DOCSLIB.ORG

An Alpine Bouquet of Algebraic Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

708 An Alpine Bouquet of Algebraic Topology Alpine Algebraic and Applied Topology Conference August 15–21, 2016 Saas-Almagell, Switzerland Christian Ausoni Kathryn Hess Brenda Johnson Ieke Moerdijk Jérôme Scherer Editors 708 An Alpine Bouquet of Algebraic Topology Alpine Algebraic and Applied Topology Conference August 15–21, 2016 Saas-Almagell, Switzerland Christian Ausoni Kathryn Hess Brenda Johnson Ieke Moerdijk Jérôme Scherer Editors EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 11S40, 18Axx, 18Gxx, 19Dxx, 55Nxx, 55Pxx, 55Sxx, 55Uxx, 57Nxx. Library of Congress Cataloging-in-Publication Data Names: Conference on Alpine Algebraic and Applied Topology (2016: Saas-Almagell, Switzerland) | Ausoni, Christian, 1968– editor. Title: An alpine bouquet of algebraic topology: Conference on Alpine Algebraic and Applied Topology, August 15–21, 2016, Saas-Almagell, Switzerland / Christian Ausoni [and four oth- ers], editors. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Contem- porary mathematics; volume 708 | Includes bibliographical references. Identifiers: LCCN 2017055549 | ISBN 9781470429119 (alk. paper) Subjects: LCSH: Algebraic topology–Congresses. | AMS: Number theory – Algebraic number theory: local and p-adic fields – Zeta functions and L-functions. msc | Category theory; homological algebra – General theory of categories and functors – General theory of categories and functors. msc | Category theory; homological algebra – Homological algebra – Homological algebra. msc | K-theory – Higher algebraic K-theory – Higher algebraic K-theory. msc | Algebraic topology – Homology and cohomology theories – Homology and cohomology theories. msc | Algebraic topology – Homotopy theory – Homotopy theory. msc | Algebraic topology – Operations and obstructions – Operations and obstructions. msc | Algebraic topology – Applied homological algebra and category theory – Applied homological algebra and category theory. msc | Manifolds and cell complexes – Topological manifolds – Topological manifolds. msc Classification: LCC QA612 .C5927 2018 | DDC 514/.2–dc23 LC record available at https://lccn.loc.gov/2017055549 DOI: http://dx.doi.org/10.1090/conm/708 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 Contents Preface v List of Conference Talks vii List of Participants ix Characteristics for E∞ ring spectra A. Baker 1 Segal objects and the Grothendieck construction P. Boavida de Brito 19 Blown-up intersection cohomology D. Chataur, M. Saralegi-Aranguren, and D. Tanre´ 45 Homotopically rigid Sullivan algebras and their applications C. Costoya, D. Mendez,´ and A. Viruel 103 A Dundas-Goodwillie-McCarthy theorem for split square-zero extensions of exact categories E. Dotto 123 Four approaches to cohomology theories with reality J. P. C. Greenlees 139 Topological Hochschild homology and the Hasse-Weil zeta function L. Hesselholt 157 The stable symplectic category and a conjecture of Kontsevich N. Kitchloo and J. Morava 181 Universal Gysin formulas for the universal Hall-Littlewood functions M. Nakagawa and H. Naruse 201 Graded multiplications on iterated bar constructions B. Stonek 245 Double homotopy (co)limits for relative categories K. Werndli 275 iii Preface After four conferences held in Arolla, it was time to search for new landscapes and summits. This 2016 edition of the Alpine conferences in algebraic topology took place farther east in the Saas Valley, in the village of Saas-Almagell, which offered many new challenging paths to explore. This led us to new perspectives on rational equivariant homotopy theory, topological Hochschild homology and K- theory, stable and unstable homotopy theory, functor calculus, and an incursion into applied algebraic topology. Our sincerest thanks go to all of the speakers and participants, whose talks and interest made this conference happen. We are very grateful for the authors, whose results are published here, and thank them for their contributions. Many thanks also to the referees for their anonymous work and to Chris Thivierge