Dissertations / Theses: 'Foundations of homotopy theory'

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Bordg, Anthony. "Modèles de l'univalence dans le cadre équivariant." Thesis, Nice, 2015. http://www.theses.fr/2015NICE4083.

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Cette thèse de doctorat a pour sujet les modèles de la théorie homotopique des types avec l'Axiome d'Univalence introduit par Vladimir Voevodsky. L'auteur prend pour cadre de travail les définitions de type-theoretic model category, type-theoretic fibration category (cette dernière étant la notion de modèle considérée dans cette thèse) et d'univers dans une type-theoretic fibration category, définitions dues à Michael Shulman. La problématique principale de cette thèse consiste à approfondir notre compréhension de la stabilité de l'Axiome d'Univalence pour les catégories de préfaisceaux, en particulier pour les groupoïdes équipés d'une involution
This PhD thesis deals with some new models of Homotopy Type Theory and the Univalence Axiom introduced by Vladimir Voevodsky. Our work takes place in the framework of the definitions of type-theoretic model categories, type-theoretic fibration categories (the notion of model under consideration in this thesis) and universe in a type-theoretic fibration category, definitions due to Michael Shulman. The goal of this thesis consists mainly in the exploration of the stability of the Univalence Axiom for categories of functors , especially for groupoids equipped with involutions

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Saleh, Bashar. "Formality and homotopy automorphisms in rational homotopy theory." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-160835.

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This licentiate thesis consists of two papers treating subjects in rational homotopy theory. In Paper I, we establish two formality conditions in characteristic zero. We prove that adg Lie algebra is formal if and only if its universal enveloping algebra is formal. Wealso prove that a commutative dg algebra is formal as a dg associative algebra if andonly if it is formal as a commutative dg algebra. We present some consequences ofthese theorems in rational homotopy theory. In Paper II, we construct a differential graded Lie model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a subspace.

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 2: Manuscript.

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Wang, Guozhen Ph D. Massachusetts Institute of Technology. "Unstable chromatic homotopy theory." Thesis, Massachusetts Institute of Technology, 2015. http://hdl.handle.net/1721.1/99321.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 57-58).
In this thesis, I study unstable homotopy theory with chromatic methods. Using the v, self maps provided by the Hopkins-Smith periodicity theorem, we can decompose the unstable homotopy groups of a space into its periodic parts, except some lower stems. For fixed n, using the Bousfield-Kuhn functor [Phi]n, we can associate to any space a spectrum, which captures the vo-periodic part of its homotopy groups. I study the homotopy type of the spectra LK(n)[Phi]nfSk, which would tell us much about the vn-periodic part of the homotopy groups of spheres provided we have a good understanding of the telescope conjecture. I make use the Goodwillie tower of the identity functor, which resolves the unstable spheres into spectra which are the Steinberg summands of classifying spaces of the additive groups of vector spaces over F,. By understanding the attaching maps of the Goodwillie tower after applying the Bousfield-Kuhn functor, we would be able to determine the homotopy type of LK(n)[Phi]nSk. As an example of how this works in concrete computations, I will compute the homotopy groups of LK(2)[Phi]nS3 at primes p >/= 5. The computations show that the unstable homotopy groups not only have finite p-torsion, their K(2)-local parts also have finite vo-torsion, which indicates there might be a more general finite v-torsion phenomena in the unstable world.
by Guozhen Wang.
Ph. D.

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Beke, Tibor 1970. "Homotopy theory and topoi." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/47465.

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Douglas, Christopher L. "Twisted stable homotopy theory." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/33095.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.
Includes bibliographical references (p. 133-137).
There are two natural interpretations of a twist of stable homotopy theory. The first interpretation of a twist is as a nontrivial bundle whose fibre is the stable homotopy category. This kind of radical global twist forms the basis for twisted parametrized stable homotopy theory, which is introduced and explored in Part I of this thesis. The second interpretation of a twist is as a nontrivial bundle whose fibre is a particular element in the stable homotopy category. This milder notion of twisting leads to twisted generalized homology and cohomology and is central to the well established field of parametrized stable homotopy theory. Part II of this thesis concerns a computational problem in parametrized stable homotopy, namely the determination of the twisted K-homology of the simple Lie groups. In more detail, the contents of the two parts of the thesis are as follows. Part I: I describe a general framework for twisted forms of parametrized stable homotopy theory. An ordinary parametrized spectrum over a space X is a map from X into the category Spec of spectra; in other words, it is a section of the trivial Spec- bundle over X. A twisted parametrized spectrum over X is a section of an arbitrary bundle whose fibre is the category of spectra. I present various ways of characterizing and classifying these twisted parametrized spectra in terms of invertible sheaves and local systems of categories of spectra. I then define homotopy-theoretic invariants of twisted parametrized spectra and describe a spectral sequence for computing these invariants.
(cont.) In a more geometric vein, I show how a polarized infinite-dimensional manifold gives rise to a twisted form of parametrized stable homotopy, and I discuss how this association should be realized explicitly in terms of semi-infinitely indexed spectra. This connection with polarized manifolds provides a foundation for applications of twisted parametrized stable homotopy to problems in symplectic Floer and Seiberg-Witten-Floer homotopy theory. Part II: I prove that the twisted K-homology of a simply connected simple Lie group G of rank n is an exterior algebra on n - 1 generators tensor a cyclic group. I give a detailed description of the order of this cyclic group in terms of the dimensions of irreducible representations of G and show that the congruences determining this cyclic order lift along the twisted index map to relations in the twisted ... bordism group of G.
by Christopher Lee Douglas.
Ph.D.

