Introduction to categorical approaches in topological data analysis - - PowerPoint PPT Presentation

Introduction to categorical approaches in topological data analysis I

Steve Oudot — Inria Saclay Dagstuhl seminar July 2017

Introduction to Persistence Theory Basics in Category Theory

Basics in Category Theory

Categories

2

Bottomline: maps between objects are as important as objects themselves

Def: A category C is made of:

• obj(C): a class of objects
• hom(C): a class of morphisms (arrows) between objects (a

f

− → b)

• ◦: a binary operation on morphisms: for all a, b, c ∈ obj(C),
• : hom(b, c) × hom(a, b) → hom(a, c).

It satisfies the axioms:

• associativity: for all a

f

− → b

g

− → c

h

− → d, h ◦ (g ◦ f) = (h ◦ g) ◦ f

• identity: ∀a ∈ obj(C) ∃!1a : a → a s.t. 1a ◦ f = f and g ◦ 1a = g

for all b

f

− → a

g

− → c

Categories

2

Top ModR Vectk Ab Grp Set

• bjects: topological spaces

morphisms: continuous maps

• bjects: sets

morphisms: set maps

• bjects: left R-modules

morphisms: R-linear maps

• bjects: k-vector spaces

morphisms: k-linear maps

• bjects: groups

morphisms: group homomorphisms

• bjects: abelian groups

morphisms: group homomorphisms

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise

example: oriented multi-graph (quiver)

• bjects = nodes

morphisms: paths (path category)

note being a thing category is the same as being a preordered class

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)

example: preordered set (S, ≤)

• bjects = elements s ∈ S

morphisms: couples (s, t) for s ≤ t ∈ S

beware that all non-oriented cycles in the graph commute for the category to be thin.

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)

example: poset (S, ≤)

• bjects = elements s ∈ S

morphisms: couples (s, t) for s ≤ t ∈ S

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

some diagram in C cone cocone

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

some diagram in C cone cocone ∃! ∃! limit colimit

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is abelian if it behaves like Ab:
• hom(a, b) ∈ Ab for all a, b ∈ obj(C), and ◦ is bilinear
• there is a zero object 0, i.e. s.t. | hom(a, 0)| = | hom(0, b)| = 1 for all

a, b ∈ obj(C) (⇒ each hom(a, b) has a unique zero morphism 0ab)

• it has all binary products (a × b) and coproducts (a ⊕ b)
• every morphism has a (unique) kernel and cokernel
• every monomorphism (w. zero kernel) is the kernel of its cokernel
• every epimorphism (w. zero cokernel) is the cokernel of its kernel
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is abelian if it behaves like Ab:
• hom(a, b) ∈ Ab for all a, b ∈ obj(C), and ◦ is bilinear
• there is a zero object 0, i.e. s.t. | hom(a, 0)| = | hom(0, b)| = 1 for all

a, b ∈ obj(C) (⇒ each hom(a, b) has a unique zero morphism 0ab)

• it has all binary products (a × b) and coproducts (a ⊕ b)
• every morphism has a (unique) kernel and cokernel
• every monomorphism (w. zero kernel) is the kernel of its cokernel
• every epimorphism (w. zero cokernel) is the cokernel of its kernel
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

(g + λh) ◦ f = (g ◦ f) + λ(h ◦ f) f ◦ (g + λh) = (f ◦ g) + λ(f ◦ h) preadditive

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is abelian if it behaves like Ab:
• hom(a, b) ∈ Ab for all a, b ∈ obj(C), and ◦ is bilinear
• there is a zero object 0, i.e. s.t. | hom(a, 0)| = | hom(0, b)| = 1 for all

a, b ∈ obj(C) (⇒ each hom(a, b) has a unique zero morphism 0ab)

• it has all binary products (a × b) and coproducts (a ⊕ b)
• every morphism has a (unique) kernel and cokernel
• every monomorphism (w. zero kernel) is the kernel of its cokernel
• every epimorphism (w. zero cokernel) is the cokernel of its kernel
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

a

• b
• (there is a unique way to factor through 0)

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is abelian if it behaves like Ab:
• hom(a, b) ∈ Ab for all a, b ∈ obj(C), and ◦ is bilinear
• there is a zero object 0, i.e. s.t. | hom(a, 0)| = | hom(0, b)| = 1 for all

a, b ∈ obj(C) (⇒ each hom(a, b) has a unique zero morphism 0ab)

• it has all binary products (a × b) and coproducts (a ⊕ b)
• every morphism has a (unique) kernel and cokernel
• every monomorphism (w. zero kernel) is the kernel of its cokernel
• every epimorphism (w. zero cokernel) is the cokernel of its kernel
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

a

a ⊕ b

a × b

• b
• (limit and colimit of diagram with nodes a and b)

additive

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is abelian if it behaves like Ab:
• hom(a, b) ∈ Ab for all a, b ∈ obj(C), and ◦ is bilinear
• there is a zero object 0, i.e. s.t. | hom(a, 0)| = | hom(0, b)| = 1 for all

a, b ∈ obj(C) (⇒ each hom(a, b) has a unique zero morphism 0ab)

• it has all binary products (a × b) and coproducts (a ⊕ b)
• every morphism has a (unique) kernel and cokernel
• every monomorphism (w. zero kernel) is the kernel of its cokernel
• every epimorphism (w. zero cokernel) is the cokernel of its kernel
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

