Game Semantics of Homotopy Type Theory

Norihiro Yamada

[email protected] University of Minnesota

Homotopy Type Theory Electronic Seminar Talks (HoTTEST) Department of Mathematics, Western University February 11, 2021

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 1 / 19 Just like axiomatic set theory is explained by sets in an informal sense, the conceptual foundation of Martin-L¨oftype theory (MLTT) is computations in an informal sense (a.k.a. the BHK-interpretation).

Proofs/objects as computations (e.g., succ : ¬ max N); MLTT as a foundation of constructive maths. On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies.

Introduction Background: MLTT vs. HoTT

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 Proofs/objects as computations (e.g., succ : ¬ max N); MLTT as a foundation of constructive maths. On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies.

Introduction Background: MLTT vs. HoTT

Just like axiomatic set theory is explained by sets in an informal sense, the conceptual foundation of Martin-L¨oftype theory (MLTT) is computations in an informal sense (a.k.a. the BHK-interpretation).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 MLTT as a foundation of constructive maths. On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies.

Introduction Background: MLTT vs. HoTT

Proofs/objects as computations (e.g., succ : ¬ max N);

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 On the other hand, homotopy type theory (HoTT) is motivated by the homotopical interpretation of MLTT. HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies.

Introduction Background: MLTT vs. HoTT

Proofs/objects as computations (e.g., succ : ¬ max N); MLTT as a foundation of constructive maths.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 HoTT = MLTT + univalence + higher inductive types (HITs); Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies.

Introduction Background: MLTT vs. HoTT

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 Homotopical interpretation: formulas as spaces, proofs/objects as points, and higher proofs/objects as paths/homotopies.

Introduction Background: MLTT vs. HoTT

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 Introduction Background: MLTT vs. HoTT

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 2 / 19 HoTT uncovers new connections between type theory, higher category theory and homotopy theory, and is a powerful foundation of maths. On the other hand, The topological view is orthogonal to the BHK-interpretation. Moreover, Are proofs given by working mathematicians points in spaces?

Motivation (The BHK-interpretation of HoTT) To extend the BHK-interpretation of MLTT to HoTT so that one can better understand HoTT as a foundation of constructive maths.

Introduction Motivation: computational understanding of HoTT

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 3 / 19 On the other hand, The topological view is orthogonal to the BHK-interpretation. Moreover, Are proofs given by working mathematicians points in spaces?

Motivation (The BHK-interpretation of HoTT) To extend the BHK-interpretation of MLTT to HoTT so that one can better understand HoTT as a foundation of constructive maths.

Introduction Motivation: computational understanding of HoTT

HoTT uncovers new connections between type theory, higher category theory and homotopy theory, and is a powerful foundation of maths.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 3 / 19 The topological view is orthogonal to the BHK-interpretation. Moreover, Are proofs given by working mathematicians points in spaces?

Motivation (The BHK-interpretation of HoTT) To extend the BHK-interpretation of MLTT to HoTT so that one can better understand HoTT as a foundation of constructive maths.

Introduction Motivation: computational understanding of HoTT

HoTT uncovers new connections between type theory, higher category theory and homotopy theory, and is a powerful foundation of maths. On the other hand,

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 3 / 19 Moreover, Are proofs given by working mathematicians points in spaces?

Motivation (The BHK-interpretation of HoTT) To extend the BHK-interpretation of MLTT to HoTT so that one can better understand HoTT as a foundation of constructive maths.

Introduction Motivation: computational understanding of HoTT

HoTT uncovers new connections between type theory, higher category theory and homotopy theory, and is a powerful foundation of maths. On the other hand, The topological view is orthogonal to the BHK-interpretation.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 3 / 19 Are proofs given by working mathematicians points in spaces?

Motivation (The BHK-interpretation of HoTT) To extend the BHK-interpretation of MLTT to HoTT so that one can better understand HoTT as a foundation of constructive maths.

Introduction Motivation: computational understanding of HoTT

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 3 / 19 Motivation (The BHK-interpretation of HoTT) To extend the BHK-interpretation of MLTT to HoTT so that one can better understand HoTT as a foundation of constructive maths.

Introduction Motivation: computational understanding of HoTT

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 3 / 19 Introduction Motivation: computational understanding of HoTT

Motivation (The BHK-interpretation of HoTT) To extend the BHK-interpretation of MLTT to HoTT so that one can better understand HoTT as a foundation of constructive maths.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 3 / 19 1 Consistency of HoTT + strict univalence: IdU (A, B) ≡ Eq(A, B); 2 Independence of Markov’s principle from this extended HoTT.

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

My game semantics of HoTT interprets Types as games between Player and Opponent; Terms as strategies (or algorithms for Player to play on games); Identity proofs as strategies that verify equality between strategies. This model can be seen as a variant of the BHK-interpretation. Corollary (Consistency and independence)

Introduction Main results

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 4 / 19 1 Consistency of HoTT + strict univalence: IdU (A, B) ≡ Eq(A, B); 2 Independence of Markov’s principle from this extended HoTT.

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

Types as games between Player and Opponent; Terms as strategies (or algorithms for Player to play on games); Identity proofs as strategies that verify equality between strategies. This model can be seen as a variant of the BHK-interpretation. Corollary (Consistency and independence)

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

My game semantics of HoTT interprets

Terms as strategies (or algorithms for Player to play on games); Identity proofs as strategies that verify equality between strategies. This model can be seen as a variant of the BHK-interpretation. Corollary (Consistency and independence)

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

My game semantics of HoTT interprets Types as games between Player and Opponent;

Identity proofs as strategies that verify equality between strategies. This model can be seen as a variant of the BHK-interpretation. Corollary (Consistency and independence)

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

My game semantics of HoTT interprets Types as games between Player and Opponent; Terms as strategies (or algorithms for Player to play on games);

This model can be seen as a variant of the BHK-interpretation. Corollary (Consistency and independence)

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

Corollary (Consistency and independence)

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 4 / 19 2 Independence of Markov’s principle from this extended HoTT.

Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

1 Consistency of HoTT + strict univalence: IdU (A, B) ≡ Eq(A, B);

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 4 / 19 Introduction Main results

Theorem (Game semantics of HoTT) There exists game semantics of HoTT (viz., MLTT equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types as well as univalent universes).

1 Consistency of HoTT + strict univalence: IdU (A, B) ≡ Eq(A, B); 2 Independence of Markov’s principle from this extended HoTT.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 4 / 19 Nevertheless, one might wonder why I choose game semantics among computational models such as realisability. My answer is the following advantages of game semantics: Effective for the study of type theory (e.g., independence of MP); Semantics of computational effects and linear logic; Applications in program verification and model checking; Rich in higher structures by its intensionality. The last point is new, and so let me explain it in the next few slides.

Introduction Why game semantics? (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19 My answer is the following advantages of game semantics: Effective for the study of type theory (e.g., independence of MP); Semantics of computational effects and linear logic; Applications in program verification and model checking; Rich in higher structures by its intensionality. The last point is new, and so let me explain it in the next few slides.

