Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Modeling mapping spaces with short hammocks
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-8066-7328
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 [en]
Mapping spaces, hammocks, homotopy theory, category theory
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-156470ISBN: 978-91-7595-384-7 (print)OAI: oai:DiVA.org:kth-156470DiVA: diva2:766901
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
In thesis
1. Homotopy Theory and TDA with a View Towards Category Theory
Open this publication in new window or tab >>Homotopy Theory and TDA with a View Towards Category Theory
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
Homotopy theory, Topological Data Analysis, Category theory, Mapping spaces, Homotopy commutative diagrams
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-186189 (URN)978-91-7729-003-2 (ISBN)
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

Open Access in DiVA

Licentiate Thesis(579 kB)170 downloads
File information
File name FULLTEXT01.pdfFile size 579 kBChecksum SHA-512
422ce43284a22067275ab01e812293ba2e8def051dd12e50d8749c3a5d38720388c094b40be84baceaebcd4a7fdf33e667af51d9f0f43cd5800c68afebabc97c
Type fulltextMimetype application/pdf

Authority records BETA

Öberg, Sebastian

Search in DiVA

By author/editor
Öberg, Sebastian
By organisation
Mathematics (Div.)
Algebra and Logic

Search outside of DiVA

GoogleGoogle Scholar
Total: 170 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 1626 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf