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Journal articles on the topic 'Foundations of homotopy theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 February 2022

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1

RIJKE, EGBERT, and BAS SPITTERS. "Sets in homotopy type theory." Mathematical Structures in Computer Science 25, no. 5 (January 30, 2015): 1172–202. http://dx.doi.org/10.1017/s0960129514000553.

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Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.

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2

Pelayo, Álvaro, and Michael A. Warren. "Homotopy type theory and Voevodsky’s univalent foundations." Bulletin of the American Mathematical Society 51, no. 4 (May 9, 2014): 597–648. http://dx.doi.org/10.1090/s0273-0979-2014-01456-9.

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3

AWODEY, STEVE, NICOLA GAMBINO, and ERIK PALMGREN. "Introduction – from type theory and homotopy theory to univalent foundations." Mathematical Structures in Computer Science 25, no. 5 (March 10, 2015): 1005–9. http://dx.doi.org/10.1017/s0960129514000474.

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We give an overview of the main ideas involved in the development of homotopy type theory and the univalent foundations of Mathematics programme. This serves as a background for the research papers published in the special issue.

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4

Thomas, Jean-Claude, and Micheline Vigué-Poirrier. "Daniel Quillen, the father of abstract homotopy theory." Journal of K-theory 11, no. 3 (March 1, 2013): 479–91. http://dx.doi.org/10.1017/is013001006jkt204.

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AbstractIn this short paper we try to describe the fundamental contribution of Quillenin the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

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5

Lumsdaine, Peter Lefanu, and Nicolas Tabareau. "Preface: Special Issue on Homotopy Type Theory and Univalent Foundations." Journal of Automated Reasoning 63, no. 2 (October 30, 2018): 157–58. http://dx.doi.org/10.1007/s10817-018-9491-3.

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6

Ahrens, Benedikt, Simon Huber, and Anders Mörtberg. "Preface to the MSCS Issue 31.1 (2021) Homotopy Type Theory and Univalent Foundations." Mathematical Structures in Computer Science 31, no. 1 (January 2021): 1–2. http://dx.doi.org/10.1017/s0960129521000244.

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7

GYLTERUD, HÅKON ROBBESTAD. "FROM MULTISETS TO SETS IN HOMOTOPY TYPE THEORY." Journal of Symbolic Logic 83, no. 3 (September 2018): 1132–46. http://dx.doi.org/10.1017/jsl.2017.84.

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AbstractWe give a model of set theory based on multisets in homotopy type theory. The equality of the model is the identity type. The underlying type of iterative sets can be formulated in Martin-Löf type theory, without Higher Inductive Types (HITs), and is a sub-type of the underlying type of Aczel’s 1978 model of set theory in type theory. The Voevodsky Univalence Axiom and mere set quotients (a mild kind of HITs) are used to prove the axioms of constructive set theory for the model. We give an equivalence to the model provided in Chapter 10 of “Homotopy Type Theory” by the Univalent Foundations Program.

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8

Joyal, André, and Myles Tierney. "On the homotopy theory of sheaves of simplicial groupoids." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 2 (August 1996): 263–90. http://dx.doi.org/10.1017/s0305004100074855.

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The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as extraordinary, sheaf cohomology.In [11] we began constructing a theory of classifying spaces for sheaves of simplicial groupoids, and that study is continued here. Such a theory is essential for the development of basic tools such as Postnikov systems, Atiyah-Hirzebruch spectral sequences, characteristic classes, and cohomology operations in extraordinary cohomology of sheaves. Thus, in some sense, we are continuing the program initiated by Illusie[7], Brown[2], and Brown and Gersten[3], though our basic homotopy theory of simplicial sheaves is different from theirs. In fact, the homotopy theory we use is the global one of [10]. As a result, there is some similarity between our theory and the theory of Jardine[8], which is also partially based on [10]

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9

Escardó, Martín Hötzel. "The Cantor–Schröder–Bernstein Theorem for $$\infty $$-groupoids." Journal of Homotopy and Related Structures 16, no. 3 (June 28, 2021): 363–66. http://dx.doi.org/10.1007/s40062-021-00284-6.

