Cubical Type Theory. Dan Licata Wesleyan University Guillaume Brunerie Université de Nice Sophia Antipolis

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1 Cubical Type Theory Dan Licata Wesleyan University Guillaume Brunerie Université de Nice Sophia Antipolis

2 Higher inductive types loop base 2

3 Higher inductive types loop base base : Circle loop : base = base 2

4 Higher inductive types 3

5 Higher inductive types f 3

6 Higher inductive types a a f a a 3

7 Higher inductive types a a p a f a p 3

8 Higher inductive types a q a p a f q a p 3

9 Higher inductive types a q a p a f q a p a : Torus p,q : a = a f : q o p = p o q 3

10 Homotopy in HoTT π1(s 1 ) = Z πk<n(s n ) = 0 Hopf fibration π2(s 2 ) = Z π3(s 2 ) = Z James Construction Freudenthal πn(s n ) = Z K(G,n) Blakers-Massey T 2 = S 1 S 1 Van Kampen Covering spaces Whitehead for n-types Cohomology axioms Mayer-Vietoris π4(s 3 ) = Z? [Brunerie, Cavallo, Favonia, Finster, Licata, Lumsdaine, Sojakova, Shulman] 4

11 Patch theories [Angiuli, Morehouse, Licata, Harper] 5

12 Problem When we implement HITs as axioms in current type theories, lots of things don t compute Can t actually run programs Complicates formalizations 6

13 T S 1 S 1 7

14 Goal A type theory with native support for higher inductive types, where everything computes 8

15 Goal A type theory with native support for higher inductive types, where everything computes Today: A type theory with native support for some higher inductive types, where some things compute (not the universe) 8

16 Globular type theory 9

17 Globular type theory Cubical type theory 9

18 Cubical models Bezem,Coquand,Huber, 2013 gave a constructive model of type theory in Kan cubical sets with symmetries; evaluator based on this Coquand, 2014: UKC for diagonals Coquand, Huber et al., 2014: model with in cubical sets with diagonals and connections; evaluator 10

19 Cubical type theories Parametrized judgement or internal inductive step Boundaries-as-terms or boundaries-as-types 11

20 Cubical type theories Parametrized judgement or internal inductive step u : n A Boundaries-as-terms or boundaries-as-types 11

21 Cubical type theories Parametrized judgement or internal inductive step u : n A u ~A v Boundaries-as-terms or boundaries-as-types 11

22 Cubical type theories Parametrized judgement or internal inductive step u : n A u ~A v Boundaries-as-terms or boundaries-as-types u:squarea 11

23 Cubical type theories Parametrized judgement or internal inductive step u : n A u ~A v Boundaries-as-terms or boundaries-as-types u:squarea u:squarea l t b r 11

24 Cubical type theories Polonsky, 2013+: internal inductive step Altenkirch&Kaposi, 2013+: internal inductive step Coquand et al., 2014: parametrized judgement, boundaries-as-terms, diagonals and connections L&B, 2014: parametrized judgement, boundaries-as-terms, just diagonals Isaev, 2014: interval type used for Id and HITs 12

25 Cubical type theory Judgements Types Kan operations 13

26 Cubical type theory Judgements Types Kan operations 14

27 Dimensions as names [Coquand,Pitts] n-dimensional cube has n dimension names free cubical operations and identities correspond to operations on names 15

28 Dimensions as variables Ψ = s1 dim, s2 dim,, sn dim A is a Ψ-cube in type u is a Ψ-cube in A, together with its boundary 16

29 Dimensions as variables Ψ = s1 dim, s2 dim,, sn dim dimension variables A is a Ψ-cube in type u is a Ψ-cube in A, together with its boundary 16

30 Points 17

31 Points point: nat : Type 17

32 Points point: element of point: nat : Type 0 : nat 17

33 Lines line in type: e:equiv A B two equivalent types A B 18

34 Lines line in type: two equivalent types e:equiv A B A B element of line: heterogeneously equal points p:e(a) = b a:a b:b 18

35 Dimensions as variables structural properties for Ψ are cubical operations and identities 19

36 Structural Properties Weakening: a variable can go unused f(x,y) := y Substitution: search and replace for variable f(x) := x 2 + 2x + 1 f(3) = (x 2 + 2x + 1)<3/x> = *

