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types ! s t e n m m o c s ' r t o c e i r d h i - - PowerPoint PPT Presentation

Apr 08, 2023 226 likes •874 views

RISE OF higher-dimensional types ! s t e n m m o c s ' r t o c e i r d h i t w favonia University of Minnesota elements some type relations elements some type a 1 b 1 a 2 b 2 p q f : A B a 2 p

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higher-dimensional

RISE OF

favonia

University of Minnesota

types

w i t h d i r e c t

  • r

' s c

  • m

m e n t s !

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elements some type

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relations elements some type

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f : A → B

a2 p q b2 ⟼ ⟼ a1 a2 p

A

b1 b2 q

B

a1 b1 ⟼

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Higher-dimensional types

provide novel abstraction that facilitates

the mechanization

  • f homotopy theory
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Higher-dimensional types

provide novel abstraction that facilitates

the mechanization

  • f homotopy theory

the abstraction

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Higher-dimensional types

provide novel abstraction that facilitates

the mechanization

  • f homotopy theory

the abstraction the work

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a b

Higher-Dimensional Types

(symmetric relations)

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a b

p

Higher-Dimensional Types

(symmetric relations)

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a b

q p

Higher-Dimensional Types

(symmetric relations)

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a b

q p

⋮ Higher-Dimensional Types

h k

(symmetric relations)

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Type Space Function Continuous Mapping Element Point Dependent Type Fibration Identiication Path A a : A f : A → B C : A → Type a =A b

Homotopy-Theoretic Interpretation

[Awodey and Warren] [Voevodsky et al] [van den Berg and Garner]
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New Features

Univalence

if e is an equivalence between types A and B, then ua(e):A=B

Higher Inductive Types

circle sphere torus

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Mechanizing theorems in univalent type theory

Higher-dimensional types

provide novel abstraction for mechanization

[experiments] [statement]

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My Thesis + Follow-Ups

Covering spaces Seifert-van Kampen theorem Blakers-Massey theorem

[Favonia and Harper] [Shulman and Favonia] [Lumsdaine, Finster, Licata, Brunerie and Favonia] [Buchholtz and Favonia]

Homotopy groups Cohomology groups

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— Algebraic Topology by Allen Hatcher

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— Algebraic Topology by Allen Hatcher

Blakers-Massey Covering spaces Seifert- van Kampen

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Homotopy Groups

{ mappings from the n-sphere }

A Sn

“higher” if n > 1

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First Homotopy Group

{ mappings from the circle }

A S1

directed loops at some point

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a

A =

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a a

{

A =

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a a a a ...

{

A =

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a a a a ... a a ...

{

A =

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a a a a ... a a ... a (much more)

{

A = A =

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Sets with Actions

Instead of computing these groups directly, consider sets with an action by the groups

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Sets with Actions

Instead of computing these groups directly, consider sets with an action by the groups

a a a ... a a ...

{ {

×

set S the group action: set S

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Sets with Actions

Instead of computing these groups directly, consider sets with an action by the groups

a a a ... a a ...

{ {

×

set S the group

Subject: sets with

action: some action by the irst homotopy group set S

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Sets with some action by the

irst homotopy group

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Sets with some action by the

irst homotopy group

≃

Covering spaces

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Covering Spaces

F : A → Set

Classical deinition

A covering space of A is a space C together with a continuous surjective map p : C → A, such that for every a ∈ A, there exists an open neighborhood U of a, such that p-1(U) is a union of disjoint open sets in A, each of which is mapped homeomorphically onto U by p.

