a ϕ b. idb refla is isovariant in a, because (interestingly!) refla=reflb ϕ:a b equivalent to commutativity of: is a ϕ b refla a ϕ b. reflb
18 The mystery of invariance, zigzags and relations Questions: Are zigzags the way to go? What are morphisms between zigzags? Well-behaved type-theoretic account of zigzags? Must not prove ζ ζ = id. Analogue of opposite for invariance?... (A B) + (A op + B) (A + B op ), (A = B) + (A loc + B) (A + B core ), f : (A core + B) is function A B that discards morphisms completely. g : (A + B loc ) is function A B that maps all structure to zigzags. Can we have a core with degenerate/infinitesimal zigzags?
19 Other results Directed function extensionality: Morphism of functions is a natural transformation, ( = a:a C(a)) = (lim C(a)) for covariant C, a:a ( = a:a C(a)) = (lim C(a)) for covariant C, a:a Some functions, such as A A op, have complicated higher variance.
20 Thanks! Thesis in pdf: ThesisAndreasNuyts.pdf Contact: andreasnuyts[at]gmail.com Questions? Opinions? Suggestions?
21 C(a) C(b) C(c) C(d) x:ac(x) F(a) F(b) F(c) F(d) x:af(x) A a b c d a b c isovariant C invariant F A d D(a) D(b) D(c) D(d) E(a) E(b) E(c) E(d) x:ae(x) x:ad(x) A a b c d a b c covariant D contravariant E A d
22 Higher variance (A + B) (A op + B op ) X X op has variance Typical variances: , , : unclear (structure on zigzags?), =: doesn t matter (equality types are groupoids).
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