2.9 KiB
2.9 KiB
## 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
<!– 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? -> 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
- 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)
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-
HoTT
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basics
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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
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-
Structures, using univalence to transport
- transporting results between isomorphic structures
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HITs, examples
- the constructed interval
- booleans and covers
- S^n
- S^1 with 2 cw structures equiv
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Homotopy n-types
-
homotopy levels being closed under type constructions, in particular Set and ETT inside HoTT
- in particular sigma types
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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.
- 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