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Heggie, Murray. "Tensor products in homotopy theory." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=72792.

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Miller, David. "Homotopy theory for stratified spaces." Thesis, University of Aberdeen, 2010. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=158352.

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There are many different notions of stratified spaces. This thesis concerns homotopically stratified spaces. These were defined by Frank Quinn in his paper Homotopically Stratified Sets ([16]). His definition of stratified space is very general and relates strata by “homotopy rather than geometric conditions”. This makes homotopically stratified spaces the ideal class of stratified spaces on which to define and study stratified homotopy theory. In the study of stratified spaces it is useful to examine spaces of popaths (paths which travel from lower strata to higher strata) and holinks (those spaces of popaths which immediately leave a lower stratum for their final stratum destination). It is not immediately clear that for adjacent strata these two path spaces are homotopically equivalent and even less clear that this equivalence can be constructed in a useful way. The first aim of this thesis is to prove such an equivalence exists for homotopically stratified spaces. We will define stratified analogues of the usual definitions of maps, homotopies and homotopy equivalences. Then we will provide an elementary criterion for deciding when a strongly stratified map is a stratified homotopy equivalence. This criterion states that a strongly stratified map is a stratified homotopy equivalence if and only if the induced maps on strata and holink spaces are homotopy equivalences. Using this criterion we will prove that any homotopically stratified space is stratified homotopy equivalent to a homotopically stratified space where neighborhoods of strata are mapping cylinders. Finally we will develop categorical descriptions of the class of homotopically stratified spaces up to stratified homotopy. The first of these categorical descriptions will involve categories with a topology on their object and morphism sets. The second categorical description will involve only categories with discrete object spaces.

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Antolini, Rosa. "Cubical structures and homotopy theory." Thesis, University of Warwick, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.338578.

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Hollander, Sharon Joy 1975. "A homotopy theory for stacks." Thesis, Massachusetts Institute of Technology, 2001. http://hdl.handle.net/1721.1/8637.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.
Includes bibliographical references (p. 69-70).
We give a homotopy theoretic characterization of stacks on a site C which allows one to think of stacks as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category, that is a formal homotopy theory, in which stacks play the special role of the fibrant objects. This allows us to compare the different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, these model structures are Quillen equivalent to the S2-nullification of Jardine's model structure on sheaves of simplicial sets on e.
by Sharon Joy Hollander.
Ph.D.

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Ochi, Yoshihiro. "Iwasawa modules via homotopy theory." Thesis, University of Cambridge, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.624327.

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Maunder, James. "Homotopy theory of moduli spaces." Thesis, Lancaster University, 2017. http://eprints.lancs.ac.uk/88429/.

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This thesis gathers three papers written by the author during PhD study at Lancaster University. In addition to these three papers, this thesis also contains two complementary sections. These two complementary sections are an introduction and a conclusion. The introduction discusses some recurring themes of the thesis and parts of the history leading to those results proven herein. The conclusion briefly comments on the outcomes of the research, its place in the current mathematical literature, and explores possibilities for further research. A common theme of this thesis is the study of Maurer-Cartan elements and their moduli spaces, that is their study up to homotopy or gauge equivalence. Within the three papers cited above (and thus within this thesis), three different applications of Maurer-Cartan elements are demonstrated. The first application constructs certain moduli spaces as models for unbased disconnected rational topological spaces. The second application constructs certain moduli spaces as those governing formal algebraic deformation problems over, not necessarily local, commutative differential graded algebras. The third application uses the presentation of L-infinity algebras as solutions to the Maurer-Cartan equation in certain commutative differential graded algebras to construct minimal models of quantum L-infinity algebras. These quantum homotopy algebras arise as the `higher genus' versions of classical (cyclic) homotopy algebras.

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Corson, Samuel M. "Applications of Descriptive Set Theory in Homotopy Theory." BYU ScholarsArchive, 2010. https://scholarsarchive.byu.edu/etd/2401.

