89 lines
3.1 KiB
Org Mode
89 lines
3.1 KiB
Org Mode
#+OPTIONS: num:nil
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#+AUTHOR: JLH
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#+AUTHOR: KL
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* Planning The HoTT Game
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** Aims of the HoTT Game
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- To get mathematicians with no experience in proof verification interested in HoTT and able to use Agda for HoTT
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- Big-ass-boss: Loop space of S^1 = Z
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- Try to balance hiding cubical implementations whilst exploiting their advantages
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** Barriers
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- HOLD Installation of emacs
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- TODO Usage of emacs
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- TODO General type theoretic foundations
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- TODO Cubical type theory
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** Format
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- [?] Everything done in .agda files
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- Partially written code with spaces for participants to fill in + answer files
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- Levels set out with mini-bosses like in Nat Num Game, but with an overall boss
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- [?] Side quests
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- References to Harper lectures and HoTT book
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** Content
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# listing topics we have pursued, NO ordering
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- emacs usage
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- agda usage + basic commands (all covered in https://agda.readthedocs.io/en/v2.6.0.1/getting-started/quick-guide.html)
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+ recommend doom emacs + implicit/explicit arguments
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+ holes and inferred types
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+ =_+_= vs =plus__=
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- type theory basics + meta (judgemental/definitional) equality vs internal (propositional) equality
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- function extensionality
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+ type formation
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- inductive types + (side Q) positive and negative constructions of Pi/Sigma types
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+ =data= and =record= + universes
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+ recursors / pattern matching + (side Q) some natural number exercises as early evidence of being able to 'do maths'?
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+ different notions of equivalence
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- fibers contractable
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- quasi-inverse
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- zig-zag + (side Q) types are infinity groupoids
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+ extra paths (univalence, fun ext, HITs)
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- HoTT + basics
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- meta interval, identity type vs path type
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+ mention identity type for compatability with other sources, but just use path type
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- path type on other types
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- dependent path type PathP vs path over
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- univalence
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- the (non)-issue of J in (Cu)TT
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- =isContr, isProp, isSet=
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- drawing pictures + Structures, using univalence to transport
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- transporting results between isomorphic structures
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+ HITs, examples
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- the constructed interval
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- booleans and covers
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- =S^n=
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- =S^1= with 2 cw structures equiv + Homotopy n-types
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- homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT
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+ in particular sigma types
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** Debriefs
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- 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.
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+ 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.
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+ difficulty is that PathP not in one fiber, but PathOver is, AND PathOver <-> PathP NON-obvious
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+ Easy to generalize situation to n-types being closed under Sigma (7.1.8 in HoTT book), we showed this assuming PathPIsoPath
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** Minibosses
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- _+_ unique on the naturals + axiomatize addition on naturals
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+ naturals is a set + fun extensionality
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+ contractability + propositions
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+ propositions closed under sigma types
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