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ICMS
15
July
disjoint work with
Favonia
2020
Evan Cavallo / Carlo Angiuli Anders Mörtberg / Andrea Vezzosi
15 July disjoint work with 2020 Evan Cavallo / Carlo Angiuli - - PowerPoint PPT Presentation
15 July disjoint work with 2020 Evan Cavallo / Carlo Angiuli - - PowerPoint PPT Presentation
ICMS 15 July disjoint work with 2020 Evan Cavallo / Carlo Angiuli Anders Mrtberg / Andrea Vezzosi Favonia floor wall floor wall wall floor comp wall wall floor cubes +) composition cubical TT major difficulty: composition
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floor
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floor
wall
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floor
wall
wall
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floor comp
wall
wall
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cubes composition
major difficulty: composition for univalent universes
+) cubical TT
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null compositions = no walls
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Brunerie's number
a program that should output 2*
*read Guillaume Brunerie's thesis
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(p = (<i> <j> <k> ((test0To4 @ j) @ k) @ i))))), i = 1))))))))))))))))))))))))))))))))))))))))))) ))) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) [])
2019.03.04-cubicaltt-fbdb422ada0287dbfc7b097c4a9355ed501be6e6-stack-lts9.5-brunerie2-brunerie_opt-2.output.gz
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nullable compositions
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nullable
not covering every corner not “true” under double negation not “true” under some closed substitutions
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kill nullable compositions!
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Plan A
reduces to floor if null?
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Plan A
reduces to floor if null?
difficult with univalence
keyword: regularity
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Plan B
ban nullable compositions?
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Plan B
ban nullable compositions?
but universes need them
in current constructions
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Plan C
a different composition based on non-nullable ones
with a different set of equations to avoid regularity
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Plan C
a different composition based on non-nullable ones
method 1: decision tree
no general construction yet
method 2: reflection
with a different set of equations to avoid regularity
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cofibrations
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method 1: decision tree
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method 1: decision tree
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method 1: decision tree
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reduced reduced
method 1: decision tree
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neocomp
See [AFH] and/or Carlo's thesis
method 1: decision tree
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neocomp
comp comp comp
See [AFH] and/or Carlo's thesis
method 1: decision tree
neocomp neocomp
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neocomp
comp comp comp
See [AFH] and/or Carlo's thesis
neocomp
method 1: decision tree
neocomp neocomp
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neocomp
limitation: the way/order to check dimension expressions needs to respect all equalities (e.g., subst.)
method 1: decision tree
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I P D S removal of inconsistent walls permutation of walls removal of duplicate walls symmetry of wall constraints
variants of [AFH]-style composition
σ symmetry for non-diagonals only
method 1: decision tree
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I P D S removal of inconsistent walls permutation of walls removal of duplicate walls symmetry of wall constraints
unsolved cases: -P+S
variants of [AFH]-style composition
σ symmetry for non-diagonals only
(no permutation, but with symmetry)
method 1: decision tree
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[AFH]-style + conjunctions
method 1: decision tree
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[AFH]-style + conjunctions
trickier with +I
how about
method 1: decision tree
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[AFH]-style + conjunctions
trickier with +I
how about [AFH], research notes, ...
solved case by case
method 1: decision tree
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[CCHM]-style composition
method 2: reflection
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[CCHM]-style composition
make intervals richer so that
method 2: reflection
is surjective
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method 2: reflection
neocomp
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method 2: reflection
neocomp
comp
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method 2: reflection
neocomp
comp
used in Cubical Agda
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Plan C
a different composition based on non-nullable ones
with a different set of equations to avoid regularity
but, is it worth it?
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none works for unknown cofibrations
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none works for unknown cofibrations
we can quantify over cofibrations in cooltt
def mycom/fun (A : 𝕁 → type) (B : 𝕁 → type) (com/A : (r : 𝕁) (φ : 𝔾) (p : (i : 𝕁) (_ : [i=r ∨ φ] (com/B : (r : 𝕁) (φ : 𝔾) (p : (i : 𝕁) (_ : [i=r ∨ φ] (r : 𝕁) (φ : 𝔾) (p : (i : 𝕁) (_ : [i=r ∨ φ]) (_ : A
no known way to kill nullable compositions
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general theory?
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general theory?
build univalent Kan universes with only these cofibrations
still very open
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further reading
[Angiuli] [VMA]
Computational Semantics of Cartesian Cubical Type Theory Cubical Agda: a dependently typed programming language with univalence and higher inductive types
thesis