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PurposeThe paper aims to determine the rational homotopy type of the total space of projectivized bundles over complex projective spaces using Sullivan minimal models, providing insights into the algebraic structure of these spaces. Design/methodology/approachThe paper utilises techniques from Sullivan’s theory of minimal models to analyse the differential graded algebraic structure of projectivized bundles. It employs algebraic methods to compute the Sullivan minimal model of and establish relationships with the base space. FindingsThe paper determines the rational homotopy type of projectivized bundles over complex projective spaces. Of great interest is how the Chern classes of the fibre space and base space, play a critical role in determining the Sullivan model ofP(E). We also provide the homogeneous space ofP(E)whenn = 2. Finally, we prove the formality ofP(E)over a homogeneous space of equal rank. Research limitations/implicationsLimitations may include the complexity of computing minimal models for higher-dimensional bundles. Practical implicationsUnderstanding the rational homotopy type of projectivized bundles facilitates computations in algebraic topology and differential geometry, potentially aiding in applications such as topological data analysis and geometric modelling. Social implicationsWhile the direct social impact may be indirect, advancements in algebraic topology contribute to broader mathematical knowledge, which can underpin developments in science, engineering, and technology with societal benefits. Originality/valueThe paper’s originality lies in its application of Sullivan minimal models to determine the rational homotopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles.
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On the ∞$\infty$‐topos semantics of homotopy type theory
Abstract Many introductions tohomotopy type theoryand theunivalence axiomgloss over the semantics of this new formal system in traditional set‐based foundations. This expository article, written as lecture notes to accompany a three‐part mini course delivered at the Logic and Higher Structures workshop at CIRM‐Luminy, attempt to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpretation of homotopy type theory with strict univalent universes in any ‐topos. As we will explain, this achievement was the product of a community effort to abstract and streamline the original arguments as well as develop new lines of reasoning.
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- Award ID(s):
- 2204304
- PAR ID:
- 10534820
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 56
- Issue:
- 2
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 461-517
- Size(s):
- p. 461-517
- Sponsoring Org:
- National Science Foundation
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