Homotopy Type Theory in Agda 17|7|7 1 Goal synthetic homotopy - - PowerPoint PPT Presentation

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Aug 14, 2022 206 likes •362 views

Homotopy Type Theory in Agda 17|7|7 1 Goal synthetic homotopy theory in Agda + other needed theories 2 Goal synthetic homotopy theory in Agda + other needed theories Agda and Coq were the only two immediately usable systems for HoTT 2

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1

Homotopy Type Theory

in Agda

17|7|7

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2

Goal

synthetic homotopy theory in Agda + other needed theories

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2

Goal

synthetic homotopy theory in Agda + other needed theories

Agda and Coq were the only two immediately usable systems for HoTT

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3

Decentralized Dev.

HoTT/Agda-HoTT favonia/homotopy [obsolete] nicolaikraus/HoTT-Agda [fork] dlicata335/hott-agda guillaumebrunerie/JamesConstruction ...

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3

Decentralized Dev.

HoTT/Agda-HoTT favonia/homotopy [obsolete] nicolaikraus/HoTT-Agda [fork] dlicata335/hott-agda guillaumebrunerie/JamesConstruction ...

porting theorems and forking are common

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3

Decentralized Dev.

HoTT/Agda-HoTT favonia/homotopy [obsolete] nicolaikraus/HoTT-Agda [fork] dlicata335/hott-agda guillaumebrunerie/JamesConstruction ...

porting theorems and forking are common

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4 HoTT/Agda-HoTT

generalized Blakers-Massey (WIP)
total space of Hopf, 3x3 lemma
Seifert-van Kampen theorem
Mayer–Vietoris sequences
cubical reasoning
Freudenthal suspension theorem
Eilenberg-MacLane spaces K(G,n)
...

Guillaume Brunerie, Kuen-Bang Hou (Favonia), Evan Cavallo, Eric Finster, Jesper Cockx, Christian Sattler, Chris Jeris and Michael Shulman

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5 Used Features

inductive-inductive & inductive-recursive
MLTT-style logic/programming languages
powerful mixfix parser
...
universe polymorphism
pattern matching

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5 Used Features

inductive-inductive & inductive-recursive
MLTT-style logic/programming languages
powerful mixfix parser
...
universe polymorphism
pattern matching
higher-order unification

Used Automation

literal overloading
FEW tactics

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6 Higher Inductive Types?

Simulated by rewriting rules in HoTT-Agda

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6 Higher Inductive Types?

Simulated by rewriting rules in HoTT-Agda

postulate S¹ : Type₀ base : S¹ loop : base == base module S¹Elim {l}{P : S¹ → Type l} (base* : P base) (loop* : base* == base* [ P ↓ loop ]) where postulate f : Π S¹ P base-β : f base ↦ base* {-# REWRITE base-β #-} postulate loop-β : apd f loop == loop*

* effectively the same as Dan's trick

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7 Semantics of Agda

NOT well-understood (as a whole)
Many individual features proved

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7 Semantics of Agda

NOT well-understood (as a whole)
Many individual features proved

Mode of Usage

Highly experimental

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8 core/ theorems/

basic synthetic homotopy theory interesting results

Structures and Stats

continuous integration through travis

[10520 code + 1024 comments] [16107 code + 1577 comments]

the entire codebase can be checked in 20-30 mins