Univalence from a Computer Science Point-of-View. Dan Licata Wesleyan University

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1 Univalence from a Computer Science Point-of-View Dan Licata Wesleyan University 1

2 Martin-Löf type theory [70s-80s] 2

3 Proofs are programs 3

4 cubicaltt [Cohen,Coquand, Huber,Mörtberg] 4

5 cubicaltt [Cohen,Coquand, Huber,Mörtberg] 4

6 cubicaltt [Cohen,Coquand, Huber,Mörtberg] exists k : nat such that n = 2k+1 4

7 Theorem: every natural number is even or odd Proof: induction on n. Base case: 0 is even Inductive case: Suppose n is even or n is odd. To show: n+1 is even or n+1 is odd. Case where n is even (n=2k): n+1 = 2k+1 is odd. Case where n is odd (n=2k+1): n+1 = 2k+2 = 2(k+1) is even. 5

8 for all n : nat, n is even or n is odd 6

9 for all n : nat, n is even or n is odd What program is this? 6

10 evenodd.ctt 7

11 for all is function or is coproduct What program is this? 8

12 coproduct injection is parity What program is this? 9

13 floor(n/2) What program is this? 10

14 proof that n = 2*floor(n/2)[+1] What program is this? 11

15 proof that n = 2*floor(n/2)[+1] (every element of Path nat k k is reflexivity/identity) 12

16 Computation elimination reduces on introduction function applied to argument reduces to body of definition projection of a pair reduces to component case distinction for coproduct reduces on injection recursion on nat reduces on zero and suc(n) 13

17 Computation elimination reduces on introduction function applied to argument reduces to body of definition projection of a pair reduces to component case distinction for coproduct reduces on injection recursion on nat reduces on zero and suc(n) definitional rather than typal equalities/paths 13

18 Computation elimination reduces on introduction function applied to argument reduces to body of definition projection of a pair reduces to component case distinction for coproduct reduces on injection recursion on nat reduces on zero and suc(n) definitional rather than typal equalities/paths equality 13

19 Computation elimination reduces on introduction function applied to argument reduces to body of definition projection of a pair reduces to component case distinction for coproduct reduces on injection recursion on nat reduces on zero and suc(n) definitional rather than typal equalities/paths equality 1-simplex 13

20 Computation elimination reduces on introduction C A C : A Type 14

21 Computation elimination reduces on introduction C C : A Type p : Path A a b A a p b 14

22 Computation elimination reduces on introduction C : A Type C c p : Path A a b c : C(a) A a p b 14

23 Computation elimination reduces on introduction C : A Type C c tr p c p : Path A a b c : C(a) then transport C p c : C(b) A a p b 14

24 Computation elimination reduces on introduction C : A Type C c tr p c p : Path A a b c : C(a) then A a p b transport C p c : C(b) and reduces to c when p is identity Path A a a 14

25 Computation elimination reduces on introduction C : A Type C c tr p c p : Path A a b c : C(a) then A a p b transport C p c : C(b) and reduces to c when p is identity Path A a a original intended model of MLTT: every path is identity 14

26 Canonicity theorem Constructive proof of: For all (closed) t:nat in MLTT, there exists a numeral k with t definitionally equal to k 15

27 Univalence Axiom (A,B : U) Equiv A B ~ Path U A B 16

28 Univalence Axiom (A,B : U) Equiv A B ~ Path U A B axioms break canonicity 16

29 Univalence Axiom (A,B : U) Equiv A B ~ Path U A B axioms break canonicity Central question for computation with univalence: what does it mean to transport along such a path? 16

30 Voevodsky s homotopy canonicity conjecture [talk in Götenborg, 2011] 17

31 Voevodsky s homotopy canonicity conjecture Constructive proof of: For all (closed) t:nat in MLTT+univalence, there exists a numeral k with a Path nat t k (potentially using univalence) 18

32 Voevodsky s homotopy canonicity conjecture Constructive proof of: For all (closed) t:nat in MLTT+univalence, there exists a numeral k with a Path nat t k (potentially using univalence) computation valid in all models 18

33 Voevodsky s homotopy canonicity conjecture Constructive proof of: For all (closed) t:nat in MLTT+univalence, there exists a numeral k with a Path nat t k (potentially using univalence) computation valid in all models ua already implies how ua computes 18

34 Progress 19

35 Progress Models of MLTT+ua in a constructive metatheory (procedure for running programs implicit in proof!) 19

36 Progress Models of MLTT+ua in a constructive metatheory (procedure for running programs implicit in proof!) Models of MLTT+ua based on programs and operational semantics 19

37 Progress Models of MLTT+ua in a constructive metatheory (procedure for running programs implicit in proof!) Models of MLTT+ua based on programs and operational semantics New type theories including MLTT+ua that satisfy (definitional) canonicity 19