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This thesis presents new theorems in homotopy theory, in particular it generalizes a theorem of Saharon Shelah. We employ a technique used by Janusz Pawlikowski to show that certain Peano continua have a least nontrivial homotopy group that is finitely presented or of cardinality continuum. We also use this technique to give some relative consistency results.

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Szumiło, Karol [Verfasser]. "Two Models for the Homotopy Theory of Cocomplete Homotopy Theories / Karol Szumiło." Bonn : Universitäts- und Landesbibliothek Bonn, 2014. http://d-nb.info/1238687156/34.

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Al, Shumrani Mohammed Ahmed Musa. "Homotopy theory in algebraic derived categories." Thesis, University of Glasgow, 2006. http://theses.gla.ac.uk/1905/.

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In this thesis, we introduce some new notions in the derived category D+(fg) (R) of bounded below chain complexes of finite type over local commutative noetherian ring R with maximal ideal m and residue field K in chapter three and study their relations to each other. Also, we set up the Adams spectral sequence for chain complexes in D+(f,g) (R) in chapter four and study its convergence. To accomplish this task, we give two background chapters. We give some good account of chain complexes in chapter one. We review some basic homological algebra and give definition and basic properties of chain complexes. Then we study the homotopy category of chain complexes and we end chapter one with section about spectral sequences. Chapter two is about the derived category of a commutative ring. Section one is about localization of categories and left and right fractions. Then in section two, we give definition of triangulated categories and some of its basic properties and we end section two with definitions of homotopy limits and colimits. In section three, we show that the derived category is a triangulated category. In section four, we give definitions of the derived functors, the derived tensor product and the derived Hom. In chapter three, we start section one by giving some facts about local rings and we end this section by showing that every bounded below chain complex of finite type has a minimal free resolution. In section two, we show a derived analog of the Whitehead Theorem. In section three, we construct Postnikov towers for chain complexes. In section four, we define the Steenrod algebra. In section five, six and seven, we define irreducible, atomic, minimal atomic, no mod m detectable homology, H*-monogenic, nuclear chain complexes and the core of a chain complex. We show some various results relating these notions to each other and give some examples. In chapter four, we set up the Adams spectral sequence in section one and study its properties. In section two, we study homology localization and local homology. In section three, we define K[0]-nilpotent completion and we show that the Adams spectral sequence for a chain complex Y converges strongly to the homology of the K[0]-nilpotent completion of Y. In section four, we study the Adams spectral sequence’s convergence where we show that the K[0]-nilpotent completion for a bounded chain complex Y consisting of finitely generated free R-modules in each degree is isomorphic to the localization of Y with respect to the H*(—, K)-theory. In section five, we present some examples.

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Raptis, Georgios. "Cobordism categories and abstract homotopy theory." Thesis, University of Oxford, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.510210.

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Hoyer, Rolf. "Two topics in stable homotopy theory." Thesis, The University of Chicago, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3627837.

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We give a definition of a norm functor from H-Mackey functors to G-Mackey functors for G a finite group and H a subgroup of G. We check that this agrees with the construction of Mazur in the case G cyclic of prime power order and also with the topological definition of norm, which has an algebraic presentation due to Ullman. We then use this norm functor to give a characterization of Tambara functors as monoids of an appropriate flavor.

The second chapter is part of a joint project with Andrew Baker. We consider what happens when we take the sphere spectrum, and kill elements of homotopy in an E_∞ fashion. This process starts with the element 2 and is repeated in order to kill all higher homotopy groups. We provide methods for identifying spherical classes and for understanding the Dyer-Lashof action at each step of the construction. We outline how this construction might be used to compute the André-Quillen homology of Eilenberg-MacLane spectra considered as algebras over the sphere spectrum.

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Lee, Wai Kei Peter. "Gröbner bases in rational homotopy theory." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/43786.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
Includes bibliographical references (leaves 38-39).
The Mayer-Vietoris sequence in cohomology has an obvious Eckmann-Hilton dual that characterizes the homotopy of a pullback, but the Eilenberg-Moore spectral sequence has no dual that characterizes the homotopy of a pushout. The main obstacle is the lack of an Eckmann-Hilton dual to the Kiinneth theorem with which to understand the homotopy of a coproduct. This difficulty disappears when working rationally, and we dualize Rector's construction of the Eilenberg-Moore spectral sequence to produce a spectral sequence converging to the homotopy of a pushout. We use Gröbner-Shirshov bases, an analogue of Gröbner bases for free Lie algebras, to compute directly the E2 term for pushouts of wedges of spheres. In particular, for a cofiber sequence A --> X --> C where A and X are wedges of spheres, we use this calculations to generalize a result of Anick by giving necessary and sufficient conditions for the map X --> C to be surjective in rational homotopy. More importantly, we are able to avoid the use of differential graded algebra and minimal models, and instead approach simple but open problems in rational homotopy theory using a simplicial perspective and the combinatorial properties of Gröbner-Shirshov bases.
by Wai Kei Peter Lee.
Ph.D.

