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

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 b 2 q a 1 b 1 A B

Higher-dimensional types provide novel abstraction that facilitates the mechanization of homotopy theory

the abstraction Higher-dimensional types provide novel abstraction that facilitates the mechanization of homotopy theory

the abstraction Higher-dimensional types provide novel abstraction that facilitates the mechanization of homotopy theory the work

Higher-Dimensional Types (symmetric relations) b a

Higher-Dimensional Types (symmetric relations) b p a

Higher-Dimensional Types (symmetric relations) b p a q

⋮ Higher-Dimensional Types (symmetric relations) b p h k a q

Homotopy-Theoretic Interpretation [Awodey and Warren] [Voevodsky et al ] [van den Berg and Garner] A Type Space a : A Element Point f : A → B Function Continuous Mapping C : A → Type Dependent Type Fibration a = A b Identi � ication Path

New Features Univalence if e is an equivalence between types A and B , then ua(e):A=B Higher Inductive Types sphere circle torus

[statement] Higher-dimensional types provide novel abstraction for mechanization [experiments] Mechanizing theorems in univalent type theory

My Thesis + Follow-Ups Homotopy groups Covering spaces [Favonia and Harper] Seifert-van Kampen theorem [Shulman and Favonia] Blakers-Massey theorem [Lumsdaine, Finster, Licata, Brunerie and Favonia] Cohomology groups [Buchholtz and Favonia]

— Algebraic Topology by Allen Hatcher

Seifert- Covering van Kampen spaces Blakers-Massey — Algebraic Topology by Allen Hatcher

Homotopy Groups { mappings from the n -sphere } S n A “higher” if n > 1

First Homotopy Group { mappings from the circle } S 1 A directed loops at some point

A = a

{ a A = a

{ ... a a a A = a

{ ... a a a A = a ... a a

{ ... a a a A = a ... a a A = a (much more)

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

Sets with Actions Instead of computing these groups directly, consider sets with an action by the groups { { ... a a a × action: ... a a the group set S set S

Sets with Actions Instead of computing these groups directly, consider sets with an action by the groups { { ... a a a × action: ... a a the group set S set S some action by the Subject: sets with � irst homotopy group

Sets with some action by the � irst homotopy group

Sets with some action by the � irst homotopy group ≃ Covering spaces

Covering Spaces Classical de � inition 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 de � inition F : A → Set

Theorem* Sets with some action by the � irst homotopy group of A ≃ Covering spaces F : A → Set More results in “Higher Groups in Homotopy Type Theory” *A is pointed and connected

Pushouts Disjoint sums + gluing A B

Pushouts Disjoint sums + gluing C A B

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

First Homotopy Groups all paths from a point to itself [ generalization ] Fundamental Groupoids all paths between any two points

C A B [ Seifert–van Kampen ] All paths are sequences of alternating paths in A and B

C A B [ Seifert–van Kampen ] Paths of the pushout can be calculated from paths of A and B and points of C

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0

Homotopy Groups of Spheres 1 2 3 4 5 6 7 8 9 10 S 1 Z 0 0 0 0 0 0 0 0 0 S 2 0 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 S 3 0 0 Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 2 Z 2 2 Z 24 ×Z 3 S 4 0 0 0 Z Z 2 Z 2 Z×Z 12 Z 2 S 5 0 0 0 0 Z Z 2 Z 2 Z 24 Z 2 Z 2 S 6 0 0 0 0 0 Z Z 2 Z 2 Z 24 0 corollary of [ Blakers-Massey ]

Blakers-Massey in homotopy type theory (2012-13) [Finster, Licata, Lumsdaine]

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia]

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk]

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk] Generalization [Anel, Biedermann, (2016 or earlier) Finster, Joyal] Generalization available on arXiv (2017) 1703.09050

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk] Generalization Mechanization [Anel, Biedermann, (2016 or earlier) Finster, Joyal] published Generalization (2016) [FFLL] available on arXiv (2017) 1703.09050

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk] Generalization Mechanization [Anel, Biedermann, (2016 or earlier) Finster, Joyal] published Generalization (2016) [FFLL] available on arXiv Mechanization of the (2017) 1703.09050 generalization in Agda? (2017-?)

Homology Groups { holes in a space } Cohomology Groups { mappings from holes in a space }

Homology Groups { holes in a space } Cohomology Groups { mappings from holes in a space } Easier than homotopy groups for many spaces of interest

Homology Groups of Spheres 1 2 3 4 5 6 S 1 Z 0 0 0 0 0 S 2 0 Z 0 0 0 0 S 3 0 0 Z 0 0 0 S 4 0 0 0 Z 0 0 S 5 0 0 0 0 Z 0 S 6 0 0 0 0 0 Z

[statement] Higher-dimensional types provide novel abstraction that facilitates the mechanization of homotopy theory [experiments] Covering spaces Seifert-van Kampen theorem Blakers-Massey theorem Cohomology groups

Post-Thesis

Post-Thesis Involved in the development of cubical type theory

Post-Thesis Involved in the development of cubical type theory Ask Bob and his students