37 Structural Properties Laws: (y 2 + 1)<7/x> = y 2 +1 (x 2 + y 2 + 5)<7/x><8/y> = (7 2 + y 2 + 5)<8/y> = ( ) = (x )<7/x> = (x 2 + y 2 + 5)<8/y><7/x> 21

38 Structural Properties Laws: (y 2 + 1)<7/x> = y 2 +1 unchanged (x 2 + y 2 + 5)<7/x><8/y> = (7 2 + y 2 + 5)<8/y> = ( ) = (x )<7/x> = (x 2 + y 2 + 5)<8/y><7/x> 21

39 Structural Properties Laws: (y 2 + 1)<7/x> = y 2 +1 (x 2 + y 2 + 5)<7/x><8/y> = (7 2 + y 2 + 5)<8/y> = ( ) = (x )<7/x> = (x 2 + y 2 + 5)<8/y><7/x> unchanged order doesn t matter 21

40 Degeneracy is weakening 22

41 Faces and diagonals are substitution 23

42 Faces and diagonals are substitution r ::= s

43 Faces x dim u : A x u<0/x> u u<1/x> 24

44 Faces x dim, y dim u : A u x y 25

45 Faces x dim, y dim u : A u<0/x> u x y 25

46 Faces x dim, y dim u : A u<0/x> u u<1/x> x y 25

47 Faces x dim, y dim u : A u<0/y> u<0/x> u u<1/x> x y 25

48 Faces x dim, y dim u : A u<0/y> u<0/x> u u<1/x> x u<1/y> y 25

49 Faces x dim, y dim u : A u<0/x><0/y> u<0/y> u<0/x> u u<1/x> x u<1/y> y 25

50 Faces x dim, y dim u : A u<0/x><0/y> u<0/y> u<0/x> u u<1/x> x u<0/x><1/y> u<1/y> y 25

51 Faces x dim, y dim u : A u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u u<1/x> x u<0/x><1/y> u<1/y> y 25

52 Faces x dim, y dim u : A u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u u<1/x> x u<0/x><1/y> u<1/y> u<1/x><1/y> y 25

53 Faces x dim, y dim u : A u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u u<1/x> x u<0/x><1/y> u<1/y> u<1/x><1/y> y 25

54 Faces x dim, y dim u : A u<0/y><1/x> u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u u<1/x> x u<0/x><1/y> u<1/y> u<1/x><1/y> y 25

55 Faces x dim, y dim u : A u<0/x><0/y> u<0/y> u<0/y><1/x> u<1/x><0/y> u<0/x> u u<1/x> x u<0/x><1/y> u<1/y> u<1/x><1/y> y substitutions for independent variables commute 25

56 Degeneracies a y x 26

57 Degeneracies a a a a x y 26

58 Degeneracies a a a a a <0/x> a x y 26

59 Degeneracies a a a <0/x> a a a a <1/x> a x y 26

60 Degeneracies a a a <0/x> a a a a <1/x> a p x y 26

61 Degeneracies a a a <0/x> a a a a <1/x> a p p p x p y 26

62 Degeneracies a a a <0/x> a a a a <1/x> a p p p p<0/y> p p<1/y> p x p y 26

63 Degeneracies a a a <0/x> a a a a <1/x> a p p p p<0/y> p p<1/y> p x p y 26

64 Degeneracies a a a <0/x> a a a a <1/x> a p p p p<0/y> p p<1/y> p x p y substitution after weakening is identity 26

65 Diagonals u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u<1/x> u<0/x><1/y> u<1/y> u<1/x><1/y> 27

66 Diagonals u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u<1/x> u<0/x><1/y> u<1/y> u<1/x><1/y> u<x/y> is a line ({x}-cube) 27

67 Diagonals u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u<1/x> u<0/x><1/y> u<1/y> u<1/x><1/y> u<x/y> is a line ({x}-cube) u<x/y><0/x> u<0/x><0/y> 27

68 Diagonals u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u<1/x> u<0/x><1/y> u<1/y> u<1/x><1/y> u<x/y> is a line ({x}-cube) u<x/y><0/x> u<0/x><0/y> 27