Imk′ ⊆ Imk ⊆ ker f

a

f

• z

k

• 0zb b

z′

k′

• 0z′b
• b/ ker k′ ֒

→ b/ ker k ֒ → b/Imf

z′ a

f

• 0az

0az′

• z
• b

k′

• k

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is abelian if it behaves like Ab:
• hom(a, b) ∈ Ab for all a, b ∈ obj(C), and ◦ is bilinear
• there is a zero object 0, i.e. s.t. | hom(a, 0)| = | hom(0, b)| = 1 for all

a, b ∈ obj(C) (⇒ each hom(a, b) has a unique zero morphism 0ab)

• it has all binary products (a × b) and coproducts (a ⊕ b)
• every morphism has a (unique) kernel and cokernel
• every monomorphism (w. zero kernel) is the kernel of its cokernel
• every epimorphism (w. zero cokernel) is the cokernel of its kernel
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

note: a small diagram is the image of a functor F : J → C for a small category J

Categories

2

Some properties:

• C is small if both obj(C) and hom(C) are sets, and large otherwise
• C is thin if | hom(a, b)| ≤ 1 for all a, b ∈ obj(C)
• C is abelian if it behaves like Ab:
• hom(a, b) ∈ Ab for all a, b ∈ obj(C), and ◦ is bilinear
• there is a zero object 0, i.e. s.t. | hom(a, 0)| = | hom(0, b)| = 1 for all

a, b ∈ obj(C) (⇒ each hom(a, b) has a unique zero morphism 0ab)

• it has all binary products (a × b) and coproducts (a ⊕ b)
• every morphism has a (unique) kernel and cokernel
• every monomorphism (w. zero kernel) is the kernel of its cokernel
• every epimorphism (w. zero cokernel) is the cokernel of its kernel
• C is complete if every (small) diagram in C has a limit
• C is co-complete if every (small) diagram in C has a colimit

Examples: Ab, ModR, Vectk, · · ·

beware that, in some cases (such as this one), the notion of inclusion

Categories

2

Top ModR Vectk Ab Grp Set

• bjects: topological spaces

morphisms: continuous maps

• bjects: sets

morphisms: set maps

• bjects: left R-modules

morphisms: R-linear maps

• bjects: k-vector spaces

morphisms: k-linear maps

• bjects: groups

morphisms: group homomorphisms

• bjects: abelian groups

morphisms: group homomorphisms

⊆ ⊆ ⊇ ⊆ ⊇

Categories

2

Def: A subcategory D of C (denoted D ⊂ C) is made of:

• a subclass of objects obj(D) ⊆ obj(C)
• a subclass of morphisms hom(D) ⊆ hom(C)

such that

• for every a

f

− → b ∈ C, if f ∈ hom(D) then a, b ∈ obj(D)

• hom(D) is closed under composition
• for all a ∈ obj(D), 1a ∈ hom(D)

Note: D ⊆ C is full if homD(a, b) = homC(a, b) for all a, b ∈ obj(D)

basically, turns commutative diagrams into commutative diagrams,

Functors

3

Def: A functor F : C → D is made of:

• a mapping between objects: obj(C) → obj(D):
• for all a, b ∈ obj(C), a mapping between morphisms:

hom(a, b) → hom(F(a), F(b)) It satisfies the functoriality axioms:

• composition: for all a

f

− → b

g

− → c ∈ C, F(g ◦ f) = F(g) ◦ F(f)

• identity: for all a ∈ obj(C), F(1a) = 1F (a)

Bottomline: formalize notion of operator between categories

note that there is no such category for all categories. Indeed, as for sets, the catego basically, turns commutative diagrams into commutative diagrams,

Functors

3

Def: A functor F : C → D is made of:

• a mapping between objects: obj(C) → obj(D):
• for all a, b ∈ obj(C), a mapping between morphisms:

hom(a, b) → hom(F(a), F(b)) It satisfies the functoriality axioms:

• composition: for all a

f

− → b

g

− → c ∈ C, F(g ◦ f) = F(g) ◦ F(f)

• identity: for all a ∈ obj(C), F(1a) = 1F (a)

Bottomline: formalize notion of operator between categories

Note: small categories and functors between them form a category Cat

Functors

3

Example: homology functor H∗ : Top → ModZ

Z ( 1

0) Z2 ( 0 1 ) Z

( 0

1) Z2 ( 1 0 0 1) Z2

(1-homology functor H1)

⊆ ⊆ ⊆ ⊆

Functors

3

k ( 1

0) k2 ( 0 1 ) k

( 0

1) k2 ( 1 0 0 1) k2

F

Example: functor F : (S, ≤) → Vectk small diagram: small cat. → Vectk

Functors

3

Examples: Def: given D ⊆ C, the inclusion functor maps each object and morphism to itself Grp

⊆

− → Set Ab

⊆

− → Grp

this one goes from pointed topological spaces (i.e. ones with a distinguished basepoint) (but there exist more general types of forgetful functors)

Functors

3

Examples: Def: given D ⊆ C, the inclusion functor maps each object and morphism to itself Grp

⊆

− → Set Ab

⊆

− → Grp Examples: Def: a forgetful functor is one that ‘forgets structure or axioms’ Grp

⊆

− → Set forgets the group structure Ab

⊆

− → Grp forgets the commutativity axiom Top• − → Top forgets about the basepoint

Natural transformations

4

Def: Given F, G : C → D, a natural transformation η : F ⇒ G is made of:

• a morphism η(a) : F(a) → G(a) ∈ D for every object a ∈ obj(C)

such that the following diagram commutes for every morphism a

f

− → b ∈ C: F(a)

F (f) η(a)

• F(b)

η(b)

• G(a)

G(f) G(b)

Bottomline: view functors as objects in some category

Natural transformations

4

Def: Given F, G : C → D, a natural transformation η : F ⇒ G is made of:

