Derivatives of the identity functor and operads
|
|
- Blaise Clarke
- 6 years ago
- Views:
Transcription
1 University of Oregon Manifolds, K-theory, and Related Topics Dubrovnik, Croatia 23 June 2014
2 Motivation We are interested in finding examples of categories in which the Goodwillie derivatives of the identity functor are the objects of an operad. Known examples: Based spaces (Ching) Spectra (Arone-Ching) L f n-local spaces (Bauer-Sadofsky) DGL r over Q (Walter)
3 Model structure on Cat We use the canonical model structure on Cat, the category of pointed small categories. Define a functor to be a weak equivalence if it is an equivalence of categories (equivalently, if it is fully faithful and essentially surjective). a cofibration if it is injective on objects. a fibration if it has the right lifting property with respect to acyclic cofibrations. The zero object in Cat is the category with one object and only the identity morphism.
4 Suspensions Definition (Quillen) Given a cofibrant object A in a pointed category M, ΣA is the colimit of the diagram A A Cyl(A) 0 where the cylinder object Cyl(A) satisfies: A A A Cyl(A) A =
5 Suspensions in Cat Proposition Cyl(C) = (C I)/( I), where I is the category with two objects and a unique isomorphism between them. Proposition Given a category C, ΣC has one object A, and End(A) = F (n 1), where n is the number of components of C. Corollary For any category C, Σ 2 C 0.
6 The category of spectra Definition (Hovey) Suppose G is a left Quillen endofunctor of a model category M. A spectrum X is a sequence {X n } n 0 of objects of M together with structure maps σ : GX n X n+1 for all n. A map of spectra f : X Y is a collection of maps f n : X n Y n so that the following diagram commutes for all n: σ X GX n X n+1 Gf n f n+1 σy GY n Y n+1 Denote the category of spectra by Sp N (M, G).
7 Model structures on Sp N (Cat, Σ) In the projective model structure on Sp N (Cat, Σ), f : X Y is a weak equivalence or fibration if and only if f n : X n Y n is a weak equivalence or fibration in Cat for all n 0. However, Σ : Sp N (Cat, Σ) Sp N (Cat, Σ) is not a Quillen equivalence. The stable model structure on Sp N (Cat, Σ) has the same cofibrations as in the projective model structure. weak equivalences are S-local equivalences, where S = {F n+1 ΣQC sqc n F n QC}. Here F n (C) is the spectrum {..., 0, C, ΣC, Σ 2 C,...}, and C is a domain or codomain of a generating cofibration of Cat.
8 Existence of Bousfield localization Remark: Hovey requires M to be left proper and cellular, but only to guarantee that the Bousfield localization exists. Sp N (Cat, Σ) is not cellular (cofibrations are not effective monomorphisms), but it is combinatorial. Theorem (Smith) If M is left proper and X-combinatorial (X some universe), and S is an X-small set of homotopy classes of morphisms of M, the left Bousfield localization L S M of M along any set representing S exists and satisfies the following conditions. 1 The cofibrations of LS M are exactly those of M. 2 The weak equivalences of LS M are the S-local equivalences.
9 Sp N (Cat, Σ) is trivial Theorem (Hovey) Σ : Sp N (Cat, Σ) Sp N (Cat, Σ) is a Quillen equivalence with respect to the stable model structure. Corollary Every object in Sp N (Cat, Σ) is weakly equivalent to the zero object in Sp N (Cat, Σ).
10 Connection to Goodwillie calculus The derivatives of the identity functor on Cat should be objects in the category of Σ n -equivariant spectra on Cat. Since Sp N (Cat, Σ) is trivial, the derivatives of the identity functor on Cat (if they exist) are trivial. Theorem (in progress) If M is a pointed, left proper, simplicial, cellular or combinatorial model category in which finite homotopy limits commute with countable directed homotopy colimits, and if there is n Z 0 such that Σ n X 0 for all X Ob(M), then the derivatives of the identity functor of M exist and are trivial.
11 Other work Proposition The category Gph of pointed directed graphs is left proper and combinatorial, and ΣX 0 for all X Ob(Gph). I am currently looking at Z/2-Mackey functors into DGL r. Knowing the derivatives of the identity functor on this category would tell us about Z/2-equivariant rational spaces.