Introduction Why game semantics? (part 1/3)

Nevertheless, one might wonder why I choose game semantics among computational models such as realisability.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19 Effective for the study of type theory (e.g., independence of MP); Semantics of computational effects and linear logic; Applications in program verification and model checking; Rich in higher structures by its intensionality. The last point is new, and so let me explain it in the next few slides.

Introduction Why game semantics? (part 1/3)

Nevertheless, one might wonder why I choose game semantics among computational models such as realisability. My answer is the following advantages of game semantics:

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19 Semantics of computational eﬀects and linear logic; Applications in program veriﬁcation and model checking; Rich in higher structures by its intensionality. The last point is new, and so let me explain it in the next few slides.

Introduction Why game semantics? (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19 Applications in program veriﬁcation and model checking; Rich in higher structures by its intensionality. The last point is new, and so let me explain it in the next few slides.

Introduction Why game semantics? (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19 Rich in higher structures by its intensionality. The last point is new, and so let me explain it in the next few slides.

Introduction Why game semantics? (part 1/3)

Nevertheless, one might wonder why I choose game semantics among computational models such as realisability. My answer is the following advantages of game semantics: Effective for the study of type theory (e.g., independence of MP); Semantics of computational effects and linear logic; Applications in program verification and model checking;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19 The last point is new, and so let me explain it in the next few slides.

Introduction Why game semantics? (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19 Introduction Why game semantics? (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 5 / 19

(E.g., q 7→ q; qqn 7→ n + 1.)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

q q q

-... ? ? ? 0 1 2 3 n m

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G.

Introduction Why game semantics? (part 2/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19

(E.g., q 7→ q; qqn 7→ n + 1.)

q q q

-... ? ? ? 0 1 2 3 n m

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G.

Introduction Why game semantics? (part 2/3)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19

(E.g., q 7→ q; qqn 7→ n + 1.)

q q

? ? n m

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G.

Introduction Why game semantics? (part 2/3)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

-... ? 0 1 2 3

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19 (E.g., q 7→ q; qqn 7→ n + 1.)

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G.

Introduction Why game semantics? (part 2/3)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

q q q

-... ? ? ? 0 1 2 3 n m

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19 (E.g., q 7→ q; qqn 7→ n + 1.)

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G.

Introduction Why game semantics? (part 2/3)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

q q q

-... ? ? ? 0 1 2 3 n m

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19 (E.g., q 7→ q; qqn 7→ n + 1.)

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G.

Introduction Why game semantics? (part 2/3)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

q q q

-... ? ? ? 0 1 2 3 n - m

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19 (E.g., q 7→ q; qqn 7→ n + 1.)

Introduction Why game semantics? (part 2/3)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

q q q

-... ? ? ? 0 1 2 3 n - m

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19 Introduction Why game semantics? (part 2/3)

Deﬁnition (Simpliﬁed games) A game is a rooted dag whose vertices (or moves) have parity O/P, and paths from a root (or positions) have parity OPOP...

q q q

-... ? ? ? 0 1 2 3 n - m

Deﬁnition (Simpliﬁed strategies) A strategy σ on a game G, written σ : G, is a map odd-length positions m1m2 . . . m2i+1 in G → P-moves m in G

s.t. m1m2 . . . m2i+1m is a position in G. (E.g., q 7→ q; qqn 7→ n + 1.)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 6 / 19 Strategies on N ⇒ N computing constant zero: q q q q

? ? ? ? n - 0 n - 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19 q q q q

? ? ? ? n - 0 n - 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero:

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19 q q

? ? n - 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q

? ? n - 0

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19 q q

? ? ? n - 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q

? n - 0

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19 ?

They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? ? n - 0 n - 0

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19

? ? -

They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? n - 0 n 0

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19

? ? -

Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? n - 0 n 0 They are extensionally the same yet intensionally diﬀerent.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19

? ? -

q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? n - 0 n 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x):

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19

? ? -

- q q qm ? - n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? n - 0 n 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q

? n - n

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19

? ? -

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? n - 0 n 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19

? ? -

In this way, the intensionality of games makes their higher structure nontrivial.

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? n - 0 n 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N .

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19

? ? -

Introduction Why game semantics? (part 3/3)

Strategies on N ⇒ N computing constant zero: q q q q

? ? n - 0 n 0 They are extensionally the same yet intensionally diﬀerent. Games IdN (n, m) and IdN⇒N (φ, ψ) := Πx:N IdN (φ ◦ x, ψ ◦ x): q q - q q qm ? ? - n - n n - n

IdN has at most one strategy, but not the case for IdN⇒N . In this way, the intensionality of games makes their higher structure nontrivial.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 7 / 19 A pioneering constructive model of HoTT is cubical sets initiated by Bezem, Coquand and Huber, and developed by many others. Homotopical in nature; Constructivity on the meta-theory. Other related work is cubical assemblies proposed by Uemura and higher dimensional meaning explanation by Angiuli and Harper. On cubical type theory; The unit interval, dimensional variables and path types. My approach: BHK-interpretation of HoTT; based on globular sets

Introduction Related work and my contributions

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 Homotopical in nature; Constructivity on the meta-theory. Other related work is cubical assemblies proposed by Uemura and higher dimensional meaning explanation by Angiuli and Harper. On cubical type theory; The unit interval, dimensional variables and path types. My approach: BHK-interpretation of HoTT; based on globular sets

Introduction Related work and my contributions

A pioneering constructive model of HoTT is cubical sets initiated by Bezem, Coquand and Huber, and developed by many others.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 Constructivity on the meta-theory. Other related work is cubical assemblies proposed by Uemura and higher dimensional meaning explanation by Angiuli and Harper. On cubical type theory; The unit interval, dimensional variables and path types. My approach: BHK-interpretation of HoTT; based on globular sets

Introduction Related work and my contributions

A pioneering constructive model of HoTT is cubical sets initiated by Bezem, Coquand and Huber, and developed by many others. Homotopical in nature;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 Other related work is cubical assemblies proposed by Uemura and higher dimensional meaning explanation by Angiuli and Harper. On cubical type theory; The unit interval, dimensional variables and path types. My approach: BHK-interpretation of HoTT; based on globular sets

Introduction Related work and my contributions

A pioneering constructive model of HoTT is cubical sets initiated by Bezem, Coquand and Huber, and developed by many others. Homotopical in nature; Constructivity on the meta-theory.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 On cubical type theory; The unit interval, dimensional variables and path types. My approach: BHK-interpretation of HoTT; based on globular sets

Introduction Related work and my contributions

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 The unit interval, dimensional variables and path types. My approach: BHK-interpretation of HoTT; based on globular sets

Introduction Related work and my contributions

A pioneering constructive model of HoTT is cubical sets initiated by Bezem, Coquand and Huber, and developed by many others. Homotopical in nature; Constructivity on the meta-theory. Other related work is cubical assemblies proposed by Uemura and higher dimensional meaning explanation by Angiuli and Harper. On cubical type theory;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 My approach: BHK-interpretation of HoTT; based on globular sets

Introduction Related work and my contributions

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 Introduction Related work and my contributions

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 8 / 19 Throughout this work, ∞-groupoids refer to strict ∞-categories whose morphisms are weakly invertible in the sense of type equivalence. Deﬁnition (Game-semantic ∞-groupoids) Deﬁne game-semantic ∞-groupoids to be ∞-groupoids internalised in the category G of games (strictly, in the subcat Gˇ ,→ G, a topos).

