Covering Spaces in Homotopy Type Theory Favonia Carnegie Mellon - - PowerPoint PPT Presentation

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Covering Spaces

inHomotopy Type Theory

Carnegie Mellon University favonia@cmu.edu This material is based upon work supported by the National Science Foundation under Grant No. 1116703. This material is based upon work supported by the National Science Foundation under Grant No. 1116703.

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Why bother?

Fundamental Groups!

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[ computer checked ]

This work is covered by Agda

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Covered Topics Classification Universality

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Part 0

Definition

7 Definition of Covering Spaces base cover covering projection

7 Definition of Covering Spaces base cover covering projection

7 Definition of Covering Spaces base cover covering projection

7 Definition of Covering Spaces base cover covering projection exact copies

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Set

Definition of Covering Spaces

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HoTT True Facts #28

Continuity is free!

Set

Definition of Covering Spaces

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HoTT True Facts #28

Continuity is free!

Set

Definition of Covering Spaces

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A → Set

Cover over A

Definition of Covering Spaces

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A → Set

fiber

• ver a

a Cover over A

Definition of Covering Spaces

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A → Set

fiber

• ver a

a Cover over A

It is a functor! Definition of Covering Spaces

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A → Set

fiber

• ver a

a Cover over A

It is a functor!

set₁ set₂ a₁ a₂

Definition of Covering Spaces

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A → Set

fiber

• ver a

a Cover over A

It is a functor!

set₁ set₂ a₁ a₂ iso

Definition of Covering Spaces

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A → Set

Cover over A

path-connected?

fiber

• ver a

a

Definition of Covering Spaces

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A → Set

Cover over A

path-connected? pointed?

fiber

• ver a

a

Definition of Covering Spaces

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circle

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circle

?

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circle

?

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Part 1

Classification

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Goal

Find representations

• f covering spaces

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path-connected

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path-connected

transport

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path-connected

a p p

transport

a

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Green part: "fixable" Yellow + Blue: inherent twists

For example…

p q r

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Green part: "fixable" Yellow + Blue: inherent twists

For example…

p p q r

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Green part: "fixable" Yellow + Blue: inherent twists

For example…

p q p q r

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Green part: "fixable" Yellow + Blue: inherent twists

For example…

p q r p q r

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Green part: "fixed" Yellow + Blue: inherent twists

For example…

p q r p q r

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Green part: "fixed" Green part: "fixed" Yellow + Blue: inherent twists

Loops

p q r p q r

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Green part: "fixed" Green part: "fixed" Yellow + Blue: inherent twists

Loops Automorphisms

p q r p q r

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1→1 1→1 2→2 1→2 2→1 1→1 2→3 3→2 It is sufficient to check the generator loop

loop

For circles…

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loop 1 . S e t X

• 2. Automorphisms by

different loops loop

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loop 1 . S e t X

• 2. Automorphisms by

different loops loop

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Fundamental Group

Sets of loops based at a point

= =

id

=

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loop elements in

fundamental group

1 . S e t X

• 2. Automorphisms by

different loops loop

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map from G to automorphisms of X

Fix G = fundamental group

A G-set is a set X with an action of G

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map from G to automorphisms of X

with functoriality… id g1 g2 g1 g2

Fix G = fundamental group

A G-set is a set X with an action of G

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G-set Classification Theorem

G-sets and covering spaces are equivalent. A set X equipped with an action, a map from G to automorphisms

Fix G = fundamental group

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It is sufficient to check the generator loop

loop

For circles…

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loop

successor

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Proof

Cover G-Set

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Set

• 1. Cover → G-set

transport

as the action

loop

loop

(restricted to loops)

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• 2. Cover → G-set → Cover

Given a G-set = a set X and an action Given a G-set = a set X and an action

Construct a cover

• 1. Every fiber is isomorphic to X
• 2. Transport is the action

(restricted to loops) (restricted to loops) such that such that

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• 2. Cover → G-set → Cover

Given a G-set = a set X and an action Given a G-set = a set X and an action

Construct a cover

Magic: Higher inductive types

• 1. Every fiber is isomorphic to X
• 2. Transport is the action

(restricted to loops) (restricted to loops) such that such that

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X

Other fibers missing

X was a fiber

• 2. Cover → G-set → Cover

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Base path p

would induce an isomorphism

p

(by “transport”)

α β

• 2. Cover → G-set → Cover

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Base path p

would induce an isomorphism

p

(by “transport”)

Fake it with a formal one! α β Point β is p α

• 2. Cover → G-set → Cover

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p

α β Point β is p α

data R (a : A) : Set : ∀ p α → R a

formal transport

• 2. Cover → G-set → Cover

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p q

Different q’s give

different copies

Needs a way to merge copies from different base paths

• 2. Cover → G-set → Cover

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p q

?

