LEADER 05612nam 22005895 450 001 9910634047703321 005 20251113195601.0 010 $a3-031-18900-0 024 7 $a10.1007/978-3-031-18900-5 035 $a(MiAaPQ)EBC7156589 035 $a(Au-PeEL)EBL7156589 035 $a(CKB)25657393400041 035 $a(PPN)268063397 035 $a(OCoLC)1356573744 035 $a(DE-He213)978-3-031-18900-5 035 $a(EXLCZ)9925657393400041 100 $a20221209d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aEffective Kan Fibrations in Simplicial Sets /$fby Benno van den Berg, Eric Faber 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (230 pages) 225 1 $aLecture Notes in Mathematics,$x1617-9692 ;$v2321 311 08$aPrint version: van den Berg, Benno Effective Kan Fibrations in Simplicial Sets Cham : Springer International Publishing AG,c2023 9783031188992 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- Related Work -- Acknowledgements -- Contents -- 1 Introduction -- 1.1 Fibrations as Structure -- 1.2 Effective Kan Fibrations -- 1.3 Summary of Contents -- Part I ?-Types from Moore Paths -- 2 Preliminaries -- 2.1 Fibred Structure -- 2.2 Double Categories of Left and Right Lifting Structures -- 2.3 Algebraic Weak Factorisation Systems -- 2.4 A Double Category of Coalgebras -- 2.5 Cofibrant Generation by a Double Category -- 2.6 Fibred Structure Revisited -- 2.7 Concluding Remark on Notation -- 3 An Algebraic Weak Factorisation System from a Dominance -- 4 An Algebraic Weak Factorisation System from a Moore Structure -- 4.1 Defining the Algebraic Weak Factorisation System -- 4.1.1 Functorial Factorisation -- 4.1.2 The Comonad -- 4.1.3 The Monad -- 4.1.4 The Distributive Law -- 4.2 Hyperdeformation Retracts -- 4.2.1 Hyperdeformation Retracts are Coalgebras -- 4.2.2 Hyperdeformation Retracts are Bifibred -- 4.3 Naive Fibrations -- 5 The Frobenius Construction -- 5.1 Naive Left Fibrations -- 5.2 The Frobenius Construction -- 6 Mould Squares and Effective Fibrations -- 6.1 A New Notion of Fibred Structure -- 6.2 Effective Fibrations -- 6.2.1 Effective Trivial Fibrations -- 6.2.2 Right and Left Fibrations -- 7 -Types -- Part II Simplicial Sets -- 8 Effective Trivial Kan Fibrations in Simplicial Sets -- 8.1 Effective Cofibrations -- 8.2 Effective Trivial Kan Fibrations -- 8.3 Local Character and Classical Correctness -- 9 Simplicial Sets as a Symmetric Moore Category -- 9.1 Polynomial Yoga -- 9.2 A Simplicial Poset of Traversals -- 9.3 Simplicial Moore Paths -- 9.4 Geometric Realization of a Traversal -- 10 Hyperdeformation Retracts in Simplicial Sets -- 10.1 Hyperdeformation Retracts Are Effective Cofibrations -- 10.2 Hyperdeformation Retracts as Internal Presheaves -- 10.3 A Small Double Category of Hyperdeformation Retracts. 327 $a10.4 Naive Kan Fibrations in Simplicial Sets -- 11 Mould Squares in Simplicial Sets -- 11.1 Small Mould Squares -- 11.2 Effective Kan Fibrations in Terms of ``Filling'' -- 12 Horn Squares -- 12.1 Effective Kan Fibrations in Terms of Horn Squares -- 12.2 Local Character and Classical Correctness -- 13 Conclusion -- 13.1 Properties of Effective Kan Fibrations -- 13.2 Directions for Future Research -- A Axioms -- A.1 Moore Structure -- A.2 Dominance -- B Cubical Sets -- C Degenerate Horn Fillers Are Unique -- D Uniform Kan Fibrations -- References -- Index. 330 $aThis book introduces the notion of an effective Kan fibration, a new mathematical structure which can be used to study simplicial homotopy theory. The main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain non-trivial properties. Here it is revealed that fundamental properties of ordinary Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, a constructive (explicit) proof is given that effective Kan fibrations are stable under push forward, or fibred exponentials. Further, it is shown that effective Kan fibrations are local, or completely determined by their fibres above representables, and the maps which can be equipped with the structure of an effective Kan fibration are precisely the ordinary Kan fibrations. Hence implicitly, both notions still describe the same homotopy theory. These new results solve an open problem in homotopy type theory and provide the first step toward giving a constructive account of Voevodsky?s model of univalent type theory in simplicial sets. 410 0$aLecture Notes in Mathematics,$x1617-9692 ;$v2321 606 $aAlgebra, Homological 606 $aLogic, Symbolic and mathematical 606 $aCategory Theory, Homological Algebra 606 $aMathematical Logic and Foundations 615 0$aAlgebra, Homological. 615 0$aLogic, Symbolic and mathematical. 615 14$aCategory Theory, Homological Algebra. 615 24$aMathematical Logic and Foundations. 676 $a780 676 $a514.24 700 $avan den Berg$b Benno$01272092 702 $aFaber$b Eric 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910634047703321 996 $aEffective Kan Fibrations in Simplicial Sets$92996483 997 $aUNINA