Categories 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 note: a small diagram is the image of a functor F : J → C for a small category J • C is co-complete if every (small) diagram in C has a colimit • 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 0 ab ) - 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 Examples: Ab , Mod R , Vect k , · · · 2
Categories Set Top ⊇ objects: sets objects: topological spaces morphisms: set maps morphisms: continuous maps ⊆ Grp Ab ⊇ objects: groups objects: abelian groups morphisms: group homomorphisms morphisms: group homomorphisms ⊆ Vect k Mod R ⊆ beware that, in some cases (such as this one), the notion of inclusion objects: k -vector spaces objects: left R -modules morphisms: k -linear maps morphisms: R -linear maps 2
Categories 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 f • for every a − → b ∈ C , if f ∈ hom( D ) then a, b ∈ obj( D ) • hom( D ) is closed under composition • for all a ∈ obj( D ) , 1 a ∈ hom( D ) Note: D ⊆ C is full if hom D ( a, b ) = hom C ( a, b ) for all a, b ∈ obj( D ) 2
Functors Bottomline: formalize notion of operator between categories 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: f g • composition: for all a − → b − → c ∈ C , basically, turns commutative diagrams into commutative diagrams, F ( g ◦ f ) = F ( g ) ◦ F ( f ) • identity: for all a ∈ obj( C ) , F (1 a ) = 1 F ( a ) 3
Functors Bottomline: formalize notion of operator between categories 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: f g • composition: for all a − → b − → c ∈ C , basically, turns commutative diagrams into commutative diagrams, F ( g ◦ f ) = F ( g ) ◦ F ( f ) • identity: for all a ∈ obj( C ) , F (1 a ) = 1 F ( a ) note that there is no such category for all categories. Indeed, as for sets, the catego Note: small categories and functors between them form a category Cat 3
Functors Example: homology functor H ∗ : Top → Mod Z ⊆ ⊆ ⊆ ⊆ (1-homology functor H 1 ) ( 1 ( 0 1 ) � Z 2 ( 1 0 0 ) � Z 2 0 1 ) � Z 2 ( 0 1 ) � Z Z 3
Functors Example: functor F : ( S, ≤ ) → Vect k � • � • � • � • • F small diagram: small cat. → Vect k ( 1 ( 0 1 ) � k 2 ( 1 0 0 ) � k 2 0 1 ) � k 2 ( 0 1 ) � k k 3
Functors Def: given D ⊆ C , the inclusion functor maps each object and morphism to itself Examples: ⊆ Grp − → Set ⊆ Ab − → Grp 3
Functors Def: given D ⊆ C , the inclusion functor maps each object and morphism to itself Examples: ⊆ Grp − → Set ⊆ Ab − → Grp Def: a forgetful functor is one that ‘forgets structure or axioms’ Examples: (but there exist more general types of forgetful functors) ⊆ Grp − → Set forgets the group structure ⊆ Ab − → Grp forgets the commutativity axiom Top • − this one goes from pointed topological spaces (i.e. ones with a distinguished basepoint) → Top forgets about the basepoint 3
� � Natural transformations Bottomline: view functors as objects in some category 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 ) f such that the following diagram commutes for every morphism a − → b ∈ C : F ( f ) � F ( a ) F ( b ) η ( a ) η ( b ) G ( f ) � G ( b ) G ( a ) 4