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AbstractWe show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or $$\infty $$ ∞ -groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean $$\infty $$ ∞ -topos.

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10

Corfield, David. "Expressing ‘the structure of’ in homotopy type theory." Synthese 197, no. 2 (November 20, 2017): 681–700. http://dx.doi.org/10.1007/s11229-017-1569-7.

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Abstract As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy which have been treated by logic. In this article, definite description, and in particular its employment within mathematics, is formulated within the type theory. Homotopy type theory has been proposed as an inherently structuralist foundational language for mathematics. Using the new formulation of definite descriptions, opportunities to express ‘the structure of’ within homotopy type theory are explored, and it is shown there is little or no need for this expression.

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11

Ladyman, James, and Stuart Presnell. "Does Homotopy Type Theory Provide a Foundation for Mathematics?" British Journal for the Philosophy of Science 69, no. 2 (June 1, 2018): 377–420. http://dx.doi.org/10.1093/bjps/axw006.

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12

Ladyman, James, and Stuart Presnell. "The Hole Argument in Homotopy Type Theory." Foundations of Physics 50, no. 4 (October 4, 2019): 319–29. http://dx.doi.org/10.1007/s10701-019-00293-9.

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Abstract The Hole Argument is primarily about the meaning of general covariance in general relativity. As such it raises many deep issues about identity in mathematics and physics, the ontology of space–time, and how scientific representation works. This paper is about the application of a new foundational programme in mathematics, namely homotopy type theory (HoTT), to the Hole Argument. It is argued that the framework of HoTT provides a natural resolution of the Hole Argument. The role of the Univalence Axiom in the treatment of the Hole Argument in HoTT is clarified.

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13

Yazdi, Ali A. "Nonlinear Flutter of Laminated Composite Plates Resting on Nonlinear Elastic Foundations Using Homotopy Perturbation Method." International Journal of Structural Stability and Dynamics 15, no. 05 (May 27, 2015): 1450072. http://dx.doi.org/10.1142/s0219455414500722.

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In this paper, the applicability of the homotopy perturbation method (HPM) in analyzing the flutter of geometrically nonlinear cross-ply rectangular laminated plates resting on nonlinear elastic foundation is investigated. The piston theory is employed to evaluate the aerodynamic pressure acting on the plate. The von Karman geometric nonlinear theory is used to construct the governing equations of the system. The Galerkin's method is used to reduce the nonlinear partial differential equations to a nonlinear second-order ordinary differential equation, and the HPM is employed to study the effect of initial deflection, aspect ratio and stacking sequence on the flutter pressure of cross-ply laminated plates. The results show that the first approximation of the HPM leads to highly accurate solutions for the geometrically nonlinear flutter of cross-ply rectangular laminated plates subjected to the aerodynamic pressure.

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14

van Oosten, Jaap. "The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. http://homotopytypetheory.org/book, Institute for Advanced Study, 2013, vii + 583 pp." Bulletin of Symbolic Logic 20, no. 4 (December 2014): 497–500. http://dx.doi.org/10.1017/bsl.2014.31.

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15

LADYMAN, JAMES, and STUART PRESNELL. "UNIVERSES AND UNIVALENCE IN HOMOTOPY TYPE THEORY." Review of Symbolic Logic 12, no. 3 (July 15, 2019): 426–55. http://dx.doi.org/10.1017/s1755020316000460.

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AbstractThe Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory (HoTT) research programme. It is said by Steve Awodey that Univalence embodies mathematical structuralism, and that Univalence may be regarded as ‘expanding the notion of identity to that of equivalence’. This article explores the conceptual, foundational and philosophical status of Univalence in Homotopy Type Theory. It extends our Types-as-Concepts interpretation of HoTT to Universes, and offers an account of the Univalence axiom in such terms. We consider Awodey’s informal argument that Univalence is motivated by the principle that reasoning should be invariant under isomorphism, and we examine whether an autonomous and rigorous justification along these lines can be given. We consider two problems facing such a justification. First, there is a difference between equivalence and isomorphism and Univalence must be formulated in terms of the former. Second, the argument as presented cannot establish Univalence itself but only a weaker version of it, and must be supplemented by an additional principle. The article argues that the prospects for an autonomous justification of Univalence are promising.

