Effective kan fibrations in simplicial sets [Documento eletrónico] / by Benno van den Berg, Eric Faber
Language: eng.Country: Switzerland, Swiss Confederation.Publication: Cham : Springer International Publishing, Springer, 2022Description: X, 230 p. : il.ISBN: 978-3-031-18900-5.Series: Lecture Notes in Mathematics, 2321Subject - Topical Name: Algebra, Homological | Mathematical logic Online Resources:Click here to access onlineItem type | Current library | Collection | Call number | Copy number | Status | Date due | Barcode | |
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QA169. FCT 98384 Hopf algebras and their generalizations from a category theoretical point of view | QA169. FCT 98390 Operads of wiring diagrams | QA169.SPR FCT Temporal type theory, a topos-theoretic approach to systems and behavior | QA169.SPR FCT Effective kan fibrations in simplicial sets | QA169.SPR FCT Simplicial methods for higher categories, segal-type models of weak n-categories | QA169.SPR FCT Higher segal spaces | QA169.SPR FCT Involutive category theory |
This book introduces the notion of an effective Kan fibration, a new mathematical structure which can be used to study simplicial homotopy theory. The main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain non-trivial properties. Here it is revealed that fundamental properties of ordinary Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, a constructive (explicit) proof is given that effective Kan fibrations are stable under push forward, or fibred exponentials. Further, it is shown that effective Kan fibrations are local, or completely determined by their fibres above representables, and the maps which can be equipped with the structure of an effective Kan fibration are precisely the ordinary Kan fibrations. Hence implicitly, both notions still describe the same homotopy theory. These new results solve an open problem in homotopy type theory and provide the first step toward giving a constructive account of Voevodsky's model of univalent type theory in simplicial sets.
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