38 Progress Models of MLTT+ua in a constructive metatheory (procedure for running programs implicit in proof!) Models of MLTT+ua based on programs and operational semantics New type theories including MLTT+ua that satisfy (definitional) canonicity Several new experimental proof assistants 19

39 Progress Models of MLTT+ua in a constructive metatheory (procedure for running programs implicit in proof!) Models of MLTT+ua based on programs and operational semantics New type theories including MLTT+ua that satisfy (definitional) canonicity Several new experimental proof assistants maybe new type theories can be interpreted in same kinds of models? 19

40 Progress Models of MLTT+ua in a constructive metatheory (procedure for running programs implicit in proof!) Models of MLTT+ua based on programs and operational semantics New type theories including MLTT+ua that satisfy (definitional) canonicity Several new experimental proof assistants maybe new type theories can be interpreted in same kinds of models? definitional equalities are easier to use 19

41 Constructive Cubical Models 20

42 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] 20

43 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] de Morgan cube category, composition filling, cofibration syntax [Cohen,Coquand,Huber,Mörtberg, 14-15] 20

44 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] de Morgan cube category, composition filling, cofibration syntax [Cohen,Coquand,Huber,Mörtberg, 14-15] cartesian cube category, homogenization, diagonal cofibs [Angiuli,Favonia,Harper,Wilson;+Brunerie,Coquand,L., 14-17] 20

45 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] de Morgan cube category, composition filling, cofibration syntax [Cohen,Coquand,Huber,Mörtberg, 14-15] cartesian cube category, homogenization, diagonal cofibs [Angiuli,Favonia,Harper,Wilson;+Brunerie,Coquand,L., 14-17] canonicity/operational models [Huber, 16; Angiuli,Favonia,Harper,Wilson, 16-17] 20

46 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] de Morgan cube category, composition filling, cofibration syntax [Cohen,Coquand,Huber,Mörtberg, 14-15] cartesian cube category, homogenization, diagonal cofibs [Angiuli,Favonia,Harper,Wilson;+Brunerie,Coquand,L., 14-17] canonicity/operational models [Huber, 16; Angiuli,Favonia,Harper,Wilson, 16-17] higher inductive types: syntax, fibrancy/operational semantics [Isaev 14,Coquand,Huber,Mörtberg, 14-17;Cavallo,Harper, 17] 20

47 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] de Morgan cube category, composition filling, cofibration syntax [Cohen,Coquand,Huber,Mörtberg, 14-15] cartesian cube category, homogenization, diagonal cofibs [Angiuli,Favonia,Harper,Wilson;+Brunerie,Coquand,L., 14-17] canonicity/operational models [Huber, 16; Angiuli,Favonia,Harper,Wilson, 16-17] higher inductive types: syntax, fibrancy/operational semantics [Isaev 14,Coquand,Huber,Mörtberg, 14-17;Cavallo,Harper, 17] type theory with non-fibrant types and exact equality as in Voevodsky s HTS [AFH 17] 20

48 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] de Morgan cube category, composition filling, cofibration syntax [Cohen,Coquand,Huber,Mörtberg, 14-15] cartesian cube category, homogenization, diagonal cofibs [Angiuli,Favonia,Harper,Wilson;+Brunerie,Coquand,L., 14-17] canonicity/operational models [Huber, 16; Angiuli,Favonia,Harper,Wilson, 16-17] higher inductive types: syntax, fibrancy/operational semantics [Isaev 14,Coquand,Huber,Mörtberg, 14-17;Cavallo,Harper, 17] type theory with non-fibrant types and exact equality as in Voevodsky s HTS [AFH 17] internal language presentations (fibrancy as type) [Coquand;Orton-Pitts 16;Spitters+ 16] + universes [Sattler;LOPS 18] 20

49 Constructive Cubical Models monoidal cube category, uniformity [Bezem,Coquand,Huber 13] de Morgan cube category, composition filling, cofibration syntax [Cohen,Coquand,Huber,Mörtberg, 14-15] cartesian cube category, homogenization, diagonal cofibs [Angiuli,Favonia,Harper,Wilson;+Brunerie,Coquand,L., 14-17] canonicity/operational models [Huber, 16; Angiuli,Favonia,Harper,Wilson, 16-17] higher inductive types: syntax, fibrancy/operational semantics [Isaev 14,Coquand,Huber,Mörtberg, 14-17;Cavallo,Harper, 17] type theory with non-fibrant types and exact equality as in Voevodsky s HTS [AFH 17] internal language presentations (fibrancy as type) [Coquand;Orton-Pitts 16;Spitters+ 16] + universes [Sattler;LOPS 18] more mathematical analyses [Awodey 13-,Gambino,Sattler 15-17] 20