at the AMS for her efficient help and guidance. We would like to thank the following institutions and organizations for their financial support and sponsorship of the 2016 Alpine Algebraic and Applied Topol- ogy Conference at Saas: - U.S. National Science Foundation, - Foundation Compositio Mathematica, -Facult´e des Sciences de la Vie, Ecole´ Polytechnique F´ed´eraledeLausanne, -Conf´erence Universitaire de Suisse Occidentale, - Centre National de la Recherche Scientifique, and - Universit´e Sorbonne Paris Cit´e. This Saas edition of the Alpine topology conferences was hosted by the Hotel Kristall Saphir in Saas-Almagell. We express our gratitude to Mrs. Rosine Breuer and her family and staff for not sparing any effort in making 70 exuberant topologists feel at home. It was a pleasure organizing this conference with their help. Ch. Ausoni, K. Hess, B. Johnson, I. Moerdijk, and J. Scherer v Fluchthorn summit, August 15, 2016 List of Conference Talks Monday, August 15, 2016 Cristina Costoya Kahn’s realizability problem Tuesday, August 16, 2016 John Greenlees Rational equivariant cohomology theories and affine formal covers of the sphere David Barnes Rational S1-equivariant ring spectra Christian Wimmer Rational global homotopy theory Adeel Khan A1-homotopy invariance in spectral algebraic geometry Vesna Stojanoska Galois extensions of motivic ring spectra Nitu Kitchloo Landweber flat real pairs and ER(n)- cohomology Tse Leung So Homotopy types of gauge groups over 4-manifolds Wednesday, August 17, 2016 Vincent Franjou Lannes’ T vs Harish-Chandra restriction Anssi Lathinen Modular characteristic classes for represen- tations over finite fields Marc Hoyois On the vanishing of negative equivariant K-theory Steffen Sagave Logarithmic topological Hochschild homology Lars Hesselholt Topological Hochschild homology and the Hasse-Weil zeta function vii viii LIST OF CONFERENCE TALKS Thursday, August 18, 2016 Agn`es Beaudry The K(2)-local Picard group at p =2 Tilman Bauer Chromatic unstable homotopy, plethories, and the Dieudonn´e correspondence David Chataur Intersection homotopy theory Friday, August 19, 2016 Vidit Nanda The discrete flow category Wo jciech Chach´olski Multi-persistence Ran Levi Topological analysis of neural networks Mark Grant The Poincar´e-Hopf theorem for line fields (revisited) Wolfgang Pitsch Relative homological algebra via truncations Saturday, August 20, 2016 Pedro Boavida de Brito Configuration categories and embedding spaces Emanuele Dotto Equivariant calculus and the tower of the identity on pointed G-spaces Kristine Bauer A combinatorial model for certain Taylor towers Marc Stephan Categorical suspension and stable Postnikov data Sune Precht Reeh Saturated fusion systems as stable retracts of groups List of Participants Arlettaz, Dominique Dotto, Emanuele University of Lausanne MIT Ausoni, Christian Dundas, Bjørn Ian University Paris 13 University of Bergen Baker, Andrew Elmendorf, Tony University of Glasgow Purdue University Calumet Bandklayder, Lauren Filakovsky, Marek Northwestern University IST Austria Barnes, David Franjou, Vincent Queen’s University Belfast University of Nantes Bauer, Kristine Fresse, Benoit University of Calgary University of Lille Grant, Mark Bauer, Tilman University of Aberdeen KTH Greenlees, John Beaudry, Agn`es University of Sheffield University of Chicago Harper, John E. Boavida de Brito, Pedro Ohio State University University of Lisbon Hess, Kathryn Bobkova, Irina EPFL University of Rochester Hesselholt, Lars Chach´olski, Wojciech Nagoya University/University of KTH Copenhagen Chataur, David Hiroshi, Naruse UPJV University of Yamanashi Ching, Michael Hoyois, Marc Amherst College MIT Costoya, Cristina Jeitziner, Rachel Universidade da Coru˜na EPFL Dlotko, Pawel Joachim, Michael Inria, Saclay University of Muenster ix x LIST OF PARTICIPANTS Johnson, Brenda Sagave, Steffen Union College Radboud University Nijmegen K¸edziorek, Magdalena Scheinmann, Nathan EPFL EPFL Khan, Adeel Yusufzai Scherer, J´erˆome University Duisburg-Essen EPFL Kitchloo, Nitu
Recommended publications
  • Fibrewise Stable Rational Homotopy