12 Thanks for your attention!
13 References G. Arone and M. Ching, Operads and chain rules for Goodwillie derivatives, Astérisque 338 (2011), 158 pp. C. Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 (2010) M. Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005) M. Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) no D. Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967, iv+156 pp. C. Rezk, A model category for categories, unpublished, B. Walter, Rational Homotopy Calculus of Functors. Dissertation, Brown University (2005).
THE HOMOTOPY CALCULUS OF CATEGORIES AND GRAPHS
THE HOMOTOPY CALCULUS OF CATEGORIES AND GRAPHS by DEBORAH A. VICINSKY A DISSERTATION Presented to the Department of Mathematics and the Graduate School of the University of Oregon in partial fulfillment
More informationGraduate algebraic K-theory seminar
Seminar notes Graduate algebraic K-theory seminar notes taken by JL University of Illinois at Chicago February 1, 2017 Contents 1 Model categories 2 1.1 Definitions...............................................
More informationFIBERWISE LOCALIZATION AND THE CUBE THEOREM
FIBERWISE LOCALIZATION AND THE CUBE THEOREM DAVID CHATAUR AND JÉRÔME SCHERER Abstract. In this paper we explain when it is possible to construct fiberwise localizations in model categories. For pointed
More informationPeriodic Localization, Tate Cohomology, and Infinite Loopspaces Talk 1
Periodic Localization, Tate Cohomology, and Infinite Loopspaces Talk 1 Nicholas J. Kuhn University of Virginia University of Georgia, May, 2010 University of Georgia, May, 2010 1 / Three talks Introduction
More informationAn Introduction to Model Categories
An Introduction to Model Categories Brooke Shipley (UIC) Young Topologists Meeting, Stockholm July 4, 2017 Examples of Model Categories (C, W ) Topological Spaces with weak equivalences f : X Y if π (X
More informationT -spectra. Rick Jardine. March 2, University of Western Ontario
University of Western Ontario March 2, 2015 T = is a pointed simplicial presheaf on a site C. A T -spectrum X consists of pointed simplicial presheaves X n, n 0 and bonding maps σ : T X n X n+1, n 0. A
More informationin path component sheaves, and the diagrams
Cocycle categories Cocycles J.F. Jardine I will be using the injective model structure on the category s Pre(C) of simplicial presheaves on a small Grothendieck site C. You can think in terms of simplicial
More informationQUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA J. DANIEL CHRISTENSEN AND MARK HOVEY Abstract. An important example of a model category is the category of unbounded chain complexes of R-modules,
More informationON THE HOMOTOPY THEORY OF ENRICHED CATEGORIES
The Quarterly Journal of Mathematics Quart. J. Math. 64 (2013), 805 846; doi:10.1093/qmath/hat023 ON THE HOMOTOPY THEORY OF ENRICHED CATEGORIES by CLEMENS BERGER (Université de Nice, Lab. J.-A. Dieudonné,
More informationLOOP SPACES IN MOTIVIC HOMOTOPY THEORY. A Dissertation MARVIN GLEN DECKER
LOOP SPACES IN MOTIVIC HOMOTOPY THEORY A Dissertation by MARVIN GLEN DECKER Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree
More informationON THE COFIBRANT GENERATION OF MODEL CATEGORIES arxiv: v1 [math.at] 16 Jul 2009
ON THE COFIBRANT GENERATION OF MODEL CATEGORIES arxiv:0907.2726v1 [math.at] 16 Jul 2009 GEORGE RAPTIS Abstract. The paper studies the problem of the cofibrant generation of a model category. We prove that,
More informationWaldhausen Additivity and Approximation in Quasicategorical K-Theory
Waldhausen Additivity and Approximation in Quasicategorical K-Theory Thomas M. Fiore partly joint with Wolfgang Lück, http://www-personal.umd.umich.edu/~tmfiore/ http://www.him.uni-bonn.de/lueck/ Motivation