Forenotice: Warren’s strict ∞-groupoid model modiﬁed and internalised in G; Challenge: To recover Warren’s method with weak inverses; Mostly generalised to ﬁnitely complete CwFs M with an NNO such that each type A ∈ TyM(Γ) is a map M(1, Γ) → Ob(M).

Game-semantic ∞-groupoids, ∞-functors and trans. Overview of my approach

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 9 / 19 Deﬁnition (Game-semantic ∞-groupoids) Deﬁne game-semantic ∞-groupoids to be ∞-groupoids internalised in the category G of games (strictly, in the subcat Gˇ ,→ G, a topos).

Game-semantic ∞-groupoids, ∞-functors and trans. Overview of my approach

Throughout this work, ∞-groupoids refer to strict ∞-categories whose morphisms are weakly invertible in the sense of type equivalence.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 9 / 19 Forenotice: Warren’s strict ∞-groupoid model modiﬁed and internalised in G; Challenge: To recover Warren’s method with weak inverses; Mostly generalised to ﬁnitely complete CwFs M with an NNO such that each type A ∈ TyM(Γ) is a map M(1, Γ) → Ob(M).

Game-semantic ∞-groupoids, ∞-functors and trans. Overview of my approach

Throughout this work, ∞-groupoids refer to strict ∞-categories whose morphisms are weakly invertible in the sense of type equivalence. Deﬁnition (Game-semantic ∞-groupoids) Deﬁne game-semantic ∞-groupoids to be ∞-groupoids internalised in the category G of games (strictly, in the subcat Gˇ ,→ G, a topos).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 9 / 19 Warren’s strict ∞-groupoid model modiﬁed and internalised in G; Challenge: To recover Warren’s method with weak inverses; Mostly generalised to ﬁnitely complete CwFs M with an NNO such that each type A ∈ TyM(Γ) is a map M(1, Γ) → Ob(M).

Game-semantic ∞-groupoids, ∞-functors and trans. Overview of my approach

Forenotice:

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 9 / 19 Challenge: To recover Warren’s method with weak inverses; Mostly generalised to ﬁnitely complete CwFs M with an NNO such that each type A ∈ TyM(Γ) is a map M(1, Γ) → Ob(M).

Game-semantic ∞-groupoids, ∞-functors and trans. Overview of my approach

Forenotice: Warren’s strict ∞-groupoid model modiﬁed and internalised in G;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 9 / 19 Mostly generalised to ﬁnitely complete CwFs M with an NNO such that each type A ∈ TyM(Γ) is a map M(1, Γ) → Ob(M).

Game-semantic ∞-groupoids, ∞-functors and trans. Overview of my approach

Forenotice: Warren’s strict ∞-groupoid model modiﬁed and internalised in G; Challenge: To recover Warren’s method with weak inverses;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 9 / 19 Game-semantic ∞-groupoids, ∞-functors and trans. Overview of my approach

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 9 / 19 Explicitly, a game-semantic ∞-category G consists of A diagram in Gˇ

s2- s1- s0- ··· - G2 - G1 - G0 t2 t1 t0

that satisﬁes sn ◦ sn+1 = sn ◦ tn+1 and tn ◦ sn+1 = tn ◦ tn+1; ˇ [n] A G-morphism ∗p : Gn ×Gp Gn → Gn for each 0 6 p < n, where

π2 - Gn ×Gp Gn Gn n−p π1 t ? ? - Gn Gp sn−p

A Gˇ-morphism in : Gn → Gn+1.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 1/2)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 10 / 19 A diagram in Gˇ

s2- s1- s0- ··· - G2 - G1 - G0 t2 t1 t0

that satisﬁes sn ◦ sn+1 = sn ◦ tn+1 and tn ◦ sn+1 = tn ◦ tn+1; ˇ [n] A G-morphism ∗p : Gn ×Gp Gn → Gn for each 0 6 p < n, where

π2 - Gn ×Gp Gn Gn n−p π1 t ? ? - Gn Gp sn−p

A Gˇ-morphism in : Gn → Gn+1.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 1/2)

Explicitly, a game-semantic ∞-category G consists of

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 10 / 19 ˇ [n] A G-morphism ∗p : Gn ×Gp Gn → Gn for each 0 6 p < n, where

π2 - Gn ×Gp Gn Gn n−p π1 t ? ? - Gn Gp sn−p

A Gˇ-morphism in : Gn → Gn+1.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 1/2)

Explicitly, a game-semantic ∞-category G consists of A diagram in Gˇ

s2- s1- s0- ··· - G2 - G1 - G0 t2 t1 t0

that satisﬁes sn ◦ sn+1 = sn ◦ tn+1 and tn ◦ sn+1 = tn ◦ tn+1;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 10 / 19 where

π2 - Gn ×Gp Gn Gn n−p π1 t ? ? - Gn Gp sn−p

A Gˇ-morphism in : Gn → Gn+1.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 1/2)

Explicitly, a game-semantic ∞-category G consists of A diagram in Gˇ

s2- s1- s0- ··· - G2 - G1 - G0 t2 t1 t0

that satisﬁes sn ◦ sn+1 = sn ◦ tn+1 and tn ◦ sn+1 = tn ◦ tn+1; ˇ [n] A G-morphism ∗p : Gn ×Gp Gn → Gn for each 0 6 p < n,

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 10 / 19 A Gˇ-morphism in : Gn → Gn+1.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 1/2)

Explicitly, a game-semantic ∞-category G consists of A diagram in Gˇ

s2- s1- s0- ··· - G2 - G1 - G0 t2 t1 t0

that satisﬁes sn ◦ sn+1 = sn ◦ tn+1 and tn ◦ sn+1 = tn ◦ tn+1; ˇ [n] A G-morphism ∗p : Gn ×Gp Gn → Gn for each 0 6 p < n, where

π2 - Gn ×Gp Gn Gn n−p π1 t ? ? - Gn Gp sn−p

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 10 / 19 Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 1/2)

Explicitly, a game-semantic ∞-category G consists of A diagram in Gˇ

s2- s1- s0- ··· - G2 - G1 - G0 t2 t1 t0

that satisﬁes sn ◦ sn+1 = sn ◦ tn+1 and tn ◦ sn+1 = tn ◦ tn+1; ˇ [n] A G-morphism ∗p : Gn ×Gp Gn → Gn for each 0 6 p < n, where

π2 - Gn ×Gp Gn Gn n−p π1 t ? ? - Gn Gp sn−p

A Gˇ-morphism in : Gn → Gn+1.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 10 / 19 They correspond to type equivalence so that we model univalence.