� � Natural transformations Bottomline: view functors as objects in some category 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 ) f such that the following diagram commutes for every morphism a − → b ∈ C : F ( f ) � F ( a ) F ( b ) η ( a ) η ( b ) G ( f ) � G ( b ) G ( a ) Prop: functors C → D and their natural transformations form a category D C (vertical composition: ( τ ◦ η )( a ) := τ ( a ) ◦ η ( a ) ) (natural isomorphisms: η ( a ) isomorphism for all a ∈ obj( C ) ) 4
� � � � � � � � � � Natural transformations Example: functors quiver → Vect k k F ( 0 1 ) 1 1 0 � k 2 k ( 1 0 ) • � � - 1 note: the actual category is given by the transitive closure of the graph η 1 � 0 0 � � • k • 0 1 ( 1 − 1 ) − 1 1 ( 1 0 ) � k 2 k ( 1 0 ) G 4
Isomorphism and equivalence of categories Bottomline: isomorphism on objects and morphisms Def: F : C → D is an isomorphism of cateories if there is G : D → C such that G ◦ F = 1 C and F ◦ G = 1 D , where 1 ∗ denotes the identity functor. 5
Isomorphism and equivalence of categories Bottomline: isomorphism ‘up to (natural) isomorphisms‘ Def: F : C → D is an isomorphism of cateories if there is G : D → C such that G ◦ F = 1 C and F ◦ G = 1 D , 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 ⇒ 1 C and F ◦ G ⇒ 1 D . (preserves most properties of a category, works up to isomorphism) Example: vect k is equivalent to Mat k but not isomorphic to it this is because every n -dim. vector space finite-dimensional k -vector spaces k -matrices 5
Isomorphism and equivalence of categories Bottomline: asymmetric sufficient conditions for equivalence 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 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 Pre( C ) is a thin category, having the same objects as C and a unique morphism for each 5
Isomorphism and equivalence of categories Bottomline: asymmetric sufficient conditions for equivalence 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 and therefore neither on morphisms across the whole category, since objects can collide 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 5
Isomorphism and equivalence of categories Bottomline: asymmetric sufficient conditions for equivalence 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 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 5
Isomorphism and equivalence of categories Bottomline: asymmetric sufficient conditions for equivalence 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 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: • Ab → Grp is an embedding • Mod Z → Ab yields an equivalence of categories 5
Isomorphism and equivalence of categories Set Top ← objects: sets objects: topological spaces morphisms: set maps morphisms: continuous maps → Grp Ab ֓ ← objects: groups objects: abelian groups morphisms: group homomorphisms morphisms: group homomorphisms → ≃ ≃ Vect ∗ Mod ∗ → beware here: for a fixed field k = R , we forget about inverses but w objects: ∗ -vector spaces objects: left ∗ -modules ֒ → Note: one can either view Vect ∗ as the category of vector spaces over a morphisms: ∗ -linear maps morphisms: ∗ -linear maps 5
� � � � � � � � � � � Two important results Thm: if C is small and D is abelian, then D C is also abelian. Note: define constructions and operations ‘pointwise’ � • � • C = D = Vect k • • • � 0 0 0 0 0 0 = � 1 � 1 � 1 0 � � � ( 0 1 ) 0 1 0 1 � k 2 � k 2 k 2 k k � 0 0 1 0 � 0 � 0 k k k = � 1 0 � 1 0 � 0 1 0 � 1 0 0 � � � � 0 0 1 0 0 0 1 0 1 0 � k 2 � k 2 k 2 k 2 k 3 6