69 Diagonals u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u<1/x> u<0/x><1/y> u<1/y> u<1/x><1/y> u<x/y> is a line ({x}-cube) u<x/y><0/x> u<0/x><0/y> u<x/y><1/x> u<1/x><1/y> 27

70 Diagonals u<0/x><0/y> u<0/y> u<1/x><0/y> u<0/x> u<1/x> u<0/x><1/y> u<1/y> u<1/x><1/y> u<x/y> is a line ({x}-cube) u<x/y><0/x> u<0/x><0/y> u<x/y><1/x> u<1/x><1/y> 27

71 Diagonals u<0/x><0/y> u<0/x> u<0/y> u<x/y> u<1/x><0/y> u<1/x> u<0/x><1/y> u<1/y> u<1/x><1/y> u<x/y> is a line ({x}-cube) u<x/y><0/x> u<0/x><0/y> u<x/y><1/x> u<1/x><1/y> 27

72 Functoriality structural properties for Γ give functoriality 28

73 Functoriality Function f : A B is an -functor : takes points to points, lines to lines, squares to squares, cubes to cubes, ap : (f : A " B) " a 0 = A " f(a 0 ) = A f() 29

74 Functoriality Function f : A B is an -functor : takes points to points, lines to lines, squares to squares, cubes to cubes, ap : (f : A " B) " a 0 = A " f(a 0 ) = A f() internalization of a substitution operation on open terms x:a u : B 29

75 Substitution 30

76 Substitution 30

77 Ap = degen + subst ; w : A u : B u 31

78 Ap = degen + subst ; w : A u : B u y dim a : A a 31

79 Ap = degen + subst ; w : A u : B u y dim; w : A u : B y dim a : A a 31

80 Ap = degen + subst ; w : A u : B y dim; w : A u : B u y dim a : A u[a<0/y>/w] a u[a/w] u[a<1/y>/w] 31

81 Cubical judgements structural properties for Ψ are cubical operations and identities structural properties for Γ give functoriality 32

82 Cubical type theory Judgements Types Kan operations 33

83 Π types 34

84 Π types 34

85 Π types 35

86 Π types 35

87 Σ types 36

88 Σ types 36

89 Identity type 37

90 Circle loop base 38

91 Circle loop base 38

92 Circle 39

93 Circle 39

94 Circle 39

95 Circle diagonal if s occurs in l 39

96 Torus a q a p a f q a p 40

97 41

98 Cubical type theory Judgements Types Kan operations 42

99 T S 1 S 1 43

100 T S 1 S 1 t2c : T " S 1 S 1 c2t : S 1 S 1 " T c2t2c : Πp:S 1 S 1. t2c (c2t p) = p t2c2t : Πt:T. c2t (t2c t) = t 43

101 t2c : T " S 1 S 1 t2c a = (base,base) t2c p z = (loop z,base) t2c q z = (base,loop z ) t2c f x,y = (loop y,loop x ) 44

102 t2c : T " S 1 S 1 t2c a = (base,base) t2c p z = (loop z,base) t2c q z = (base,loop z ) t2c f x,y = (loop y,loop x ) z dim (loop z,base) : S 1 S 1 because z dim loopz : S 1 z dim base : S 1 44

103 t2c : T " S 1 S 1 t2c a = (base,base) t2c p z = (loop z,base) t2c q z = (base,loop z ) t2c f x,y = (loop y,loop x ) x,y dim (loop y,loop x ) : S 1 S 1 45

104 t2c : T " S 1 S 1 t2c a = (base,base) t2c p z = (loop z,base) t2c q z = (base,loop z ) t2c f x,y = (loop y,loop x ) x,y dim (loop y,loop x ) : S 1 S 1 base loop y loop y loop y base 45

105 t2c : T " S 1 S 1 t2c a = (base,base) t2c p z = (loop z,base) t2c q z = (base,loop z ) t2c f x,y = (loop y,loop x ) x,y dim (loop y,loop x ) : S 1 S 1 base loop x loop y loop y loop y base loop x base base loop x 45

106 t2c : T " S 1 S 1 t2c a = (base,base) t2c p z = (loop z,base) t2c q z = (base,loop z ) t2c f x,y = (loop y,loop x ) (loop z,base)<0/z> (base,base) (base,loop z )<0/z> (base,base) (loop y,loop x )<0/x> (loop y,base) (loop y,loop x )<0/y> (base, loop x ) similarly for 1 46