These data are required to satisfy the axioms of strict ∞-categories in terms of commutative diagrams in Gˇ. A game-semantic ∞-category G is a game-semantic ∞-groupoid if it is equipped with Gˇ-morphisms with the ‘expected’ sources and targets

invn retn secn trin Gn −→ Gn Gn −→ Gn+1 Gn −→ Gn+1 Gn −→ Gn+2

that are functorial (on the nose). ‘Expected’ sources and targets: Given an n-cell f ∈ G(1,Gn), one gets

−1 −1 f := inv ◦ f : t ◦ f → s ◦ f ηf := ret ◦ f : f ∗ f → i ◦ s ◦ f

−1 f := sec ◦ f : f ∗ f → i ◦ t ◦ f δf := tri ◦ f :(i ◦ f) ∗ ηf → f ∗ (i ◦ f)

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 2/2)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 11 / 19 They correspond to type equivalence so that we model univalence.

A game-semantic ∞-category G is a game-semantic ∞-groupoid if it is equipped with Gˇ-morphisms with the ‘expected’ sources and targets

invn retn secn trin Gn −→ Gn Gn −→ Gn+1 Gn −→ Gn+1 Gn −→ Gn+2

that are functorial (on the nose). ‘Expected’ sources and targets: Given an n-cell f ∈ G(1,Gn), one gets

−1 −1 f := inv ◦ f : t ◦ f → s ◦ f ηf := ret ◦ f : f ∗ f → i ◦ s ◦ f

−1 f := sec ◦ f : f ∗ f → i ◦ t ◦ f δf := tri ◦ f :(i ◦ f) ∗ ηf → f ∗ (i ◦ f)

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 2/2)

These data are required to satisfy the axioms of strict ∞-categories in terms of commutative diagrams in Gˇ.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 11 / 19 They correspond to type equivalence so that we model univalence.

‘Expected’ sources and targets: Given an n-cell f ∈ G(1,Gn), one gets

−1 −1 f := inv ◦ f : t ◦ f → s ◦ f ηf := ret ◦ f : f ∗ f → i ◦ s ◦ f

−1 f := sec ◦ f : f ∗ f → i ◦ t ◦ f δf := tri ◦ f :(i ◦ f) ∗ ηf → f ∗ (i ◦ f)

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 2/2)

invn retn secn trin Gn −→ Gn Gn −→ Gn+1 Gn −→ Gn+1 Gn −→ Gn+2

that are functorial (on the nose).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 11 / 19 They correspond to type equivalence so that we model univalence.

Given an n-cell f ∈ G(1,Gn), one gets

−1 −1 f := inv ◦ f : t ◦ f → s ◦ f ηf := ret ◦ f : f ∗ f → i ◦ s ◦ f

−1 f := sec ◦ f : f ∗ f → i ◦ t ◦ f δf := tri ◦ f :(i ◦ f) ∗ ηf → f ∗ (i ◦ f)

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 2/2)

invn retn secn trin Gn −→ Gn Gn −→ Gn+1 Gn −→ Gn+1 Gn −→ Gn+2

that are functorial (on the nose). ‘Expected’ sources and targets:

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 11 / 19 They correspond to type equivalence so that we model univalence.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 2/2)

invn retn secn trin Gn −→ Gn Gn −→ Gn+1 Gn −→ Gn+1 Gn −→ Gn+2

that are functorial (on the nose). ‘Expected’ sources and targets: Given an n-cell f ∈ G(1,Gn), one gets

−1 −1 f := inv ◦ f : t ◦ f → s ◦ f ηf := ret ◦ f : f ∗ f → i ◦ s ◦ f

−1 f := sec ◦ f : f ∗ f → i ◦ t ◦ f δf := tri ◦ f :(i ◦ f) ∗ ηf → f ∗ (i ◦ f)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 11 / 19 Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-groupoids (part 2/2)

invn retn secn trin Gn −→ Gn Gn −→ Gn+1 Gn −→ Gn+1 Gn −→ Gn+2

that are functorial (on the nose). ‘Expected’ sources and targets: Given an n-cell f ∈ G(1,Gn), one gets

−1 −1 f := inv ◦ f : t ◦ f → s ◦ f ηf := ret ◦ f : f ∗ f → i ◦ s ◦ f

−1 f := sec ◦ f : f ∗ f → i ◦ t ◦ f δf := tri ◦ f :(i ◦ f) ∗ ηf → f ∗ (i ◦ f)

They correspond to type equivalence so that we model univalence.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 11 / 19 Define a (set-theoretic) ∞-groupoid |G| by |G|n := G(1,Gn). Definition (Game-semantic ∞-functors) Define game-semantic ∞-functors between game-semantic G ∞-groupoids G and H to be strategies φ ∈ TmG(N,H ), where G Gn ? ? Gn H (n) := Hn , s.t. φ := (φn := φ ◦ n : Hn )n∈G(1,N) forms ∞-functors |G| → |H| internalised in G that preserve the data of inverses.

? ? ? Explicitly, the functoriality of the family φ means: sn ◦ φn+1 = φn ◦ sn, ? ? ? [n] [n] ? ? ? ? tn ◦ φn+1 = φn ◦ tn, φn ◦ ∗p = ∗p ◦ (φn ×Bp φn) and φn+1 ◦ in = in ◦ φn. Lemma (∞-category of game-semantic ∞-categories) The category ∞GGpd of game-semantic ∞-groupoids and ∞-functors gives rise to a (set-theoretic) ∞-category.

The map | | extends to a functor ∞GGpd → ∞Gpd := ∞SetGpd.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-functors

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 12 / 19 Deﬁnition (Game-semantic ∞-functors) Deﬁne game-semantic ∞-functors between game-semantic G ∞-groupoids G and H to be strategies φ ∈ TmG(N,H ), where G Gn ? ? Gn H (n) := Hn , s.t. φ := (φn := φ ◦ n : Hn )n∈G(1,N) forms ∞-functors |G| → |H| internalised in G that preserve the data of inverses.

The map | | extends to a functor ∞GGpd → ∞Gpd := ∞SetGpd.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-functors

Deﬁne a (set-theoretic) ∞-groupoid |G| by |G|n := G(1,Gn).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 12 / 19 ? ? ? Explicitly, the functoriality of the family φ means: sn ◦ φn+1 = φn ◦ sn, ? ? ? [n] [n] ? ? ? ? tn ◦ φn+1 = φn ◦ tn, φn ◦ ∗p = ∗p ◦ (φn ×Bp φn) and φn+1 ◦ in = in ◦ φn. Lemma (∞-category of game-semantic ∞-categories) The category ∞GGpd of game-semantic ∞-groupoids and ∞-functors gives rise to a (set-theoretic) ∞-category.