    Fibrewise Stable Rational Homotopy

    Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Fibrewise Stable Rational Homotopy Yves F´elix, Aniceto Murillo and Daniel Tanr´e Abstract In this paper, for a given space B we establish a correspondence between differential graded ∗ modules over C (B; Q) and fibrewise rational stable spaces over B. This correspondence opens the door for topological translations of algebraic constructions made with modules over a commutative differential graded algebra. More precisely, given fibrations E → B and ′ E → B, the set of stable rational homotopy classes of maps over B is isomorphic to ∗ ∗ ′ ∗ ExtC∗(B;Q)`C (E ; Q),C (E; Q)´. In particular, a nilpotent, finite type CW-complex X is a q rational Poincar´ecomplex if there exist non trivial stable maps over XQ from (X × S )Q to q+N (X ∨ S )Q for exactly one N. 1. Introduction Sullivan’s approach to rational homotopy theory is based on an adjoint pair of functors between the category of simplicial sets and the category CDGA of commutative differential graded algebras over Q (henceforth called cdga’s ). This correspondence induces an equivalence between the homotopy category of rational (nilpotent and with finite Betti numbers) simplicial sets and a homotopy category of cdga’s. The fact that any algebraic construction or property in CDGA has a topological translation is the most important feature of rational homotopy theory. Until now, however, there was no known procedure for realizing differential graded modules (henceforth called dgm’s) over cdga’s topologically, even though a number of important topological invariants have been interpreted in terms of dgm’s over the years.
  • HE WANG Abstract. a Mini-Course on Rational Homotopy Theory

    HE WANG Abstract. a Mini-Course on Rational Homotopy Theory

    RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition.
  • Remarks on Combinatorial and Accessible Model Categories

    Remarks on Combinatorial and Accessible Model Categories

    Theory and Applications of Categories, Vol. 37, No. 9, 2021, pp. 266{275. REMARKS ON COMBINATORIAL AND ACCESSIBLE MODEL CATEGORIES JIRˇ´I ROSICKY´ Abstract. Using full images of accessible functors, we prove some results about com- binatorial and accessible model categories. In particular, we give an example of a weak factorization system on a locally presentable category which is not accessible. 1. Introduction Twenty years ago, M. Hovey asked for examples of model categories which are not cofi- brantly generated. This is the same as asking for examples of weak factorization systems which are not cofibrantly generated. One of the first examples was given in [1]: it is a weak factorization system (L; R) on the locally presentable category of posets where L consists of embeddings. The reason is that posets injective to embeddings are precisely complete lattices which do not form an accessible category. Hence L is not cofibrantly generated. Since then, the importance of accessible model categories and accessible weak factorization systems has emerged, and, the same question appears again, i.e., to give an example of a weak factorization system (L; R) on a locally presentable category which is not accessible. Now, L-injective objects do not necessarily form an accessible category but only a full image of an accessible functor. Such full images are accessible only under quite restrictive assumptions (see [5]). But, for an accessible weak factorization system, L-injective objects form the full image of a forgetful functor from algebraically L-injective objects (see Corollary 3.3). Moreover, such full images are closed under reduced prod- ucts modulo κ-complete filters for some regular cardinal κ (see Theorem 2.5).
  • CURRICULUM VITAE Jonathan Andrew Scott

    CURRICULUM VITAE Jonathan Andrew Scott

    CURRICULUM VITAE Jonathan Andrew Scott [email protected] Address: 6-53 Maclaren Street, Ottawa ON K2P 0K3 Canada Current Affiliation: Department of Mathematics and Statistics, University of Ottawa Telephone (office): (613) 562-5800 ext. 2082 Telephone (home): (613) 236-4812 Education 2000 Ph.D. in Mathematics, University of Toronto. Supervisor: Steve Halperin. 1994 M.Sc. in Mathematics, University of Toronto. 1993 B.Sc. (Honours) in Mathematical Physics, Queen’s University at Kingston. Research Interests Operads, category theory, E∞ algebras, cohomology operations, loop space homology, Bockstein spectral sequence. Publications 1. K. Hess, P.-E. Parent, and J. Scott, A chain coalgebra model for the James map, Homology, Homotopy Appl. 9 (2007) 209–231, arXiv:math.AT/0609444. 2. K. Hess, P.-E. Parent, J. Scott, and A. Tonks, A canonical enriched Adams-Hilton model for simplicial sets, Adv. Math. 207 (2006) 847–875, arXiv:math.AT/0408216. 3. J. Scott, Hopf algebras up to homotopy and the Bockstein spectral sequence, Algebr. Geom. Topol. 5 (2005) 119–128, arXiv:math.AT/0412207. 4. J. Scott, A torsion-free Milnor-Moore theorem, J. London Math. Soc. (2) 67 (2003) 805–816, arXiv:math.AT/0103223. 5. J. Scott, Algebraic structure in the loop space homology Bockstein spectral sequence, Trans. Amer. Math. Soc. 354 (2002) 3075–3084, arXiv: math.AT/9912106. 6. J. Scott, A factorization of the homology of a differential graded Lie algebra, J. Pure Appl. Alg. 167 (2002) 329–340. Submitted Manuscripts 1. K. Hess and J. Scott, Homotopy morphisms and Koszul resolutions of operads 2. K. Hess, P.-E. Parent, and J.
  • Fundamental Algebraic Geometry