� � � � � � � � � � � � � � � � � � Two important results Thm: if C is small and D is abelian, then D C is also abelian. Note: define constructions and operations ‘pointwise’ � • � • C = D = Vect k • • • � 0 0 0 0 0 0 = � 1 � 1 � 1 0 � � � ( 0 1 ) 0 1 0 1 k 2 k 2 k 2 k k − 1 ( 0 − 1 ) 1 0 0 0 0 1 0 � 0 � 0 k k k � 1 � 0 0 0 0 � k 2 � k 2 ker = 0 0 coker = 0 k 6
� � � � Two important results Thm: if C is small and D is abelian, then D C is also abelian. Thm: (Mitchell’s embedding) moreover, the embedding is exact, so that it preserves monomorphisms and epimo Every small abelian category embeds into some module category. Note: in our case, D C is not small, however we can construct an embedding exists � • � • C = D = Vect k • • • Vect C → Mod k C k ֒ where k C is the category algebra generated by morphisms (finite paths) in C (the product in k C being given by composition of morphisms / paths) � 1 � 1 � 1 0 � � � ( 0 1 ) 0 1 0 1 Embedding on objects: � k 2 � k 2 k 2 k k �→ k ⊕ k 2 ⊕ k ⊕ k 2 ⊕ k 2 equipped with k C -mod structure given by the linear maps at this stage this is just a k -vector space associated with morphisms in C (paths in graph) 6
Two important results and a definition Thm: if C is small and D is abelian, then D C is also abelian. Thm: (Mitchell’s embedding) moreover, the embedding is exact, so that it preserves monomorphisms and epimo Every small abelian category embeds into some module category. Def: Given C, D additive, F : C → D is additive if F (0) = 0 and F ( a ⊕ b ) = F ( a ) ⊕ F ( b ) . at this stage this is just a k -vector space 6
Introduction to Persistence Theory
Topological Data Analysis (TDA) C’est de ce cette constatation qu’est nee l’analyse topologique des donnees, dont le topological invariants for classification β 0 = β 2 = 1 like homology groups, or the dimension of their free part (called Betti numbers) β 1 = 2 triangulation Algebraic topology Applied algebraic topology compact set topological descriptors for inference and comparison β 0 β 1 β 2 point cloud 2
Topological Data Analysis (TDA) C’est de ce cette constatation qu’est nee l’analyse topologique des donnees, dont le Properties of topological descriptors: • invariant under coordinate changes • stable with respect to perturbations • informative • versatile Applied algebraic topology compact set topological descriptors for inference and comparison β 0 β 1 β 2 point cloud 2
Topological Data Analysis (TDA) C’est de ce cette constatation qu’est nee l’analyse topologique des donnees, dont le 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.) Applied algebraic topology compact set topological descriptors for inference and comparison β 0 β 1 β 2 point cloud 2
Topological Persistence in a Nutshell R X topological space f : X → R f X persistence ∞ Dg f signature: persistence diagram encodes the topological structure of the pair ( X, f ) 3
Topological Persistence in a Nutshell 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 R f t F t := f − 1 (( −∞ , t ]) R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R f R 4
Topological Persistence in a Nutshell 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 R • Alternate representation as a (multi-) set of points in the plane ( diagram ). f ∞ β α β α R 4
