An Introduction to Model Categories
|
|
- Anthony Robinson
- 6 years ago
- Views:
Transcription
1 An Introduction to Model Categories Brooke Shipley (UIC) Young Topologists Meeting, Stockholm July 4, 2017
2 Examples of Model Categories (C, W ) Topological Spaces with weak equivalences f : X Y if π (X ) = π (Y ).
3 Examples of Model Categories (C, W ) Topological Spaces with weak equivalences f : X Y if π (X ) = π (Y ). Chain complexes with quasi-isomorphisms f : C D if H (C) = H (D).
4 Examples of Model Categories (C, W ) Topological Spaces with weak equivalences f : X Y if π (X ) = π (Y ). Chain complexes with quasi-isomorphisms f : C D if H (C) = H (D). Simplicial abelian groups with weak equivalences f : A B if H (NA) = H (NB).
5 Definition of Model Categories Definition: A model category is a category C with 3 classes of maps W, C, and F, satisfying 5 axioms as below. Weak equivalences, denoted, Cofibrations, denoted, and Fibrations, denoted.
6 Definition of Model Categories Definition: A model category is a category C with 3 classes of maps W, C, and F, satisfying 5 axioms as below. Weak equivalences, denoted, Cofibrations, denoted, and Fibrations, denoted. closed under composition
7 Definition of Model Categories Definition: A model category is a category C with 3 classes of maps W, C, and F, satisfying 5 axioms as below. Weak equivalences, denoted, Cofibrations, denoted, and Fibrations, denoted. closed under composition acyclic cofibrations A B acyclic fibrations X Y
8 Axioms for Model Categories C has all finite colimits and limits.
9 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third.
10 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third. W, C, F are closed under retracts.
11 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third. W, C, F are closed under retracts. Lifting: Lifts exist in the following squares: A X A X B Y B Y
12 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third. W, C, F are closed under retracts. Lifting: Lifts exist in the following squares: A X A X B Y B Y Factorization: Any map f : X Y factors in two ways X Z Y X W Y.
13 Homotopy Category, Quillen Pair, Quillen Equivalence The homotopy category of a model category (C, W ) is defined by inverting the weak equivalences. Ho(C) = C[W 1 ]
14 Homotopy Category, Quillen Pair, Quillen Equivalence The homotopy category of a model category (C, W ) is defined by inverting the weak equivalences. Ho(C) = C[W 1 ] Given C, D model categories and an adjunction: C F D, then (F, G) is a Quillen pair if F preserves cofibrations and G preserves fibrations. Then there is an induced adjunction: Ho(C) LF Ho(D) RG G
15 Homotopy Category, Quillen Pair, Quillen Equivalence The homotopy category of a model category (C, W ) is defined by inverting the weak equivalences. Ho(C) = C[W 1 ] Given C, D model categories and an adjunction: C F D, then (F, G) is a Quillen pair if F preserves cofibrations and G preserves fibrations. Then there is an induced adjunction: Ho(C) LF Ho(D) RG If (LF, RG) induces an equivalence on the homotopy categories, then (F, G) is a Quillen equivalence. C QE D and Ho(C) = Ho(D). G
16 Examples The projective model structure on ch + : W = quasi-isomoprhisms F = epimorphisms in positive degree C = monomorphisms with projective cokernel.
17 Examples The projective model structure on ch + : W = quasi-isomoprhisms F = epimorphisms in positive degree C = monomorphisms with projective cokernel. The injective model structure on ch : W = quasi-isomoprhisms C = monomorphisms in negative degree F = epimorphisms with injective kernel.
18 Examples The projective model structure on ch + : W = quasi-isomoprhisms F = epimorphisms in positive degree C = monomorphisms with projective cokernel. The injective model structure on ch : W = quasi-isomoprhisms C = monomorphisms in negative degree F = epimorphisms with injective kernel. Both extend to model structures on Ch: Ch Proj QE Ch Inj and Ho(Ch Proj ) = Ho(Ch Inj )
19 Counter-examples and Examples There are examples of model categories C, D with Ho(C) = Ho(D), but there is no Quillen pair inducing this equivalence. So, C QE D.