The map | | extends to a functor ∞GGpd → ∞Gpd := ∞SetGpd.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-functors

Define a (set-theoretic) ∞-groupoid |G| by |G|n := G(1,Gn). Definition (Game-semantic ∞-functors) Define game-semantic ∞-functors between game-semantic G ∞-groupoids G and H to be strategies φ ∈ TmG(N,H ), where G Gn ? ? Gn H (n) := Hn , s.t. φ := (φn := φ ◦ n : Hn )n∈G(1,N) forms ∞-functors |G| → |H| internalised in G that preserve the data of inverses.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 12 / 19 Lemma (∞-category of game-semantic ∞-categories) The category ∞GGpd of game-semantic ∞-groupoids and ∞-functors gives rise to a (set-theoretic) ∞-category.

The map | | extends to a functor ∞GGpd → ∞Gpd := ∞SetGpd.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-functors

? ? ? Explicitly, the functoriality of the family φ means: sn ◦ φn+1 = φn ◦ sn, ? ? ? [n] [n] ? ? ? ? tn ◦ φn+1 = φn ◦ tn, φn ◦ ∗p = ∗p ◦ (φn ×Bp φn) and φn+1 ◦ in = in ◦ φn.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 12 / 19 The map | | extends to a functor ∞GGpd → ∞Gpd := ∞SetGpd.

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-functors

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 12 / 19 Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic ∞-functors

The map | | extends to a functor ∞GGpd → ∞Gpd := ∞SetGpd.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 12 / 19 This functor | | : ∞GGpd → ∞Gpd sends game-semantic ∞-functors φ : G → H to the (set-theoretic) ∞-functors |φ| : |G| → |H| given by ? |φ|n : γ ∈ G(1,Gn) 7→ φn ◦ γ ∈ G(1,Hn). In ∞GGpd, n-cells (n > 1) are game-semantic (n − 1)-transformations: Deﬁnition (Game-semantic transformations) Deﬁne game-semantic transformations between game-semantic ∞-functors φ, ψ : G → H to be transformations |φ| → |ψ| internalised in G, and similarly game-semantic n-transformations for all n > 0.

Explicitly, a game-semantic 1-transformation φ → ψ is a G-morphism ? ? α : G0 → H1 with s ◦ α = φ0, t ◦ α = ψ0 and naturality. The functor | | extends to an ∞-functor ∞GGpd → ∞Gpd by

|α|γ := α ◦ γ ∈ G(1,Hn+1)(γ ∈ G(1,G0)).

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic transformations

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 13 / 19 In ∞GGpd, n-cells (n > 1) are game-semantic (n − 1)-transformations: Deﬁnition (Game-semantic transformations) Deﬁne game-semantic transformations between game-semantic ∞-functors φ, ψ : G → H to be transformations |φ| → |ψ| internalised in G, and similarly game-semantic n-transformations for all n > 0.

|α|γ := α ◦ γ ∈ G(1,Hn+1)(γ ∈ G(1,G0)).

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic transformations

This functor | | : ∞GGpd → ∞Gpd sends game-semantic ∞-functors φ : G → H to the (set-theoretic) ∞-functors |φ| : |G| → |H| given by ? |φ|n : γ ∈ G(1,Gn) 7→ φn ◦ γ ∈ G(1,Hn).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 13 / 19 Deﬁnition (Game-semantic transformations) Deﬁne game-semantic transformations between game-semantic ∞-functors φ, ψ : G → H to be transformations |φ| → |ψ| internalised in G, and similarly game-semantic n-transformations for all n > 0.

|α|γ := α ◦ γ ∈ G(1,Hn+1)(γ ∈ G(1,G0)).

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic transformations

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 13 / 19 Explicitly, a game-semantic 1-transformation φ → ψ is a G-morphism ? ? α : G0 → H1 with s ◦ α = φ0, t ◦ α = ψ0 and naturality. The functor | | extends to an ∞-functor ∞GGpd → ∞Gpd by

|α|γ := α ◦ γ ∈ G(1,Hn+1)(γ ∈ G(1,G0)).

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic transformations

This functor | | : ∞GGpd → ∞Gpd sends game-semantic ∞-functors φ : G → H to the (set-theoretic) ∞-functors |φ| : |G| → |H| given by ? |φ|n : γ ∈ G(1,Gn) 7→ φn ◦ γ ∈ G(1,Hn). In ∞GGpd, n-cells (n > 1) are game-semantic (n − 1)-transformations: Deﬁnition (Game-semantic transformations) Deﬁne game-semantic transformations between game-semantic ∞-functors φ, ψ : G → H to be transformations |φ| → |ψ| internalised in G, and similarly game-semantic n-transformations for all n > 0.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 13 / 19 The functor | | extends to an ∞-functor ∞GGpd → ∞Gpd by

|α|γ := α ◦ γ ∈ G(1,Hn+1)(γ ∈ G(1,G0)).

Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic transformations

Explicitly, a game-semantic 1-transformation φ → ψ is a G-morphism ? ? α : G0 → H1 with s ◦ α = φ0, t ◦ α = ψ0 and naturality.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 13 / 19 Game-semantic ∞-groupoids, ∞-functors and trans. Game-semantic transformations

|α|γ := α ◦ γ ∈ G(1,Hn+1)(γ ∈ G(1,G0)).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 13 / 19 Now, one can sketch the game semantics G of HoTT in ∞GGpd: J K ` Γ ctx 7→ Γ G ∈ ∞GGpd0; J K Γ ` A ty 7→ A G ∈ ∞Gpd1(| Γ G|, ∞GGpd) where we later extend the ∞J -categoryK ∞GGpdJ K to an ∞-groupoid;

Γ ` a : A 7→ a G ∈ ∞GGpd1( Γ G, Γ G. A G) that is a section of J K J K J K J K the ﬁrst projection Γ G. A G → Γ G (of Grothendieck const.); J K J K J K Γ ` p : a1 =A a2 7→ p G ∈ ∞GGpd2( a1 G, a2 G); J K J K J K Γ ` q : p1 =a1=Aa2 p2 7→ q G ∈ ∞GGpd3( p1 G, p2 G), and so on. J K J K J K We interpret One-, Zero- and N-types by discrete game-semantic ∞-groupoids, and Id-type by IdA(γ, α1, α2) := A(γ)(α1, α2) ,→ A(γ). In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 ` Γ ctx 7→ Γ G ∈ ∞GGpd0; J K Γ ` A ty 7→ A G ∈ ∞Gpd1(| Γ G|, ∞GGpd) where we later extend the ∞J -categoryK ∞GGpdJ K to an ∞-groupoid;

Game semantics of HoTT Overview of the interpretation

Now, one can sketch the game semantics G of HoTT in ∞GGpd: J K

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 Γ ` A ty 7→ A G ∈ ∞Gpd1(| Γ G|, ∞GGpd) where we later extend the ∞J -categoryK ∞GGpdJ K to an ∞-groupoid;