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16

Muranov, Yuri V., and Anna Szczepkowska. "Path homology theory of edge-colored graphs." Open Mathematics 19, no. 1 (January 1, 2021): 706–23. http://dx.doi.org/10.1515/math-2021-0049.

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Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

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17

Yu, Qiang, and Hang Xu. "A homotopy-based wavelet approach for large deflection of a circular plate on nonlinear foundations with parameterized boundaries." Computers & Mathematics with Applications 90 (May 2021): 80–95. http://dx.doi.org/10.1016/j.camwa.2021.03.015.

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18

CRANE, LOUIS. "RELATIONAL TOPOLOGY AS THE FOUNDATION FOR QUANTUM GRAVITY." Modern Physics Letters A 20, no. 17n18 (June 14, 2005): 1261–69. http://dx.doi.org/10.1142/s0217732305017731.

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We propose a new approach to the quantum theory of gravitation, in which the point sets of regions of spacetime are replaced by objects in a category. The objects can be constructed as limits of spin foam models, thus directly connecting to approximations to general relativity which have already been studied, and leading to a mathematically natural interpretation of the meaning of the triangulations in the spin foam models. The physical motivation for our proposal is a connection between ideas about the limited amount of information which can flow from one region to another in General Relativity, an analysis of the problem of the infinities in QFT, and the relational or categorical approach to topology in abstract homotopy theory and algebraic geometry.

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19

Yazdi, Ali A. "Large amplitude flutter analysis of functionally graded carbon nanotube reinforced composite plates with piezoelectric layers on nonlinear elastic foundation." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, no. 2 (October 25, 2017): 533–44. http://dx.doi.org/10.1177/0954410017736546.

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This paper presents a study of geometrical nonlinear flutter of functionally graded carbon nanotube-reinforced composite plates embedded with piezoelectric layers subjected to supersonic flow on nonlinear elastic foundation. The governing equations of the system are obtained using the classical plate theory and von Karman geometric nonlinearity. The linear piston theory is utilized to evaluate the aerodynamic pressure. Galerkin method is used to reduce the governing equations to an ordinary differential equation with respect to time in the form of Duffing equation. The homotopy perturbation method is employed to study the effect of large amplitude on the nondimensional flutter pressure. It is assumed that carbon nano-tubes are distributed along thickness in two different manners namely uniform distribution and functionally graded. The effects of volume fraction of carbon nanotubes, large amplitude, different distribution types, piezoelectric layers, and applied voltage on flutter pressure are studied.

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20

Smith, Abraham, Paul Bendich, and John Harer. "Persistent obstruction theory for a model category of measures with applications to data merging." Transactions of the American Mathematical Society, Series B 8, no. 1 (February 2, 2021): 1–38. http://dx.doi.org/10.1090/btran/56.

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Collections of measures on compact metric spaces form a model category (“data complexes”), whose morphisms are marginalization integrals. The fibrant objects in this category represent collections of measures in which there is a measure on a product space that marginalizes to any measures on pairs of its factors. The homotopy and homology for this category allow measurement of obstructions to finding measures on larger and larger product spaces. The obstruction theory is compatible with a fibrant filtration built from the Wasserstein distance on measures. Despite the abstract tools, this is motivated by a widespread problem in data science. Data complexes provide a mathematical foundation for semi-automated data-alignment tools that are common in commercial database software. Practically speaking, the theory shows that database JOIN operations are subject to genuine topological obstructions. Those obstructions can be detected by an obstruction cocycle and can be resolved by moving through a filtration. Thus, any collection of databases has a persistence level, which measures the difficulty of JOINing those databases. Because of its general formulation, this persistent obstruction theory also encompasses multi-modal data fusion problems, some forms of Bayesian inference, and probability couplings.

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21

Vallette, Bruno. "Homotopy theory of homotopy algebras." Annales de l'Institut Fourier 70, no. 2 (2020): 683–738. http://dx.doi.org/10.5802/aif.3322.