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Decker, Marvin Glen. "Loop spaces in motivic homotopy theory." [College Station, Tex. : Texas A&M University, 2006. http://hdl.handle.net/1969.1/ETD-TAMU-1808.

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Silva, Júnior João Alves. "First steps in homotopy type theory." Universidade Federal de Pernambuco, 2014. https://repositorio.ufpe.br/handle/123456789/13853.

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Submitted by Natalia de Souza Gonçalves (natalia.goncalves@ufpe.br) on 2015-05-08T13:12:46Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) dissertation.pdf: 1398032 bytes, checksum: ba6c27cf093110dd1dcf9fea1b529c41 (MD5)
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CNPq
Em abril de 2013, o Programa de Fundamentos Univalentes do IAS, Princeton, lançou o primeiro livro em teoria homotópica de tipos, apresentando várias provas de resultados da teoria da homotopia em “um novo estilo de ‘teoria de tipos informal’ que pode ser lida e entendida por um ser humano, como um complemento à prova formal que pode ser checada por uma máquina”. O objetivo desta dissertação é dar uma abordagem mais detalhada e acessível a algumas dessas provas. Escolhemos como leitmotiv uma versão tipoteórica (originalmente proposta por Michael Shulman) de uma prova padrão de 1(S1) = Z usando espaços de recobrimento. Um ponto crucial dela é o uso do “lema do achatamento” (flattening lemma), primeiramente formulado em generalidade por Guillaume Brunerie, cujo enunciado é bem complicado e cuja a prova é difícil, muito técnica e extensa. Enunciamos e provamos um caso particular desse lema, restringindo-o à mínima generalidade exigida pela demonstração de 1(S1) = Z. Também simplificamos outros resultados auxiliares, adicionamos detalhes a algumas provas e incluímos algumas provas originais de lemas simples como “composição de mapas preserva homotopia”, “contrabilidade é uma invariante homotópica”, “todo mapa entre tipos contráteis é uma equivalência”, etc.

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Kraus, Nicolai. "Truncation levels in homotopy type theory." Thesis, University of Nottingham, 2015. http://eprints.nottingham.ac.uk/28986/.

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Homotopy type theory (HoTT) is a branch of mathematics that combines and benefits from a variety of fields, most importantly homotopy theory, higher dimensional category theory, and, of course, type theory. We present several original results in homotopy type theory which are related to the truncation level of types, a concept due to Voevodsky. To begin, we give a few simple criteria for determining whether a type is 0-truncated (a set), inspired by a well-known theorem by Hedberg, and these criteria are then generalised to arbitrary n. This naturally leads to a discussion of functions that are weakly constant, i.e. map any two inputs to equal outputs. A weakly constant function does in general not factor through the propositional truncation of its domain, something that one could expect if the function really did not depend on its input. However, the factorisation is always possible for weakly constant endofunctions, which makes it possible to define a propositional notion of anonymous existence. We additionally find a few other non-trivial special cases in which the factorisation works. Further, we present a couple of constructions which are only possible with the judgmental computation rule for the truncation. Among these is an invertibility puzzle that seemingly inverts the canonical map from Nat to the truncation of Nat, which is perhaps surprising as the latter type is equivalent to the unit type. A further result is the construction of strict n-types in Martin-Lof type theory with a hierarchy of univalent universes (and without higher inductive types), and a proof that the universe U(n) is not n-truncated. This solves a hitherto open problem of the 2012/13 special year program on Univalent Foundations at the Institute for Advanced Study (Princeton). The main result of this thesis is a generalised universal property of the propositional truncation, using a construction of coherently constant functions. We show that the type of such coherently constant functions between types A and B, which can be seen as the type of natural transformations between two diagrams over the simplex category without degeneracies (i.e. finite non-empty sets and strictly increasing functions), is equivalent to the type of functions with the truncation of A as domain and B as codomain. In the general case, the definition of natural transformations between such diagrams requires an infinite tower of conditions, which exists if the type theory has Reedy limits of diagrams over the ordinal omega. If B is an n-type for some given finite n, (non-trivial) Reedy limits are unnecessary, allowing us to construct functions from the truncation of A to B in homotopy type theory without further assumptions. To obtain these results, we develop some theory on equality diagrams, especially equality semi-simplicial types. In particular, we show that the semi-simplicial equality type over any type satisfies the Kan condition, which can be seen as the simplicial version of the fundamental result by Lumsdaine, and by van den Berg and Garner, that types are weak omega-groupoids. Finally, we present some results related to formalisations of infinite structures that seem to be impossible to express internally. To give an example, we show how the simplex category can be implemented so that the categorical laws hold strictly. In the presence of very dependent types, we speculate that this makes the Reedy approach for the famous open problem of defining semi-simplicial types work.

