differential cohesive type theory - github pages · real-cohesive homotopy type theory shulman...
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Differential Cohesive Type Theory
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< a = a &[ [
a [ a ]
OverviewWant a syntax for
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• Modal Type Theory
• Simple Differential Cohesive Type Theory
• Towards Differential Cohesive Homotopy Type Theory
Overview
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Modal Type Theory
• Why do we need Modal Type Theory?
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No-Go Theorem for Comonads
(Shulman ’15) Any internal comonadic modality is of the form
⇤A ⇠= A⇥ Ufor some proposition U
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No-Go Theorem for Comonads
(Shulman ’15) Any internal comonadic modality is of the form
⇤A ⇠= A⇥ Ufor some proposition U
The solution: comonads can’t be applied in every context, so add a syntax for restricted contexts.
Change the Judgmental structure!
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Modal Logic
Γ ⊢ ☐A
∅ ⊢ A
♢A ⊢ ♢C
A ⊢ ♢C
A : A is true in the current world
⇤A : A is true in all possible worlds
⇧A : A is true in some possible world
Traditionally: encode different modes of “truth”
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Modal Type Theory
• Encode different modes of “proof” or construction:
• Smooth variation
• Continuous variation
• Discontinuity
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Real-Cohesive Homotopy Type Theory Shulman ‘15
a [ a ]• Extends Homotopy Type Theory with an extra
context of “discontinuous dependency”.
• flat, sharp defined using a modal type theory
• Shape is defined as localization at the reals.
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Simple Differential Cohesive Type Theory
• A non-dependent type theory that includes all of the modalities of differential cohesion.
• Dependent type theory rules will be generalizations of these.
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Simple DCTT
Types A,B,C ::= A | [A | ]A |<A |=A |&A|A ! B | 1 |A⇥B |X
Modes m,n ::= [ |< | 1 |= |
compare real-cohesion which only has flat, 1.
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Modes
� ctx A type m 2 {[,<, 1,=, }�, x :m A ctx
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Modes
< a = a &[ [
a [ a ]
Modes are the left adjoints, monadic or comonadic
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Modes
[ < 1 = Monads
Comonads
Easier to Use Harder to Use
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[ < 1 =
� |� |⇥ |⇤ |⌅ ` t : A
Alternative notation for contexts
Modes
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[ < 1 =
� |� |⇥ |⇤ |⌅ ` t : A
Alternative notation for contexts
Modes
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�, x :m A ` t : Bdenotes
JtK : J�K ⇥ JmKJAK ! JBK
Meaning of the Modes
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�, x :m A ` t : B
Meaning of the Modes
[ : is possibly discontinuous
< : possibly ignores infinitesimal extensions
1 : is smooth
= : has 0 derivatives
: is constant on each connected component
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VariablesVariable rule is a counit
m 1
�, x :m A ` x : A
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VariablesCan’t directly use m > 1 vars
m 1
�, x :m A ` x : A
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VariablesModal Promotion
m n �, x :n A ` t : B
�, x :m A ` t : B
Examples: 1. Locally constant functions have 0 derivative 2. Functions with 0 derivative are smooth 3. Smooth functions are “at most discontinuous”
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CutCan only cut against restricted
substitutions
� `mu : A �, x :m A ` t : C
� ` t[u/x] : C
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Rule out substitutions like
� `mu : A �, x :m A ` t : C
� ` t[u/x] : C
y :1 B ` u : A x :[ A, y :1 B ` t : C