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Example: distance function R 2 → R f P : x �→ min p ∈ P � x − p � 2 0 4 8 12 16 20 24 28 32 5
Formalism (1-d persistence) Fix an indexing set T ⊆ R and a field k . Def: A filtration over T is a functor F : ( T, ≤ ) → Top Canonical example: given f : X → R , take the sublevel-sets filtration over R : F ( t ) := f − 1 (( −∞ , t ]) and F ( s ≤ t ) := ( F ( s ) ⊆ F ( t )) for all s ≤ t ∈ R then: F ( s ≤ s ) = 1 F ( s ) and F ( s ≤ u ) = F ( t ≤ u ) ◦ F ( s ≤ t ) for all s ≤ t ≤ u ∈ R 6
Formalism (1-d persistence) Fix an indexing set T ⊆ R and a field k . Def: A filtration over T is a functor F : ( T, ≤ ) → Top Canonical example: given f : X → R , take the sublevel-sets filtration over R : F ( t ) := f − 1 (( −∞ , t ]) and F ( s ≤ t ) := ( F ( s ) ⊆ F ( t )) for all s ≤ t ∈ R then: F ( s ≤ s ) = 1 F ( 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, ≤ ) → Vect k Example: given f : X → R and its sublevel-sets filtration F : ( R , ≤ ) → Top , let M := H ∗ ◦ F : ( R , ≤ ) → Vect k , where H ∗ is singular homology over k 6
Formalism (1-d persistence) Fix an indexing set T ⊆ R and a field k . Def: A filtration over T is a functor F : ( T, ≤ ) → Top Canonical example: given f : X → R , take the sublevel-sets filtration over R : F ( t ) := f − 1 (( −∞ , t ]) and F ( s ≤ t ) := ( F ( s ) ⊆ F ( t )) for all s ≤ t ∈ R then: F ( s ≤ s ) = 1 F ( 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, ≤ ) → Vect k Example: given f : X → R and its sublevel-sets filtration F : ( R , ≤ ) → Top , let M := H ∗ ◦ F : ( R , ≤ ) → Vect k , where H ∗ is singular homology over k Notes: • Vect k is abelian ⇒ so is Vect ( T, ≤ ) hence we can talk about decompositions into direct sums k hence the name. Note that, although the functor category itself is not small, it is indexed • Vect ( T, ≤ ) ֒ → Mod R where R is the algebra generated by ( s, t ) s ≤ t ∈ T k 6
Formalism (1-d persistence) Example: → T ⊂ R T = [5] f : − 1 ≤ 2 ≤ 3 ≤ 4 ≤ 5 F ⊆ ⊆ ⊆ ⊆ H 1 ( − ; k ) ( 1 ( 0 1 ) � k 2 ( 1 0 0 ) � k 2 0 1 ) � k 2 ( 0 1 ) � k k 6
Decompositions Let M be a persistence module over some index set T ⊆ R . Theorem. Then, M decomposes as a direct sum of interval modules k [ b ∗ ,d ∗ ] : 0 0 0 1 1 0 0 0 � · · · � 0 � · · · � 0 � · · · � 0 � k � k 0 � �� � � �� � � �� � i<b ∗ [ b ∗ , d ∗ ] i>d ∗ in the following cases: � M ≃ k [ b ∗ j ,d ∗ j ] • T is finite [Gabriel 1972] [Auslander 1974] , j ∈ J • M is pointwise finite-dimensional (pfd), i.e. M : ( T, ≤ ) → vect k forward means that all arrows i → j satisfy i ≤ j , with the relation i ≤ j ≤ k ˜ i ≤ k [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 Let M be a persistence module over some index set T ⊆ R . Theorem. Then, M decomposes as a direct sum of interval modules k [ b ∗ ,d ∗ ] : 0 0 0 1 1 0 0 0 � · · · � 0 � · · · � 0 � · · · � 0 � k � k 0 � �� � � �� � � �� � 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, ≤ ) → vect k forward means that all arrows i → j satisfy i ≤ j , with the relation i ≤ j ≤ k ˜ i ≤ k [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 Example: 1 ≤ 2 ≤ 3 ≤ 4 ≤ 5 F ⊆ ⊆ ⊆ ⊆ H 1 ( − ; k ) ( 1 ( 0 1 ) � k 2 ( 1 0 0 ) � k 2 0 1 ) � k 2 ( 0 1 ) � k k 7