50 Kan filling p r q 21

51 Kan filling p -1 ;q;r p r q 21

52 Kan filling p -1 ;q;r p r q 21

53 Kan filling p -1 ;q;r p r q 21

54 Kan filling p -1 ;q;r p r q 21

55 Kan filling p -1 ;q;r p r q 21

56 Kan filling p -1 ;q;r p r q 21

57 Kan filling p -1 ;q;r p r q 21

58 Kan filling p -1 ;q;r p r q 21

59 Main Ideas 22

60 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals 22

61 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems 22

62 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems algorithms for filling in,, Path, universe, univalence 22

63 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems algorithms for filling in,, Path, universe, univalence definition of fibration chosen carefully stable under change of base (uniformity), (trivial) cofibrations in harmony with choice of cube category 22

64 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems algorithms for filling in,, Path, universe, univalence definition of fibration chosen carefully stable under change of base (uniformity), (trivial) cofibrations in harmony with choice of cube category known methods use P A := A(- I) or A yi and its right adjoint to define universes and filling in them 22

65 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems algorithms for filling in,, Path, universe, univalence definition of fibration chosen carefully stable under change of base (uniformity), (trivial) cofibrations in harmony with choice of cube category known methods use P A := A(- I) or A yi and its right adjoint to define universes and filling in them unclear if any type theoretic model structures are Quillen-equiv to sset/top; some are not [Sattler] 22

66 Main Ideas 23

67 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals 23

68 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems 23

69 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems algorithms for filling in,, Path, universe, univalence 23

70 Main Ideas Sets Cop for C free semicartesian category on 0,1 : * I; free cartesian category; with connections/reversals fibration: algebraic/specified solutions to filling problems algorithms for filling in,, Path, universe, univalence definition of fibration chosen carefully stable under change of base (uniformity), (trivial) cofibrations in harmony with choice of cube category 23

71 Relation to sset? known methods use P A := A(- I) or A yi and its right adjoint to define universes and filling in them unclear if any type theoretic model structures are Quillen-equiv to sset/top; some are not [Sattler] 24

72 Recommender System faculty/kapulkin/seminars/ hottest.html Last spring: Coquand Angiuli October 11: Favonia research/summerschool/ summer18/topics.php Harper 25

73 Computation with univalence in 26

74 Computation with univalence in definitions of Z 26

75 Computation with univalence in definitions of Z fundamental groups of S 1 and T 26

76 Computation with univalence in definitions of Z fundamental groups of S 1 and T calculation of π4(s 3 ) 26

77 Running the equivalence principle 27

78 Z in type theory (1) nat + nat negative non-negative inl 1 inl 0 inr 0 inr 1 inr 2 28

79 Z in type theory (2) nat +(0,0) nat non-positive non-negative inl 2 inl 1 inr 0 inr 1 inr 2 inl 0 29

80 Z in type theory (3) (nat nat) / (a+b =nat a +b) (0,1) (0,0) (1,0) (1,2) (1,1) (2,1) 30

81 Z in type theory (4) free (set-level) group on one generator pred(zero) zero suc(zero) pred(succ(zero)) succ(pred(zero)) 31

82 Z in type theory (5) loops in S loop -1 id loop loop;loop loop -1 ;loop loop;loop -1 32

83 addition (1) 33

84 addition (1) -1-a + -1-b = -2-(a+b) inl(a) + inl(b) = inl(1+a+b) 33

85 addition (1) -1-a + b = (b - (1+a)) sub : nat nat # Z 34

86 addition (1) -1-a + b = (b - (1+a)) sub : nat nat # Z 34

87 addition (3) add ((a,b),(a,b )) = (a+a,b+b ) plus proof that respects quotient assoc: ((a1,b1)+(a2,b2))+(a3,b3) = ((a1+a2)+a3,(b1+b2)+b3) = (a1+(a2+a3),b1+(b2+b3)) = (a1,b1)+((a2,b2)+(a3,b3)) 35

88 Equivalence of (1) and (3) plus proof mutually inverse 36

89 Using univalence Therefore: any construction on types that can be defined for Zd can be transferred to Z, and vice versa 37

90 Group structure 38

91 Without univalence Given e : A B 39

92 Without univalence Given e : A B GroupStr : U # U 39

93 Without univalence Given e : A B GroupStr : U # U define GroupStr A GroupStr B 39

94 Without univalence Given e : A B GroupStr : U # U define GroupStr A GroupStr B e.g. b1 B b2 = e(e -1 (b1) A e -1 (b2)) 39

95 Without univalence Given e : A B GroupStr : U # U define GroupStr A GroupStr B e.g. b1 B b2 = e(e -1 (b1) A e -1 (b2)) No definable construction on types differentiates equivalent types 39

