68 lines
2.6 KiB
Markdown
68 lines
2.6 KiB
Markdown
# 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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- [?] Work towards showing an interesting result in HoTT
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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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- `data` and `record`
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- 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? -> basic doom usage and command differences with nude agda.
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- implicit/explicit arguments
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- holes and inferred types
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- `_+_` and `plus__`
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- type theory basics
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- meta (judgemental/definitional) equality vs internal (propositional) equality
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- constructing types in universes
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- universes
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- recursors / pattern matching
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- side quest: 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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a) fibers contractable
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b) quasi-inverse
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c) zig-zag
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- types are infinity groupoids
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- positive and negative constructions of Pi/Sigma types
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- HoTT
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- basics
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a) meta interval, identity type vs path type
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b) path type on other types
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c) dependent path type PathP vs path over
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d) univalence
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e) the (non)-issue of J in (Cu)TT
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f) isContr, isProp, isSet
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- Structures, univalence and transport
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a) transporting results between isomorphic structures
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- HITs, examples
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a) the constructed interval
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b) booleans and covers
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c) S^n
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d)
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- Homotopy n-types
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a) 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
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- 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,
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- 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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