A Homotopical Categorification of the Euler Calculus Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Cordelia Laura Elizabeth Henderson Moggach Submitted: Liverpool, 28 January 2020 Minor modifications: Grenoble, 28 July 2020 Abstract Euler calculus is an analogue of the theory of integration for constructible func- tions rather than measurable ones. Due to its computationally accessible nature, Euler calculus plays a central role in aspects of applied algebraic topology, for example in enumeration problems involving networks of sensors. A geometric description of the constructible functions is given by the Grothendieck group of the constructible derived category. This sheaf-theoretic categorification of the constructible functions is well-known. We present an alternative geometric cate- gorification of the constructible functions given by a suitable homotopy category; an analogue of the classical Spanier{Whitehead category but for suitably `tame' spaces over the source space of the constructible functions. To do so, we develop an axiomatic method for constructing Spanier{Whitehead categories given some ambient category with certain basic properties. The lifting of the operations of the Euler calculus to functors between these Spanier{Whitehead categories should illuminate homotopical aspects of the Euler calculus. ii In memory of my father, Anthony Austin Moggach, 1946 { 2015, who hoped so much to ride the Mersey ferry. iv Acknowledgements This PhD thesis was funded by the UK Engineering and Physical Sciences Re- search Council [EPSRC Doctoral Training Studentship, Award 1577502]. I am very grateful for their generous support. I would like to thank, first and foremost, my supervisor, Dr Jon Woolf. Sec- ondly, my grandmother, Mary Henderson, to whom I am indebted for passing on her own love of mathematics. I would also like to thank the Department of Mathematical Sciences at the University of Liverpool, in particular, Professor Victor Goryunov, Professor John Gracey, Dr Vladimir Guletskii, Dr Toby Hall, Professor Alexander Movchan, Professor Natalia Movchan, Dr Anna Pratousse- vitch, Professor Mary Rees, Professor Lasse Rempe-Gillen, Dr Ozgur Selsil, and Professor Vladimir Zakalyukin. In addition, I am especially grateful to the following people for their inspi- ration and support: Brel, Lucy D. Brett, Magali Delacoste, J.L. Durand, Jo Forbes-Turko, Raphael Hamilton, Faustine Leyrat, Jean-Michel Morland, Mohan P.R, Romain Paulhan, B´en´edictePoggi-Chanu, Beno^ıtVerrier, Caroline Ziani, and my mother, Emma Henderson. vi Contents Introduction ix 1 Definable spaces 1 1.1 Definitions and elementary properties . .3 1.2 The CW-structure of a compact definable space . 10 1.3 The category Def of compact definable spaces . 16 1.4 The constructible functions . 31 2 The classical Spanier{Whitehead category 35 2.1 A brief introduction to triangulated categories . 36 2.2 The pointed category of finite CW-complexes, CW∗ ........ 38 h 2.3 The homotopy category of pointed finite CW-complexes, CW∗ .. 40 2.4 The Spanier{Whitehead category, SW(CW∗)........... 43 3 An axiomatic approach to Spanier{Whitehead categories 47 3.1 An ambient category C ........................ 48 3.2 The slice-coslice category CZ .................... 61 3.3 Coexact mapping cone sequences in CZ ............... 90 3.4 Slice-coslice categories relative to different base-objects . 103 3.5 The SW-category SW(CZ ) of the slice-coslice category . 113 3.6 Functoriality . 119 4 The Spanier{Whitehead categories 123 4.1 A weaker set of assumptions on the ambient category . 124 4.2 The Spanier{Whitehead category SW(CW∗)........... 126 4.3 The categories Def and DefZ .................... 131 4.4 The definable SW-category, SW(Def∗)............... 134 4.5 The Grothendieck group of SW(Def∗)............... 136 4.6 The relative definable SW-category, SW(DefZ ).......... 142 4.7 The Grothendieck group of SW(DefZ )............... 143 4.8 Lifting the Euler calculus to SW(DefZ ).............. 146 vii viii Contents Bibliography 149 Introduction Euler calculus was developed in the late 1980s by Viro [Vir88] and by Schapira [Sch91, Sch95]. It provides an integration theory for constructible functions which allows one to study the topology of constructible sets and functions. Viro started from the observation that compactly supported Euler characteristic χc, is addi- tive and so is almost a measure, the only difference being that it is not necessarily positive. From this perspective he developed the Euler integral by analogy with integration with respect to a measure. His main applications were to complex ge- ometry and singularity theory | [GZ10] is a survey of this circle of ideas and its more recent relations with motivic measure and other topics in algebraic geome- try. Schapira started from the