107 c2t : S 1 " (S 1 " T) c2t base = c2t1 c2t loop y = c2t2 47

108 c2t : S 1 " (S 1 " T) c2t base = c2t1 c2t loop y = c2t2 c2t1 : S 1 " T c2t1 base = a c2t1 loop x = q x 47

109 c2t : S 1 " (S 1 " T) c2t base = c2t1 c2t loop y = c2t2 c2t1 : S 1 " T c2t1 base = a c2t1 loop x = q x y dim c2t2 : S 1 " T c2t2 base = py c2t2 loop x = f x,y 47

110 c2t : S 1 " (S 1 " T) c2t base = c2t1 c2t loop y = c2t2 c2t1 : S 1 " T c2t1 base = a c2t1 loop x = q x y dim c2t2 : S 1 " T c2t2 base = py c2t2 loop x = f x,y c2t2<0/y> c2t1 because py<0/y> a f x,y <0/y> q x 47

111 t2c2t : Πt:T. c2t (t2c t) = t t2c2t a = refl a t2c2t p z = refl p z t2c2t q z = refl q z t2c2t f x,y = refl f x,y 48

112 t2c2t : Πt:T. c2t (t2c t) = t t2c2t a = refl a t2c2t p z = refl p z t2c2t q z = refl q z t2c2t f x,y = refl f x,y c2t (t2c a) c2t (base,base) c2t base base c2t1 base a 48

113 t2c2t : Πt:T. c2t (t2c t) = t t2c2t a = refl a t2c2t p z = refl p z t2c2t q z = refl q z t2c2t f x,y = refl f x,y c2t (t2c p z ) c2t (loopz,base) c2t loop z base c2t2<z/y> base p z 49

114 t2c2t : Πt:T. c2t (t2c t) = t t2c2t a = refl a t2c2t p z = refl p z t2c2t q z = refl q z t2c2t f x,y = refl f x,y c2t (t2c p z ) c2t (base,loopz) c2t base loop z c2t1 loop z q z 50

115 t2c2t : Πt:T. c2t (t2c t) = t t2c2t a = refl a t2c2t p z = refl p z t2c2t q z = refl q z t2c2t f x,y = refl f x,y c2t (t2c loopx,y) c2t (loopy,loopx) c2t loop y loop x c2t2 loop x f x,y 51

116 c2t2c : Πb,b :S 1. t2c (c2t b b ) = (b,b ) c2t2c base = c2t2c1 t2c2t loop y = c2t2c2 52

117 c2t2c : Πb,b :S 1. t2c (c2t b b ) = (b,b ) c2t2c base = c2t2c1 t2c2t loop y = c2t2c2 c2t2c1 : Πb :S 1. t2c (c2t1 b ) = (base,b ) c2t2c1 base = refl (base,base) c2t2c1 loop x = refl (base,loop x ) 52

118 c2t2c : Πb,b :S 1. t2c (c2t b b ) = (b,b ) c2t2c base = c2t2c1 t2c2t loop y = c2t2c2 c2t2c1 : Πb :S 1. t2c (c2t1 b ) = (base,b ) c2t2c1 base = refl (base,base) c2t2c1 loop x = refl (base,loop x ) y c2t2c2 : Πb :S 1. t2c (c2t2 b ) = (loop y,b ) c2t2c2 base = refl (loopy,base) c2t2c2 loop x = refl (loop y,loop x ) 52

119 c2t2c : Πb,b :S 1. t2c (c2t b b ) = (b,b ) c2t2c base = c2t2c1 t2c2t loop y = c2t2c2 c2t2c1 : Πb :S 1. t2c (c2t1 b ) = (base,b ) c2t2c1 base = refl (base,base) c2t2c1 loop x = refl (base,loop x ) y c2t2c2 : Πb :S 1. t2c (c2t2 b ) = (loop y,b ) c2t2c2 base = refl (loopy,base) c2t2c2 loop x = refl (loop y,loop x ) c2t2c2<0/y> c2t2c1 52