Decompositions Proof in the case T finite ( T = [ n ] ) and M pfd: Lemma: M decomposes into indecomposables: M ≃ � j ∈ J M j where M j ≃ 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 M j where M j ≃ 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 M j where M j ≃ A ⊕ B ⇒ A = 0 or B = 0 . Lemma: Every indecomposable is an interval module. 7
� � Decompositions Proof in the case T finite ( T = [ n ] ) and M pfd: Lemma: M decomposes into indecomposables: M ≃ � j ∈ J M j where M j ≃ 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 | rk M ( i ≤ j ) � = 0 } . � · · · � 0 � · · · � M ( b ) � M ( d ) � ⋆ M = 0 · · · 0 Let K ⊆ M (submodule) be defined by: 0 if t < b K ( t ) := ker M ( t ≤ d ) if b ≤ t ≤ d this is well-defined because kernels are sent to kernels, by the composition law M ( t ) if t > d 7
� � Decompositions Proof in the case T finite ( T = [ n ] ) and M pfd: Lemma: M decomposes into indecomposables: M ≃ � j ∈ J M j where M j ≃ 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 | rk M ( i ≤ j ) � = 0 } . � · · · � 0 � · · · � M ( b ) � M ( d ) � ⋆ M = 0 · · · 0 Let K ⊆ M (submodule) be defined by: 0 if t < b K ( t ) := ker M ( t ≤ d ) if b ≤ t ≤ d this is well-defined because kernels are sent to kernels, by the composition law M ( t ) if t > d 7
� � Decompositions Proof in the case T finite ( T = [ n ] ) and M pfd: Lemma: M decomposes into indecomposables: M ≃ � j ∈ J M j where M j ≃ A ⊕ B ⇒ A = 0 or B = 0 . Lemma: Every indecomposable is an interval module. Proof: take M � = 0 indecomposable. Note: L b � = 0 because by definition rk M ( b ≤ d ) � = 0 Choose L b � = 0 s.t. M ( b ) = K ( b ) ⊕ L b and define L ⊆ M (submodule) by: Let b := min { i | M ( i ) � = 0 } then d := max { j | rk M ( i ≤ j ) � = 0 } . � 0 if t < b or t > d L ( t ) := � · · · � 0 � · · · � M ( b ) � M ( d ) � ⋆ M = 0 · · · Im M ( b ≤ t ) | L b if b ≤ t ≤ d 0 Let K ⊆ M (submodule) be defined by: 0 if t < b K ( t ) := ker M ( t ≤ d ) if b ≤ t ≤ d this is well-defined because kernels are sent to kernels, by the composition law M ( t ) if t > d 7
� � Decompositions Proof in the case T finite ( T = [ n ] ) and M pfd: Lemma: M decomposes into indecomposables: M ≃ � j ∈ J M j where M j ≃ A ⊕ B ⇒ A = 0 or B = 0 . Lemma: Every indecomposable is an interval module. Proof: take M � = 0 indecomposable. Note: L b � = 0 because by definition rk M ( b ≤ d ) � = 0 Choose L b � = 0 s.t. M ( b ) = K ( b ) ⊕ L b and define L ⊆ M (submodule) by: Let b := min { i | M ( i ) � = 0 } then d := max { j | rk M ( i ≤ j ) � = 0 } . � 0 if t < b or t > d L ( t ) := � · · · � 0 � · · · � M ( b ) � M ( d ) � ⋆ M = 0 · · · Im M ( b ≤ t ) | L b if b ≤ t ≤ d Then: ∃ N ⊆ M (submodule) s.t. M = K ⊕ L ⊕ N ( N defined pointwise by induction) 0 Let K ⊆ M (submodule) be defined by: 0 if t < b K ( t ) := ker M ( t ≤ d ) if b ≤ t ≤ d this is well-defined because kernels are sent to kernels, by the composition law M ( t ) if t > d 7
� � Decompositions Proof in the case T finite ( T = [ n ] ) and M pfd: Lemma: M decomposes into indecomposables: M ≃ � j ∈ J M j where M j ≃ A ⊕ B ⇒ A = 0 or B = 0 . Lemma: Every indecomposable is an interval module. Proof: take M � = 0 indecomposable. Note: L b � = 0 because by definition rk M ( b ≤ d ) � = 0 Choose L b � = 0 s.t. M ( b ) = K ( b ) ⊕ L b and define L ⊆ M (submodule) by: Let b := min { i | M ( i ) � = 0 } then d := max { j | rk M ( i ≤ j ) � = 0 } . � 0 if t < b or t > d L ( t ) := � · · · � 0 � · · · � M ( b ) � M ( d ) � ⋆ M = 0 · · · Im M ( b ≤ t ) | L b if b ≤ t ≤ d It is easily seen that K ∩ L = 0 , because L transports L b from b up to d (included) therefore it cannot Then: ∃ N ⊆ M (submodule) s.t. M = K ⊕ L ⊕ N ( N defined pointwise by induction) 0 Let K ⊆ M (submodule) be defined by: Now, first isomorphism theorem ⇒ L ≃ k r [ b,d ] where r = rk M ( b ≤ d ) > 0 0 if t < b M indecomposable ⇒ K = N = 0 and r = 1 . � K ( t ) := ker M ( t ≤ d ) if b ≤ t ≤ d this is well-defined because kernels are sent to kernels, by the composition law M ( t ) if t > d 7
Structure and Stability at the functional level X topological space, f : X → R pfd function (i.e. s.t. H ∗ ◦ F is pfd) the stability property is easiest to describe in the functional setting sublevel-sets filtration → barcode / diagram barcode ≡ multiset of intervals R f diagram ≡ multiset of points ∞ β α β α X 8
Structure and Stability at the functional level Theorem: For any pfd functions f, g : X → R , d ∞ b (Dg f, Dg g ) ≤ � f − g � ∞ R f g ∞ X 8
Structure and Stability at the functional level Theorem: For any pfd functions f, g : X → R , d ∞ b (Dg f, Dg g ) ≤ � f − g � ∞ Note: there are variants where f, g do not have the same domain X but in the generalized case the statement is more subtle to state, so let me stick to this simpler version R f g ∞ X 8
Structure and Stability at the functional level Persistence diagram ≡ locally finite multiset in the closed half-plane ∆ × R + → Y ′ ⊆ Y ): γ Given a partial matching γ : X ↔ Y (i.e. a bijection X ⊇ X ′ − 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 : � � ∆ (2) c ( γ ) := max y = γ ( x ) c ( x, y ) , max z ∈ ( X \ X ′ ) ⊔ ( Y \ Y ′ ) c ( z ) max z y x bottleneck distance: z ¯ d b ( X, Y ) := γ : X ↔ Y c ( γ ) inf 8
Soft stability Let F, G : ( R , ≤ ) → Top be the sublevel-sets filtrations of f, g Obs: For all t ∈ R , F ( t ) ⊆ G ( t + ε ) ⊆ F ( t + 2 ε ) , where ε := � f − g � ∞ . R t + ε t X 9
� � � � � � � � � � � Soft stability Let F, G : ( R , ≤ ) → Top be the sublevel-sets filtrations of f, g Obs: For all t ∈ R , F ( t ) ⊆ G ( t + ε ) ⊆ F ( t + 2 ε ) , where ε := � f − g � ∞ . Hence the commutative diagrams for all s ≤ t ∈ R : F ( t ) F ( t + 2 ε ) F ( s ) F ( t ) � G ( t + ε ) G ( t + ε ) G ( s + ε ) (inclusion maps) � F ( t + ε ) F ( t + ε ) F ( s + ε ) � G ( t + 2 ε ) G ( t ) G ( s ) G ( t ) 9
� � � � � � � � � � � Soft stability Let F, G : ( R , ≤ ) → Top be the sublevel-sets filtrations of f, g Obs: For all t ∈ R , F ( t ) ⊆ G ( t + ε ) ⊆ F ( t + 2 ε ) , where ε := � f − g � ∞ . Hence the commutative diagrams for all s ≤ t ∈ R : (post-composition with H ∗ ) H ∗ ◦ F ( s ) H ∗ ◦ F ( t ) H ∗ ◦ F ( t ) H ∗ ◦ F ( t + 2 ε ) � H ∗ ◦ G ( t + ε ) H ∗ ◦ G ( s + ε ) H ∗ ◦ G ( t + ε ) (linear maps) � H ∗ ◦ F ( t + ε ) H ∗ ◦ F ( s + ε ) H ∗ ◦ F ( t + ε ) � H ∗ ◦ G ( t + 2 ε ) H ∗ ◦ G ( s ) H ∗ ◦ G ( t ) H ∗ ◦ G ( t ) ε -interleaving between H ∗ ◦ F and H ∗ ◦ G 9
� � � Soft stability 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 : M ( s ) M ( t ) � N ( t + ε ) N ( s + ε ) 9
� � � Soft stability 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 : M ( s ) M ( t ) � N ( t + ε ) N ( s + ε ) Def: ( ε -shift endofunctor) D ( R , ≤ ) D ( R , ≤ ) − → � � � � M [ ε ]( t ) := M ( t + ε ) ∀ t ∈ R � � � �− → M [ ε ] s.t. M � � − [ ε ] : � M [ ε ]( s ≤ t ) := M ( s + ε ≤ t + ε ) ∀ s ≤ t ∈ R � � � � � � ( φ [ ε ] : M [ ε ] ⇒ N [ ε ]) s.t. φ [ ε ]( t ) := φ ( t + ε ) ∀ t ∈ R ( φ : M ⇒ N ) �− → � 9