y :1 B ` t[u/x] : C
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Rule out substitutions like
� `mu : A �, x :m A ` t : C
� ` t[u/x] : C
y :1 B ` u : A x :[ A, y :1 B ` t : C
y :1 B ` t[u/x] : C
u is smooth in y, but t is discontinuous in x, so the composition is not necessarily smooth in y
DISALLOWED
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� `mu : A �, x :m A ` t : C
� ` t[u/x] : C
� |� |⇥ |⇤ |⌅ `[ A :⌘ � | · | · | · |⌅ ` A
� |� |⇥ |⇤ |⌅ `< A :⌘ � |� | · | · |⌅ ` A
� |� |⇥ |⇤ |⌅ `1 A :⌘ � |� |⇥ |⇤ |⌅ ` A
� |� |⇥ |⇤ |⌅ `= A :⌘ � |⇤ | · | · |⌅ ` A
� |� |⇥ |⇤ |⌅ ` A :⌘ � |⌅ | · | · | · ` A
would not keep Xi in real-cohesion
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Example: Reduction
� |� | · |⇥,⇤ |⌅ ` e : A
� |� |⇥ |⇤ |⌅ ` e< : <A<I
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Example: Reduction
Comonadic introduction: variables become *harder* to use
� |� | · |⇥,⇤ |⌅ ` e : A
� |� |⇥ |⇤ |⌅ ` e< : <A<I
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Example: Reduction
� |� | · |⇥,⇤ |⌅ ` t : A
t : [�⇥<�⇥=⇥⇥=⇤⇥ ⌅ ! A m 6= 1 =) <m = m
<t : [�⇥<�⇥=⇥⇥=⇤⇥ ⌅ ! <A 1 =<t � (· · · ) : [�⇥<�⇥⇥⇥=⇤⇥ ⌅ ! <A
� |� |⇥ |⇤ |⌅ ` t< : <A
Justified by functoriality
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Example: Reduction
• same semantics as substitution • producing RA under different restrictions means
you can use A at different modes
� `mu : <A �, x :m< A ` t : C
� ` let x< :m= u in t : C<E
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Example: Shapehow do we use monadically modal variables? Monadic introduction rules promote variables
� |�,⇥,⇤,⌅ | · | · | · ` e : A
� |� |⇥ |⇤ |⌅ ` e : AI
Increased expressivity over cohesion In cohesion, they would be promoted to 1 but in differential cohesion promoted to R
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Example: Shape
� |�,⇥,⇤,⌅ | · | · | · ` t : A
t : [�⇥<�⇥<⇥⇥<⇤⇥<⌅ ! A [ = [ < = <t : [�⇥ �⇥ ⇥⇥ ⇤⇥ ⌅ ! A < 1 =
t � (· · · ) : [�⇥<�⇥⇥⇥=⇤⇥ ⌅ ! A
� |� |⇥ |⇤ |⌅ ` t : A
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Example: Shape
� | �,⇥,⇤,⌅ | · | · | · ` e : A � | � | ⇥ | ⇤ | ⌅, x : A ` e
0 : C
� | � | ⇥ | ⇤ | ⌅ ` let x := e in e
0 : CE
Unlike Reduction, one mode of cut is optimal, so we only need one rule.
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Simple DCTT
• Linear order on modes makes it easy to pick a “best” mode for a rule.
• Of 12 intro/elim rules for modalities, only 2: reduction/co-reduction elimination need mode annotations
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Complicated…
• 6 modalities, 5 modes, too complex to do “manually”
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FSCD 2017
Fibrational Framework
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• Provide a Mode Theory:
• 0-cells ~ Categories
• 1-cells ~ Adjunctions
• 2-cells ~ Morphisms of Adjunctions
• Get an “Assembly code” modal logic
• simplify manually after
Mode Theory =>
Modal Type Theory
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Fibrational FrameworkUniform rules for left adjoints
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Fibrational FrameworkUniform rules for right adjoints
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Mode Theory • Multiple possibilities for a mode theory
• Split monads/comonads into adjunctions, relationship encoded by assembling these adjunctions in various ways, add in equalities for (co)-reflections.
• 3 categories (Discrete, Smooth, Formal Smooth)
• At least 9 functors.
• Treat the monads/comonads *directly* as the adjoints of the theory, add in 2-cells to make them monads/comonads
• 1 category (Formal Smooth)
• 4 adjunctions
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Mode Theory • Multiple possibilities for a mode theory
• Split monads/comonads into adjunctions, relationship encoded by assembling these adjunctions in various ways, add in equalities for (co)-reflections.
• 3 categories (Discrete, Smooth, Formal Smooth)
• At least 9 functors.
