Unifying Cubical Models of Univalent Type Theory Evan Cavallo - PowerPoint PPT Presentation

Unifying Cubical Models of Univalent Type Theory Evan Cavallo Anders Mörtberg Carnegie Mellon University Stockholm University Andrew W Swan University of Amsterdam CSL 2020 · JAN 16 0

Univalent Type Theory ▧ Dependent type theory CSL 2020 · JAN 16 1

Univalent Type Theory ▧ Dependent type theory func � on/implica � on/ ∀ product/ ∃ induc � ve types equality universe(s) of types CSL 2020 · JAN 16 1

Univalent Type Theory ▧ Dependent type theory func � on/implica � on/ ∀ product/ ∃ induc � ve types equality universe(s) of types CSL 2020 · JAN 16 1

Univalent Type Theory ▧ Iden � ty + ◈ Least re fl exive rela � on ( ⇒ symmetric, transi � ve, etc.) ◈ “ Underdetermined” CSL 2020 · JAN 16 2

Univalent Type Theory ▧ Univalence Axiom (Voevodsky) ▧ Equivalence ◈ CSL 2020 · JAN 16 3

Univalent Type Theory ▧ Iden �� es are not unique CSL 2020 · JAN 16 4

Univalent Type Theory ▧ Iden �� es are not unique CSL 2020 · JAN 16 4

Univalent Type Theory ▧ Iden �� es are not unique ▧ More: add higher induc � ve types ◈ Quo � ents for proof-relevant iden � ty ◈ Language for synthe � c homotopy theory CSL 2020 · JAN 16 4

Models of Univalent Type Theory ▧ Simplicial set model (Kapulkin & Lumsdaine ’12/’18, a � er Voevodsky) ◈ Classical se � ng for homotopy theory ◈ Essen � ally non-construc � ve (Bezem, Coquand, & Parmann ’15) ▧ Cubical set model (Bezem, Coquand, & Huber ’13) ◈ First construc � ve model of univalence ◈ Problems with higher induc � ve types resolved in Cohen, Coquand, Huber, & Mörtberg ’15 and Angiuli, Favonia, & Harper ’18 models CSL 2020 · JAN 16 5

Cubical Set Models ▧ Interpret contexts as cubical sets ◈ family of sets indexed by interval variable contexts { } { } { } CSL 2020 · JAN 16 6

Cubical Set Models ▧ Interpret contexts as cubical sets ◈ family of sets indexed by interval variable contexts ◈ { } { } CSL 2020 · JAN 16 7

Cubical Set Models ▧ Interpret contexts as cubical sets ◈ family of sets indexed by interval variable contexts ◈ ▧ Interpret types as fi bra � ons CSL 2020 · JAN 16 8

Fibra � ons ▧ Part 1 (coercion): “if then ” CSL 2020 · JAN 16 9

Fibra � ons ▧ Part 1 (coercion): “if then ” CSL 2020 · JAN 16 9

Fibra � ons ▧ Part 1 (coercion): “if then ” CSL 2020 · JAN 16 9

Fibra � ons ▧ Part 1 (coercion): “if then ” CSL 2020 · JAN 16 9

Fibra � ons ▧ Part 1 (coercion): “if then ” ▧ Part 2 (composi � on): a cube in A can be adjusted CSL 2020 · JAN 16 10

Fibra � ons ▧ Part 1 (coercion): “if then ” ▧ Part 2 (composi � on): a cube in A can be adjusted CSL 2020 · JAN 16 10

Fibra � ons ▧ Part 1 (coercion): “if then ” ▧ Part 2 (composi � on): a cube in A can be adjusted CSL 2020 · JAN 16 10

Fibra � ons ▧ Part 1 (coercion): “if then ” ▧ Part 2 (composi � on): a cube in A can be adjusted CSL 2020 · JAN 16 10

Fibra � ons ▧ Part 1 (coercion): “if then ” ▧ Part 2 (composi � on): a cube in A can be adjusted CSL 2020 · JAN 16 10

Fibra � ons ▧ Part 1 (coercion): “if then ” ▧ Part 2 (composi � on): a cube in A can be adjusted ▧ A fi bra � on is a family suppor � ng these opera � ons CSL 2020 · JAN 16 11

Two approaches ▧ Cohen, Coquand, Huber, & Mörtberg ’15 ▧ Result: fi bra � ons closed under type formers CSL 2020 · JAN 16 12

Two approaches ▧ Angiuli, Favonia, & Harper ’18 ▧ Result: fi bra � ons closed under type formers CSL 2020 · JAN 16 13

Two approaches ▧ CCHM ◈ ◈ ▧ AFH ◈ ◈ Is there a unifying construc � on that generalizes these? CSL 2020 · JAN 16 14

Unifying construc � on Q: A: CSL 2020 · JAN 16 15

Unifying construc � on Q: A: CSL 2020 · JAN 16 15

Unifying construc � on Q: A: CSL 2020 · JAN 16 15

Unifying construc � on Q: A: CSL 2020 · JAN 16 15

Unifying construc � on Q: A: IDEA (CMS): CSL 2020 · JAN 16 15

Unifying construc � on ▧ Fibra � ons are closed under type formers ▧ Fibra � ons par � cipate in a model structure CSL 2020 · JAN 16 16

Unifying construc � on ▧ Parameterized by category C with 𝕁 and Φ (+ axioms) AFH = CMS ( ) , , 𝕁 CCHM = CMS ( ) , , 𝕁 ▧ Also new models, e.g. cartesian w/ only faces in Φ CSL 2020 · JAN 16 17

Unifying construc � on ▧ Formulated following Orton & Pi � s ’16 (for CCHM), Angiuli, Brunerie, Coquand, Favonia, Harper, & Licata ’18 (for AFH) ◈ assume C interprets ordinary type theory ◈ describe axioms and construc � on in internal language ◈ enables straigh � orward formaliza � on (ours in Agda ) CSL 2020 · JAN 16 18

Unifying construc � on ▧ Model structure ◈ se � ng for homotopy theory ◈ following Sa � ler ’17 (for CCHM) ◈ use Swan ’18 to translate coercion r → s { C (co fi bra � ons): generated by Φ W (weak equivalences): equivalences F ( fi bra � ons): fi bra � ons ◈ Our ( C , W , F ) has F maximal such that families in F have coercion 0 → r CSL 2020 · JAN 16 19

Future work ▧ Original cubical model: Bezem, Coquand, & Huber ’13 ◈ Substructural: no diagonal maps between cubes { } { } ◈ De fi ni � ons of fi bra � on structure for types rely on the absence of diagonals ▧ How do cubical models relate to other models? CSL 2020 · JAN 16 20