TheHoTTGame/Plan.org

Planning The HoTT Game

• Aims of the HoTT Game
  • To get mathematicians with no experience in proof verification interested in HoTT and able to use Agda for HoTT
  • [?] Work towards showing an interesting result in HoTT
  • Try to balance hiding cubical implementations whilst exploiting their advantages
• Barriers
  • HOLD Installation of emacs
  • Usage of emacs
  • General type theoretic foundations
  • Cubical type theory
• Format
  • [?] Everything done in .agda files
  • Partially written code with spaces for participants to fill in + answer files
  • Levels set out with mini-bosses like in Nat Num Game, but with an overall boss
  • [?] Side quests
  • References to Harper lectures and HoTT book
• Content
• Debriefs

Aims of the HoTT Game

To get mathematicians with no experience in proof verification interested in HoTT and able to use Agda for HoTT

[?] Work towards showing an interesting result in HoTT

Try to balance hiding cubical implementations whilst exploiting their advantages

Barriers

HOLD Installation of emacs

TODO Usage of emacs

TODO General type theoretic foundations

TODO Cubical type theory

Format

[?] Everything done in .agda files

Partially written code with spaces for participants to fill in + answer files

Levels set out with mini-bosses like in Nat Num Game, but with an overall boss

[?] Side quests

References to Harper lectures and HoTT book

Content

• emacs usage

  • `data` and `record`
  • basic commands (all covered in https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html)
  • recommend doom emacs? -> basic doom usage and command differences with nude agda.
  • implicit/explicit arguments
  • holes and inferred types
  • `_+_` and `plus__`

• type theory basics

  • meta (judgemental/definitional) equality vs internal (propositional) equality
  • constructing types in universes
  • universes
  • recursors / pattern matching
  • side quest: some natural number exercises as early evidence of being able to 'do maths'?

  • different notions of equivalence

    1. fibers contractable
    2. quasi-inverse
    3. zig-zag
  • types are infinity groupoids
  • positive and negative constructions of Pi/Sigma types

• HoTT

  • basics

    1. meta interval, identity type vs path type
    2. path type on other types
    3. dependent path type PathP vs path over
    4. univalence
    5. the (non)-issue of J in (Cu)TT
    6. isContr, isProp, isSet

  • Structures, univalence and transport

    1. transporting results between isomorphic structures

  • HITs, examples

    1. the constructed interval
    2. booleans and covers
    3. S^n

  • Homotopy n-types

    1. homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT

       • in particular sigma types

Debriefs

• 2021 July 15; Homotopy n-types

  • watched (Harper) lecture 15 on Sets being closed under type formations ->- motivates showing in Agda Sets closed under Sigma.
  • Harper does product case, claiming sigma case follows analogously,
  • attempt proof in Cubical Agda but highly non-obvious how to use that fibers are Sets.
  • difficulty is that PathP not in one fiber, but PathOver is, AND PathOver <-> PathP NON-obvious
  • Easy to generalize situation to n-types being closed under Sigma (7.1.8 in HoTT book), we showed this assuming PathPIsoPath