120 t2c : T " S 1 S 1 t2c a = (base,base) t2c p z = (loop z,base) t2c q z = (base,loop z ) t2c f x,y = (loop y,loop x ) c2t : S 1 " (S 1 " T) c2t base = base a loopx qx c2t loop y = base p y loopx fx,y Composites: induct; reflexivity in each case 53

121 54

122 55

123 Research directions Kan operations with diagonals: Tomorrow: Π Σ Id N S 1 T To do: universe, HITCs Definitional canonicity for cubical type theory Semantics: Is cubical type theory too strict? Can cubical type theory be translated back to book HoTT in general? (homotopy canonicity) 56

124 Identity type Heterogeneous identity type as primitive: a 0 a 1 A 0 A A 1 57

125 Identity type Heterogeneous identity type as primitive: a 0 a 1 A 0 A A 1 57

126 Identity type intro 58

127 Identity type elim 59

128 Identity type elim 59

129 Identity type elim 59

130 Identity type elim [c.f. Stone-Harper singleton calculus] 60

131 Refl 61

132 Refl really _.A : degenerate line 61

133 Refl degeneracy because s doesn t occur really _.A : degenerate line 61

134 Ap 62

135 Ap 62

136 Definitional Equalities 63

137 Dependent Ap 64

138 Dependent Ap path in B over p@s 64

139 Dependent Ap path in B over p@s non-dependent ap is literally a special case! 64

140 Transport ap + coerce along a line in the universe: 65

141 Transport ap + coerce along a line in the universe: coes.a 0"1 : A<0/s> " A<1/s> 65

142 Coercion is type-directed coey.a B 0"1 : A<0/y> B<0/y> " A<1/y> B<1/y> 66

143 Coercion is type-directed coey.a B 0"1 : A<0/y> B<0/y> " A<1/y> B<1/y> coey.a 0"1 : A<0/y> " A<1/y> 66

144 Coercion is type-directed coey.a B 0"1 : A<0/y> B<0/y> " A<1/y> B<1/y> coey.a 0"1 : A<0/y> " A<1/y> coey.b 0"1 : B<0/y> " B<1/y> 66

145 Coercion is type-directed coey.a B 0"1 : A<0/y> B<0/y> " A<1/y> B<1/y> coey.a B 0"1 p = (coey.a 0"1 (fst p), coey.b 0"1 (snd p)) coey.a 0"1 : A<0/y> " A<1/y> coey.b 0"1 : B<0/y> " B<1/y> 66

146 Coercion is type-directed coey.a"b 0"1 : A<0/y> " B<0/y> " A<1/y> " B<1/y> A<0/y> B<0/y> A<1/y> B<1/y> 67

147 Coercion is type-directed coey.a"b 0"1 : A<0/y> " B<0/y> " A<1/y> " B<1/y> A<0/y> B<0/y> coey.a 0"1 A<1/y> B<1/y> 67

148 Coercion is type-directed coey.a"b 0"1 : A<0/y> " B<0/y> " A<1/y> " B<1/y> A<0/y> B<0/y> coey.a 0"1 coey.b 0"1 A<1/y> B<1/y> 67

149 Coercion is type-directed coey.a"b 0"1 : A<0/y> " B<0/y> " A<1/y> " B<1/y> A<0/y> B<0/y> coey.a 1"0 coey.b 0"1 A<1/y> B<1/y> 68

150 Coercion is type-directed coex.id_a(,) 0"1 : IdA(<0/x>,<0/x>) " IdA(<1/x>,<1/x>) <0/x> p <0/x> <1/x> <1/x> 69

151 Groupoid Structure Reformulate identity, inverses, composition as: identity + one ternary operation p : a = b q : a = c r : c = d a p b q c r d 70

152 Groupoid Structure Reformulate identity, inverses, composition as: identity + one ternary operation p : a = b q : a = c r : c = d a p b q c r d also type-directed 70

153 71

154 (x, x ) (l,l ) (y, y ) (t,t ) A A (z, z ) (r,r ) (w, w ) 71

155 (x, x ) (l,l ) (t,t ) A A (z, z ) (r,r ) x t (y, y ) z (w, w ) l A r y w 71

156 (x, x ) (l,l ) (t,t ) A A (z, z ) (r,r ) x t (y, y ) z (w, w ) l A r y b w 71