• a morphism η(a) : F(a) → G(a) ∈ D for every object a ∈ obj(C)

such that the following diagram commutes for every morphism a

f

− → b ∈ C: F(a)

F (f) η(a)

• F(b)

η(b)

• G(a)

G(f) G(b)

Bottomline: view functors as objects in some category

Prop: functors C → D and their natural transformations form a category DC (vertical composition: (τ ◦ η)(a) := τ(a) ◦ η(a)) (natural isomorphisms: η(a) isomorphism for all a ∈ obj(C))

note: the actual category is given by the transitive closure of the graph

Natural transformations

4

k k

1

• 1
• ( 1

0)

k2

• ( 0 1 )
• k

k

−1

• 1
• ( 1

0)

k2

( 1 0 )

• ( 1 −1 )
• 0 0

0 1

• 1

1

Example: functors quiver → Vectk

F G η

Isomorphism and equivalence of categories

5

Def: F : C → D is an isomorphism of cateories if there is G : D → C such that G ◦ F = 1C and F ◦ G = 1D, where 1∗ denotes the identity functor.

Bottomline: isomorphism on objects and morphisms

this is because every n-dim. vector space

Isomorphism and equivalence of categories

5

Def: F : C → D is an isomorphism of cateories if there is G : D → C such that G ◦ F = 1C and F ◦ G = 1D, where 1∗ denotes the identity functor. Def: F : C → D and G : D → C form an equivalence of cateories if there are natural isomorphisms G ◦ F ⇒ 1C and F ◦ G ⇒ 1D.

Bottomline: isomorphism ‘up to (natural) isomorphisms‘

(preserves most properties of a category, works up to isomorphism) Example: vectk is equivalent to Matk but not isomorphic to it finite-dimensional k-vector spaces k-matrices

Pre(C) is a thin category, having the same objects as C and a unique morphism for each

Isomorphism and equivalence of categories

5

Def: F : C → D is:

• faithful if hom(a, b) → hom(F(a), F(b)) is injective for all a, b ∈ obj(C)
• full if hom(a, b) → hom(F(a), F(b)) is surjective for all a, b ∈ obj(C)
• fully faithful if both full and faithful

Bottomline: asymmetric sufficient conditions for equivalence

Examples:

• Grp → Set is faithful but not full (missing set maps)
• Ab → Grp is fully faithful
• C → Pre(C) (preorder reflection) is full but not faithful unless C is thin

and therefore neither on morphisms across the whole category, since objects can collide

Isomorphism and equivalence of categories

5

Def: F : C → D is:

• faithful if hom(a, b) → hom(F(a), F(b)) is injective for all a, b ∈ obj(C)
• full if hom(a, b) → hom(F(a), F(b)) is surjective for all a, b ∈ obj(C)
• fully faithful if both full and faithful

Bottomline: asymmetric sufficient conditions for equivalence

Note: full (resp. faithful) does not imply surjective (resp. injective) on objects Note: fully faithful implies injective on objects up to isomorphism: F(a) ≃ F(b) ⇒ a ≃ b

Isomorphism and equivalence of categories

5

Def: F : C → D is:

• faithful if hom(a, b) → hom(F(a), F(b)) is injective for all a, b ∈ obj(C)
• full if hom(a, b) → hom(F(a), F(b)) is surjective for all a, b ∈ obj(C)
• fully faithful if both full and faithful

Bottomline: asymmetric sufficient conditions for equivalence

Prop: If F : C → D is fully faithful then it yields an equivalence of categories between C and its image F(C). F is then called an embedding of C into D. Examples:

• Ab → Grp is an embedding

Isomorphism and equivalence of categories

5

Def: F : C → D is:

• faithful if hom(a, b) → hom(F(a), F(b)) is injective for all a, b ∈ obj(C)
• full if hom(a, b) → hom(F(a), F(b)) is surjective for all a, b ∈ obj(C)
• fully faithful if both full and faithful

Bottomline: asymmetric sufficient conditions for equivalence

Prop: If F : C → D is fully faithful then it yields an equivalence of categories between C and its image F(C). F is then called an embedding of C into D. Prop: F : C → D yields an equivalence of categories between C and D iff F fully faithful and essentially (i.e. up to isomorphism) surjective Examples:

• ModZ → Ab yields an equivalence of categories
• Ab → Grp is an embedding

beware here: for a fixed field k = R, we forget about inverses but w

Note: one can either view Vect∗ as the category of vector spaces over a

Isomorphism and equivalence of categories

5

Top Mod∗ Vect∗ Ab Grp Set

• bjects: topological spaces

morphisms: continuous maps

• bjects: sets

morphisms: set maps

• bjects: left ∗-modules

morphisms: ∗-linear maps

• bjects: ∗-vector spaces

morphisms: ∗-linear maps

• bjects: groups

morphisms: group homomorphisms

• bjects: abelian groups

morphisms: group homomorphisms

֒ →

≃

→ ← ֓ → ←

≃

→

Two important results

6

Thm: if C is small and D is abelian, then DC is also abelian. Note: define constructions and operations ‘pointwise’

C = D = Vectk

• 0 =

k

1

• k2

k

1 1

• k2

( 0 1 )

• 1 0

0 1

• k2
• k

k

• k

1

• =

k2

1 0 0 0

• k2

k2

1 0 1 0

• k3

0 1 0 0 0 1

• 1 0 0

0 1 0

• k2

Two important results

6

Thm: if C is small and D is abelian, then DC is also abelian. Note: define constructions and operations ‘pointwise’

C = D = Vectk

• 0 =

k

1

• 1
• k2
• k

1 1

• −1
• k2

( 0 1 )

• 1 0

0 1

• ( 0 −1 )
• k2
• k

k

• k

1

• ker =

k2

• k
• 1
• k2

coker = 0

Note: in our case, DC is not small, however we can construct an embedding exists at this stage this is just a k-vector space moreover, the embedding is exact, so that it preserves monomorphisms and epimo

Two important results

6

Thm: if C is small and D is abelian, then DC is also abelian. Thm: (Mitchell’s embedding) Every small abelian category embeds into some module category.

