# 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
- 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
 <!-- listing topics we have pursued, NO ordering -->
- 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
    a) fibers contractable
    b) quasi-inverse
    c) zig-zag
  - types are infinity groupoids
  - positive and negative constructions of Pi/Sigma types
- HoTT
  - basics
    a) meta interval, identity type vs path type
    b) path type on other types
    c) dependent path type PathP vs path over
    d) univalence
    e) the (non)-issue of J in (Cu)TT
    f) isContr, isProp, isSet
  - Structures, univalence and transport
    a) transporting results between isomorphic structures
  - HITs, examples
    a) the constructed interval
    b) booleans and covers
    c) S^n
    d)
  - Homotopy n-types
    a) 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