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Ali, Seema. "Colouring generalized Kneser graphs and homotopy theory." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0014/MQ34938.pdf.

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Gupta, Neha. "Homotopy quantum field theory and quantum groups." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/38110/.

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The thesis is divided into two parts one for dimension 2 and the other for dimension 3. Part one (Chapter 3) of the thesis generalises the definition of an n-dimensional HQFT in terms of a monoidal functor from a rigid symmetric monoidal category X-Cobn to any monoidal category A. In particular, 2-dimensional HQFTs with target K(G,1) taking values in A are generated from any Turaev G-crossed system in A and vice versa. This is the generalisation of the theory given by Turaev into a purely categorical set-up. Part two (Chapter 4) of the thesis generalises the concept of a group-coalgebra, Hopf group-coalgebra, crossed Hopf group-coalgebra and quasitriangular Hopf group-coalgebra in the case of a group scheme. Quantum double of a crossed Hopf group-scheme coalgebra is constructed in the affine case and conjectured for the more general non-affine case. We can construct 3-dimensional HQFTs from modular crossed G-categories. The category of representations of a quantum double of a crossed Hopf group-coalgebra is a ribbon (quasitriangular) crossed group-category, and hence can generate 3-dimensional HQFTs under certain conditions if the category becomes modular. However, the problem of systematic finding of modular crossed G-categories is largely open.

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Hellstrøm-Finnsen, Magnus. "The Homotopy Theory of (∞,1)-Categories." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for fysikk, 2014. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-24362.

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The homotopy category of a stable (∞,1)-category can be endowed with a triangulated structure. The main objective of this thesis is to give a proof of this fact. First it will be discussed some ideas of higher category theory, before (∞,1)-categories and models of (∞,1)-categories will be studied. In particular, topological categories and simplicial categories will be mentioned, but the main focus will be on quasi-categories, which all are models for (∞,1)-categories. The theory of (∞,1)-categories, which is required in order to define stable (∞,1)-categories, is then discussed, in particular functors, subcategories, join constructions, undercategories, overcategories, initial objects, terminal objects, limits and colimits are formally discussed for quasi-categories. Finally, the definition of a stable (∞,1)-category will be discussed. Then the main theorem will be proved, after the required properties of stable (∞,1)-categories are discussed. Background theory from ordinary categories and simplicial sets are collected in the appendices.

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Sutton, Thomas. "Rational homotopy theory and derived commutative algebra." Thesis, University of Sheffield, 2016. http://etheses.whiterose.ac.uk/17667/.

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This thesis presents work relating to the rich connections between Rational Homotopy Theory and Commutative Algebra, and builds on the classical work of Quillen and Sullivan, and more recent work of Greenlees, Hess and Shamir.

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Quirin, Kevin. "Lawvere-Tierney sheafification in Homotopy Type Theory." Thesis, Nantes, Ecole des Mines, 2016. http://www.theses.fr/2016EMNA0298/document.

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Le but principal de cette thèse est de définir une extension de la traduction de double-négation de Gödel à tous les types tronqués, dans le contexte de la théorie des types homotopique. Ce but utilisera des théories déjà existantes, comme la théorie des faisceaux de Lawvere-Tierney, quenous adapterons à la théorie des types homotopiques. En particulier, on définira le fonction de faisceautisation de Lawvere-Tierney, qui est le principal théorème présenté dans cette thèse.Pour le définir, nous aurons besoin de concepts soit déjà définis en théorie des types, soit non existants pour l’instant. En particulier, on définira une théorie des colimits sur des graphes, ainsi que leur version tronquée, et une notion de modalités tronquées basée sur la définition existante de modalité.Presque tous les résultats présentés dans cette thèse sont formalisée avec l’assistant de preuve Coq, muni de la librairie [HoTT/Coq]
The main goal of this thesis is to define an extension of Gödel not-not translation to all truncated types, in the setting of homotopy type theory. This goal will use some existing theories, like Lawvere-Tierney sheaves theory in toposes, we will adapt in the setting of homotopy type theory. In particular, we will define a Lawvere-Tierney sheafification functor, which is the main theorem presented in this thesis.To define it, we will need some concepts, either already defined in type theory, either not existing yet. In particular, we will define a theory of colimits over graphs as well as their truncated version, and the notion of truncated modalities, based on the existing definition of modalities.Almost all the result presented in this thesis are formalized with the proof assistant Coq together with the library [HoTT/Coq]

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Smyth, Conor. "Transversal homotopy theory of Whitney stratified manifolds." Thesis, University of Liverpool, 2012. http://livrepository.liverpool.ac.uk/6653/.