C = D = Vectk VectC

k ֒

→ ModkC where kC is the category algebra generated by morphisms (finite paths) in C (the product in kC being given by composition of morphisms / paths) Embedding on objects:

k

1

• k2

k

1 1

• k2

( 0 1 )

• 1 0

0 1

• k2

k ⊕ k2 ⊕ k ⊕ k2 ⊕ k2 equipped with kC-mod structure given by the linear maps

→

associated with morphisms in C (paths in graph)

at this stage this is just a k-vector space moreover, the embedding is exact, so that it preserves monomorphisms and epimo

Two important results

6

Thm: if C is small and D is abelian, then DC is also abelian. Thm: (Mitchell’s embedding) Every small abelian category embeds into some module category.

and a definition

Def: Given C, D additive, F : C → D is additive if F(0) = 0 and F(a ⊕ b) = F(a) ⊕ F(b).

Introduction to Persistence Theory

C’est de ce cette constatation qu’est nee l’analyse topologique des donnees, dont le

like homology groups, or the dimension of their free part (called Betti numbers)

Topological Data Analysis (TDA)

2 topological invariants for classification

β0 = β2 = 1 β1 = 2

Algebraic topology Applied algebraic topology

β0 β1 β2 compact set triangulation point cloud topological descriptors for inference and comparison

C’est de ce cette constatation qu’est nee l’analyse topologique des donnees, dont le

Topological Data Analysis (TDA)

2

Applied algebraic topology

β0 β1 β2 compact set point cloud topological descriptors for inference and comparison

Properties of topological descriptors:

• invariant under coordinate changes
• stable with respect to perturbations
• informative
• versatile

C’est de ce cette constatation qu’est nee l’analyse topologique des donnees, dont le

Topological Data Analysis (TDA)

2

Applied algebraic topology

β0 β1 β2 compact set point cloud topological descriptors for inference and comparison

4 pillars to the theory (topological persistence):

• decomposition theorems (∃ barcodes)
• algorithms (computation of barcodes)
• stability theorems (barcodes as stable descriptors)
• statistics (means, cvgence rates, etc.)

3

f : X → R persistence Dg f X topological space

∞

X R f signature: persistence diagram encodes the topological structure of the pair (X, f)

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

t

Ft := f−1((−∞, t])

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family

f

Topological Persistence in a Nutshell

4

R R

Inside the black box:

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family
• Finite set of intervals (barcode) encodes births/deaths of topological features

f

Topological Persistence in a Nutshell

4

α β

R R

Inside the black box:

α β ∞

• Nested family (filtration) of sublevel-sets f−1((−∞, t]) for t ranging over R
• Track the evolution of the topology throughout the family
• Finite set of intervals (barcode) encodes births/deaths of topological features

f

• Alternate representation as a (multi-)

set of points in the plane (diagram).

Topological Persistence in a Nutshell

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

5

fP : R2 → R x → minp∈P x − p2

4 8 12 16 20 24 28 32

Example: distance function

6

Formalism (1-d persistence)

Def: A filtration over T is a functor F : (T, ≤) → Top Fix an indexing set T ⊆ R and a field k.

Canonical example: given f : X → R, take the sublevel-sets filtration over R: then: F(s ≤ s) = 1F (s) and F(s ≤ u) = F(t ≤ u) ◦ F(s ≤ t) for all s ≤ t ≤ u ∈ R F(t) := f−1((−∞, t]) and F(s ≤ t) := (F(s) ⊆ F(t)) for all s ≤ t ∈ R

6

Formalism (1-d persistence)

Def: A filtration over T is a functor F : (T, ≤) → Top Fix an indexing set T ⊆ R and a field k.

Canonical example: given f : X → R, take the sublevel-sets filtration over R: then: F(s ≤ s) = 1F (s) and F(s ≤ u) = F(t ≤ u) ◦ F(s ≤ t) for all s ≤ t ≤ u ∈ R

Def: A persistence module over T is a functor M : (T, ≤) → Vectk

Example: given f : X → R and its sublevel-sets filtration F : (R, ≤) → Top, F(t) := f−1((−∞, t]) and F(s ≤ t) := (F(s) ⊆ F(t)) for all s ≤ t ∈ R let M := H∗ ◦ F : (R, ≤) → Vectk, where H∗ is singular homology over k

hence we can talk about decompositions into direct sums hence the name. Note that, although the functor category itself is not small, it is indexed

6

Formalism (1-d persistence)

Def: A filtration over T is a functor F : (T, ≤) → Top Fix an indexing set T ⊆ R and a field k.

Canonical example: given f : X → R, take the sublevel-sets filtration over R: then: F(s ≤ s) = 1F (s) and F(s ≤ u) = F(t ≤ u) ◦ F(s ≤ t) for all s ≤ t ≤ u ∈ R

Def: A persistence module over T is a functor M : (T, ≤) → Vectk

Example: given f : X → R and its sublevel-sets filtration F : (R, ≤) → Top, F(t) := f−1((−∞, t]) and F(s ≤ t) := (F(s) ⊆ F(t)) for all s ≤ t ∈ R let M := H∗ ◦ F : (R, ≤) → Vectk, where H∗ is singular homology over k

Notes:

• Vectk is abelian ⇒ so is Vect(T,≤)

k

• Vect(T,≤)

k

֒ → ModR where R is the algebra generated by (s, t)s≤t∈T

6

Formalism (1-d persistence)

k ( 1

0) k2 ( 0 1 ) k

( 0

1) k2 ( 1 0 0 1) k2

Example:

H1(−; k)

⊆ ⊆ ⊆ ⊆

T = [5] 1 2 3 4 5 ≤ ≤ ≤ ≤ f : − → T ⊂ R

F

Decompositions

forward means that all arrows i → j satisfy i ≤ j, with the relation i ≤ j ≤ k˜ i ≤ k

Theorem.