157 (x, x ) (l,l ) (t,t ) A A (z, z ) (r,r ) x t (y, y ) z (w, w ) x t z l A r l A r y b w y w 71

158 (x, x ) (l,l ) (t,t ) A A (z, z ) (r,r ) x t (y, y ) z (w, w ) x t z l A r l A r y b w y b w 71

159 (x, x ) (t,t ) (z, z ) (l,l ) A A (r,r ) x t (y, y ) z (b,b ) (w, w ) x t z l A r l A r y b w y b w 71

160 p t u l = A r q v l : p = = q p q u v : = A t : p = = u r : u = = v 72

161 refl p l q p t u refl l = A r q v l : p = = q p q u v : = A t : p = = u r : u = = v 72

162 refl p t u refl refl p l q p t u refl l = A r q v l : p = = q p q u v : = A t : p = = u r : u = = v 72

163 refl p t u refl refl refl p l q p t u u r v refl l = A r refl q v l : p = = q p q u v : = A t : p = = u r : u = = v 72

164 p t u l = A r q v refl refl refl p l q p t u u r v refl refl refl 73

165 p t u l = A r q v refl refl refl p l q p t u u r v refl refl refl 73

166 p t u l l = A r q p q v refl refl refl p l q p t u u r v refl refl refl 73

167 p t u l t l = A r q p q v refl refl refl p l q p t u u r v refl refl refl 73

168 p t u l t r l = A r q p v u q v refl refl refl p l q p t u u r v refl refl refl 73

169 73 p t q u v l r = A p q l refl refl p u t refl refl u v r refl refl l t q r v p u

170 73 p t q u v l r = A p q l refl refl p u t refl refl u v r refl refl l t q r v p u

171 73 p t q u v l r = A p q l refl refl p u t refl refl u v r refl refl l t q r v p u

172 73 p t q u v l r = A p q l refl refl p u t refl refl u v r refl refl l t q r v p u

173 73 p t q u v l r = A p q l refl refl p u t refl refl u v r refl refl l t q r v p u b

174 73 p t q u v l r = A p q l refl refl p u t refl refl u v r refl refl l t q r v p u b b

175 Kan condition: any n-dimensional open box has a lid, and an inside 74

176 Cubical models Bezem,Coquand,Huber, 2013 gave a constructive model of type theory in Kan cubical sets with symmetries; evaluator based on this Coquand, 2014: UKC for diagonals Coquand, Huber et al., 2014: model with in cubical sets with diagonals and connections; evaluator 75

177 Kan composition 76

178 Kan composition tube dimensions 76

179 Kan composition tube dimensions filling dimension 76

180 Kan composition sides of tube tube dimensions filling dimension 76

181 Kan composition sides of tube transverse face tube dimensions filling dimension 76

182 Kan composition sides of tube transverse face adjacentcompatible tube dimensions filling dimension 76

183 Kan composition sides of tube transverse face adjacentcompatible tube-fitting tube dimensions filling dimension 76

184 Kan composition sides of tube transverse face adjacentcompatible tube-fitting from tube dimensions filling dimension 76

185 Kan composition sides of tube transverse face adjacentcompatible tube-fitting from to tube dimensions filling dimension 76

186 Kan 0 A<0/x> b A A<1/x> tube: empty filling dimension: x transverse side: r = 0 77

187 Kan 0 A<0/x> b A A<1/x> coea x:0"1 (b) tube: empty filling dimension: x transverse side: r = 0 77

188 Kan 0 A<0/x> b A A<1/x> coea x:0"1 (b) b tube: empty filling dimension: x transverse side: r = 0 tube: empty filling dimension: x transverse side: r = 1 77

189 Kan 0 A<0/x> b A A<1/x> coea x:0"1 (b) tube: empty filling dimension: x transverse side: r = 0 coea x:1"0 (b) b tube: empty filling dimension: x transverse side: r = 1 77