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22

Goerss, Paul, John Greenlees, and Stefan Schwede. "Homotopy Theory." Oberwolfach Reports 12, no. 1 (2015): 731–81. http://dx.doi.org/10.4171/owr/2015/14.

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23

Grodal, Jesper, Michael Hill, and Birgit Richter. "Homotopy Theory." Oberwolfach Reports 16, no. 3 (September 9, 2020): 2183–256. http://dx.doi.org/10.4171/owr/2019/36.

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24

Shitanda, Yoshimi. "Abstract homotopy theory and homotopy theory of functor category." Hiroshima Mathematical Journal 19, no. 3 (1989): 477–97. http://dx.doi.org/10.32917/hmj/1206129287.

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25

Rezk, Charles. "A model for the homotopy theory of homotopy theory." Transactions of the American Mathematical Society 353, no. 3 (June 20, 2000): 973–1007. http://dx.doi.org/10.1090/s0002-9947-00-02653-2.

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26

Abbott, Steve, Paul G. Goerss, and John F. Jardine. "Simplicial Homotopy Theory." Mathematical Gazette 84, no. 500 (July 2000): 360. http://dx.doi.org/10.2307/3621718.

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27

Jardine, J. F. "Categorical homotopy theory." Homology, Homotopy and Applications 8, no. 1 (2006): 71–144. http://dx.doi.org/10.4310/hha.2006.v8.n1.a3.

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28

Awodey, Steve, and Robert Harper. "Homotopy type theory." ACM SIGLOG News 2, no. 1 (January 28, 2015): 37–44. http://dx.doi.org/10.1145/2728816.2728825.

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29

Levine, Marc. "Motivic Homotopy Theory." Milan Journal of Mathematics 76, no. 1 (June 18, 2008): 165–99. http://dx.doi.org/10.1007/s00032-008-0088-x.

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30

Scheerer, Hans, and Daniel Tanré. "R Local Homotopy Theory as Part of Tame Homotopy Theory." Bulletin of the London Mathematical Society 22, no. 6 (November 1990): 591–98. http://dx.doi.org/10.1112/blms/22.6.591.

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31

Kajiura, Hiroshige. "Cyclicity in homotopy algebras and rational homotopy theory." Georgian Mathematical Journal 25, no. 4 (December 1, 2018): 545–70. http://dx.doi.org/10.1515/gmj-2018-0058.

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AbstractKadeishvili proposes a minimal{C_{\infty}}-algebra as a rational homotopy model of a space. We discuss a cyclic version of this Kadeishvili{C_{\infty}}-model and apply it to classifying rational Poincaré duality spaces. We classify 1-connected minimal cyclic{C_{\infty}}-algebras whose cohomology algebras are those of{(S^{p}\times S^{p+2q-1})\sharp(S^{q}\times S^{2p+q-1})}, where{2\leq p\leq q}. We also include a proof of the decomposition theorem for cyclic{A_{\infty}}and{C_{\infty}}-algebras.

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32

Bondarko, Mikhail, and Frédéric Déglise. "Dimensional homotopy t-structures in motivic homotopy theory." Advances in Mathematics 311 (April 2017): 91–189. http://dx.doi.org/10.1016/j.aim.2017.02.003.

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33

Grigor’yan, Alexander, Yong Lin, Yuri Muranov, and Shing-Tung Yau. "Homotopy Theory for Digraphs." Pure and Applied Mathematics Quarterly 10, no. 4 (2014): 619–74. http://dx.doi.org/10.4310/pamq.2014.v10.n4.a2.

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34

Muro, Fernando, and Constanze Roitzheim. "Homotopy theory of bicomplexes." Journal of Pure and Applied Algebra 223, no. 5 (May 2019): 1913–39. http://dx.doi.org/10.1016/j.jpaa.2018.08.007.

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35

Duma, Adrian. "A generalised homotopy theory." Nonlinear Analysis: Theory, Methods & Applications 30, no. 8 (December 1997): 4937–48. http://dx.doi.org/10.1016/s0362-546x(97)00467-7.

