#+OPTIONS: num:nil
#+AUTHOR: JLH
#+AUTHOR: KL

* 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
  - Big-ass-boss: Loop space of S^1 = Z
  - 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
- agda usage
  + basic commands (all covered in https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html)
  + recommend doom emacs
  + implicit/explicit arguments
  + holes and inferred types
  + =_+_= vs =plus__=
- type theory basics
  + meta (judgemental/definitional) equality vs internal (propositional) equality
    - function extensionality
  + type formation
    - inductive types
      + (side Q) positive and negative constructions of Pi/Sigma types
      + =data= and =record=
  + universes
  + recursors / pattern matching
  + (side Q) some natural number exercises as early evidence of being able to 'do maths'?
  + different notions of equivalence
    - fibers contractable
    - quasi-inverse
    - zig-zag
  + (side Q) types are infinity groupoids
  + extra paths (univalence, fun ext, HITs)
- HoTT
  + basics
    - meta interval, identity type vs path type
      + mention identity type for compatability with other sources, but just use path type
    - path type on other types
    - dependent path type PathP vs path over
    - univalence
    - the (non)-issue of J in (Cu)TT
    - =isContr, isProp, isSet=
    - drawing pictures
  + Structures, using univalence to transport
    - transporting results between isomorphic structures
  + HITs, examples
    - the constructed interval
    - booleans and covers
    - =S^n=
    - =S^1= with 2 cw structures equiv
  + Homotopy n-types
    - 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

** Mixolydian Bosses

+ universe classifies bundles

** SuperUltraMegaHyperLydianBosses
+ natural number object unique and `_+_` unique on any nat num obj
  + nat num obj unique
  + `_+_` unique on a model of nat num obj
    - axiomatize addition on naturals
    - naturals is a set
    - fun extensionality
    - contractability
    - propositions
    - propositions closed under sigma types
  + univalence

** Top 100 (set theoretic) misconceptions about type theory
+ Propositions
+ Proof relevance
+ Propositions are _inside the theory_
+ Membership not the same as :
  + typing is unique (doesn't make sense to intersect two types)
  +
+ Though set theory had fewer axioms type theory's assumptions are
  more intuitive (hence intionistic type theory).
  There is no fiddling about with membership to construct things
  e.g. cartesian product
+ 'we cannot use LEM' ~ not assuming law of excluded middle _globally_ means
  type theory theorems are stronger!

**  Fundamental group of S1
+ def of S¹
+ a bunch of stuff about ℤ
+ isoToPath to make ℤ ≡ ℤ
+ subst
+ funExt
+ set-truncation
+ paths as equality