    Fundamental Algebraic Geometry

    http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
  • Curriculum Vitae - Rogier Bos

    Curriculum Vitae - Rogier Bos

    Curriculum Vitae - Rogier Bos Personal Full name: Rogier David Bos. Date and place of birth: August 7, 1978, Ter Aar, The Netherlands. Email: rogier bos [at] hotmail.com. Education 2010 M. Sc. Mathematics Education at University of Utrecht, 2007 Ph. D. degree in Mathematics at the Radboud University Nijmegen, thesis: Groupoids in geometric quantization, advisor: Prof. dr. N. P. Landsman. Manuscript committee: Prof. dr. Gert Heckman, Prof. dr. Ieke Moerdijk (Universiteit Utrecht), Prof. dr. Jean Renault (Universit´ed'Orl´eans), Prof. dr. Alan Weinstein (University of California), Prof. dr. Ping Xu (Penn State University). 2001 M. Sc. (Doctoraal) in Mathematics at the University of Utrecht, thesis: Operads in deformation quantization, advisor: Prof. dr. I. Moerdijk; minors in philosophy and physics. 1997 Propedeuse in Mathematics at the University of Utrecht. 1997 Propedeuse in Physics at the University of Utrecht. 1996 Gymnasium at Erasmus College Zoetermeer. General Working experience 2010-now Co-author dutch mathematics teaching series \Wagen- ingse Methode". 2010-now Lecturer at Master Mathematics Education at Hogeschool Utrecht. 2009-now Mathematics and general science teacher at the Chris- telijk Gymnasium Utrecht. 2007-2008 Post-doctoral researcher at the Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior T´ecnicoin Lisbon (Portugal). 2002-2007 Ph.D.-student mathematical physics at the University of Amsterdam (Radboud University Nijmegen from 2004); funded by FOM; promotor: Prof. dr. N. P. Landsman. 2002 march-august Mathematics teacher at the highschool Minkema College in Woerden. 1 1999-2001 Teacher of mathematics exercise classes at the Univer- sity of Utrecht. University Teaching Experience 2007 Mathematics deficiency repair course for highschool students (in preparation of studying mathematics or physics), Radboud Univer- sity Nijmegen.
  • Rational Homotopy Theory: a Brief Introduction

    Rational Homotopy Theory: a Brief Introduction

    Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article.
  • An Outline of Algebraic Set Theory

    An Outline of Algebraic Set Theory

    An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16].
  • A Monoidal Dold-Kan Correspondence for Comodules

