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Homotopy Theory and TDA with a View Towards Category Theory
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-8066-7328
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains three papers. Paper A and Paper B deal with homotopy theory and Paper C deals with Topological Data Analysis. All three papers are written from a categorical point of view.

In Paper A we construct categories of short hammocks and show that their weak homotopy type is that of mapping spaces. While doing this we tackle the problem of applying the nerve to large categories without the use of multiple universes. The main tool in showing the connection between hammocks and mapping spaces is the use of homotopy groupoids, homotopy groupoid actions and the homotopy fiber of their corresponding Borel constructions.

In Paper B we investigate the notion of homotopy commutativity. We show that the fundamental category of a simplicial set is the localization of a subset of the face maps in the corresponding simplex category. This is used to define ∞-homotopy commutative diagrams as functors that send these face maps to weak equivalences. We show that if the simplicial set is the nerve of a small category then such functors are weakly equivalent to functors sending the face maps to isomorphisms. Lastly we show a connection between ∞-homotopy commutative diagrams and mapping spaces of model categories via hammock localization.

In Paper C we study multidimensional persistence modules via tame functors. By defining noise systems in the category of tame functors we get a pseudo-metric topology on these functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

Abstract [sv]

Denna avhandling innehåller tre artiklar. Artikel A och Artikel B handlar om homotopiteori och Artikel C handlar om topologisk dataanalys. Alla tre artiklar är skrivna från en kategorisk synvinkel.

I Artikel A konstruerar vi kategorier av korta hängmattor och visar att dess svaga homotopityper är ekvivalenta med avbildningsrum. Samtidigt som vi gör detta så tacklar vi även problemet med att applicera nerv-funktorn till stora kategorier utan att använda sig av multipla universum. Huvudverktyget för att visa kopplingen mellan hängmattor och avbildningsrum är användandet av homotopigruppoider, deras verkan samt den homotopiska fibern av deras respektive Borel-konstruktioner.

I Artikel B undersöker vi konceptet homotopisk kommutativitet. Vi visar att fundamentalkategorin hos en simpliciell mängd är lokaliseringen av en delmängd av sido-avbildningarna i den korresponderande simpliciella kategorin. Detta används för att definiera ∞-homotopiskt kommuterande diagram som funktorer som skickar dessa sido-avbildningar till svaga ekvivalenser. Vi visar att om den simpliciella mängden är nerven av en liten kategori så är sådana funktorer svagt ekivalenta till funktorer som skickar sido-avbildningarna till isomorfier. Slutligen så visar vi på en koppling mellan ∞-homotopiskt kommuterande diagram och avbildningsrum hos modellkategorier via hängmatte-lokalisering.

I Artikel C studerar multidimensionella persistensmoduler via tama funktorer. Genom att definiera brussystem i kategorin av tama funktorer så får vi en pseudo-metrisk topologi på dessa funktorer. Vi visar hur denna pseduo-metrik kan användas för att identifiera persistenta egenskaper hos kompakta multidimensionella persistensmoduler. För att räkna antalet sådana persistenta egenskaper så introducerar vi karakteristik-räknings-invarianten och visar att tilldelandet av denna variant till kompakta tama funktorer är en 1-Lipschitz operation. För endimensionell persistens så förklarar vi hur, genom att välja lämpigt brussystem, karakteristik-räknings-invarianten identifierar samma persistenta egenskaper som den streckkods-konstruktionen.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. , 23 p.
Series
TRITA-MAT-A, 2016:05
Keyword [en]
Homotopy theory, Topological Data Analysis, Category theory, Mapping spaces, Homotopy commutative diagrams
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-186189ISBN: 978-91-7729-003-2OAI: oai:DiVA.org:kth-186189DiVA: diva2:926055
Public defence
2016-06-07, Kollegiesalen, Brinellvägen 8, Stockholm, 15:00 (English)
Opponent
Supervisors
Note

QC 20160516

Available from: 2016-05-16 Created: 2016-05-04 Last updated: 2016-05-16Bibliographically approved
List of papers
1. Modeling mapping spaces with short hammocks
Open this publication in new window or tab >>Modeling mapping spaces with short hammocks
2014 (English)Licentiate thesis, monograph (Other academic)
Abstract [en]

We construct a category of short hammocks and show that it has the weak homotopy type of mapping spaces. In doing so we tackle the problem of applying the nerve to large categories without the use of multiple universes. We also explore what the mapping space is. The main tool in showing the connection between hammocks and mapping spaces will be the use of homotopy groupoids, homotopy groupoid actions and the homotopy fiber of their corresponding bar constructions.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. vi, 56 p.
Series
TRITA-MAT-A, 2014:14
Keyword
Mapping spaces, hammocks, homotopy theory, category theory
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-156470 (URN)978-91-7595-384-7 (ISBN)
Presentation
2014-12-18, 3418, Lindstedtsvägen 25, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20141208

Available from: 2014-12-08 Created: 2014-11-28 Last updated: 2016-05-16Bibliographically approved
2. Rigidifying homotopy commutative diagrams
Open this publication in new window or tab >>Rigidifying homotopy commutative diagrams
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we investigate functors indexed by simplex categories that send certain face maps to weak equivalences. We explain why such functors can be regarded as homotopy commutative diagrams. The key question we consider is related to rigidifications of such functors: under what circumstances is such a functor weakly equivalent to a functor that send these face maps to isomorphisms? We show that if the simplicial set is the nerve of a small category then such an homotopy commutative diagram can indeed be rigidified. We conjecture that this is also true whenever the simplicial set is a quasi-category. Lastly we show a connection between our homotopy commutative diagrams and mapping spaces of model categories via hammock localization.

Keyword
homotopy commutative diagrams, homotopy theory, category theory, rigidification
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-186188 (URN)
Note

QCR 20160531

Available from: 2016-05-04 Created: 2016-05-04 Last updated: 2016-05-31Bibliographically approved
3. Multidimensional Persistence and Noise
Open this publication in new window or tab >>Multidimensional Persistence and Noise
Show others...
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-168011 (URN)
Note

QS 2015

Available from: 2015-05-25 Created: 2015-05-25 Last updated: 2016-05-16Bibliographically approved

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Thesis Introduction(576 kB)91 downloads
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