On the theory of higher Segal spaces
On the theory of higher Segal spaces
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DissertationDate of Exam
14.07.2020Date of Publication
21.10.2020Advisor
Stroppel, CatharinaCo-Referee
Dyckerhoff, TobiasMetadata
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Abstract
This thesis contains three chapters, each dealing with one particular aspect of the theory of higher Segal spaces introduced by Dyckerhoff and Kapranov:
(1) By exhibiting the simplex category as an ∞-categorical localization of the dendrex category of Moerdijk and Weiss, we identify the homotopy theory of 2-Segal spaces with that of invertible ∞-operads.
(2) Inspired by a heuristic analogy with the manifold calculus of Goodwillie and Weiss, we characterize the various higher Segal conditions in terms of purely categorical conditions of higher weak excision on the simplex category and on Connes’ cyclic category.
(3) We establish a large class of ∞-categorical Morita-equivalences of Dold–Kan type. As an application we describe higher Segal simplicial objects in the additive context as truncated coherent chain complexes; in the stable context, we identify higher Segal Γ-objects with polynomial functors in the sense of Goodwillie.
(1) By exhibiting the simplex category as an ∞-categorical localization of the dendrex category of Moerdijk and Weiss, we identify the homotopy theory of 2-Segal spaces with that of invertible ∞-operads.
(2) Inspired by a heuristic analogy with the manifold calculus of Goodwillie and Weiss, we characterize the various higher Segal conditions in terms of purely categorical conditions of higher weak excision on the simplex category and on Connes’ cyclic category.
(3) We establish a large class of ∞-categorical Morita-equivalences of Dold–Kan type. As an application we describe higher Segal simplicial objects in the additive context as truncated coherent chain complexes; in the stable context, we identify higher Segal Γ-objects with polynomial functors in the sense of Goodwillie.
Subjects
higher structures, higher Segal objects, 2-Segal, simplicial objects, cyclic objects, infinity-operads, cyclic operads, manifold calculus, excisive functors, Dold-Kan correspondence, Goodwillie calculus
Classification (DDC)
510 MathematikWalde, Tashi: On the theory of higher Segal spaces. - Bonn, 2020. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-59387
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-59387
@phdthesis{handle:20.500.11811/8705,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-59387,
author = {{Tashi Walde}},
title = {On the theory of higher Segal spaces},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2020,
month = oct,
note = {This thesis contains three chapters, each dealing with one particular aspect of the theory of higher Segal spaces introduced by Dyckerhoff and Kapranov:
(1) By exhibiting the simplex category as an ∞-categorical localization of the dendrex category of Moerdijk and Weiss, we identify the homotopy theory of 2-Segal spaces with that of invertible ∞-operads.
(2) Inspired by a heuristic analogy with the manifold calculus of Goodwillie and Weiss, we characterize the various higher Segal conditions in terms of purely categorical conditions of higher weak excision on the simplex category and on Connes’ cyclic category.
(3) We establish a large class of ∞-categorical Morita-equivalences of Dold–Kan type. As an application we describe higher Segal simplicial objects in the additive context as truncated coherent chain complexes; in the stable context, we identify higher Segal Γ-objects with polynomial functors in the sense of Goodwillie.},
url = {https://hdl.handle.net/20.500.11811/8705}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-59387,
author = {{Tashi Walde}},
title = {On the theory of higher Segal spaces},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2020,
month = oct,
note = {This thesis contains three chapters, each dealing with one particular aspect of the theory of higher Segal spaces introduced by Dyckerhoff and Kapranov:
(1) By exhibiting the simplex category as an ∞-categorical localization of the dendrex category of Moerdijk and Weiss, we identify the homotopy theory of 2-Segal spaces with that of invertible ∞-operads.
(2) Inspired by a heuristic analogy with the manifold calculus of Goodwillie and Weiss, we characterize the various higher Segal conditions in terms of purely categorical conditions of higher weak excision on the simplex category and on Connes’ cyclic category.
(3) We establish a large class of ∞-categorical Morita-equivalences of Dold–Kan type. As an application we describe higher Segal simplicial objects in the additive context as truncated coherent chain complexes; in the stable context, we identify higher Segal Γ-objects with polynomial functors in the sense of Goodwillie.},
url = {https://hdl.handle.net/20.500.11811/8705}
}
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