� � � Soft stability 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 : M ( s ) M ( t ) ε -shift natural transformation: � M ⇒ M [ ε ] � � � t �→ M ( t ≤ t + ε ) � M ( t + ε ) � M ( s + ε ) Def: ( ε -shift endofunctor) D ( R , ≤ ) D ( R , ≤ ) − → � � � � M [ ε ]( t ) := M ( t + ε ) ∀ t ∈ R � � � �− → M [ ε ] s.t. M � � − [ ε ] : � M [ ε ]( s ≤ t ) := M ( s + ε ≤ t + ε ) ∀ s ≤ t ∈ R � � � � � � ( φ [ ε ] : M [ ε ] ⇒ N [ ε ]) s.t. φ [ ε ]( t ) := φ ( t + ε ) ∀ t ∈ R ( φ : M ⇒ N ) �− → � 9
� � � � � � � � Soft stability 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 M [ ε ] M [2 ε ] φ φ [ ε ] ψ [ ε ] ψ N N [ ε ] N [2 ε ] 9
� � � � � � � � Soft stability 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 M [ ε ] M [2 ε ] φ φ [ ε ] ψ [ ε ] ψ N N [ ε ] N [2 ε ] Def: d i ( M, N ) := inf { ε | M, N are ε -interleaved } 9
� � � � � � � � Soft stability 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 M [ ε ] M [2 ε ] φ φ [ ε ] ψ [ ε ] ψ N N [ ε ] N [2 ε ] Def: d i ( 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 , d i ( H ◦ F, H ◦ G ) ≤ � f − g � ∞ . 9
� � � � � � � � Soft stability 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 M [ ε ] M [2 ε ] φ φ [ ε ] ψ [ ε ] ψ N N [ ε ] N [2 ε ] - take H = H ∗ : Top → vect k Def: d i ( M, N ) := inf { ε | M, N are ε -interleaved } - can we build vect ( R , ≤ ) → Bar ? k Thm: (soft stability) For any functions f, g : X → R and any functor H : Top → D , d i ( H ◦ F, H ◦ G ) ≤ � f − g � ∞ . 9
The category of barcodes Def: [Kashiwara, Schapira 2017] The category Bar KS is composed of: • objects: barcode ≡ ( A, I ) where: – A is a finite or countable indexing set – I = ( I α ) α ∈ A is a locally finite collection of intervals I α = [ b ∗ α , d ∗ α ] α ∈ ¯ with b ∗ α ≤ d ∗ R • morphisms: given barcodes ( A, I ) and ( B, J ) : � hom(( A, I ) , ( B, J )) := k ( α,β ) ( α,β ) | α ∈ A,β ∈ B,b ∗ α ≤ d ∗ β ≤ a ∗ β ≤ b ∗ β where k ( α,β ) = k . matrices with zero entries the entry u αβ is associated with interval for incomparable intervals u =( u αβ ) v =( v βγ ) � ( B, J ) � ( C, K ) : • given ( A, I ) � matrix product ( v ◦ u ) αγ := u αβ v βγ ( v ◦ u ) αγ is the matrix entry associated with interval I α being mapp β ∈ B 10
The category of barcodes Thm: [Kashiwara, Schapira 2017] • Bar KS is additive: it has a zero object (empty barcode) and direct sums • The functor vect ( R , ≤ ) Bar KS Ψ : − → k � ( A, I ) �− → α ∈ A k I α yields an equivalence of additive categories. proof sketch: - fully faithful and additive by construction - essentially surjective thanks to the decomposition theorem for pfd modules � 10
The category of barcodes Thm: [Kashiwara, Schapira 2017] • Bar KS is additive: it has a zero object (empty barcode) and direct sums • The functor vect ( R , ≤ ) Bar KS Ψ : − → k � ( A, I ) �− → α ∈ A k I α yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R , Ψ − 1 denotes the pseudo-inverse of Ψ up to isomo d i (Ψ − 1 ( H ∗ ◦ F ) , Ψ − 1 ( H ∗ ◦ G )) ≤ � f − g � ∞ . 10