More informationGraph stable equivalences and operadic constructions in equivariant spectra
Graph stable equivalences and operadic constructions in equivariant spectra Markus Hausmann, Luís A. Pereira October 23, 2015 Abstract One of the key insights of the work of Blumberg and Hill in [2] is
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Brown representability in A 1 -homotopy theory Niko Naumann and Markus Spitzweck Preprint Nr. 18/2009 Brown representability in A 1 -homotopy theory Niko Naumann and Markus
More informationHOMOTOPY THEORY ADAM KAYE
HOMOTOPY THEORY ADAM KAYE 1. CW Approximation The CW approximation theorem says that every space is weakly equivalent to a CW complex. Theorem 1.1 (CW Approximation). There exists a functor Γ from the
More informationTHE CELLULARIZATION PRINCIPLE FOR QUILLEN ADJUNCTIONS
THE CELLULARIZATION PRINCIPLE FOR QUILLEN ADJUNCTIONS J. P. C. GREENLEES AND B. SHIPLEY Abstract. The Cellularization Principle states that under rather weak conditions, a Quillen adjunction of stable
More informationON THE HOMOTOPY THEORY OF ENRICHED CATEGORIES
ON THE HOMOTOPY THEORY OF ENRICHED CATEGORIES CLEMENS BERGER AND IEKE MOERDIJK Abstract. We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given
More informationModel Structures on the Category of Small Double Categories
Model Structures on the Category of Small Double Categories CT2007 Tom Fiore Simona Paoli and Dorette Pronk www.math.uchicago.edu/ fiore/ 1 Overview 1. Motivation 2. Double Categories and Their Nerves
More informationCategorical models of homotopy type theory
Categorical models of homotopy type theory Michael Shulman 12 April 2012 Outline 1 Homotopy type theory in model categories 2 The universal Kan fibration 3 Models in (, 1)-toposes Homotopy type theory
More informationMODEL-CATEGORIES OF COALGEBRAS OVER OPERADS
Theory and Applications of Categories, Vol. 25, No. 8, 2011, pp. 189 246. MODEL-CATEGORIES OF COALGEBRAS OVER OPERADS JUSTIN R. SMITH Abstract. This paper constructs model structures on the categories
More informationAndré Quillen spectral sequence for THH
Topology and its Applications 29 (2003) 273 280 www.elsevier.com/locate/topol André Quillen spectral sequence for THH Vahagn Minasian University of Illinois at Urbana-Champaign, 409 W. Green St, Urbana,
More informationAXIOMS FOR GENERALIZED FARRELL-TATE COHOMOLOGY
AXIOMS FOR GENERALIZED FARRELL-TATE COHOMOLOGY JOHN R. KLEIN Abstract. In [Kl] we defined a variant of Farrell-Tate cohomology for a topological group G and any naive G-spectrum E by taking the homotopy
More informationCommutative ring objects in pro-categories and generalized Moore spectra
Commutative ring objects in pro-categories and generalized Moore spectra Daniel G. Davis, Tyler Lawson June 30, 2013 Abstract We develop a rigidity criterion to show that in simplicial model categories
More informationSPECTRAL ENRICHMENTS OF MODEL CATEGORIES
SPECTRAL ENRICHMENTS OF MODEL CATEGORIES DANIEL DUGGER Abstract. We prove that every stable, presentable model category can be enriched in a natural way over symmetric spectra. As a consequence of the
More informationHOMOTOPY 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
More informationarxiv: v1 [math.at] 17 Apr 2008
DG CATEGORIES AS EILENBERG-MAC LANE SPECTRAL ALGEBRA arxiv:0804.2791v1 [math.at] 17 Apr 2008 GONÇALO TABUADA Abstract. We construct a zig-zag of Quillen equivalences between the homotopy theories of differential
More informationMODEL CATEGORIES OF DIAGRAM SPECTRA
MODEL CATEGORIES OF DIAGRAM SPECTRA M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY Abstract. Working in the category T of based spaces, we give the basic theory of diagram spaces and diagram spectra.