Type-theoretic deinition

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Theorem*

Covering spaces F : A → Set

*A is pointed and connected

Sets with some action by the

irst homotopy group of A

≃

More results in “Higher Groups in Homotopy Type Theory”

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Pushouts

A B Disjoint sums + gluing

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Pushouts

A B Disjoint sums + gluing C

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Pushouts

A B Disjoint sums + gluing c f(c) g(c) C

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First Homotopy Groups

all paths from a point to itself all paths between any two points

Fundamental Groupoids

[ generalization ]

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A B C All paths are sequences of alternating paths in A and B

[ Seifert–van Kampen ]

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A B C

[ Seifert–van Kampen ]

Paths of the pushout can be calculated from paths of A and B and points of C

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

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Homotopy Groups of Spheres

S1 0 Z Z2 Z Z2 Z12 Z2 Z2 Z Z2 Z12 Z2 Z 0 0 0 S2 S3 S4 S5 S6 0 0 0 Z2 Z Z2 Z×Z12 Z2

2 Z2 2 Z24×Z3

Z2 Z3 Z15 Z2 Z3 Z15 0 0 0 Z2 Z Z2 Z24 Z2 Z2 0 0 0 Z2 Z Z2 Z24 1 2 3 4 5 6 7 8 9 10

[ Blakers-Massey ]

corollary of

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Blakers-Massey in homotopy type theory

(2012-13)

[Finster, Licata, Lumsdaine]

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Blakers-Massey in homotopy type theory Full mechanization of Blakers-Massey in Agda

(2012-13) (2013)

[Finster, Licata, Lumsdaine] [Licata?] [Favonia]

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Blakers-Massey in homotopy type theory Full mechanization of Blakers-Massey in Agda Un-mechanization into classical theory

(2012-13) (2013) (2014)

unpublished [Rezk] [Finster, Licata, Lumsdaine] [Licata?] [Favonia]

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Blakers-Massey in homotopy type theory Full mechanization of Blakers-Massey in Agda Un-mechanization into classical theory Generalization

(2012-13) (2013) (2014) (2016 or earlier)

Generalization available on arXiv

(2017)

1703.09050 unpublished [Rezk] [Finster, Licata, Lumsdaine] [Licata?] [Favonia]

[Anel, Biedermann, Finster, Joyal]

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Blakers-Massey in homotopy type theory Full mechanization of Blakers-Massey in Agda Un-mechanization into classical theory Generalization

(2012-13) (2013) (2014) (2016 or earlier)

Generalization available on arXiv

(2017)

Mechanization published

(2016)

1703.09050 unpublished [Rezk] [Finster, Licata, Lumsdaine] [Licata?] [Favonia] [FFLL]

[Anel, Biedermann, Finster, Joyal]

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Blakers-Massey in homotopy type theory Full mechanization of Blakers-Massey in Agda Un-mechanization into classical theory Generalization Mechanization of the generalization in Agda?

(2012-13) (2013) (2014) (2016 or earlier)

Generalization available on arXiv

(2017)

Mechanization published

(2016)

1703.09050

(2017-?)

unpublished [Rezk] [Finster, Licata, Lumsdaine] [Licata?] [Favonia] [FFLL]

[Anel, Biedermann, Finster, Joyal]

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Homology Groups

{ holes in a space }

Cohomology Groups

{ mappings from holes in a space }

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Homology Groups

{ holes in a space }

Cohomology Groups

{ mappings from holes in a space }

Easier than homotopy groups for many spaces of interest

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Homology Groups of Spheres

S1 Z 0 0 Z S2 Z S3 S4 0 Z S5 0 Z S6 Z 1 2 3 4 5 6

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Higher-dimensional types

provide novel abstraction that facilitates the mechanization of homotopy theory

[experiments] [statement]

Covering spaces Seifert-van Kampen theorem Blakers-Massey theorem Cohomology groups

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Post-Thesis

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Post-Thesis

Involved in the development of cubical type theory

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Post-Thesis

Involved in the development of cubical type theory

Ask Bob and his students

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Thanks to Bob, the PoP group, the HoTT community, CMU sta (esp. Deborah and Catherine), my spouse, and many, many people

...also Mark Rothko for artistic inspiration

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Have fun!