Let M be a persistence module over some index set T ⊆ R. Then, M decomposes as a direct sum of interval modules k[b∗,d∗]:

· · · k

1

· · ·

1

k · · ·

• i<b∗
• [b∗, d∗]
• i>d∗

in the following cases:

• T is finite [Gabriel 1972] [Auslander 1974],
• M is pointwise finite-dimensional (pfd), i.e. M : (T, ≤) → vectk

[Webb 1985] [Crawley-Boevey 2012].

Moreover, when it exists, the decomposition is unique up to isomorphism and permutation of the terms [Azumaya 1950].

(the barcode is a complete descriptor of the algebraic structure of M)

M ≃

• j∈J

k[b∗

j ,d∗ j ]

7

Decompositions

forward means that all arrows i → j satisfy i ≤ j, with the relation i ≤ j ≤ k˜ i ≤ k

Theorem.

Let M be a persistence module over some index set T ⊆ R. Then, M decomposes as a direct sum of interval modules k[b∗,d∗]:

· · · k

1

· · ·

1

k · · ·

• i<b∗
• [b∗, d∗]
• i>d∗

in the following cases:

• T is finite [Gabriel 1972] [Auslander 1974],
• M is pointwise finite-dimensional (pfd), i.e. M : (T, ≤) → vectk

[Webb 1985] [Crawley-Boevey 2012].

Moreover, when it exists, the decomposition is unique up to isomorphism and permutation of the terms [Azumaya 1950].

(the barcode is a complete descriptor of the algebraic structure of M) 7

Decompositions

k ( 1

0) k2 ( 0 1 ) k

( 0

1) k2 ( 1 0 0 1) k2

Example:

H1(−; k)

⊆ ⊆ ⊆ ⊆

1 2 3 4 5 ≤ ≤ ≤ ≤

F

7

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0.

7

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0.

Proof by induction on total dimension dim M := n

i=1 dim M(i):

• trivially true for dim M = 0
• if dim M > 0 then either M itself is indecomposable, or M ≃ A ⊕ B

with A = 0 = B hence dim A, dim B < dim M. Apply then IH to A, B.

• 7

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0. Lemma: Every indecomposable is an interval module.

7

this is well-defined because kernels are sent to kernels, by the composition law

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0. Lemma: Every indecomposable is an interval module.

Proof: take M = 0 indecomposable. Let b := min{i | M(i) = 0} then d := max{j | rkM(i ≤ j) = 0}. M = 0

· · · M(b)

• · · ·

M(d) ⋆ · · ·

Let K ⊆ M (submodule) be defined by: K(t) :=      0 if t < b ker M(t ≤ d) if b ≤ t ≤ d M(t) if t > d 7

this is well-defined because kernels are sent to kernels, by the composition law

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0. Lemma: Every indecomposable is an interval module.

Proof: take M = 0 indecomposable. Let b := min{i | M(i) = 0} then d := max{j | rkM(i ≤ j) = 0}. M = 0

· · · M(b)

• · · ·

M(d) ⋆ · · ·

Let K ⊆ M (submodule) be defined by: K(t) :=      0 if t < b ker M(t ≤ d) if b ≤ t ≤ d M(t) if t > d 7

Note: Lb = 0 because by definition rkM(b ≤ d) = 0 this is well-defined because kernels are sent to kernels, by the composition law

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0. Lemma: Every indecomposable is an interval module.

Proof: take M = 0 indecomposable. Let b := min{i | M(i) = 0} then d := max{j | rkM(i ≤ j) = 0}. M = 0

· · · M(b)

• · · ·

M(d) ⋆ · · ·

Let K ⊆ M (submodule) be defined by: K(t) :=      0 if t < b ker M(t ≤ d) if b ≤ t ≤ d M(t) if t > d Choose Lb = 0 s.t. M(b) = K(b) ⊕ Lb and define L ⊆ M (submodule) by: L(t) :=

• 0 if t < b or t > d

ImM(b ≤ t)|Lb if b ≤ t ≤ d 7

Note: Lb = 0 because by definition rkM(b ≤ d) = 0 this is well-defined because kernels are sent to kernels, by the composition law

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0. Lemma: Every indecomposable is an interval module.

Proof: take M = 0 indecomposable. Let b := min{i | M(i) = 0} then d := max{j | rkM(i ≤ j) = 0}. M = 0

· · · M(b)

• · · ·

M(d) ⋆ · · ·

Let K ⊆ M (submodule) be defined by: K(t) :=      0 if t < b ker M(t ≤ d) if b ≤ t ≤ d M(t) if t > d Choose Lb = 0 s.t. M(b) = K(b) ⊕ Lb and define L ⊆ M (submodule) by: L(t) :=

• 0 if t < b or t > d

ImM(b ≤ t)|Lb if b ≤ t ≤ d Then: ∃N ⊆ M (submodule) s.t. M = K ⊕ L ⊕ N (N defined pointwise by induction) 7

It is easily seen that K ∩ L = 0, because L transports Lb from b up to d (included) therefore it cannot Note: Lb = 0 because by definition rkM(b ≤ d) = 0 this is well-defined because kernels are sent to kernels, by the composition law

Decompositions

Proof in the case T finite (T = [n]) and M pfd: Lemma: M decomposes into indecomposables: M ≃

j∈J Mj

where Mj ≃ A ⊕ B ⇒ A = 0 or B = 0. Lemma: Every indecomposable is an interval module.