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36

James, I. M. "Lectures on Homotopy Theory." Bulletin of the London Mathematical Society 25, no. 2 (March 1993): 203. http://dx.doi.org/10.1112/blms/25.2.203a.

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37

Smirnov, V. A. "HOMOTOPY THEORY OF COALGEBRAS." Mathematics of the USSR-Izvestiya 27, no. 3 (June 30, 1986): 575–92. http://dx.doi.org/10.1070/im1986v027n03abeh001194.

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38

Miller, David A. "Strongly stratified homotopy theory." Transactions of the American Mathematical Society 365, no. 9 (March 4, 2013): 4933–62. http://dx.doi.org/10.1090/s0002-9947-2013-05795-9.

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39

Raptis, George. "Homotopy theory of posets." Homology, Homotopy and Applications 12, no. 2 (2010): 211–30. http://dx.doi.org/10.4310/hha.2010.v12.n2.a7.

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40

Hovey, Mark, John H. Palmieri, and Neil P. Strickland. "Axiomatic stable homotopy theory." Memoirs of the American Mathematical Society 128, no. 610 (1997): 0. http://dx.doi.org/10.1090/memo/0610.

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41

Chachólski, Wojciech, and Jérôme Scherer. "Homotopy theory of diagrams." Memoirs of the American Mathematical Society 155, no. 736 (2002): 0. http://dx.doi.org/10.1090/memo/0736.

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42

Babson, Eric, Hélène Barcelo, Mark de Longueville, and Reinhard Laubenbacher. "Homotopy theory of graphs." Journal of Algebraic Combinatorics 24, no. 1 (August 2006): 31–44. http://dx.doi.org/10.1007/s10801-006-9100-0.

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43

Antolini, Rosa. "Cubical structures, homotopy theory." Annali di Matematica Pura ed Applicata 178, no. 1 (December 2000): 317–24. http://dx.doi.org/10.1007/bf02505901.

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44

Cordier, Jean-Marc, and Timothy Porter. "Homotopy coherent category theory." Transactions of the American Mathematical Society 349, no. 1 (1997): 1–54. http://dx.doi.org/10.1090/s0002-9947-97-01752-2.

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45

Drummond-Cole, Gabriel C., Jae-Suk Park, and John Terilla. "Homotopy probability theory I." Journal of Homotopy and Related Structures 10, no. 3 (November 19, 2013): 425–35. http://dx.doi.org/10.1007/s40062-013-0067-y.

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46

Drummond-Cole, Gabriel C., Jae-Suk Park, and John Terilla. "Homotopy probability theory II." Journal of Homotopy and Related Structures 10, no. 3 (April 6, 2014): 623–35. http://dx.doi.org/10.1007/s40062-014-0078-3.

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47

Kriz, Igor. "p-adic homotopy theory." Topology and its Applications 52, no. 3 (October 1993): 279–308. http://dx.doi.org/10.1016/0166-8641(93)90108-p.

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48

Lazarev, Andrey, and Martin Markl. "Disconnected rational homotopy theory." Advances in Mathematics 283 (October 2015): 303–61. http://dx.doi.org/10.1016/j.aim.2015.07.009.

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49

Groth, Moritz, and Jan Šťovíček. "Tilting theory via stable homotopy theory." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 743 (October 1, 2018): 29–90. http://dx.doi.org/10.1515/crelle-2015-0092.

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Abstract We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillie’s strongly (co)cartesian n-cubes. As applications we construct abstract Auslander–Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi–Yau property. This is used to offer a new perspective on May’s axioms for monoidal, triangulated categories.

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50

Kılıçman, Adem, and Amin Saif. "Homotopy Extension Property in Homotopy Theory for Topological Semigroups." ISRN Geometry 2012 (March 4, 2012): 1–9. http://dx.doi.org/10.5402/2012/359403.

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The purpose of this paper is to extend the concept of homotopy extension property in homotopy theory for topological spaces to its analogical structure in homotopy theory for topological semigroups. In this extension, we also give some results concerning on absolutely retract and its properties.

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