Game semantics of HoTT Overview of the interpretation

Now, one can sketch the game semantics G of HoTT in ∞GGpd: J K ` Γ ctx 7→ Γ G ∈ ∞GGpd0; J K ,

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 where we later extend the ∞-category ∞GGpd to an ∞-groupoid;

Game semantics of HoTT Overview of the interpretation

Now, one can sketch the game semantics G of HoTT in ∞GGpd: J K ` Γ ctx 7→ Γ G ∈ ∞GGpd0; J K Γ ` A ty 7→ A G ∈ ∞Gpd1(| Γ G|, ∞GGpd), J K J K

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 Γ ` a : A 7→ a G ∈ ∞GGpd1( Γ G, Γ G. A G) that is a section of J K J K J K J K the ﬁrst projection Γ G. A G → Γ G (of Grothendieck const.); J K J K J K Γ ` p : a1 =A a2 7→ p G ∈ ∞GGpd2( a1 G, a2 G); J K J K J K Γ ` q : p1 =a1=Aa2 p2 7→ q G ∈ ∞GGpd3( p1 G, p2 G), and so on. J K J K J K We interpret One-, Zero- and N-types by discrete game-semantic ∞-groupoids, and Id-type by IdA(γ, α1, α2) := A(γ)(α1, α2) ,→ A(γ). In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

Now, one can sketch the game semantics G of HoTT in ∞GGpd: J K ` Γ ctx 7→ Γ G ∈ ∞GGpd0; J K Γ ` A ty 7→ A G ∈ ∞Gpd1(| Γ G|, ∞GGpd), where we later extend the ∞J -categoryK ∞GGpdJ K to an ∞-groupoid;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 Γ ` p : a1 =A a2 7→ p G ∈ ∞GGpd2( a1 G, a2 G); J K J K J K Γ ` q : p1 =a1=Aa2 p2 7→ q G ∈ ∞GGpd3( p1 G, p2 G), and so on. J K J K J K We interpret One-, Zero- and N-types by discrete game-semantic ∞-groupoids, and Id-type by IdA(γ, α1, α2) := A(γ)(α1, α2) ,→ A(γ). In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

Γ ` a : A 7→ a G ∈ ∞GGpd1( Γ G, Γ G. A G) that is a section of J K J K J K J K the ﬁrst projection Γ G. A G → Γ G (of Grothendieck const.); J K J K J K

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 Γ ` q : p1 =a1=Aa2 p2 7→ q G ∈ ∞GGpd3( p1 G, p2 G), and so on. J K J K J K We interpret One-, Zero- and N-types by discrete game-semantic ∞-groupoids, and Id-type by IdA(γ, α1, α2) := A(γ)(α1, α2) ,→ A(γ). In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

Γ ` a : A 7→ a G ∈ ∞GGpd1( Γ G, Γ G. A G) that is a section of J K J K J K J K the ﬁrst projection Γ G. A G → Γ G (of Grothendieck const.); J K J K J K Γ ` p : a1 =A a2 7→ p G ∈ ∞GGpd2( a1 G, a2 G); J K J K J K

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 and so on. We interpret One-, Zero- and N-types by discrete game-semantic ∞-groupoids, and Id-type by IdA(γ, α1, α2) := A(γ)(α1, α2) ,→ A(γ). In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 We interpret One-, Zero- and N-types by discrete game-semantic ∞-groupoids, and Id-type by IdA(γ, α1, α2) := A(γ)(α1, α2) ,→ A(γ). In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 and Id-type by IdA(γ, α1, α2) := A(γ)(α1, α2) ,→ A(γ). In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 In the rest of the talk, I focus on Pi-type and univalent universes.

Game semantics of HoTT Overview of the interpretation

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 Game semantics of HoTT Overview of the interpretation

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 14 / 19 In the following, I omit the semantic bracket G. J K I interpret Pi-type by Π(Γ,A) ∈ ∞GGpd0: 0-cells are sections Γ → Γ.A of the projection Γ.A → Γ; n-cells σ → τ (n > 0) are game-semantic n-transformations σ → τ;

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 I interpret Pi-type by Π(Γ,A) ∈ ∞GGpd0: 0-cells are sections Γ → Γ.A of the projection Γ.A → Γ; n-cells σ → τ (n > 0) are game-semantic n-transformations σ → τ;

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

In the following, I omit the semantic bracket G. J K

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 0-cells are sections Γ → Γ.A of the projection Γ.A → Γ; n-cells σ → τ (n > 0) are game-semantic n-transformations σ → τ;

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

In the following, I omit the semantic bracket G. J K I interpret Pi-type by Π(Γ,A) ∈ ∞GGpd0:

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 n-cells σ → τ (n > 0) are game-semantic n-transformations σ → τ;

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

In the following, I omit the semantic bracket G. J K I interpret Pi-type by Π(Γ,A) ∈ ∞GGpd0: 0-cells are sections Γ → Γ.A of the projection Γ.A → Γ;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

In the following, I omit the semantic bracket G. J K I interpret Pi-type by Π(Γ,A) ∈ ∞GGpd0: 0-cells are sections Γ → Γ.A of the projection Γ.A → Γ; n-cells σ → τ (n > 0) are game-semantic n-transformations σ → τ;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 and similarly for the data of inverses: invn, retn, secn and trin.

Game semantics of HoTT Game semantics of Pi-type (part 1/2)

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 Game semantics of HoTT Game semantics of Pi-type (part 1/2)

The composition ∗p : Π(Γ,A)n ×Π(Γ,A)p Π(Γ,A)n → Π(Γ,A)n internalises the following algorithm in G:

σ,τ ∆ 2 σ×τ 2 ∗p Γ0 −→ (Γ.A)n 7→ Γ0 −→ Γ0 −→ (Γ.A)n −→ (Γ.A)n ;

The identity in : Π(Γ,A)n → Π(Γ,A)n+1 internalises the following algorithm in G:

σ σ in Γ0 −→ (Γ.A)n 7→ Γ0 −→ (Γ.A)n −→ (Γ.A)n+1 ,

and similarly for the data of inverses: invn, retn, secn and trin.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 15 / 19 Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

Validates strict funext: IdΠ(Γ,A)(φ, ψ) = Πx:ΓIdA(x)(φ(x), ψ(x)); Nontrivial computations that verify fun(φ) = fun(ψ); The oracle O is crucial to handle the inﬁnitary domain N. Some higher cells in Π(Γ,A) are only weakly invertible, and so we must weaken the strict invertibility anyway (not only for univalence). Identities in Π(Γ,A) never visit the domain N; Compositions in Π(Γ,A) cannot undo a visit to the domain N.

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 Validates strict funext: IdΠ(Γ,A)(φ, ψ) = Πx:ΓIdA(x)(φ(x), ψ(x)); Nontrivial computations that verify fun(φ) = fun(ψ); The oracle O is crucial to handle the inﬁnitary domain N. Some higher cells in Π(Γ,A) are only weakly invertible, and so we must weaken the strict invertibility anyway (not only for univalence). Identities in Π(Γ,A) never visit the domain N; Compositions in Π(Γ,A) cannot undo a visit to the domain N.