Proof: take M = 0 indecomposable. Let b := min{i | M(i) = 0} then d := max{j | rkM(i ≤ j) = 0}. M = 0

· · · M(b)

• · · ·

M(d) ⋆ · · ·

Let K ⊆ M (submodule) be defined by: K(t) :=      0 if t < b ker M(t ≤ d) if b ≤ t ≤ d M(t) if t > d Choose Lb = 0 s.t. M(b) = K(b) ⊕ Lb and define L ⊆ M (submodule) by: L(t) :=

• 0 if t < b or t > d

ImM(b ≤ t)|Lb if b ≤ t ≤ d Then: ∃N ⊆ M (submodule) s.t. M = K ⊕ L ⊕ N (N defined pointwise by induction) 7 Now, first isomorphism theorem ⇒ L ≃ kr

[b,d] where r = rkM(b ≤ d) > 0

M indecomposable ⇒ K = N = 0 and r = 1.

the stability property is easiest to describe in the functional setting

Structure and Stability at the functional level

α β

X R

f

α β ∞

sublevel-sets filtration → barcode X topological space, f : X → R pfd function (i.e. s.t. H∗ ◦ F is pfd)

8 diagram ≡ multiset of points

/ diagram

barcode ≡ multiset of intervals

Structure and Stability at the functional level

X R

f

∞

Theorem: For any pfd functions f, g : X → R, d∞

b (Dg f, Dg g) ≤ f − g∞

g

8

but in the generalized case the statement is more subtle to state, so let me stick to this simpler version

Structure and Stability at the functional level

X R

f

∞

Theorem: For any pfd functions f, g : X → R, d∞

b (Dg f, Dg g) ≤ f − g∞

g

Note: there are variants where f, g do not have the same domain X 8

Structure and Stability at the functional level

8

Persistence diagram ≡ locally finite multiset in the closed half-plane ∆ × R+ cost of a matched pair y = γ(x): c(x, y) := x − y∞ cost of an unmatched point z ∈ (X \ X′) ⊔ (Y \ Y ′): c(z) := z − ¯ z∞ cost of the matching γ: Given a partial matching γ : X ↔ Y (i.e. a bijection X ⊇ X′

γ

− → Y ′ ⊆ Y ):

x y z ¯ z

∆

(2)

bottleneck distance: db(X, Y ) := inf

γ:X↔Y c(γ)

c(γ) := max

• max

y=γ(x) c(x, y),

max

z∈(X\X′)⊔(Y \Y ′) c(z)

Soft stability

9

Obs: For all t ∈ R, F(t) ⊆ G(t + ε) ⊆ F(t + 2ε), where ε := f − g∞.

t t + ε X R

Let F, G : (R, ≤) → Top be the sublevel-sets filtrations of f, g

Soft stability

9

Obs: For all t ∈ R, F(t) ⊆ G(t + ε) ⊆ F(t + 2ε), where ε := f − g∞. Hence the commutative diagrams for all s ≤ t ∈ R: F(t)

• F(t + 2ε)

G(t + ε)

• Let F, G : (R, ≤) → Top be the sublevel-sets filtrations of f, g

(inclusion maps)

F(t + ε)

• G(t)
• G(t + 2ε)

F(s)

• F(t)
• G(s + ε)

G(t + ε)

F(s + ε)

F(t + ε)

G(s)

• G(t)

Soft stability

9

Obs: For all t ∈ R, F(t) ⊆ G(t + ε) ⊆ F(t + 2ε), where ε := f − g∞. Hence the commutative diagrams for all s ≤ t ∈ R: Let F, G : (R, ≤) → Top be the sublevel-sets filtrations of f, g

(linear maps) (post-composition with H∗)

H∗ ◦ F(t)

• H∗ ◦ F(t + 2ε)

H∗ ◦ G(t + ε)

• H∗ ◦ F(t + ε)
• H∗ ◦ G(t)
• H∗ ◦ G(t + 2ε)

H∗ ◦ F(s)

• H∗ ◦ F(t)
• H∗ ◦ G(s + ε)

H∗ ◦ G(t + ε)

H∗ ◦ F(s + ε)

H∗ ◦ F(t + ε)

H∗ ◦ G(s)

• H∗ ◦ G(t)
• ε-interleaving between H∗ ◦ F and H∗ ◦ G

Soft stability

9

M(s)

• M(t)
• N(s + ε)

N(t + ε)

Note: the following commutative diagram for all s ≤ t ∈ R is that of a natural transformation M ⇒ N[ε] where indices in N are shifted by ε:

Soft stability

9

M(s)

• M(t)
• N(s + ε)

N(t + ε)

Note: the following commutative diagram for all s ≤ t ∈ R is that of a natural transformation M ⇒ N[ε] where indices in N are shifted by ε: Def: (ε-shift endofunctor)

−[ε] :

• D(R,≤)

− → D(R,≤) M − → M[ε] s.t.

• M[ε](t) := M(t + ε) ∀t ∈ R

M[ε](s ≤ t) := M(s + ε ≤ t + ε) ∀s ≤ t ∈ R (φ : M ⇒ N) − → (φ[ε] : M[ε] ⇒ N[ε]) s.t. φ[ε](t) := φ(t + ε) ∀t ∈ R

Soft stability

9

Note: the following commutative diagram for all s ≤ t ∈ R is that of a natural transformation M ⇒ N[ε] where indices in N are shifted by ε: Def: (ε-shift endofunctor)

−[ε] :

• D(R,≤)

− → D(R,≤) M − → M[ε] s.t.