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We modify the theory of homotopy groups to obtain invariants of Whitney stratified spaces by considering smooth maps which are transversal to all strata, and smooth homotopies through such maps. Using this idea we obtain transversal homotopy monoids with duals for any Whitney stratified space. Just as in ordinary homotopy theory we may also define higher categorical invariants of spaces. Here instead of groupoids we obtain categories with duals. We concentrate on examples involving the sphere, stratified by a point and its complement, and complex projective space stratified in a natural way. We also suggest a definition for n-category with dual, which we call a Whitney category. This is defined as a presheaf on a certain category of Whitney stratified spaces, that resticts to a sheaf on a certain subcategory. We show in detail that this definition matches the accepted notion of n-category with duals, at least for small n. It also allows us to prove a version of the Tangle Hypothesis, due to Baez and Dolan, which states that "The n-category of framed codimension k-tangles is equivalent to the free k-tuply monoidal n-category with duals on one object".

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Mohamad, Nadia. "Coarse version of homotopy theory (axiomatic structure)." Thesis, University of Sheffield, 2013. http://etheses.whiterose.ac.uk/4304/.

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In topology, homotopy theory can be put into an algebraic framework. The most complete such framework is that of a Quillen model Category [[15], [5]]. The usual class of coarse spaces appears to be too small to be a Quillen model category. For example, it lacks a good notion of products. However, there is a weaker notion of a cofibration category due to Baues [[1], [2]]. The aim in this thesis is to look at notions of cofibration category within the world of coarse geometry. In particular, there are several sensible notions of the structure of a coarse version of a cofibration category that we define here. Later we compare these notions and apply them to computations. To be precise, there are notions of homotopy groups in a Baues cofibration category. So we compare these groups as well for the different structures we have defined, and to the more concrete notion of coarse homotopy groups defined also in [10]. Going further, there is an abstract notion of a cell complex defined in the context of a cofibration category. In the coarse setting, we prove such cell complexes have a more geometric definition, and precisely we prove that a coarse CW-complex is a cell complex. The ultimate goal of such computations is a version of the Whitehead theorem relating coarse homotopy groups and coarse homotopy equivalences for cell complexes. Abstract versions of the Whitehead theorem are known for cofibration categories [1], so we relate these abstract results to something more geometric. Another direction of the thesis involves Quillen model categories. As already mentioned, there are obstructions to the class of coarse spaces being a Quillen model category; there is no apparent way to define category-theoretic products of coarse spaces. However, such obvious objections vanish if we add extra spaces to the coarse category. These extra spaces are termed non-unital coarse spaces in [9]. We have proved most of Quillen axioms but the existence of limits in one of our categories.

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28

Moriya, Shunji. "Rational homotopy theory and differential graded category." 京都大学 (Kyoto University), 2010. http://hdl.handle.net/2433/120625.

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29

Haydon, James Henri. "Étale homotopy sections of algebraic varieties." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:88019ba2-a589-4179-ad7f-1eea234d284c.

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We define and study the fundamental pro-finite 2-groupoid of varieties X defined over a field k. This is a higher algebraic invariant of a scheme X, analogous to the higher fundamental path 2-groupoids as defined for topological spaces. This invariant is related to previously defined invariants, for example the absolute Galois group of a field, and Grothendieck’s étale fundamental group. The special case of Brauer-Severi varieties is considered, in which case a “sections conjecture” type theorem is proved. It is shown that a Brauer-Severi variety X has a rational point if and only if its étale fundamental 2-groupoid has a special sort of section.

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30

Johnson, Mark William. "Enriched sheaf theory as a framework for stable homotopy theory /." Thesis, Connect to this title online; UW restricted, 1999. http://hdl.handle.net/1773/5775.

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31

Kestin, Gregory Michael. "Light-Shell Theory Foundations." Thesis, Harvard University, 2014. http://dissertations.umi.com/gsas.harvard:11596.

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We study the motivation and groundwork for the construction of a Light-Shell Effective Theory, an effective field theory for describing the matter emerging from high-energy collisions and the accompanying radiation. We begin in chapter 2 with a simple electrodynamics calculation to motivate the picture of the ``light-shell," in which all electric and magnetic fields lie on a spherical shell that moves outward at the speed of light. The result turns out to do more than motivate, as it also hints at an important feature of the theory, namely the gauge in which we subsequently choose to do calculations, called Light-Shell Gauge.
Physics

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32

Ge, Yuzhen. "Homotopy algorithms for the H² and the combined H²/H[infinity] model order reduction problems /." This resource online, 1993. http://scholar.lib.vt.edu/theses/available/etd-09292009-020303/.

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Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1993.
On t.p. "[infinity]" appears as the infinity symbol and is superscript. Vita. Abstract. Includes bibliographical references (leaves 51-54). Also available via the Internet.

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33

Patchkoria, Irakli [Verfasser]. "Rigidity in equivariant stable homotopy theory / Irakli Patchkoria." Bonn : Universitäts- und Landesbibliothek Bonn, 2013. http://d-nb.info/1044971703/34.

