In this article, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. If the set of transformations acting is a group, we obtain the so-called GENEOs. We then study the spaces of measurements, whose domains are subjected to the action of certain self-maps and the space of P-GENEOs between these spaces. We define pseudo-metrics on them and show some properties of the resulting spaces. In particular, we show how such spaces have convenient approximation and convexity properties.

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for R2 -valued regular functions defined on a closed Riemannian manifold. In particular, we prove that in this case the matching distance is realised either at special values or at values corresponding to vertical, horizontal or slope 1 lines.

This thesis is a compilation of results that can be framed within the field of applied topology. The starting point of our study is objects presenting a possibly complex intrinsic geometry. The main goal is then to simplify, without trivializing, the geometric information characterising these objects by choosing an appropriate representation. Thus, besides being simple and compact, the chosen representation should maintain the wealth of features of the initial object.In Topological Data Analysis (TDA), this simplification process can be done by assigning to each geometric object a functor indexed by a suitable poset.The most important fact about these functors is that, under appropriate hypotheses on the geometric object, they are discretisable. Being discretisable in this context means that they can be finitely encoded by a finite poset mapping to the original indexing poset. It is then possible to make one step further by computing invariants for the representations obtained. Desirable features for such invariants are to be effectively computable and suitable to describe metrics on. Comparing them gives, in fact, a good approximation of the comparison of the underlying geometric objects which are our primary interest. 

Paper A studies decompositions of simplicial complexes that are induced by coverings of their vertices. These decompositions are inspired by data analysis where commonly the data is given by a distance space, to which a filtered simplicial complex can be associated. We study how the homotopy type of a decomposed complex differs from the initial one, both for generic and for metric simplicial complexes.

Another model to perform data analysis from a topological perspective is given by the theory of group equivariant nonexpansive operators.In Paper B, we show that such operators form a complementary tool to persistent homology in the context of TDA. We propose a categorical structure incorporating both models and then we study the functoriality of persistence. 

In Paper C we investigate suitable indexing posets for tame functors. The attention is focused on upper semilattices, which are particularly well suited for this purpose. Another class of posets that have similar properties to upper semilattices is the one of realisations, which we introduce here. Their similarities are both combinatorial, in particular concerning a notion of dimension that we introduce, and related to homological algebra for the tame functors indexed by them. In Paper C we also propose a method based on Koszul complexes to compute homological invariants for tame functors indexed either by upper semilattices or realisations. This question is then expanded in Paper D, where we study homological invariants relative to a chosen class of projectives, possibly different to the standard ones. We propose a framework to translate from the relative to the standard setting, where Koszul complexes are available to perform the computations. We also identify an obstruction for such translation to be possible and characterise it for several examples of relative projectives. 

In Paper E we study the geometrical properties of a well-established metric in 2-parameter persistent homology, called the matching distance. Motivated by the need for effectiveness in the computation of such metric, we study its geometric properties.In particular, we show how to take advantage of the differential geometric structure of the underlying objects to understand the properties of the metric. 

In Paper F we study the category of discretisable functors with values in non-negative chain complexes. In this category, we are particularly interested in cofibrant indecomposables, which require a model structure to be defined. Thus, we first identify a new class of posets indexing the functors for which a projective model structure exists and give a characterisation of cofibrant indecomposables there. In the case, the indexing poset is not of this type, we outline a technique to construct arbitrarily complicated cofibrant indecomposables.

Denna avhandling är en sammanställning av resultat inom tillämpad topologi.Utgångspunkten för vår studie är objekt som presenterar en möjligen komplex inneboende geometri.Huvudmålet är då att förenkla den geometriska informationen som kännetecknar dessa objekt, utan att trivialisera den.Således, förutom att vara enkel och kompakt, bör den valda representationen bibehålla rikedomen av egenskaper hos det ursprungliga objektet.I Topologisk Data Analys (TDA) kan denna förenklingsprocess göras genom att tilldela varje geometriskt objekt en funktor indexerad av en lämplig pomängd.Det viktigaste med dessa funktorer är att de är diskretiserbara under lämpliga antaganden om det geometriska objektet.Att vara diskretiserbar i detta sammanhang innebär att de kan ändligt kodas genom en finit pomängd-mappning till den ursprungliga indexeringspomängden.Det är då möjligt att ta ytterligare steg genom att beräkna invarianter av de representationer som erhålls.Önskvärda egenskaper för sådana invarianter är att vara effektivt beräkningsbara ochlämplig att beskriva metriker på.Att jämföra invarianterna ger då en bra approximation av jämföra de underliggande geometriska objekten, som är vårt primära intresse.

Artikel A studerar dekompositioner av simpliciala komplex som induceras av täckningar av deras hörn.Dessa dekompositioner är inspirerade av dataanalys där datan vanligtvis ges av ett metriskt utrymme, till vilket ett filtrerat simplicialt komplex kan associeras.Vi studerar hur homotopitypen för ett nedbrutet komplex skiljer sig från det initiala, både för generiska och för metriska simpliciala komplex.