More informationarxiv: v1 [math.at] 18 Nov 2015
AN ALGEBRAIC MODEL FOR RATIONAL G SPECTRA OVER AN EXCEPTIONAL SUBGROUP MAGDALENA KȨDZIOREK arxiv:1511.05993v1 [math.at] 18 Nov 2015 Abstract. We give a simple algebraic model for rational G spectra over
More informationo ALGEBRAIC K-THEORY AND HIGHER CATEGORIES ANDREW J. BLUMBERG Abstract. The outline of the talk. 1. Setup Goal: Explain algebraic K-theory as a functor from the homotopical category of homotopical categories
More informationarxiv: v2 [math.at] 18 Sep 2008
TOPOLOGICAL HOCHSCHILD AND CYCLIC HOMOLOGY FOR DIFFERENTIAL GRADED CATEGORIES arxiv:0804.2791v2 [math.at] 18 Sep 2008 GONÇALO TABUADA Abstract. We define a topological Hochschild (THH) and cyclic (TC)
More informationTopological Logarithmic Structures
Department of Mathematics University of Oslo 25th Nordic and 1st British Nordic Congress of Mathematicians Oslo, June 2009 Outline 1 2 3 Outline Pre-log rings Pre-log S-algebras Repleteness 1 2 3 Pre-log
More informationON LEFT AND RIGHT MODEL CATEGORIES AND LEFT AND RIGHT BOUSFIELD LOCALIZATIONS
Homology, Homotopy and Applications, vol. 12(2), 2010, pp.245 320 ON LEFT AND RIGHT MODEL CATEGORIES AND LEFT AND RIGHT BOUSFIELD LOCALIZATIONS CLARK BARWICK (communicated by Brooke Shipley) Abstract We
More informationBOUSFIELD LOCALIZATION AND ALGEBRAS OVER OPERADS
BOUSFIELD LOCALIZATION AND ALGEBRAS OVER OPERADS DAVID WHITE Thanks for the invitation and thanks to Peter May for all the helpful conversations over the years. More details can be found on my website:
More informationCommutative ring objects in pro-categories and generalized Moore spectra
1 Commutative ring objects in pro-categories and generalized Moore spectra DANIEL G DAVIS TYLER LAWSON We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric
More informationAlgebraic model structures
Algebraic model structures Emily Riehl Harvard University http://www.math.harvard.edu/~eriehl 18 September, 2011 Homotopy Theory and Its Applications AWM Anniversary Conference ICERM Emily Riehl (Harvard
More informationHigher Prop(erad)s. Philip Hackney, Marcy Robertson*, and Donald Yau UCLA. July 1, 2014
Higher Prop(erad)s Philip Hackney, Marcy Robertson*, and Donald Yau UCLA July 1, 2014 Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, 2014 1 / 1 Intro: Enriched Categories
More informationAn introduction to calculus of functors
An introduction to calculus of functors Ismar Volić Wellesley College International University of Sarajevo May 28, 2012 Plan of talk Main point: One can use calculus of functors to answer questions about
More information2 BROOKE SHIPLEY up to homotopy" information as differential graded R-algebras. Verifying this conjecture is the subject of this paper. To formulate t
HZ-ALGEBRA SPECTRA ARE DIFFERENTIAL GRADED ALGEBRAS BROOKE SHIPLEY Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely,
More informationLoday construction in functor categories
Wayne State University AMS Bloomington April, 1st 2017 Motivation Goal: Describe a computational tool (using some categorical machinery) for computing higher order THH, which is an approximation to iterated
More informationSOME EXERCISES. This is not an assignment, though some exercises on this list might become part of an assignment. Class 2
SOME EXERCISES This is not an assignment, though some exercises on this list might become part of an assignment. Class 2 (1) Let C be a category and let X C. Prove that the assignment Y C(Y, X) is a functor
More informationModuli Problems for Structured Ring Spectra
Moduli Problems for Structured Ring Spectra P. G. Goerss and M. J. Hopkins 1 1 The authors were partially supported by the National Science Foundation (USA). In this document we make good on all the assertions
More informationA CLOSED MODEL CATEGORY FOR (n 1)-CONNECTED SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 11, November 1996 A CLOSED MODEL CATEGORY FOR (n 1)-CONNECTED SPACES J. IGNACIO EXTREMIANA ALDANA, L. JAVIER HERNÁNDEZ PARICIO, AND M.