96 Using univalence Z Zd : Path U Z Zd univalence 40

97 Using univalence Z Zd : Path U Z Zd univalence GroupStr : U # U 40

98 Using univalence Z Zd : Path U Z Zd univalence GroupStr : U # U define GroupStr Zd 40

99 Using univalence Z Zd : Path U Z Zd univalence GroupStr : U # U define GroupStr Zd mechanically derive GroupStr Z by transporting along the equivalence 40

100 intdiff.ctt 41

101 Higher inductive types and synthetic homotopy theory 42

102 Circle Circle S 1 is a higher inductive type generated by base 43

103 Circle Circle S 1 is a higher inductive type generated by base : S 1 loop : Path S 1 base base base 43

104 Circle Circle S 1 is a higher inductive type generated by base : S 1 loop : Path S 1 base base loop -1 id base Free type ( -groupoid/uniform Kan cubical set): id loop -1 inv : loop;loop -1 = id... loop;loop 43

105 Universal Cover wind : Ω(S 1 ) # Z defined by lifting a loop to the cover, and giving the other endpoint of 0 lifting loop adds 1 lifting loop -1 subtracts 1 44

106 Universal Cover Helix : S 1 # U Helix(base) := Z Helix(loop) := ua(x x+1 : Z Z) lifting loop adds 1 lifting loop -1 subtracts 1 45

107 circletalk.ctt 46

108 Torus a q a p a s q a p a : Torus p,q : Path a a s : Square q q p p 47

109 T S 1 S 1 free type: suffices to specify images of generators 48

110 Ω(T) # Ω(S 1 x S 1 ) # Ω(S 1 ) x Ω(S 1 ) 49

111 q q p p p p q q q p p p p p q q q 50

112 t p q s q s p q s p 51

113 t p q s q s p q s p 51

114 torustalk.ctt 52

115 Suspension north south merid(a) a 53

116 Join A push B 54

117 Synth homotopy theory 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 (Co)homology Mayer-Vietoris π4(s 3 ) = Z2 [Brunerie, Cavallo, Favonia, Finster, Licata, Lumsdaine, Sojakova, Shulman] 55

118 Synth homotopy theory Serre Spectral Sequence [Avigad, Awodey, Buchholtz, van Doorn, Newstead, Rijke, Shulman] Cellular Cohomology [Favonia, Buchholtz] Higher Groups [Buchholtz, van Doorn, Rijke] Cayley-Dickson, Quaternionic Hopf [Buchholtz, Rijke] Real projective spaces [Buchholtz, Rijke] Free Higher Groups [Kraus, Altenkirch] 56

119 Brunerie s number Constructive proof in type theory of: There exists a k such that π4(s 3 ) Z/kZ 57

120 Brunerie s number Constructive proof in type theory of: There exists a k such that π4(s 3 ) Z/kZ generator of π3(s 3 ) 57

121 Brunerie s number Constructive proof in type theory of: There exists a k such that π4(s 3 ) Z/kZ generator of π3(s 3 ) view S 3 as join 57

122 Brunerie s number Constructive proof in type theory of: There exists a k such that π4(s 3 ) Z/kZ generator of π3(s 3 ) view S 3 as join main map 57

123 Brunerie s number Constructive proof in type theory of: There exists a k such that π4(s 3 ) Z/kZ generator of π3(s 3 ) view S 3 as join main map π3(s 2 ) is Z 57

124 Proof Assistants cubicaltt [Cohen,Coquand,Huber,Mörtberg] cubical Agda [Vezzosi] redtt [Angiuli,Cavallo,Favonia,Harper,Mörtberg,Sterling] yacctt [Angiuli,Mörtberg] redprl [Angiuli,Cavallo,Favonia,Harper,Sterling] different cube categories, filling operations optimized definitions of filling operations term representations, evaluation strategies, def. equality non-fibrant types and exact equalities 58

125 CS Applications guarded recursion [Birkedal,Bizjak,Clouston,Spitters,Vezzosi] relational parametricity [Bernardy,Coquand,Moulin; Nuyts,Vezzosi,Devriese] effects in computational cubical type theory? [Angiuli,Cavallo,Favonia,Harper,Sterling,Wilson] transporting along functions in directed type theories? [Riehl,Shulman;Riehl,Sattler;L.,Weaver] 59

126 Questions homotopy theories of cubical sets models? [Sattler;Kapulkin,Voevodsky 18] interpret cubical type theory in other models? homotopy canonicity for MLTT+ua? Path U A B definitionally equal to Equiv A B? [Altenkrich,Kaposi] regularity? A I + transport id def. equal to id [Awodey] 60

127 Thanks! 61

redtt cartesian cubical proof assistant Carlo Angiuli Evan Cavallo Favonia Robert Harper Anders Mörtberg Jon Sterling

redtt cartesian cubical proof assistant Carlo Angiuli Evan Cavallo Favonia Robert Harper Anders Mörtberg Jon Sterling homotopy type theory dependent type-theoretic language for higher dimensional mathematics.