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34

So, Tse Leung. "Homotopy theory of gauge groups over 4-manifolds." Thesis, University of Southampton, 2018. https://eprints.soton.ac.uk/428056/.

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Given a principal G-bundle P over a space X, the gauge group G(P) of P is the topological group of G-equivariant automorphisms of P which fix X. The study of gauge groups has a deep connection to topics in algebraic geometry and the topology of 4-manifolds. Topologists have been studying the topology of gauge groups of principal G-bundles over 4-manifolds for a long time. In this thesis, we investigate the homotopy types of gauge groups when X is an orientable, connected, closed 4-manifold. In particular, we study the homotopy types of gauge groups when X is a non-simply-connected 4-manifold or a simply-connected non-spin 4-manifold. Furthermore, we calculate the orders of the Samelson products on low rank Lie groups, which help determine the classification of gauge groups over S⁴.

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35

Lee, Doobum. "Contributions to rational homotopy theory of S¹-spaces /." The Ohio State University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487676847117724.

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36

Wierstra, Felix. "Hopf Invariants in Real and Rational Homotopy Theory." Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-146246.

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In this thesis we use the theory of algebraic operads to define a complete invariant of real and rational homotopy classes of maps of topological spaces and manifolds. More precisely let f,g : M -> N be two smooth maps between manifolds M and N. To construct the invariant, we define a homotopy Lie structure on the space of linear maps between the homology of M and the homotopy groups of N, and a map mc from the set of based maps from M to N, to the set of Maurer-Cartan elements in the convolution algebra between the homology and homotopy. Then we show that the maps f and g are real (rational) homotopic if and only if mc(f) is gauge equivalent to mc(g), in this homotopy Lie convolution algebra. In the last part we show that in the real case, the map mc can be computed by integrating certain differential forms over certain subspaces of M. We also give a method to determine in certain cases, if the Maurer-Cartan elements mc(f) and mc(g) are gauge equivalent or not.

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Manuscript.

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37

Chakraborty, Amal. "Parallel homotopy curve tracking on a hypercube." Diss., This resource online, 1990. http://scholar.lib.vt.edu/theses/available/etd-09162005-115011/.

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38

Schmidt, Birgit. "Spin 4-manifolds and Pin(2)-equivariant homotopy theory." [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=968719244.

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39

Cruickshank, James. "Twisted cobordism and its relationship to equivariant homotopy theory." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf.

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40

Beattie, Malcolm I. C. "Towers, modules and Moore spaces in proper homotopy theory." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240518.

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41

Ullman, Harry Edward. "The equivariant stable homotopy theory around isometric linear maps." Thesis, University of Sheffield, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.537997.

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42

Orton, Richard Ian. "Cubical models of homotopy type theory : an internal approach." Thesis, University of Cambridge, 2019. https://www.repository.cam.ac.uk/handle/1810/289441.

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This thesis presents an account of the cubical sets model of homotopy type theory using an internal type theory for elementary topoi. Homotopy type theory is a variant of Martin-Lof type theory where we think of types as spaces, with terms as points in the space and elements of the identity type as paths. We actualise this intuition by extending type theory with Voevodsky's univalence axiom which identifies equalities between types with homotopy equivalences between spaces. Voevodsky showed the univalence axiom to be consistent by giving a model of homotopy type theory in the category of Kan simplicial sets in a paper with Kapulkin and Lumsdaine. However, this construction makes fundamental use of classical logic in order to show certain results. Therefore this model cannot be used to explain the computational content of the univalence axiom, such as how to compute terms involving univalence. This problem was resolved by Cohen, Coquand, Huber and Mortberg, who presented a new model of type theory in Kan cubical sets which validated the univalence axiom using a constructive metatheory. This meant that the model provided an understanding of the computational content of univalence. In fact, the authors present a new type theory, cubical type theory, where univalence is provable using a new "glueing" type former. This type former comes with appropriate definitional equalities which explain how the univalence axiom should compute. In particular, Huber proved that any term of natural number type constructed in this new type theory must reduce to a numeral. This thesis explores models of type theory based on the cubical sets model of Cohen et al. It gives an account of this model using the internal language of toposes, where we present a series of axioms which are sufficient to construct a model of cubical type theory, and hence a model of homotopy type theory. This approach therefore generalises the original model and gives a new and useful method for analysing models of type theory. We also discuss an alternative derivation of the univalence axiom and show how this leads to a potentially simpler proof of univalence in any model satisfying the axioms mentioned above, such as cubical sets. Finally, we discuss some shortcomings of the internal language approach with respect to constructing univalent universes. We overcome these difficulties by extending the internal language with an appropriate modality in order to manipulate global elements of an object.

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43

Membrillo, Solis Ingrid Amaranta. "Homotopy theory of gauge groups over certain 7-manifolds." Thesis, University of Southampton, 2017. https://eprints.soton.ac.uk/424732/.