More informationTHE HEART OF A COMBINATORIAL MODEL CATEGORY
Theory and Applications of Categories, Vol. 31, No. 2, 2016, pp. 31 62. THE HEART OF A COMBINATORIAL MODEL CATEGORY ZHEN LIN LOW Abstract. We show that every small model category that satisfies certain
More informationHall Algebras and Bridgeland Stability Conditions for Stable -categories
Hall Algebras and Bridgeland Stability Conditions for Stable -categories Hiro Lee Tanaka, Harvard University Goodwillie Birthday Conference, June 25, 2014 Dubrovnik, Croatia Today s topic: Factorization
More informationTangent categories of algebras over operads
Yonatan Harpaz, Joost Nuiten and Matan Prasma Abstract The abstract cotangent complex formalism, as developed in the -categorical setting by Lurie, brings together classical notions such as parametrized
More informationHomology of dendroidal sets
Homology of dendroidal sets Matija Bašić and Thomas Nikolaus September 2, 2015 Abstract We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor.
More informationDerived Algebraic Geometry III: Commutative Algebra
Derived Algebraic Geometry III: Commutative Algebra May 1, 2009 Contents 1 -Operads 4 1.1 Basic Definitions........................................... 5 1.2 Fibrations of -Operads.......................................
More informationHomotopy Theory of Topological Spaces and Simplicial Sets
Homotopy Theory of Topological Spaces and Simplicial Sets Jacobien Carstens May 1, 2007 Bachelorthesis Supervision: prof. Jesper Grodal KdV Institute for mathematics Faculty of Natural Sciences, Mathematics
More informationMath Homotopy Theory Hurewicz theorem
Math 527 - Homotopy Theory Hurewicz theorem Martin Frankland March 25, 2013 1 Background material Proposition 1.1. For all n 1, we have π n (S n ) = Z, generated by the class of the identity map id: S
More informationLOCALIZATION, UNIVERSAL PROPERTIES, AND HOMOTOPY THEORY
LOCLIZTION, UNIVERSL PROPERTIES, ND HOMOTOPY THEORY DVID WHITE Localization in lgebra Localization in Category Theory ousfield localization 1. The right way to think about localization in algebra Localization
More informationIterated Bar Complexes of E-infinity Algebras and Homology Theories
Iterated Bar Complexes of E-infinity Algebras and Homology Theories BENOIT FRESSE We proved in a previous article that the bar complex of an E -algebra inherits a natural E -algebra structure. As a consequence,
More informationLocal higher category theory
February 27, 2017 Path category The nerve functor B : cat sset is def. by BC n = hom(n, C), where n is the poset 0 1 n. The path category functor P : sset cat is the left adjoint of the nerve: P(X ) =
More informationarxiv:math/ v1 [math.at] 21 Aug 2001
CLASSIFICATION OF STABLE MODEL CATEGORIES arxiv:math/0108143v1 [math.at] 21 Aug 2001 STEFAN SCHWEDE AND BROOKE SHIPLEY Abstract: A stable model category is a setting for homotopy theory where the suspension
More informationTHE STABLE SHAPE OF COMPACT SPACES WITH COUNTABLE COHOMOLOGY GROUPS. S lawomir Nowak University of Warsaw, Poland
GLASNIK MATEMATIČKI Vol. 42(62)(2007), 189 194 THE STABLE SHAPE OF COMPACT SPACES WITH COUNTABLE COHOMOLOGY GROUPS S lawomir Nowak University of Warsaw, Poland Dedicated to Professor Sibe Mardešić on the
More informationSECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS
SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon
More informationAutour de la Géométrie Algébrique Dérivée
Autour de la Géométrie Algébrique Dérivée groupe de travail spring 2013 Institut de Mathématiques de Jussieu Université Paris Diderot Organised by: Gabriele VEZZOSI Speakers: Pieter BELMANS, Brice LE GRIGNOU,
More informationHOMOTOPY THEORY OF S-BIMODULES, NAIVE RING SPECTRA AND STABLE MODEL CATEGORIES
HOMOTOPY THEORY OF S-BIMODULES, NAIVE RING SPECTRA AND STABLE MODEL CATEGORIES Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen
More informationSUSPENSION SPECTRA AND HIGHER STABILIZATION