190 Kan 1 ty 0 b ty 1 tube dimension: y filling dimension: x transverse side: 0 78

191 Kan 1 ty 0 b coex.a;ty0,ty1 0"1 (b) ty 1 tube dimension: y filling dimension: x transverse side: 0 78

192 Kan 1 ty 0 b ty 1 tube dimension: y filling dimension: x transverse side: y 79

193 Kan 1 ty 0 coex.a;ty0,ty1 y"0 (b) b ty 1 tube dimension: y filling dimension: x transverse side: y 79

194 Kan 1 ty 0 b ty 1 tube dimension: y filling dimension: x transverse side: y 80

195 Kan 1 ty 0 b coex.a;ty0,ty1 0"y (b) ty 1 tube dimension: y filling dimension: x transverse side: y 80

196 Kan 2 z y x tube dimensions: x,z filling dimension: y transverse side: 1 81

197 Kan 2 z y x tube dimensions: x,z filling dimension: y transverse side: 1 81

198 Kan 2 z y x tube dimensions: x,z filling dimension: y transverse side: 1 81

199 Kan 2 z y x tube dimensions: x,z filling dimension: y transverse side: 1 81

200 Uniform Kan Condition 82

201 Uniform Kan Condition 82

202 Uniform Kan Condition q l p 82

203 Uniform Kan Condition q l p t 82

204 Uniform Kan Condition l t r q p v u 82

205 Uniform Kan Condition q l b p t v r u 82

206 Uniform Kan Condition <0/y> q l b p t v r u 82

207 Uniform Kan Condition <0/y> t<0/y> q l b p t v r u l<0/y> b r<0/y> 82

208 Generalized open boxes x,y,z dim coey.a;t x;y:1"0 (b) z y x tube dimensions: x filling dimension: y transverse side: 1 83

209 Generalized open boxes x,y,z dim coey.a;t x;y:1"0 (b) z y x tube dimensions: x filling dimension: y transverse side: 1 83

210 Generalized open boxes x,y,z dim coey.a;t x;y:1"0 (b) z y x tube dimensions: x filling dimension: y transverse side: 1 83

211 Kan composition 84

212 Uniformity Naturality in extra dimensions part of coe( )<r 0 /s 0 > 85

213 Coercion is type-directed = define uniform Kan structure on each type 86

214 Coercion reduces coey.a B 0"1 : A<0/y> B<0/y> " A<1/y> B<1/y> coey.a B 0"1 p = (coey.a 0"1 (fst p), coey.b 0"1 (snd p)) coey.a 0"1 : A<0/y> " A<1/y> coey.b 0"1 : B<0/y> " B<1/y> 87

215 (x, x ) (t,t ) (z, z ) (l,l ) A A (r,r ) t (y, y ) z (b,b ) (w, w ) x t z A r l A r b w y b w 88

216 (x, x ) (t,t ) (z, z ) (l,l ) A A (r,r ) x t (y, y ) z (b,b ) (w, w ) x t z l A r l A r y b w y b w 88

217 (x, x ) (t,t ) (z, z ) (l,l ) A A (r,r ) x t (y, y ) z (b,b ) (w, w ) x t z l A r l A r y b w y b w 88

218 coey.a"b 0"1 : A<0/y> " B<0/y> " A<1/y> " B<1/y> A<0/y> B<0/y> coey.a 1"0 coey.b 0"1 A<1/y> B<1/y> 89

219 coey.a"b 0"1 : A<0/y> " B<0/y> " A<1/y> " B<1/y> A<0/y> B<0/y> coey.a 1"0 coey.b 0"1 A<1/y> B<1/y> 89

220 p t u l t r l = A r q p v u q v 90

221 p t u l t r l = A r q p v u q v 90

222 p l t = A u r q l b p t v r u q v 90

223 p l t = A u r q l b p t v r u q b v 90

224 p l t = A u r q l b p t v r u q b v 90

225 Coercion is a constructor In ITs/HITs, coe is canonical and elim reduces on it ap Cover (loop o loop) = ap Cover loop o ap Cover loop 91

226 Coercion is a constructor In ITs/HITs, coe is canonical and elim reduces on it ap Cover (loop o loop) = ap Cover loop o ap Cover loop 91

227 Discussion Topics Kan filling from Kan composition Regularity: Definitional rule for J on refl Why diagonals for HITs Univalence Kan composition for univalence/the universe Issues with HITCs 92

Functional programs that prove theorems about spaces

Functional programs that prove theorems about spaces Dan Licata Wesleyan University Department of Mathematics and Computer Science Kepler Conjecture (1611) No way to pack equally-sized spheres in space