En annan modell för att utföra dataanalys ur ett topologiskt perspektiv ges av teorin om gruppekvivarianta icke-expansiva operatorer.I Paper B visar vi att sådana operatörer utgör ett komplementärt verktyg till ihållande homologi i samband med TDA.Vi föreslår en kategorisk struktur som inkluderar båda modellerna och sedan studerar vi funktorialiteten av persistens.

I Paper C undersöker vi lämpliga indexeringspositioner för tama funktorer.Fokus ligger på övre semigitter, som är särskilt väl lämpade för detta ändamål.En annan klass av pomängder som har liknande egenskaper som övre semigitter är den av realisationer, som vi introducerar här.Deras likheter är både kombinatoriska, särskilt när det gäller en dimensionsuppfattning som vi introducerar, och relaterade till homologisk algebra för de tama funktorer som indexeras av dem.I Paper C föreslår vi också en metod baserad på Koszul-komplex för att beräkna homologiska invarianter för tama funktorer indexerade antingen med övre semigitter eller realisationer.Denna fråga utökas sedan i Paper D, där vi studerar homologiska invarianter i förhållande till en vald klass av projektiva objekt, möjligen olika de vanliga.Vi föreslår ett ramverk för att översätta från den relativa till standardfallet, där Koszul-komplex är tillgängliga för att utföra beräkningarna.Vi identifierar också ett hinder för att en sådan översättning ska vara möjlig och karakteriserar den för flera exempel på relativa projektiv.

I Paper E studerar vi de geometriska egenskaperna hos en väletablerad metrik i 2-parameter ihållande homologi, kallad matchningsavståndet.Motiverade av behovet av effektivitet vid beräkningen av sådan metrik studerar vi dess geometriska egenskaper.I synnerhet visar vi hur man drar fördel av den differentiella geometriska strukturen hos de underliggande objekten för att förstå metrikens egenskaper.

I Paper F studerar vi kategorin av diskretiserbara funktioner med värden i icke-negativa kedjekomplex.I den här kategorin är vi särskilt intresserade av kofibranter odelbara, som kräver en modellstruktur för att definieras.Sålunda identifierar vi först en ny klass av pomängder som indexerar de funktioner för vilka det finns en projektiv modellstruktur och ger en karakterisering av kofibranter som är odelbara där.Om indexeringsposen inte är av denna typ,vi skisserar en teknik för att konstruera godtyckligt komplicerade kofibranter odelbara.

Motivated by applications in Topological Data Analysis, we consider decompositionsof a simplicial complex induced by a cover of its vertices. We study how the homotopytype of such decompositions approximates the homotopy of the simplicial complexitself. The difference between the simplicial complex and such an approximationis quantitatively measured by means of the so called obstruction complexes. Ourgeneral machinery is then specialized to clique complexes, Vietoris-Rips complexesand Vietoris-Rips complexes of metric gluings.

In this work, we provide a model structure on full subcategories of tame objects in functor categories indexed by continuous realizations of posets of dimension 1. We also characterize the indecomposable cofibrant objects when the landing category is the one of bounded non-negative chain complexes. In addition, we present a general method to construct indecomposables in a functor category by a gluing technique.

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for ℝ2-valued regular functions defined on a Riemannian closed manifold. In particular, we prove that in this case the matching distance is realised either at special values or at values corresponding to vertical, horizontal or slope 1 lines.

In relative homological algebra of vector space valued functors indexed by a poset, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in relative minimal resolutions. In this article we show that, under certain conditions, grading the chosen family of functors by an upper semilattice guarantees the existence of relative minimal resolutions and the uniqueness of direct sum decompositions in these resolutions. These conditions are necessary for defining relative Betti diagrams and computing these diagrams using Koszul complexes.

We introduce a construction called realisation whichtransforms posets into posets. We show that realisations shareseveral key features with upper semilattices which are essentialin persistence. For example, we define local dimensions of pointsin a poset and show that these numbers for realisations behavein a similar way as they do for upper semilattices. Furthermore,similarly to upper semilattices, realisations have well behaved discrete approximations which are suitable for capturing homologicalproperties of functors indexed by them. These discretisations areconvenient and effective for describing tameness of functors. Homotopical and homological properties of tame functors, particularlythose indexed by realisations, are discussed.

The aim of this article is to describe a new perspective on functoriality ofpersistent homology and explain its intrinsic symmetry that is often overlooked. A dataset for us is a finite collection of functions, called measurements, with a finite domain. Sucha data set might contain internal symmetries which are effectively captured by the actionof a set of the domain endomorphisms. Different choices of the set of endomorphismsencode different symmetries of the data set. We describe various category structureson such enriched data sets and prove some of their properties such as decompositionsand morphism formations. We also describe a data structure, based on coloured directedgraphs, which is convenient to encode the mentioned enrichment. We show that persistenthomology preserves only some aspects of these collection of enriched data sets however notall. In other words persistent homology is not a functor on the entire category of enricheddata sets. Nevertheless we show that persistent homology is functorial locally. We usethe concept of set equivariant operator (SEO) to capture some of the information missedby persistent homology. Moreover, we provide examples and give ways to construct suchSEOs.