• M[ε](t) := M(t + ε) ∀t ∈ R

M[ε](s ≤ t) := M(s + ε ≤ t + ε) ∀s ≤ t ∈ R (φ : M ⇒ N) − → (φ[ε] : M[ε] ⇒ N[ε]) s.t. φ[ε](t) := φ(t + ε) ∀t ∈ R

ε-shift natural transformation:

• M ⇒ M[ε]

t → M(t ≤ t + ε)

M(s)

• M(t)
• M(s + ε)

M(t + ε)

Soft stability

9

Def: An ε-interleaving between M, N : (R, ≤) → D is given by φ : M ⇒ N[ε] and ψ : N ⇒ M[ε] such that the following diagram commutes, where the horizontal natural transformations are induced by ε-shifts: M

• φ
• M[ε]
• φ[ε]
• M[2ε]

N

• ψ
• N[ε]
• ψ[ε]
• N[2ε]

Soft stability

9

Def: An ε-interleaving between M, N : (R, ≤) → D is given by φ : M ⇒ N[ε] and ψ : N ⇒ M[ε] such that the following diagram commutes, where the horizontal natural transformations are induced by ε-shifts: M

• φ
• M[ε]
• φ[ε]
• M[2ε]

N

• ψ
• N[ε]
• ψ[ε]
• N[2ε]

Def: di(M, N) := inf{ε | M, N are ε-interleaved}

Soft stability

9

Def: An ε-interleaving between M, N : (R, ≤) → D is given by φ : M ⇒ N[ε] and ψ : N ⇒ M[ε] such that the following diagram commutes, where the horizontal natural transformations are induced by ε-shifts: M

• φ
• M[ε]
• φ[ε]
• M[2ε]

N

• ψ
• N[ε]
• ψ[ε]
• N[2ε]

Def: di(M, N) := inf{ε | M, N are ε-interleaved} Thm: (soft stability) For any functions f, g : X → R and any functor H : Top → D, di(H ◦ F, H ◦ G) ≤ f − g∞.

Soft stability

9

Def: An ε-interleaving between M, N : (R, ≤) → D is given by φ : M ⇒ N[ε] and ψ : N ⇒ M[ε] such that the following diagram commutes, where the horizontal natural transformations are induced by ε-shifts: M

• φ
• M[ε]
• φ[ε]
• M[2ε]

N

• ψ
• N[ε]
• ψ[ε]
• N[2ε]

Def: di(M, N) := inf{ε | M, N are ε-interleaved} Thm: (soft stability) For any functions f, g : X → R and any functor H : Top → D, di(H ◦ F, H ◦ G) ≤ f − g∞.

• take H = H∗ : Top → vectk
• can we build vect(R,≤)

k

→ Bar?

the entry uαβ is associated with interval (v ◦ u)αγ is the matrix entry associated with interval Iα being mapp

The category of barcodes

10

Def: [Kashiwara, Schapira 2017] The category BarKS is composed of:

• objects: barcode ≡ (A, I) where:

– A is a finite or countable indexing set – I = (Iα)α∈A is a locally finite collection of intervals Iα = [b∗

α, d∗ α]

with b∗

α ≤ d∗ α ∈ ¯

R

• morphisms: given barcodes (A, I) and (B, J):

hom((A, I), (B, J)) :=

• (α,β)|α∈A,β∈B,b∗

β≤b∗ α≤d∗ β≤a∗ β

k(α,β) where k(α,β) = k.

• given (A, I)

u=(uαβ)

(B, J)

v=(vβγ)

(C, K) :

(v ◦ u)αγ :=

• β∈B

uαβvβγ

matrices with zero entries for incomparable intervals matrix product

The category of barcodes

10

Thm: [Kashiwara, Schapira 2017]

• BarKS is additive: it has a zero object (empty barcode) and direct sums
• The functor

Ψ : BarKS − → vect(R,≤)

k

(A, I) − →

• α∈A kIα

yields an equivalence of additive categories. proof sketch:

• fully faithful and additive by construction
• essentially surjective thanks to the decomposition theorem for pfd modules

Ψ−1 denotes the pseudo-inverse of Ψ up to isomo

The category of barcodes

10

Thm: [Kashiwara, Schapira 2017]

• BarKS is additive: it has a zero object (empty barcode) and direct sums
• The functor

Ψ : BarKS − → vect(R,≤)

k

(A, I) − →

• α∈A kIα

yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R, di(Ψ−1(H∗ ◦ F), Ψ−1(H∗ ◦ G)) ≤ f − g∞.

Ψ−1 denotes the pseudo-inverse of Ψ up to isomo this is because di allows for more general mappings than just partial matchings (i.e. matrices

The category of barcodes

10

Thm: [Kashiwara, Schapira 2017]

• BarKS is additive: it has a zero object (empty barcode) and direct sums
• The functor

Ψ : BarKS − → vect(R,≤)

k

(A, I) − →

• α∈A kIα

yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R, di(Ψ−1(H∗ ◦ F), Ψ−1(H∗ ◦ G)) ≤ f − g∞. Pb: di ≤ db in BarKL

Ψ−1 denotes the pseudo-inverse of Ψ up to isomo this is because di allows for more general mappings than just partial matchings (i.e. matrices

The category of barcodes

10

Thm: [Kashiwara, Schapira 2017]

• BarKS is additive: it has a zero object (empty barcode) and direct sums
• The functor

Ψ : BarKS − → vect(R,≤)

k

(A, I) − →

• α∈A kIα

yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R, di(Ψ−1(H∗ ◦ F), Ψ−1(H∗ ◦ G)) ≤ f − g∞. Pb: di ≤ db in BarKL Def: [Bauer, Lesnick 2016] BarBL: same objects, hom-sets reduced to diagonal matrices up to reordering Thm: di = db in BarBL.

