Persistent Homology & Category Theory. November 8, 2016

Transcription

1 Persistent Homology & Category Theory November 8, 2016

2 Persistent Homology There are many introductory papers written on the topic of persistent homology...

3 Persistent Homology Here is the Topology & Data by Gunnar Carlsson

4 Persistent Homology What is Persistent Homology by Shmuel Weinberger

5 Persistent Homology Barcodes: the persistent topology of data by Robert Ghrist

6 Why category theory? There are various views on what category theory is about, and what it is good for. Here are some. As a language, it offers economy of thought and expression It reveals common ideas in (ostensibly) unrelated areas of mathematics A single result proved in category theory generates many results in different areas of mathematics.

7 Why category theory? There are various views on what category theory is about, and what it is good for. Here are some. As a language, it offers economy of thought and expression It reveals common ideas in (ostensibly) unrelated areas of mathematics A single result proved in category theory generates many results in different areas of mathematics. To each species of mathematical structure, there corresponds a category, whose objects have that structure, and whose morphisms preserve it - Goguen

8 Why category theory? There are various views on what category theory is about, and what it is good for. Here are some. As a language, it offers economy of thought and expression It reveals common ideas in (ostensibly) unrelated areas of mathematics A single result proved in category theory generates many results in different areas of mathematics. To each species of mathematical structure, there corresponds a category, whose objects have that structure, and whose morphisms preserve it - Goguen Less flatteringly it is often referred to as abstract nonsense.

9 Why category theory?

10 Why category theory?

11 Why category theory? David Spivak Research Scientist Department of Mathematics MIT Office: Building 2, Room dspivak--math/mit/edu Curriculum Vitae. Current research projects Technical Proposal: "Pixel matrices and other compositional analyses of interconnected systems". This is a proposal for another awarded AFOSR grant. Technical Proposal: "Categorical approach to agent interaction". This is the proposal for the awarded AFOSR grant FA Technical Proposal: Category-theoretic Approaches for the Analysis of Distributed Systems. This is the proposal for the awarded NASA grant NNH13ZEA001N-SSAT. Other grants. These are several grant proposals, some funded, some in the pipeline, others not funded, that explain various facets of my research. Categorical informatics. What is the underlying mathematical structure of information itself? FQL. Functorial query language. Build your own categories, functors, and database instances, and push them around with data migration functors. (Joint with Ryan Wisnesky). Category theory book MIT Press has published "Category theory for the sciences". The book can also be purchased on Amazon. Here are reviews by the MAA, by the AMS, and by SIAM. Free HTML version: not as nice to read, but free to the world, on the MIT Press website. An older version, entitled "Category theory for scientists", can be found here. Course (Spring 2013). Course webpage.

12 Categories A category C is specified by the following data: a class C 0 = Obj(C), the objects of the category;

13 Categories A category C is specified by the following data: a class C 0 = Obj(C), the objects of the category; a class C 1 = Arr(C) = Mor(C), the arrows or morphisms of the category. We write C (x, y) or MorC(x, y) or Mor(x, y) to denote the set of arrows x y.

14 Categories A category C is specified by the following data: a class C 0 = Obj(C), the objects of the category; a class C 1 = Arr(C) = Mor(C), the arrows or morphisms of the category. We write C (x, y) or MorC(x, y) or Mor(x, y) to denote the set of arrows x y. There is a composition operation : C (x, y) C (y, z) C (x, z).

15 Categories Composition is associative: (hg)f = h(gf ) when either side is defined.

16 Categories Composition is associative: (hg)f = h(gf ) when either side is defined. Every object x C 0 has an element 1 x C (x, x), which is an identity in the sense that f = f 1 x and g = 1 xg.

17 The Category of Sets objects are sets (alice) morphisms are functions between sets

18 Further Examples

19 The Category of Sets Any partial order (P, ) Objects are the elements of the partial order Morphisms represent the relation. Composition works because of the transitivity of.

20 Functors Let C, D be categories. A functor F : C D is specified by the following data: Every object x C 0 is assigned an object F (x) D 0.

21 Functors Let C, D be categories. A functor F : C D is specified by the following data: Every object x C 0 is assigned an object F (x) D 0. Every arrow f MorC (x, y) is assigned an arrow F (f ) Mor D (F (x), F (y)).

22 Functors Let C, D be categories. A functor F : C D is specified by the following data: Every object x C 0 is assigned an object F (x) D 0. Every arrow f MorC (x, y) is assigned an arrow F (f ) Mor D (F (x), F (y)). The functor respects composition: F (f g) = F (f ) F (g).

23 Functors Let C, D be categories. A functor F : C D is specified by the following data: Every object x C 0 is assigned an object F (x) D 0. Every arrow f MorC (x, y) is assigned an arrow F (f ) Mor D (F (x), F (y)). The functor respects composition: F (f g) = F (f ) F (g). The functor respects identities: F (1 x) = 1 F (x).

24 Functors Let C, D be categories. A functor F : C D is specified by the following data: Every object x C 0 is assigned an object F (x) D 0. Every arrow f MorC (x, y) is assigned an arrow F (f ) Mor D (F (x), F (y)). The functor respects composition: F (f g) = F (f ) F (g). The functor respects identities: F (1 x) = 1 F (x). Examples forgetful functor Top Set which takes a topological space to its underlying set, and which takes each continuous function to itself. There is a forgetful functor Vect Set which takes a vector space to its underlying set, and which takes each linear map to itself. H k : Top Group is a functor from the category of topological spaces to the category of groups.

25 The mechanism for comparing two functors or diagrams is the natural transformation.

26 The mechanism for comparing two functors or diagrams is the natural transformation. Natural Transformations Let C, D be categories and F, G : C D functors. A natural transformation ν from F to G, written ν : F G is defined as follows. To each object x C we assign an arrow ν x : F (x) G(x) of D.

27 The mechanism for comparing two functors or diagrams is the natural transformation. Natural Transformations Let C, D be categories and F, G : C D functors. A natural transformation ν from F to G, written ν : F G is defined as follows. To each object x C we assign an arrow ν x : F (x) G(x) of D. For each arrow f : x y of C we require that the diagram F (x) F (f ) F (y) ν x G(x) G(f ) ν y G(y) commutes.

28 Applications of Category Theory to Persistent Homology There is one important category that you ve come across in this class..

29 Applications of Category Theory to Persistent Homology There is one important category that you ve come across in this class.. Let Vect be a category of vector spaces, and P a partially ordered set.

30 Applications of Category Theory to Persistent Homology There is one important category that you ve come across in this class.. Let Vect be a category of vector spaces, and P a partially ordered set. Recall that we regard P as a category with object set P, and with a unique morphism from x to y whenever x y. The Category of Persistence Vector Spaces A P-persistence vector space is a functor φ: P Vect.

31 Applications of Category Theory to Persistent Homology There is one important category that you ve come across in this class.. Let Vect be a category of vector spaces, and P a partially ordered set. Recall that we regard P as a category with object set P, and with a unique morphism from x to y whenever x y. The Category of Persistence Vector Spaces A P-persistence vector space is a functor φ: P Vect. (In class P = R +. This coincides with our definition: it means a family of F-vector spaces {V r } r [0, ), together with linear transformations L V (r, r ): V r V r whenever r r, so that L V (r, r ) L V (r, r ) = L V (r, r ) for all r r r.).

32 Applications of Category Theory to Persistent Homology P-persistence vector spaces form a category in their own right, where a morphism F from φ to ψ is a natural transformation.

33 Applications of Category Theory to Persistent Homology P-persistence vector spaces form a category in their own right, where a morphism F from φ to ψ is a natural transformation. In more concrete terms, a morphism from one persistence vector space into another is a persistence linear map, i.e. is a family of linear transformations f r : V r W r, so that for all r r, all the diagrams V r L V (r,r ) V r f r f r W r L W (r,r ) W r commute in the sense that f r L V (r, r ) = L W (r, r ) f r.

34 Applications of Category Theory to Persistent Homology In class we showed from scratch that we have a decomposition theorem for finitely presented persistence vector spaces (there is no such theorem for general R-persistence vector spaces). This is not how it is done in most survey papers on the topic.

35 Applications of Category Theory to Persistent Homology In class we showed from scratch that we have a decomposition theorem for finitely presented persistence vector spaces (there is no such theorem for general R-persistence vector spaces). This is not how it is done in most survey papers on the topic. The main ingredient is to note that the category of N-persistence vector spaces over F is equivalent to another category, one of non-negatively graded vector spaces over F[t]. (Z/2Z[t] consists of polynomials f (t) with coefficients in Z/2Z in variable t. For example, t + 1 and t 7 + t 2 are both polynomials with coefficients in Z/2Z.)

36 Applications of Category Theory to Persistent Homology Let {V n} be any N-persistence vector space over F. We define an associated graded module θ({v n}) over the graded polynomial ring F[t] as follows: θ({v n}) = s 0 V s, where the n-th graded part is the vector space V n. The action of the polynomial generator t is given by t (v 0, v 1, v 2,...) = (0, φ 01v 0, φ 12v 1,...). It is readily checked that θ is a functor from the category of N-persistence vector spaces over F to the category of graded F[t]-modules. It is in fact an equivalence of categories, since an inverse functor can be given by V {V n}, where the morphisms φ mn are given by multiplication by t n m.

37 Applications of Category Theory to Persistent Homology There is a classification theorem for finitely generated graded F[t]-modules (Structure theorem for finitely generated modules over a principal ideal domain). Classification theorem for finitely generated graded F[t]-modules Let V denote any finitely generated non-negatively graded F[t]-module. Then there are integers {i 1,..., i m}, {j 1,..., j n}, {l 1,..., l n}, and an isomorphism V = m t is F[t] s=1 k=1 n t j k (F[t]/t l k F[t]), The decomposition is unique up to permutation of factors.

38 Applications of Category Theory to Persistent Homology There is a classification theorem for finitely generated graded F[t]-modules (Structure theorem for finitely generated modules over a principal ideal domain). Classification theorem for finitely generated graded F[t]-modules Let V denote any finitely generated non-negatively graded F[t]-module. Then there are integers {i 1,..., i m}, {j 1,..., j n}, {l 1,..., l n}, and an isomorphism V = m t is F[t] s=1 k=1 n t j k (F[t]/t l k F[t]), The decomposition is unique up to permutation of factors. This classification theorem has a natural interpretation. The free portions are in bijective correspondence with those homology generators which come into existence at parameter i s and which persist for all future parameter values. The torsional elements correspond to those homology generators which appear at parameter j k and disappear at parameter j k + l k. Finitely presented persistence F-vector spaces mapped under θ are finitely generated non-negatively generated F[t]-modules.