Seifert-van Kampen Theorem in Homotopy Type Theory [ Toronto - - PowerPoint PPT Presentation

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Jun 07, 2023 353 likes •800 views

Seifert-van Kampen Theorem in Homotopy Type Theory [ Toronto version ] Michael Shulman @ U of San Diego * Favonia @ CMU 1 Homotopy Type Theory * Type theory <-> topology - types ~= spaces - terms ~= points - functions ~= continuous

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Seifert-van Kampen Theorem in Homotopy Type Theory

* Favonia @ CMU Michael Shulman @ U of San Diego

[ Toronto version ]

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Homotopy Type Theory

types ~= spaces
terms ~= points
functions ~= continuous maps
identifications ~= paths

* Non-trivial identifications * Type theory <-> topology

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a b

terms

Iterated Paths

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a b

terms paths

Iterated Paths

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a b

terms paths

Iterated Paths

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a b

terms paths paths of paths

Iterated Paths

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a b f(a) f(b) ~> f A B Functorial

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Subject of Study

fundamental groups of pushouts

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Subject of Study

"structure of loops" fundamental groups of pushouts

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Subject of Study

"structure of loops" fundamental groups of pushouts "disjoint union added with bridges"

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Fundamental Groups a

(unique) ways to travel from a to a

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Fundamental Groups a

(unique) ways to travel from a to a

here they correspond to integers

positive <--> clockwise negative <--> counter zero <--> staying

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Fundamental Groups a

(unique) ways to travel from a to a

here they correspond to integers

positive <--> clockwise negative <--> counter zero <--> staying

Trunc 0 (a == a) ~= Z

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Fundamental Groups a

(unique) ways to travel from a to a

much more if a new path is added

Trunc 0 (a == a) ~= Z * Z

(free product)

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(Homotopy) Pushouts A B C

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(Homotopy) Pushouts A B C

data Pushout (A B C : Type) (f : C -> A) (g : C -> B) : Type where left : A -> Pushout A B C f g right : B -> Pushout A B C f g glue : (c : C) -> left (f c) == right (g c)

c f(c) g(c)

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A B C

ways to travel from to ?

(Homotopy) Pushouts

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A B C (Homotopy) Pushouts

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A B C

alternative paths in A and B

(Homotopy) Pushouts

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Theorem Statement

fund.grp(pushout) ~= ?(??(A), ??(B), C)

??: paths between any two points for any A, B, C, f and g,

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Fundamental Groupoids a

(unique) ways to travel from a to b

b

Trunc 0 (a == b)

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Theorem Statement

fund.groupoid(pushout) ~= ?(fund.groupoid(A), fund.groupoid(B), C) ?: "seqs of alternative elems" for any A, B, C, f and g,

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Alternative Sequences

induction on both ends: A to A, A to B, B to A, B to B [p1, p2, ..., pn] A B

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Alternative Sequences

A B = A B A B = A B quotients of alternative sequences by killing trivial identifications

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Alternative Sequences

each case is a quotient

f alternative sequences

induction on both ends: A to A, A to B, B to A, B to B [p1, p2, ..., pn] A B

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Alternative Sequences

next: unify four cases into

ne type family alt.seq

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Alternative Sequences

next: unify four cases into

ne type family alt.seq

alt.seq a (f c) ~= alt.seq a (g c)

~=

show respects for bridges by C. ex:

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alt.seq a (f c) ~= alt.seq a (g c)

~=

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18 [..., p] |--> [..., p, trivial]

alt.seq a (f c) ~= alt.seq a (g c)

~=

|-->

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18 [..., p] |--> [..., p, trivial]

alt.seq a (f c) ~= alt.seq a (g c)

[..., p, trivial] <--| [..., p]

~=

|--> <--|

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Alternative Sequences

A -> A A -> B B -> A B -> B

seq a (f c) ~= seq a (g c) seq b (f c) ~= seq b (g c) seq (f c) a ~= seq (g c) a seq (f c) b ~= seq (g c) b

commutes

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Theorem

fund.groupoid(pushout) ~= alt.seqs(fund.groupoid(A), fund.groupoid(B), C)

(zero pages left before the proofs)

for any A, B, C, f and g,

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Recipe of Equivalences

* two functions back and forth ("decode" and "encode") * round-trips are identity

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fund.groupoid -> alt.seqs

(all paths)

encode

Trunc 0 (p == q) -> alt.seqs p q

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fund.groupoid -> alt.seqs

path induction: consider only trivial paths

(all paths)

encode

Trunc 0 (p == q) -> alt.seqs p q (p : Pushout) -> alt.seqs p p

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fund.groupoid -> alt.seqs

path induction: consider only trivial paths

(all paths)

encode

Trunc 0 (p == q) -> alt.seqs p q (p : Pushout) -> alt.seqs p p

pushout induction

A B A A B B

bridges by C (next page)

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A B A B A B

applying the diagonal in coherence square

=?

case A case B

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A B A B A B

applying the diagonal in coherence square

=?

case A case B witnessed by the quotient

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alt.seq -> fund.groupoid

just compositions!

decode

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alt.seq -> fund.groupoid

just compositions!

decode

grpd -> seqs -> grpd

again by path induction (similar to "encode")

decode encode

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alt.seq -> fund.groupoid

just compositions!

decode

grpd -> seqs -> grpd

again by path induction (similar to "encode")

seqs -> grpd -> seqs

induction on sequences

lemma: encode(decode[p1,p2,...]) = p1 :: encode(decode[p2,...])

decode encode encode decode

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Theorem

fund.groupoid(pushout) = alt.seqs(fund.groupoid(A), fund.groupoid(B), C) for any A, B, C, f and g,

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Final Notes

* Refined version: Can focus on just the set of base points of C covering its components.

github.com/HoTT/HoTT-Agda/blob/1.0/Homotopy/VanKampen.agda

* All mechanized in Agda

www.cs.cmu.edu/~kuenbanh/files/vankampen.pdf