fact that (under suitable conditions) constructible functions are the Grothendieck group of the constructible derived category of sheaves. The operations of the Euler calculus then arise as `de-categorifications’ of well-known operations on constructible sheaves. His applications were mainly in real analytic geometry, particularly to tomography and questions initiated from robotics. The survey paper [CGR12] focusses on yet other applications to sensing which have been developed by Baryshnikov, Ghrist and others; it also contains an extensive bibliography. The main objective of this thesis is to provide an alterna- tive geometric categorification of the constructible functions via an appropriate homotopy category, and lift the operations of the Euler calculus to the underlying triangulated category of this homotopy category. A calculus is a collection of rules for computation. The rules of the Euler calculus can be summarised as follows. The bounded constructible functions on a compact definable space X form a ring CF (X) which is generated by indicator functions of definable subsets. A continuous definable map β : X ! Y induces functorial homomorphisms of abelian groups ∗ β∗ : CF (X) ! CF (Y ) and β : CF (Y ) ! CF (X) ; ∗ moreover β is a ring homomorphism. Here, by `functorial' we mean that β∗ ∗ = ∗ ∗ ∗ (β )∗ and β = (β ) where β and are composable definable maps. These satisfy, and are determined by, (a) β∗(1A) = χ(A) where χ is the Euler characteristic, 1A is the indicator function of definable A ⊂ X and β : X ! pt is the unique map to a point; ix x Introduction (b) β∗(f) = f ◦ β; ∗ ∗ (c) (base change) b a∗ = α∗β whenever W α X β b Y a Z is a cartesian diagram; ∗ (d) (projection formula) β∗(f · β g) = β∗f · g; The key properties of the Euler calculus are captured by the list above. Following [Vir88, Sch91] we use the notation Z β∗(f) = f dχ X when β : X ! pt is the map to a point, and refer to pushforward to a point as taking the Euler integral or integral with respect to the Euler characteristic of the constructible function f. The Euler characteristic is a topological invariant which is well-defined for spaces that have finite cell decompositions of some form. Accordingly, we want to focus on a suitably nice class of spaces which are well-behaved and have a natural cell decomposition into a finite number of cells. The tame spaces that we want to consider are the definable spaces. A space is said to be definable if it belongs to some o-minimal structure on R and is given the subspace topology of the usual Euclidean topology on Rm. n An o-minimal structure R = fRng on R is a collection of subsets of R for each n 2 N such that n S.1 Rn is a boolean algebra of subsets of R ; S.2 R is closed under cartesian products; n S.3 Rn contains the diagonals f(x1; :::; xn) 2 R j xi = xjg for any i < j; S.4 R is closed under projections π : Rn ! Rn−1 onto the first n−1 coordinates; 2 O.1 R2 contains the subdiagonal f(x; y) 2 R : x < yg; O.2 R1 consists of the finite unions of open intervals and points. The first four axioms define the notion of a structure, and the final two axioms guarantee the o-minimality of the structure. Logicians are at the origin of o- minimal structures. However, it was in the 1980s that o-minimal structures began xi to be studied geometrically as a generalisation of classes such as the semianalytic and semialgebraic sets. In particular by the mathematicians Pillay and Steinhorn in [PS84] where the term \o-minimal structure" was used to highlight similarities with the model theoretic notion of a \strongly minimal structure". The term o-minimal is short for \order minimal." Over the last thirty years the subject has continued to grow, spurred on by a proof that the real exponential field is o-minimal provided by Wilkie in 1991 in [Wil96]. An o-minimal expansion of R, which is an o-minimal structure on R that also contains the graphs of addition and multiplication, was first defined by van den Dries in [vdD84] and, in his own words, is `an excellent framework for develop- ing tame topology, or topologie mod´er´ee,' as laid out in Grothendieck's \Esquisse d'un Programme" of 1984, [Gro97]. We set definability to mean definable in some fixed o-minimal expansion R of R. One of the key theorems on definable spaces is the Cell Decomposition Theorem, which tells us that nice cell decompositions of definable spaces exist (where the closure of any cell is a union of cells). It is important to realise that the notion of `cell decomposition' in an o-minimal expansion of R is not the same as a CW-structure; however, Theorem 1.2.16 will relate these two concepts (as will be discussed shortly). In particular and as called for, a definable space has a well-defined Euler characteristic which is independent of the chosen cell decomposition.