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The gauge groups of principal G-bundles over low dimensional spaces have been extensively studied in homotopy theory due to their connections to other areas in mathematics, such as the Yang-Mills gauge theory in mathematical physics. In 2011 Donaldson and Segal established the mathematical set-up to construct new gauge theories over high dimensional spaces. In this thesis we study the homotopy theory of gauge groups over 7-manifolds that arise as total spaces of S 3 -bundles over S 4 and their connected sums. We classify principal G-bundles over manifolds M up to isomorphism in the following cases: (1) M is an S 3 -bundle over S 4 with torsion-free homology; (2) M is an S 3 -bundle over S 4 with non-torsion-free homology and π6(G) = 0; (3) M is a connected sum of S 3 -bundles over S 4 with torsion-free homology and π6(G) = 0. We obtain integral homotopy decomposition of the gauge groups in the cases for which the manifold is either a product of spheres, or a twisted product of spheres, or a connected sum of those. We obtain p-local homotopy decompositions of the loop spaces of the gauge groups in the cases for which the manifold has torsion in homology. Gauge groups of principal G-bundles over manifolds homotopy equivalent to S 7 are classified up to a p-local homotopy equivalence.

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44

Dreckmann, Winfried. "Distributivgesetze in der Homotopietheorie." Bonn : [s.n.], 1993. http://catalog.hathitrust.org/api/volumes/oclc/31453541.html.

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45

Cheng, E. L. G. "Higher-dimensional category theory : opetopic foundations." Thesis, University of Cambridge, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597569.

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The problem of defining a weak n-category has been approached in various different ways, but so far the relationship between these approaches has not been fully understood. The subject of this thesis is the 'opetopic' theory of n-categories, embracing a group of definitions based on the theory of 'opetopes'. This approach was first proposed by Baez and Dolan, and further approaches to the theory have been proposed by Hermida, Makkai and Power, and Leinster. The opetopic definition of n-category has two stages. First, the language for describing k-cells is set up; this, in the language of Baez and Dolan, is the theory of opetopes. Then, a concept of universality is introduced, to deal with composition and coherence. We first exhibit an equivalence between the three theories of opetopes as far as they have been proposed. We then give an explicit description of the category Opetope of opetopes. We also give an alternative presentation of the construction of opetopes using the 'allowable graphs' of Kelly and MacLane. The underlying data for an opetopic n-category is given by an opetopic set. The category of opetopic sets is described explicitly by Baez and Dolan; we prove that this category is in fact equivalent to the category of presheaves on Opetope. We then turn our attention to the fully definition of (weak) n-categories. We define for each n a category Opic-n-Cat of opetopic n-categories and 'lax n-functors'. We then examine low-dimensional cases, and exhibit an equivalence between the opetopic and classical theories for the cases n £ 2, giving in particular an equivalence between the opetopic and classical approaches to bicategories.

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46

Maibom, Heidi Lene. "Philosophical foundations of the Theory Theory of folk psychology." Thesis, University College London (University of London), 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.343900.

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47

Stanculescu, Alexandru. "Homotopy theories on enriched categories and on comonoids." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=115852.

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The main purpose of this work is to study model category structures (in the sense of Quillen) on the categories of small categories and small symmetric multicategories enriched over an arbitrary monoidal model category. Among these model structures, there is one of the greatest importance in applications. We call it the Dwyer-Kan model structure (for enriched categories or enriched symmetric multicategories), and a large amount of this work is dedicated to establishing it for different choices of monoidal model categories. Another model structure that we study is what we call the fibred model structure, again for both small categories and small symmetric multicategories enriched over a suitable monoidal model category.
The other purpose of this work is to study model category structures on the category of comonoids in a monoidal model category.

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48

Vicinsky, Deborah. "The Homotopy Calculus of Categories and Graphs." Thesis, University of Oregon, 2015. http://hdl.handle.net/1794/19283.

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We construct categories of spectra for two model categories. The first is the category of small categories with the canonical model structure, and the second is the category of directed graphs with the Bisson-Tsemo model structure. In both cases, the category of spectra is homotopically trivial. This implies that the Goodwillie derivatives of the identity functor in each category, if they exist, are weakly equivalent to the zero spectrum. Finally, we give an infinite family of model structures on the category of small categories.

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49

Wimmer, Christian [Verfasser]. "Rational global homotopy theory and geometric fixed points / Christian Wimmer." Bonn : Universitäts- und Landesbibliothek Bonn, 2017. http://d-nb.info/1149744863/34.

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50

Vişinescu, Bogdan C. "K-Theory and Homotopy Type of Certain Infinite C*-Algebras." University of Cincinnati / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1178909005.

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Dissertations / Theses on the topic 'Foundations of homotopy theory'

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