• Treat the monads/comonads *directly* as the adjoints of the theory, add in 2-cells to make them monads/comonads
• 1 category (Formal Smooth)
• 4 adjunctions
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Mode Theory 0 p
1
<,=, , [ : p ! pand a product that the above respect
⇥ : (p, p) ! p > : () ! p2 [ ) < ) 1 ) = )
eq
m[ = [m =
m= = mm< = m
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Mode Theory 0 p
1
<,=, , [ : p ! pand a product that the above respect
⇥ : (p, p) ! p > : () ! p2 [ ) < ) 1 ) = )
eq
m[ = [m =
m= = mm< = m
Every 1 cell generates left and right adjoint (named for left)
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Mode Theory 0 p
1
<,=, , [ : p ! pand a product that the above respect
⇥ : (p, p) ! p > : () ! p2 [ ) < ) 1 ) = )
eq
m[ = [m =
m= = mm< = m
Note: product means we function types too
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Mode Theory 0 p
1
<,=, , [ : p ! pand a product that the above respect
⇥ : (p, p) ! p > : () ! p2 [ ) < ) 1 ) = )
eq
m[ = [m =
m= = mm< = m
Note: The “Infinitesimal” modalities don’t survive interaction with a cohesive modality
because the infinitesimal structure is destroyed
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• Gives us meta-theoretic properties: substitution, variables working properly together
• F -| U, so all of our adjunctions hold
• Manually we have verified that the F,U presentation of middle adjoints are equivalent.
Fibrational Framework
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Simple DCTT
• Nice simple type theory, but not much use for Differential Geometry
• Can prove that the modalities have the right relationship and preserve products, but not much else.
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Towards Differential Cohesive Homotopy Type Theory
• How to make a dependent modal type theory with differential cohesive \infty-toposes as models?
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Comonadic Modalities
• We have a nice description of monadic modalities (see previous talk), but less worked out on the comonadic side.
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Who can you depend on?
• Summary:
• flat can depend on flat (assumptions)
• & can depend on flat, &
• Reduction can depend on flat, but not itself.
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Differential Cohesive HoTT
• Try adding dependent types to our simple type theory.
• Problems even with just cohesion:
• Monadic assumptions don’t work out so nicely.
a [ a ]
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Real-Differential Cohesion
• In the models, the shape modality is usually a localization, can define them internally and add axioms (Shulman ’15)
• Working on doing the same thing for inf. shape
• May avoid weird dependency issues by avoiding monadic modes.
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Progress on Real-DCoHoTTPrevious work: Shulman ’15, “Real Cohesion”
defined
< a = a &[ [
a [ a ]
defined
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Works
defined
< a = a &[ [
a [ a ]
defined
Progress on Real-DCoHoTT
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Needs semantics work
defined
< a = a &[ [
a [ a ]
defined
Progress on Real-DCoHoTT
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???
defined
< a = a &[ [
a [ a ]
defined
Progress on Real-DCoHoTT
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Conclusion• Simple Type Theory for full Diff. Cohesion.
• “Abstract” DC-HoTT requires a more sophisticated understanding of modal dependency.
• Extending Shulman’s Real-Cohesion may be easier, less general.
• https://github.com/jpaykin/DifferentialCohesiveHoTT
• Full rules, WIP Agda formalization
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Shapely Dependence
� ` A type
� ` A type
� ctx � ` mA type m 2 { , [}�, x :m A ctx
� ` A type � discrete
�,� ` [A type
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Shapely Dependence
• Can’t order the context since dependency structure and natural transformations don’t align.
• Shape and flat both define the discrete types, but since ∫A can depend on non-discrete types, need to look at transitive dependence
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Shapely DependenceNo best choice sometimes
? ` t : C
x : A, y :[ B[x] ` t : C
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Shapely DependenceNo best choice sometimes
x :1 A ` t : C
x : A, y :[ B[x] ` t : C
x : A, y :[ B[x] ` t : C
x : A, y :[ B[x] ` t : C
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Shapely DependenceMike’s proposal: keep track of *how*
types depend on terms
unclear how to justify in general
x : A, y :[ B[x ] ` t : C [ ⇠ 1
x : A, y :[ B[x ] ` t : C