The category of barcodes Thm: [Kashiwara, Schapira 2017] • Bar KS is additive: it has a zero object (empty barcode) and direct sums • The functor vect ( R , ≤ ) Bar KS Ψ : − → k � ( A, I ) �− → α ∈ A k I α yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R , Ψ − 1 denotes the pseudo-inverse of Ψ up to isomo d i (Ψ − 1 ( H ∗ ◦ F ) , Ψ − 1 ( H ∗ ◦ G )) ≤ � f − g � ∞ . Pb: d i ≤ d b in Bar KL this is because d i allows for more general mappings than just partial matchings (i.e. matrices 10
The category of barcodes Thm: [Kashiwara, Schapira 2017] • Bar KS is additive: it has a zero object (empty barcode) and direct sums • The functor vect ( R , ≤ ) Bar KS Ψ : − → k � ( A, I ) �− → α ∈ A k I α yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R , Ψ − 1 denotes the pseudo-inverse of Ψ up to isomo d i (Ψ − 1 ( H ∗ ◦ F ) , Ψ − 1 ( H ∗ ◦ G )) ≤ � f − g � ∞ . Pb: d i ≤ d b in Bar KL this is because d i allows for more general mappings than just partial matchings (i.e. matrices Def: [Bauer, Lesnick 2016] Bar BL : same objects, hom-sets reduced to diagonal matrices up to reordering Thm: d i = d b in Bar BL . 10
The category of barcodes Thm: [Kashiwara, Schapira 2017] • Bar KS is additive: it has a zero object (empty barcode) and direct sums • The functor vect ( R , ≤ ) Bar KS Ψ : − → k � ( A, I ) �− → α ∈ A k I α yields an equivalence of additive categories. Corollary: (soft stability with barcodes) For any pfd functions f, g : X → R , Ψ − 1 denotes the pseudo-inverse of Ψ up to isomo d i (Ψ − 1 ( H ∗ ◦ F ) , Ψ − 1 ( H ∗ ◦ G )) ≤ � f − g � ∞ . Pb: d i ≤ d b in Bar KL this is because d i allows for more general mappings than just partial matchings (i.e. matrices Def: [Bauer, Lesnick 2016] Bar BL : same objects, hom-sets reduced to diagonal matrices up to reordering Pb.: Ψ : Bar BL → vect ( R , ≤ ) Thm: d i = d b in Bar BL . not equiv. of categories ∄ Φ : vect ( R , ≤ ) → Bar BL k k 10
Hard stability Thm: (isometry) For any pfd modules M, N : ( R , ≤ ) → vect k , d b (Dg M, Dg N ) = d i ( M, N ) . 11
Hard stability Thm: (isometry) For any pfd modules M, N : ( R , ≤ ) → vect k , d b (Dg M, Dg N ) = d i ( M, N ) . Proof of ≥ (converse stability): [Lesnick 2011] [Chazal et al. 2016] → X ′ ⊆ Dg N . γ Given ε > d b (Dg M, Dg N ) , take partial matching Dg M ⊇ X this is a matching between the summands − Sort summands of M, N such that the latter decompose as follows: � � M ≃ M j N ≃ N j j ∈ J j ∈ J where each pair ( M j , N j ) is either: j ) with matched intervals γ ( I j ) = I ′ • a pair of summands ( k I j , k I ′ j • ( k I j , 0) with I j / ∈ X unmatched ∈ X ′ unmatched j ) with I ′ • (0 , k I ′ j / 11
Hard stability Thm: (isometry) For any pfd modules M, N : ( R , ≤ ) → vect k , d b (Dg M, Dg N ) = d i ( M, N ) . Proof of ≥ (converse stability): [Lesnick 2011] [Chazal et al. 2016] → X ′ ⊆ Dg N . γ Given ε > d b (Dg M, Dg N ) , take partial matching Dg M ⊇ X this is a matching between the summands − Sort summands of M, N such that the latter decompose as follows: � � M ≃ M j N ≃ N j j ∈ J j ∈ J where each pair ( M j , N j ) is either: j ) with matched intervals γ ( I j ) = I ′ • a pair of summands ( k I j , k I ′ j • ( k I j , 0) with I j / ∈ X unmatched - when � I j − I ′ j � ∞ ≤ ε , k I j and ∈ X ′ unmatched j ) with I ′ • (0 , k I ′ j / j can be ε -interleaved k I ′ - when � I j � ∞ ≤ ε , k I j and 0 can - take direct sum � 11 be ε -interleaved
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