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

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Unifying Cubical Models of Univalent Type Theory Evan Cavallo Anders Mrtberg Carnegie Mellon University Stockholm University Andrew W Swan University of Amsterdam CSL 2020 JAN 16 0 Univalent Type Theory Dependent type theory CSL

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CSL 2020 · JAN 16

Unifying Cubical Models

  • f Univalent Type Theory

Anders Mörtberg

Stockholm University

Evan Cavallo

Carnegie Mellon University

Andrew W Swan

University of Amsterdam

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CSL 2020 · JAN 16 1

▧ Dependent type theory

Univalent Type Theory

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CSL 2020 · JAN 16 1

▧ Dependent type theory

Univalent Type Theory

funcon/implicaon/∀ product/∃ inducve types equality universe(s) of types

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CSL 2020 · JAN 16 1

▧ Dependent type theory

Univalent Type Theory

funcon/implicaon/∀ product/∃ inducve types equality universe(s) of types

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CSL 2020 · JAN 16 2

▧ Identy

+

◈ Least reflexive relaon (⇒ symmetric, transive, etc.) ◈ “ Underdetermined”

Univalent Type Theory

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▧ Equivalence

Univalent Type Theory

▧ Univalence Axiom (Voevodsky)

◈

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CSL 2020 · JAN 16 4

▧ Idenes are not unique

Univalent Type Theory

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CSL 2020 · JAN 16 4

▧ Idenes are not unique

Univalent Type Theory

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CSL 2020 · JAN 16 4

▧ Idenes are not unique

Univalent Type Theory

▧ More: add higher inducve types

◈ Quoents for proof-relevant identy ◈ Language for synthec homotopy theory

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CSL 2020 · JAN 16 5

◈ Classical seng for homotopy theory

Models of Univalent Type Theory

▧ Simplicial set model (Kapulkin & Lumsdaine ’12/’18, aer Voevodsky)

◈ Essenally non-construcve (Bezem, Coquand, & Parmann ’15) ◈ First construcve model of univalence

▧ Cubical set model (Bezem, Coquand, & Huber ’13)

◈ Problems with higher inducve types resolved in Cohen, Coquand, Huber, & Mörtberg ’15 and Angiuli, Favonia, & Harper ’18 models

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CSL 2020 · JAN 16 6

Cubical Set Models

▧ Interpret contexts as cubical sets

{ } { } { }

◈ family of sets indexed by interval variable contexts

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CSL 2020 · JAN 16 7

Cubical Set Models

{ } { }

▧ Interpret contexts as cubical sets

◈ family of sets indexed by interval variable contexts ◈

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CSL 2020 · JAN 16 8

Cubical Set Models

▧ Interpret contexts as cubical sets

◈ family of sets indexed by interval variable contexts ◈

▧ Interpret types as fibraons

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CSL 2020 · JAN 16 9

Fibraons

▧ Part 1 (coercion): then “if ”

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CSL 2020 · JAN 16 9

Fibraons

▧ Part 1 (coercion): then “if ”

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CSL 2020 · JAN 16 9

Fibraons

▧ Part 1 (coercion): then “if ”

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CSL 2020 · JAN 16 9

Fibraons

▧ Part 1 (coercion): then “if ”

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CSL 2020 · JAN 16 10

Fibraons

▧ Part 1 (coercion): ▧ Part 2 (composion): a cube in A can be adjusted then “if ”

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CSL 2020 · JAN 16 10

Fibraons

▧ Part 1 (coercion): ▧ Part 2 (composion): a cube in A can be adjusted then “if ”

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CSL 2020 · JAN 16 10

Fibraons

▧ Part 1 (coercion): ▧ Part 2 (composion): a cube in A can be adjusted then “if ”

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CSL 2020 · JAN 16 10

Fibraons

▧ Part 1 (coercion): ▧ Part 2 (composion): a cube in A can be adjusted then “if ”

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CSL 2020 · JAN 16 10

Fibraons

▧ Part 1 (coercion): ▧ Part 2 (composion): a cube in A can be adjusted then “if ”

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CSL 2020 · JAN 16 11

Fibraons

▧ A fibraon is a family supporng these operaons ▧ Part 1 (coercion): ▧ Part 2 (composion): a cube in A can be adjusted then “if ”

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CSL 2020 · JAN 16 12

Two approaches

Cohen, Coquand, Huber, & Mörtberg ’15 ▧ Result: fibraons closed under type formers ▧

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CSL 2020 · JAN 16 13

Two approaches

Angiuli, Favonia, & Harper ’18 ▧ Result: fibraons closed under type formers ▧

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CSL 2020 · JAN 16 14

Two approaches

CCHM ▧

◈

AFH ▧

◈ ◈ ◈

Is there a unifying construcon that generalizes these?

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CSL 2020 · JAN 16 15

Unifying construcon

Q: A:

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CSL 2020 · JAN 16 15

Unifying construcon

Q: A:

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CSL 2020 · JAN 16 15

Unifying construcon

Q: A:

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CSL 2020 · JAN 16 15

Unifying construcon

Q: A:

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CSL 2020 · JAN 16 15

Unifying construcon

Q: A: IDEA (CMS):

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CSL 2020 · JAN 16 16

Unifying construcon

Fibraons are closed under type formers ▧ Fibraons parcipate in a model structure ▧

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CSL 2020 · JAN 16 17

Unifying construcon

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 ▧

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CSL 2020 · JAN 16 18

Unifying construcon

◈ assume C interprets ordinary type theory

Formulated following Orton & Pis ’16 (for CCHM),

Angiuli, Brunerie, Coquand, Favonia, Harper, & Licata ’18 (for AFH)

▧

◈ describe axioms and construcon in internal language ◈ enables straighorward formalizaon (ours in Agda)

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CSL 2020 · JAN 16 19

Unifying construcon

Model structure ▧

◈ seng for homotopy theory ◈ following Saler ’17 (for CCHM) ◈ use Swan ’18 to translate coercion r → s

{

C (cofibraons): generated by Φ W (weak equivalences): equivalences F (fibraons): fibraons ◈ Our (C, W, F) has F maximal such that families in F have coercion 0 → r

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CSL 2020 · JAN 16 20

Future work

Original cubical model: Bezem, Coquand, & Huber ’13 ▧

◈ Substructural: no diagonal maps between cubes

{ } { }

◈ Definions of fibraon structure for types rely on the absence of diagonals

How do cubical models relate to other models? ▧