SUSPENSION SPECTRA AND HIGHER STABILIZATION JACOBSON R. BLOMQUIST AND JOHN E. HARPER Abstract. We prove that the stabilization of spaces functor the classical construction of associating a spectrum to
More informationarxiv: v1 [math.at] 21 Apr 2014
MONOIDAL BOUSFIELD LOCALIZATIONS AND ALGEBRAS OVER OPERADS arxiv:1404.5197v1 [math.at] 21 Apr 2014 DAVID WHITE Abstract. We give conditions on a monoidal model categorymand on a set of mapscso that the
More informationJournal of Pure and Applied Algebra
Journal of Pure and Applied Algebra 214 (2010) 1384 1398 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa Homotopy theory of
More informationLecture 2: Spectra and localization
Lecture 2: Spectra and localization Paul VanKoughnett February 15, 2013 1 Spectra (Throughout, spaces means pointed topological spaces or simplicial sets we ll be clear where we need one version or the
More informationLECTURE 5: v n -PERIODIC HOMOTOPY GROUPS
LECTURE 5: v n -PERIODIC HOMOTOPY GROUPS Throughout this lecture, we fix a prime number p, an integer n 0, and a finite space A of type (n + 1) which can be written as ΣB, for some other space B. We let
More informationarxiv: v2 [math.at] 29 Nov 2007
arxiv:0708.3435v2 [math.at] 29 Nov 2007 ON THE DREADED RIGHT BOUSFIELD LOCALIZATION by Clark Barwick Abstract. I verify the existence of right Bousfield localizations of right semimodel categories, and
More informationFUNCTOR CALCULUS FOR UNDER CATEGORIES AND THE DE RHAM COMPLEX
FUNCTOR CALCULUS FOR UNDER CATEGORIES AND THE DE RHAM COMPLEX K. BAUER, R. ELDRED, B. JOHNSON, AND R. MCCARTHY Abstract. Goodwillie s calculus of homotopy functors associates a tower of polynomial approximations,
More information28 The fundamental groupoid, revisited The Serre spectral sequence The transgression The path-loop fibre sequence 23
Contents 8 The fundamental groupoid, revisited 1 9 The Serre spectral sequence 9 30 The transgression 18 31 The path-loop fibre sequence 3 8 The fundamental groupoid, revisited The path category PX for
More informationGENERAL THEORY OF LOCALIZATION
GENERAL THEORY OF LOCALIZATION DAVID WHITE Localization in Algebra Localization in Category Theory Bousfield localization Thank them for the invitation. Last section contains some of my PhD research, under
More informationLECTURE X: KOSZUL DUALITY
LECTURE X: KOSZUL DUALITY Fix a prime number p and an integer n > 0, and let S vn denote the -category of v n -periodic spaces. Last semester, we proved the following theorem of Heuts: Theorem 1. The Bousfield-Kuhn
More informationHOMOTOPY APPROXIMATION OF MODULES
Journal of Algebra and Related Topics Vol. 4, No 1, (2016), pp 13-20 HOMOTOPY APPROXIMATION OF MODULES M. ROUTARAY AND A. BEHERA Abstract. Deleanu, Frei, and Hilton have developed the notion of generalized
More informationALGEBRAIC K-THEORY: DEFINITIONS & PROPERTIES (TALK NOTES)
ALGEBRAIC K-THEORY: DEFINITIONS & PROPERTIES (TALK NOTES) JUN HOU FUNG 1. Brief history of the lower K-groups Reference: Grayson, Quillen s Work in Algebraic K-Theory 1.1. The Grothendieck group. Grothendieck
More informationRigidity and algebraic models for rational equivariant stable homotopy theory
Rigidity and algebraic models for rational equivariant stable homotopy theory Brooke Shipley UIC March 28, 2013 Main Question Question: Given stable model categories C and D, if Ho(C) and Ho(D) are triangulated
More informationThe abstract cotangent complex and Quillen cohomology of enriched categories
The abstract cotangent complex and Quillen cohomology of enriched categories Yonatan Harpaz, Joost Nuiten and Matan Prasma Abstract In his fundamental work, Quillen developed the theory of the cotangent
More informationTHE INFINITE SYMMETRIC PRODUCT AND HOMOLOGY THEORY
THE INFINITE SYMMETRIC PRODUCT AND HOMOLOGY THEORY ANDREW VILLADSEN Abstract. Following the work of Aguilar, Gitler, and Prieto, I define the infinite symmetric product of a pointed topological space.
