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Effective Kan Fibrations in Simplicial Sets (Paperback, 1st ed. 2022)
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Effective Kan Fibrations in Simplicial Sets (Paperback, 1st ed. 2022)
Series: Lecture Notes in Mathematics, 2321
Expected to ship within 10 - 15 working days
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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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