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 Nontrivial computations that verify fun(φ) = fun(ψ); The oracle O is crucial to handle the inﬁnitary domain N. Some higher cells in Π(Γ,A) are only weakly invertible, and so we must weaken the strict invertibility anyway (not only for univalence). Identities in Π(Γ,A) never visit the domain N; Compositions in Π(Γ,A) cannot undo a visit to the domain N.

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

Validates strict funext: IdΠ(Γ,A)(φ, ψ) = Πx:ΓIdA(x)(φ(x), ψ(x));

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 The oracle O is crucial to handle the inﬁnitary domain N. Some higher cells in Π(Γ,A) are only weakly invertible, and so we must weaken the strict invertibility anyway (not only for univalence). Identities in Π(Γ,A) never visit the domain N; Compositions in Π(Γ,A) cannot undo a visit to the domain N.

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

Validates strict funext: IdΠ(Γ,A)(φ, ψ) = Πx:ΓIdA(x)(φ(x), ψ(x)); Nontrivial computations that verify fun(φ) = fun(ψ);

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 Some higher cells in Π(Γ,A) are only weakly invertible, and so we must weaken the strict invertibility anyway (not only for univalence). Identities in Π(Γ,A) never visit the domain N; Compositions in Π(Γ,A) cannot undo a visit to the domain N.

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

Validates strict funext: IdΠ(Γ,A)(φ, ψ) = Πx:ΓIdA(x)(φ(x), ψ(x)); Nontrivial computations that verify fun(φ) = fun(ψ); The oracle O is crucial to handle the inﬁnitary domain N.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 Identities in Π(Γ,A) never visit the domain N; Compositions in Π(Γ,A) cannot undo a visit to the domain N.

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 Compositions in Π(Γ,A) cannot undo a visit to the domain N.

Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 Game semantics of HoTT Game semantics of Pi-type (part 2/2)

Game-semantic trans. make sense as identity proofs in Pi-type. - q q IdN⇒N (φ, ψ) = qn ? - fun(φ)(n) - fun(ψ)(n)

2 where fun(φ) := { (n, m) ∈ N | φ ◦ n = m }, and similarly for fun(ψ).

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 16 / 19 1 Extend the (set-theoretic) ∞-category ∞GGpd to an ∞-groupoid, in which higher cells are type equivalence; 2 Internalise U ,→ ∞GGpd in G, where plays on U0 encode freely generated objects in ∞GGpd0, and Un (n > 0) restricts ∞GGpdn. Lemma (∞-groupoid of game-semantic ∞-groupoids) The following extends the ∞-category ∞GGpd to an ∞-groupoid.

0-cells are game-semantic ∞-groupoids; A 1-cell (φ, ψ, σ, τ, µ): G → H consists of game-semantic ∞-functors φ : G → H and ψ : H → G, trans. σ : ψ ∗ φ → i(G) and τ : φ ∗ ψ → i(H), and 2-trans. µ : i(φ) ∗ σ → τ ∗ i(ψ); (n > 1) An n-cell (α, β, η, , δ):(φ, ψ, σ, τ, µ) → (φ0, ψ0, σ0, τ 0, µ0) consists of game-semantic (n − 1)-trans. α : φ → φ0 and β : φ0 → φ, n-trans. η : β ∗ α → i(φ) and : α ∗ β → i(φ0), and an (n + 1)-trans. δ : i(α) ∗ η → ∗ i(α);

Game semantics of HoTT Game semantics of univalent universes (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 17 / 19 2 Internalise U ,→ ∞GGpd in G, where plays on U0 encode freely generated objects in ∞GGpd0, and Un (n > 0) restricts ∞GGpdn. Lemma (∞-groupoid of game-semantic ∞-groupoids) The following extends the ∞-category ∞GGpd to an ∞-groupoid.

Game semantics of HoTT Game semantics of univalent universes (part 1/3)

1 Extend the (set-theoretic) ∞-category ∞GGpd to an ∞-groupoid, in which higher cells are type equivalence;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 17 / 19 Lemma (∞-groupoid of game-semantic ∞-groupoids) The following extends the ∞-category ∞GGpd to an ∞-groupoid.

Game semantics of HoTT Game semantics of univalent universes (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 17 / 19 0-cells are game-semantic ∞-groupoids; A 1-cell (φ, ψ, σ, τ, µ): G → H consists of game-semantic ∞-functors φ : G → H and ψ : H → G, trans. σ : ψ ∗ φ → i(G) and τ : φ ∗ ψ → i(H), and 2-trans. µ : i(φ) ∗ σ → τ ∗ i(ψ); (n > 1) An n-cell (α, β, η, , δ):(φ, ψ, σ, τ, µ) → (φ0, ψ0, σ0, τ 0, µ0) consists of game-semantic (n − 1)-trans. α : φ → φ0 and β : φ0 → φ, n-trans. η : β ∗ α → i(φ) and : α ∗ β → i(φ0), and an (n + 1)-trans. δ : i(α) ∗ η → ∗ i(α);

Game semantics of HoTT Game semantics of univalent universes (part 1/3)

1 Extend the (set-theoretic) ∞-category ∞GGpd to an ∞-groupoid, in which higher cells are type equivalence; 2 Internalise U ,→ ∞GGpd in G, where plays on U0 encode freely generated objects in ∞GGpd0, and Un (n > 0) restricts ∞GGpdn. Lemma (∞-groupoid of game-semantic ∞-groupoids) The following extends the ∞-category ∞GGpd to an ∞-groupoid.

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 17 / 19 A 1-cell (φ, ψ, σ, τ, µ): G → H consists of game-semantic ∞-functors φ : G → H and ψ : H → G, trans. σ : ψ ∗ φ → i(G) and τ : φ ∗ ψ → i(H), and 2-trans. µ : i(φ) ∗ σ → τ ∗ i(ψ); (n > 1) An n-cell (α, β, η, , δ):(φ, ψ, σ, τ, µ) → (φ0, ψ0, σ0, τ 0, µ0) consists of game-semantic (n − 1)-trans. α : φ → φ0 and β : φ0 → φ, n-trans. η : β ∗ α → i(φ) and : α ∗ β → i(φ0), and an (n + 1)-trans. δ : i(α) ∗ η → ∗ i(α);

Game semantics of HoTT Game semantics of univalent universes (part 1/3)

0-cells are game-semantic ∞-groupoids;

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 17 / 19 (n > 1) An n-cell (α, β, η, , δ):(φ, ψ, σ, τ, µ) → (φ0, ψ0, σ0, τ 0, µ0) consists of game-semantic (n − 1)-trans. α : φ → φ0 and β : φ0 → φ, n-trans. η : β ∗ α → i(φ) and : α ∗ β → i(φ0), and an (n + 1)-trans. δ : i(α) ∗ η → ∗ i(α);

Game semantics of HoTT Game semantics of univalent universes (part 1/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 17 / 19 Game semantics of HoTT Game semantics of univalent universes (part 1/3)