More informationOVERVIEW OF SPECTRA. Contents
OVERVIEW OF SPECTRA Contents 1. Motivation 1 2. Some recollections about Top 3 3. Spanier Whitehead category 4 4. Properties of the Stable Homotopy Category HoSpectra 5 5. Topics 7 1. Motivation There
More informationA Hodge decomposition spectral sequence for E n-homology
A Hodge decomposition spectral sequence for E n -homology Royal Institute of Technology, Stockholm Women in Homotopy Theory and Algebraic Geometry, Berlin, 14.09.2016 1 2 3 A Hodge spectral sequence for
More informationErrata to Model Categories by Mark Hovey
Errata to Model Categories by Mark Hovey Thanks to Georges Maltsiniotis, maltsin@math.jussieu.fr, for catching most of these errors. The one he did not catch, on the non-smallness of topological spaces,
More informationA CARTAN-EILENBERG APPROACH TO HOMOTOPICAL ALGEBRA F. GUILLÉN, V. NAVARRO, P. PASCUAL, AND AGUSTÍ ROIG. Contents
A CARTAN-EILENBERG APPROACH TO HOMOTOPICAL ALGEBRA F. GUILLÉN, V. NAVARRO, P. PASCUAL, AND AGUSTÍ ROIG Abstract. In this paper we propose an approach to homotopical algebra where the basic ingredient is
More informationTHE HOMOTOPY THEORY OF EQUIVARIANT POSETS
THE HOMOTOPY THEORY OF EQUIVARIANT POSETS PETER MAY, MARC STEPHAN, AND INNA ZAKHAREVICH Abstract. Let G be a discrete group. We prove that the category of G-posets admits a model structure that is Quillen
More informationREPRESENTATIONS OF SPACES
REPRESENTATIONS OF SPACES WOJCIECH CHACHÓLSKI AND JÉRÔME SCHERER Abstract. We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial.
More informationHOMOTOPY THEORY FOR ALGEBRAS OVER POLYNOMIAL MONADS
Theory and Applications of Categories, Vol. 32, No. 6, 2017, pp. 148 253. HOMOTOPY THEORY FOR ALGEBRAS OVER POLYNOMIAL MONADS M. A. BATANIN AND C. BERGER Abstract. We study the existence and left properness
More informationAlgebraic Models for Homotopy Types III Algebraic Models in p-adic Homotopy Theory
Algebraic Models for Homotopy Types III Algebraic Models in p-adic Homotopy Theory Michael A. Mandell Indiana University Young Topologists Meeting 2013 July 11, 2013 M.A.Mandell (IU) Models in p-adic Homotopy
More informationTHE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA
THE EILENBERG-WATTS THEOREM IN HOMOTOPICAL ALGEBRA MARK HOVEY Abstract. The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial
More informationHomology and Cohomology of Stacks (Lecture 7)
Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic
More informationCombinatorial Models for M (Lecture 10)
Combinatorial Models for M (Lecture 10) September 24, 2014 Let f : X Y be a map of finite nonsingular simplicial sets. In the previous lecture, we showed that the induced map f : X Y is a fibration if
More informationKathryn Hess. Conference on Algebraic Topology, Group Theory and Representation Theory Isle of Skye 9 June 2009
Institute of Geometry, lgebra and Topology Ecole Polytechnique Fédérale de Lausanne Conference on lgebraic Topology, Group Theory and Representation Theory Isle of Skye 9 June 2009 Outline 1 2 3 4 of rings:
More informationAbstracting away from cell complexes
Abstracting away from cell complexes Michael Shulman 1 Peter LeFanu Lumsdaine 2 1 University of San Diego 2 Stockholm University March 12, 2016 Replacing big messy cell complexes with smaller and simpler
More informationarxiv:math/ v1 [math.at] 13 Nov 2001
arxiv:math/0111151v1 [math.at] 13 Nov 2001 Miller Spaces and Spherical Resolvability of Finite Complexes Abstract Jeffrey Strom Dartmouth College, Hanover, NH 03755 Jeffrey.Strom@Dartmouth.edu www.math.dartmouth.edu/~strom/
More informationEquivariant Trees and G-Dendroidal Sets