    A Monoidal Dold-Kan Correspondence for Comodules

    A MONOIDAL DOLD-KAN CORRESPONDENCE FOR COMODULES MAXIMILIEN PEROUX´ Abstract. We present the notion of fibrantly generated model categories. Cofibrantly generated model categories are generalizations of CW-approximations which provide an inductive cofibrant replacement. We provide examples of inductive fibrant replacements constructed as Postnikov towers. These provide new types of arguments to compute homotopy limits in model categories. We provide examples for simplicial and differential graded comodules. Our main application is to show that simplicial comodules and connective differential graded comodules are Quillen equivalent and their derived cotensor products correspond. 1. Introduction Let R be a commutative ring. The Dold-Kan correspondence established in [Dol58, Kan58] is an equiv- ≥0 alence between the category sModR of simplicial R-modules and the category ChR of non-negative chain ≥0 complexes over R. The normalization functor N : sModR → ChR is symmetric monoidal with respect to the graded tensor product of chains, the levelwise tensor product of simplicial modules and the Eilenberg-Zilber ≥0 map. The inverse equivalence Γ : ChR → sModR also has a (non-symmetric) monoidal structure coming from the Alexander-Whitney map. The categories are not isomorphic as symmetric monoidal categories. ≥0 Nevertheless, the homotopy categories of sModR and ChR have isomorphic symmetric monoidal structures when endowed with their respective derived tensor product. Formally, it was shown in [SS03] that there is a weak symmetric monoidal Quillen equivalence between the symmetric monoidal model categories of sModR ≥0 and ChR . The equivalence lifts to the associated categories of algebras and modules. In [P´er20a, 4.1], ≥0 we proved that the underlying ∞-categories of sModR and ChR (in the sense of [Lur17, 1.3.4.15, 4.1.7.4]) are equivalent as symmetric monoidal ∞-categories.
  • Normal and Conormal Maps in Homotopy Theory 3

    Normal and Conormal Maps in Homotopy Theory 3

    NORMAL AND CONORMAL MAPS IN HOMOTOPY THEORY EMMANUEL D. FARJOUN AND KATHRYN HESS Abstract. Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets or the category of chain complexes over a commutative ring. Introduction In a recent article [3], it was observed that a crossed module of groups could be viewed as a up-to-homotopy version of the inclusion of a normal subgroup. Mo- tivated by this observation, the authors of [3] formulated a definition of h-normal (homotopy normal) maps of simplicial groups, noting in particular that the con- jugation map from a simplicial group into its simplicial automorphism group is h-normal. The purpose of this paper is to provide a general, homotopical framework for the results in [3]. In particular, given a monoidal category M that is also a ho- motopical category [2], under reasonable compatability conditions between the two structures, we define and study notions of h-normality for maps of monoids and of h-conormality for maps of comonoids (Definition 2.4), observing that the two notions are dual in a strong sense.
  • Homotopy Completion and Topological Quillen Homology of Structured Ring Spectra

    Homotopy Completion and Topological Quillen Homology of Structured Ring Spectra

    HOMOTOPY COMPLETION AND TOPOLOGICAL QUILLEN HOMOLOGY OF STRUCTURED RING SPECTRA JOHN E. HARPER AND KATHRYN HESS Abstract. Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (−1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theo- rems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topo- logical Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in re- verse form). We also prove absolute and relative Hurewicz theorems and a corresponding Whitehead theorem for topological Quillen homology. Further- more, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQ- completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan’s completion of spaces with respect to homology, and Carlsson's and Arone-Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in un- bounded chain complexes.
  • On The´Etale Fundamental Group of Schemes Over the Natural Numbers

    On The´Etale Fundamental Group of Schemes Over the Natural Numbers

    On the Etale´ Fundamental Group of Schemes over the Natural Numbers Robert Hendrik Scott Culling February 2019 A thesis submitted for the degree of Doctor of Philosophy of the Australian National University It is necessary that we should demonstrate geometrically, the truth of the same problems which we have explained in numbers. | Muhammad ibn Musa al-Khwarizmi For my parents. Declaration The work in this thesis is my own except where otherwise stated. Robert Hendrik Scott Culling Acknowledgements I am grateful beyond words to my thesis advisor James Borger. I will be forever thankful for the time he gave, patience he showed, and the wealth of ideas he shared with me. Thank you for introducing me to arithmetic geometry and suggesting such an interesting thesis topic | this has been a wonderful journey which I would not have been able to go on without your support. While studying at the ANU I benefited from many conversations about al- gebraic geometry and algebraic number theory (together with many other inter- esting conversations) with Arnab Saha. These were some of the most enjoyable times in my four years as a PhD student, thank you Arnab. Anand Deopurkar helped me clarify my own ideas on more than one occasion, thank you for your time and help to give me new perspective on this work. Finally, being a stu- dent at ANU would not have been nearly as fun without the chance to discuss mathematics with Eloise Hamilton. Thank you Eloise for helping me on so many occasions. To my partner, Beth Vanderhaven, and parents, Cherry and Graeme Culling, thank you for your unwavering support for my indulgence of my mathematical curiosity.