(n > 0) The identity on each n-cell (α, β, η, , δ) is the quintuple

i(α, β, η, , δ) := (i(α), i(α), i2(α), i2(α), i3(α));

The identity on each 0-cell G is the quintuple

i(G) := (i(G), i(G), i2(G), i2(G), i3(G));

The inverse of each n-cell (α, β, η, , δ) is the quadruple

inv(α, β, η, , δ) := (β, α, , η, dual(δ)),

where dual(µ): i(β) ∗ → η ∗ i(β) is too technical to present here;

Game semantics of HoTT Game semantics of univalent universes (part 2/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 18 / 19 ret(α, β, η, , δ) := (η, inv ◦ η, ret ◦ η, sec ◦ η, tri ◦ η), and similarly for sec(α, β, η, , δ) and tri(α, β, η, , δ);

(n > 0) The identity on each n-cell (α, β, η, , δ) is the quintuple

i(α, β, η, , δ) := (i(α), i(α), i2(α), i2(α), i3(α));

The identity on each 0-cell G is the quintuple

i(G) := (i(G), i(G), i2(G), i2(G), i3(G));

Game semantics of HoTT Game semantics of univalent universes (part 2/3)

The inverse of each n-cell (α, β, η, , δ) is the quadruple

inv(α, β, η, , δ) := (β, α, , η, dual(δ)),

where dual(µ): i(β) ∗ → η ∗ i(β) is too technical to present here;

(n > 0) The identity on each n-cell (α, β, η, , δ) is the quintuple

i(α, β, η, , δ) := (i(α), i(α), i2(α), i2(α), i3(α));

Game semantics of HoTT Game semantics of univalent universes (part 2/3)

The inverse of each n-cell (α, β, η, , δ) is the quadruple

inv(α, β, η, , δ) := (β, α, , η, dual(δ)),

where dual(µ): i(β) ∗ → η ∗ i(β) is too technical to present here; The identity on each 0-cell G is the quintuple

i(G) := (i(G), i(G), i2(G), i2(G), i3(G));

Game semantics of HoTT Game semantics of univalent universes (part 2/3)

The inverse of each n-cell (α, β, η, , δ) is the quadruple

inv(α, β, η, , δ) := (β, α, , η, dual(δ)),

where dual(µ): i(β) ∗ → η ∗ i(β) is too technical to present here; The identity on each 0-cell G is the quintuple

i(G) := (i(G), i(G), i2(G), i2(G), i3(G));

(n > 0) The identity on each n-cell (α, β, η, , δ) is the quintuple

i(α, β, η, , δ) := (i(α), i(α), i2(α), i2(α), i3(α));

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 18 / 19 Game semantics of HoTT Game semantics of univalent universes (part 2/3)

The inverse of each n-cell (α, β, η, , δ) is the quadruple

inv(α, β, η, , δ) := (β, α, , η, dual(δ)),

where dual(µ): i(β) ∗ → η ∗ i(β) is too technical to present here; The identity on each 0-cell G is the quintuple

i(G) := (i(G), i(G), i2(G), i2(G), i3(G));

(n > 0) The identity on each n-cell (α, β, η, , δ) is the quintuple

i(α, β, η, , δ) := (i(α), i(α), i2(α), i2(α), i3(α));

ret(α, β, η, , δ) := (η, inv ◦ η, ret ◦ η, sec ◦ η, tri ◦ η), and similarly for sec(α, β, η, , δ) and tri(α, β, η, , δ);

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 18 / 19 For µ :Γ0 → U0 ⇔ fun(µ) ∈ TyG(Γ)0 , deﬁne the game U0 by

N Σ(A, B) IdA(α1, α2) q q q - ](Id)

? ? ? - - ](N) A ](Σ) B A α1 : A α2 : A

The ∗p-composition of composable n-cells (α, β, η, , δ) and (α0, β0, η0, 0, δ0) is the quintuple

0 0 0 0 0 (α , β , η , , δ ) ∗p (α, β, η, , δ) ( (α0 ∗ α, β ∗ β0, η ∗ (i(β) ∗ η0 ∗ i(α)),... ) if p = n − 1; := n n n+1 n n 0 0 0 0 0 (α ∗p α, β ∗p β, η ∗p η, ∗p , δ ∗p δ) if p < n − 1,

Game semantics of HoTT Game semantics of univalent universes (part 3/3)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 19 / 19 For µ :Γ0 → U0 ⇔ fun(µ) ∈ TyG(Γ)0 , deﬁne the game U0 by

N Σ(A, B) IdA(α1, α2) q q q - ](Id)

? ? ? - - ](N) A ](Σ) B A α1 : A α2 : A

Game semantics of HoTT Game semantics of univalent universes (part 3/3)

The ∗p-composition of composable n-cells (α, β, η, , δ) and (α0, β0, η0, 0, δ0) is the quintuple

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 19 / 19 N Σ(A, B) IdA(α1, α2) q q q - ](Id)

? ? ? - - ](N) A ](Σ) B A α1 : A α2 : A

Game semantics of HoTT Game semantics of univalent universes (part 3/3)

The ∗p-composition of composable n-cells (α, β, η, , δ) and (α0, β0, η0, 0, δ0) is the quintuple

0 0 0 0 0 (α , β , η , , δ ) ∗p (α, β, η, , δ) ( (α0 ∗ α, β ∗ β0, η ∗ (i(β) ∗ η0 ∗ i(α)),... ) if p = n − 1; := n n n+1 n n 0 0 0 0 0 (α ∗p α, β ∗p β, η ∗p η, ∗p , δ ∗p δ) if p < n − 1, For µ :Γ0 → U0 ⇔ fun(µ) ∈ TyG(Γ)0 , deﬁne the game U0 by

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 19 / 19 Σ(A, B) IdA(α1, α2) q q - ](Id)

? ? - - A ](Σ) B A α1 : A α2 : A

Game semantics of HoTT Game semantics of univalent universes (part 3/3)

The ∗p-composition of composable n-cells (α, β, η, , δ) and (α0, β0, η0, 0, δ0) is the quintuple

? ](N)

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 19 / 19 IdA(α1, α2) q - ](Id)

- ? A α1 : A α2 : A

Game semantics of HoTT Game semantics of univalent universes (part 3/3)

The ∗p-composition of composable n-cells (α, β, η, , δ) and (α0, β0, η0, 0, δ0) is the quintuple

q q

? ? ](N) A ](Σ) - B

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 19 / 19 Game semantics of HoTT Game semantics of univalent universes (part 3/3)

The ∗p-composition of composable n-cells (α, β, η, , δ) and (α0, β0, η0, 0, δ0) is the quintuple

N Σ(A, B) IdA(α1, α2) q q q - ](Id)

? ? ? - - ](N) A ](Σ) B A α1 : A α2 : A

N. Yamada (Univ. of Minnesota) Game semantics of HoTT Feb. 11, 2021, Western 19 / 19