Equivariant Trees and -Dendroidal Sets 31st Summer Conference in Topology and Applications: Algebraic Topology Special Session Leicester, UK August 5, 2016 Joint with Luis Pereira Motivating Questions
More informationEndomorphism Rings of Abelian Varieties and their Representations
Endomorphism Rings of Abelian Varieties and their Representations Chloe Martindale 30 October 2013 These notes are based on the notes written by Peter Bruin for his talks in the Complex Multiplication
More informationMODULAR REPRESENTATION THEORY AND PHANTOM MAPS
MODULAR REPRESENTATION THEORY AND PHANTOM MAPS RICHARD WONG Abstract. In this talk, I will introduce and motivate the main objects of study in modular representation theory, which leads one to consider
More informationCellularity, composition, and morphisms of algebraic weak factorization systems
Cellularity, composition, and morphisms of algebraic weak factorization systems Emily Riehl University of Chicago http://www.math.uchicago.edu/~eriehl 19 July, 2011 International Category Theory Conference
More informationLEFT-INDUCED MODEL STRUCTURES AND DIAGRAM CATEGORIES
LEFT-INDUCED MODEL STRUCTURES AND DIAGRAM CATEGORIES MARZIEH BAYEH, KATHRYN HESS, VARVARA KARPOVA, MAGDALENA KȨDZIOREK, EMILY RIEHL, AND BROOKE SHIPLEY Abstract. We prove existence results for and verify
More informationHOMOTOPY THEORY OF MODULES OVER OPERADS AND NON-Σ OPERADS IN MONOIDAL MODEL CATEGORIES
HOMOTOPY THEORY OF MODULES OVER OPERADS AND NON-Σ OPERADS IN MONOIDAL MODEL CATEGORIES JOHN E. HARPER Abstract. We establish model category structures on algebras and modules over operads and non-σ operads
More informationDerived Algebraic Geometry I: Stable -Categories
Derived Algebraic Geometry I: Stable -Categories October 8, 2009 Contents 1 Introduction 2 2 Stable -Categories 3 3 The Homotopy Category of a Stable -Category 6 4 Properties of Stable -Categories 12 5
More informationEquivalence of the Combinatorial Definition (Lecture 11)
Equivalence of the Combinatorial Definition (Lecture 11) September 26, 2014 Our goal in this lecture is to complete the proof of our first main theorem by proving the following: Theorem 1. The map of simplicial
More informationA model-independent theory of -categories
Emily Riehl Johns Hopkins University A model-independent theory of -categories joint with Dominic Verity Joint International Meeting of the AMS and the CMS Dominic Verity Centre of Australian Category
More informationAXIOMATIC STABLE HOMOTOPY THEORY
AXIOMATIC STABLE HOMOTOPY THEORY MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND Abstract. We define and investigate a class of categories with formal properties similar to those of the homotopy category
More informationThe Goodwillie-Weiss Tower and Knot Theory - Past and Present
The Goodwillie-Weiss Tower and Knot Theory - Past and Present Dev P. Sinha University of Oregon June 24, 2014 Goodwillie Conference, Dubrovnik, Croatia New work joint with Budney, Conant and Koytcheff
More informationCohomology operations and the Steenrod algebra
Cohomology operations and the Steenrod algebra John H. Palmieri Department of Mathematics University of Washington WCATSS, 27 August 2011 Cohomology operations cohomology operations = NatTransf(H n ( ;
More informationHomotopy type theory: a new connection between logic, category theory and topology
Homotopy type theory: a new connection between logic, category theory and topology André Joyal UQÀM Category Theory Seminar, CUNY, October 26, 2018 Classical connections between logic, algebra and topology
More informationCLASS NOTES MATH 527 (SPRING 2011) WEEK 5
CLASS NOTES MATH 527 (SPRING 2011) WEEK 5 BERTRAND GUILLOU 1. Mon, Feb. 14 The same method we used to prove the Whitehead theorem last time also gives the following result. Theorem 1.1. Let X be CW and
More information