108 lines
3.5 KiB
Markdown
108 lines
3.5 KiB
Markdown
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# Table of Contents
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- [Planning The HoTT Game](#org3bb90ed)
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- [Aims of the HoTT Game](#orga8d795d)
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- [Barriers](#org122a1d0)
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- [Format](#org3ea389f)
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- [Content](#org70d2231)
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- [Debriefs](#org37fbeb9)
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<a id="org3bb90ed"></a>
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# Planning The HoTT Game
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<a id="orga8d795d"></a>
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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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<a id="org122a1d0"></a>
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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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<a id="org3ea389f"></a>
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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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<a id="org70d2231"></a>
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## Content
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- emacs usage
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- agda usage
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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
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- implicit/explicit arguments
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- holes and inferred types
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- `_+_` vs `plus__`
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- type theory basics
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- 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
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- (side Q) positive and negative constructions of Pi/Sigma types
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- `data` and `record`
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- universes
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- recursors / pattern matching
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- (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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1. fibers contractable
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2. quasi-inverse
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3. zig-zag
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- (side Q) types are infinity groupoids
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- extra paths (univalence, fun ext, HITs)
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- HoTT
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- basics
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1. 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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2. path type on other types
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3. dependent path type PathP vs path over
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4. univalence
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5. the (non)-issue of J in (Cu)TT
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6. isContr, isProp, isSet
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7. drawing pictures
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- Structures, using univalence to transport
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1. transporting results between isomorphic structures
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- HITs, examples
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1. the constructed interval
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2. booleans and covers
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3. S<sup>n</sup>
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4. S<sup>1</sup> with 2 cw structures equiv
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- Homotopy n-types
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1. 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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<a id="org37fbeb9"></a>
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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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