α

If it will be some cover…

• 2. Cover → G-set → Cover

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p q

q must be (q p⁻¹) p

?

loop

α

loop = If it will be some cover…

• 2. Cover → G-set → Cover

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p q

q must be (q p⁻¹) p

loop

α

loop =

q α = (loop p) α = p (loop α)

If it will be some cover…

Key: functoriality

• 2. Cover → G-set → Cover

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p q loop

α

We mimic functoriality

q α = (loop p) α = p (loop α) q α = (loop p) α = p (loop α) action is transport for loops

Going back to the construction…

• 2. Cover → G-set → Cover

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p q loop

α

data R (a : A) : Set : ∀ p α → R a : ∀ l p α → (l p) α = p (l α) We mimic functoriality

q α = (loop p) α = p (loop α)

• 2. Cover → G-set → Cover

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→ (l p) α data R (a : A) : Set : ∀ p α → R a : ∀ l p α

R is equivalent to the original cover R is equivalent to the original cover

Acknowledgements: Thanks to Guillaume Brunerie, Daniel Grayson and Chris Kapulkin for helping me state and prove this.

Theorem Theorem

= p (l α)

• 2. Cover → G-set → Cover

39 39

Classification Theorem

If G is the fundamental group G-sets and covering spaces are equivalent.

[ recap ] [ recap ]

40 40 Technical Notes

All truncations were omied. You want this lemma: WARNING: NASTY MATH AHEAD

Given a constant (pointwise-equal) function f : A → B where B is a set find a g : ||A|| → B such that f = g · | - | A B ||A|| f g | - |

41 41

Part 2

Universality

covers that cover every cover covers that cover every cover

42 42 Universality

Assumption: Everything is path-connected and pointed Universal

42 42 Universality

Assumption: Everything is path-connected and pointed Universal

u n i q u e

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set of paths with one end fixed

A simple universal cover

pointed

Assumption: Everything is pointed and path-connected.

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set of paths with one end fixed

Technical Notes

WARNING: NASTY MATH AHEAD

λ x . || = x ||₀

The simple universal cover is x

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set of paths with one end fixed

Technical Notes

WARNING: NASTY MATH AHEAD

λ x . || = x ||₀

The simple universal cover is x Path induction!

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Theorem

It is inital.

Simply Connected

• ne and only one

It is equivalent to any simply connected cover. It is inital.

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circle

46 46

circle

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circle

Z1 Z3 Z2

Z

Group quotients!

47 47

Theorem

It is inital.

Simply Connected

• ne and only one

It is equivalent to any simply connected cover.

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• ne to many

p q p q

transport

pointed

Weak Initiality

48 48

• ne to many

p q p q

transport

pointed

Weak Initiality

quotient

=

49 49 Strong Initiality

p

cover₁ cover₂

Sufficient to consider p = identity path and ( collide) =?

pointed

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Theorem

It is inital. It is equivalent to any simply connected cover.

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• ne to one

simply connected

pointed

p q p q If lied p and q are the same…

51 51

• ne to one

simply connected

pointed

p q p q

s.c. identifies lied paths

If lied p and q are the same…

=

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• ne to one

simply connected

pointed

p q p q

projection is retraction of liing s.c. identifies lied paths

If lied p and q are the same…

= = =

52 52

Theorem

It is inital. It is equivalent to any simply connected cover.

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fiber over

fundamental group

=

54 54

Agda code

github.com/HoTT/HoTT-Agda/blob/2.0 github.com/HoTT/HoTT-Agda/blob/2.0

54 54

Agda code

github.com/HoTT/HoTT-Agda/blob/2.0 github.com/HoTT/HoTT-Agda/blob/2.0

Thanks

55 55 Definition of Path Homotopy continuous deformation

Path of paths

55 55 Definition of Path Homotopy

Identity of identity types

x y q p p : x = y q : x = y h : p = q