∄ Φ : vect(R,≤)

k

→ BarBL Ψ−1 denotes the pseudo-inverse of Ψ up to isomo this is because di allows for more general mappings than just partial matchings (i.e. matrices

The category of barcodes

10

Thm: [Kashiwara, Schapira 2017]

• BarKS is additive: it has a zero object (empty barcode) and direct sums
• The functor

Ψ : BarKS − → vect(R,≤)

k

(A, I) − →

• α∈A kIα

yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R, di(Ψ−1(H∗ ◦ F), Ψ−1(H∗ ◦ G)) ≤ f − g∞. Pb: di ≤ db in BarKL Def: [Bauer, Lesnick 2016] BarBL: same objects, hom-sets reduced to diagonal matrices up to reordering Thm: di = db in BarBL. Pb.: Ψ : BarBL → vect(R,≤)

k

not equiv. of categories

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N).

11

this is a matching between the summands

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). Proof of ≥ (converse stability): [Lesnick 2011] [Chazal et al. 2016] Given ε > db(Dg M, Dg N), take partial matching Dg M ⊇ X

γ

− → X′ ⊆ Dg N. Sort summands of M, N such that the latter decompose as follows: M ≃

• j∈J

Mj N ≃

• j∈J

Nj where each pair (Mj, Nj) is either:

• a pair of summands (kIj, kI′

j) with matched intervals γ(Ij) = I′

j

• (kIj, 0) with Ij /

∈ X unmatched

• (0, kI′

j) with I′

j /

∈ X′ unmatched

11

this is a matching between the summands

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). Proof of ≥ (converse stability): [Lesnick 2011] [Chazal et al. 2016] Given ε > db(Dg M, Dg N), take partial matching Dg M ⊇ X

γ

− → X′ ⊆ Dg N. Sort summands of M, N such that the latter decompose as follows: M ≃

• j∈J

Mj N ≃

• j∈J

Nj where each pair (Mj, Nj) is either:

• a pair of summands (kIj, kI′

j) with matched intervals γ(Ij) = I′

j

• (kIj, 0) with Ij /

∈ X unmatched

• (0, kI′

j) with I′

j /

∈ X′ unmatched

• when Ij − I′

j∞ ≤ ε, kIj and

kI′

j can be ε-interleaved

• when Ij∞ ≤ ε, kIj and 0 can

be ε-interleaved

• take direct sum

11

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). Proof of ≤ (stability):

• 1. interpolation at functional level (soft+hard) [Cohen-Steiner et al. 2005]
• 2. discretizations (loose bound) [Chazal et al. 2009]
• 3. interpolation between modules [Chazal et al. 2009-2016]
• 4. matchings induced from morphisms [Bauer, Lesnick 2014]
• 5. Hall’s marriage theorem [Bjerkevik 2016]

11

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). interpolation between modules: Given M, N : (R, ≤) → Vectk with di(M, N) = ε, find (Uα)0≤α≤ε such that:

• U0 ≃ M
• Uε ≃ N
• ∀0 ≤ α ≤ β ≤ ε, di(Uα, Uβ) ≤ β − α

Proof of ≤ (stability):

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φ ∆ε ∆0 ψ ∆ε ∆0

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). interpolation between modules: Proof of ≤ (stability): Embed (R, ≤) into (R2, ≤) (w. product order) as ∆t for an arbitrary t ∈ R+ M : (∆0, ≤) → Vectk N : (∆ε, ≤) → Vectk

11

φ ∆ε ∆0 ψ ∆ε ∆0

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). interpolation between modules: Proof of ≤ (stability): Embed (R, ≤) into (R2, ≤) (w. product order) as ∆t for an arbitrary t ∈ R+ M : (∆0, ≤) → Vectk N : (∆ε, ≤) → Vectk ε-interleaving (φ, ψ) yields functor F : (∆0 ∪ ∆ε, ≤) → Vectk

11

φ ∆ε ∆0 ψ ∆ε ∆0

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). interpolation between modules: Proof of ≤ (stability): Embed (R, ≤) into (R2, ≤) (w. product order) as ∆t for an arbitrary t ∈ R+ M : (∆0, ≤) → Vectk N : (∆ε, ≤) → Vectk ε-interleaving (φ, ψ) yields functor F : (∆0 ∪ ∆ε, ≤) → Vectk interpolating family (Uα)0≤α≤ε

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≡ functor G : (∆[0,ε], ≤) → Vectk

∆ε ∆0

Hard stability

Thm: (isometry) For any pfd modules M, N : (R, ≤) → vectk, db(Dg M, Dg N) = di(M, N). interpolation between modules: Proof of ≤ (stability): Embed (R, ≤) into (R2, ≤) (w. product order) as ∆t for an arbitrary t ∈ R+ M : (∆0, ≤) → Vectk N : (∆ε, ≤) → Vectk ε-interleaving (φ, ψ) yields functor F : (∆0 ∪ ∆ε, ≤) → Vectk interpolating family (Uα)0≤α≤ε use left Kan extension of ∆0 ∪ ∆ε ֒ → ∆[0,ε]: G(t) := lim − → F|s≤t∈∆0∪∆ε G(s ≤ t) given by universality of colimit s t

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≡ functor G : (∆[0,ε], ≤) → Vectk