20 Counter-examples and Examples There are examples of model categories C, D with Ho(C) = Ho(D), but there is no Quillen pair inducing this equivalence. So, C QE D. (N, Γ) form a Quillen pair and a Quillen equivalence sab QE ch + and Ho(sAb) = Ho(ch + )
21 Counter-examples and Examples There are examples of model categories C, D with Ho(C) = Ho(D), but there is no Quillen pair inducing this equivalence. So, C QE D. (N, Γ) form a Quillen pair and a Quillen equivalence sab QE ch + and Ho(sAb) = Ho(ch + ) (Schwede-S. 2003) N induces a functor on simplicial rings, and is part of a Quillen equivalence, s Rings QE DGA + and Ho(s Rings) = Ho(DGA + )
Graduate 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 informationK-theory and derived equivalences (Joint work with D. Dugger) Neeman proved the above result for regular rings.
K-theory and derived equivalences (Joint work with D. Dugger) Main Theorem: If R and S are two derived equivalent rings, then K (R) = K (S). Neeman proved the above result for regular rings. Main Example
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 informationAN INTRODUCTION TO MODEL CATEGORIES
N INTRODUCTION TO MODEL CTEGORIES JUN HOU FUNG Contents 1. Why model categories? 1 2. Definitions, examples, and basic properties 3 3. Homotopy relations 5 4. The homotopy category of a model category
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 informationGrothendieck duality for affine M 0 -schemes.
Grothendieck duality for affine M 0 -schemes. A. Salch March 2011 Outline Classical Grothendieck duality. M 0 -schemes. Derived categories without an abelian category of modules. Computing Lf and Rf and
More informationDerivatives of the identity functor and operads
University of Oregon Manifolds, K-theory, and Related Topics Dubrovnik, Croatia 23 June 2014 Motivation We are interested in finding examples of categories in which the Goodwillie derivatives of the identity
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 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 informationRELATIVE HOMOLOGICAL ALGEBRA VIA TRUNCATIONS
RELATIVE HOMOLOGICAL ALGEBRA VIA TRUNCATIONS WOJCIECH CHACHÓLSKI, AMNON NEEMAN, WOLFGANG PITSCH, AND JÉRÔME SCHERER Abstract. To do homological algebra with unbounded chain complexes one needs to first
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 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 informationAlgebras and modules in monoidal model categories
Algebras and modules in monoidal model categories Stefan Schwede and Brooke E. Shipley 1 arxiv:math/9801082v1 [math.at] 19 Jan 1998 1 Summary Abstract: We construct model category structures for monoids
More informationHomotopy Limits and Colimits in Nature A Motivation for Derivators Summer School on Derivators Universität Freiburg August 2014
F. Hörmann Homotopy Limits and Colimits in Nature A Motivation for Derivators Summer School on Derivators Universität Freiburg August 2014 preliminary version 21.10.14 An introduction to the notions of
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 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 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 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 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 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 informationarxiv: v1 [math.kt] 24 Nov 2007
DIFFERENTIAL GRADED VERSUS SIMPLICIAL CATEGORIES arxiv:0711.3845v1 [math.kt] 24 Nov 2007 GONÇALO TABUADA Abstract. We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential
More informationAN ALGEBRAIC MODEL FOR RATIONAL TORUS-EQUIVARIANT SPECTRA. Contents. Part 1. Overview Introduction Outline of the argument.
AN ALGEBRAIC MODEL FOR RATIONAL TORUS-EQUIVARIANT SPECTRA J. P. C. GREENLEES AND B. SHIPLEY Abstract. We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit
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 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 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 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 informationA UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY
A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY STEFAN SCHWEDE AND BROOKE SHIPLEY 1. Introduction Roughly speaking, the stable homotopy category of algebraic topology is obtained from the homotopy category
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 informationCLUSTER CATEGORIES FOR TOPOLOGISTS
CLUSTER CATEGORIES FOR TOPOLOGISTS JULIA E. BERGNER AND MARCY ROBERTSON Abstract. We consider triangulated orbit categories, with the motivating example of cluster categories, in their usual context of
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 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 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 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 informationarxiv: v1 [math.kt] 19 Apr 2013
Contemporary Mathematics arxiv:1304.5314v1 [math.kt] 19 Apr 2013 Derived Representation Schemes and Noncommutative Geometry Yuri Berest, Giovanni Felder, and Ajay Ramadoss Abstract. Some 15 years ago M.
More informationarxiv:math/ v2 [math.kt] 2 Oct 2003
A remark on K-theory and S-categories arxiv:math/0210125v2 [math.kt] 2 Oct 2003 Bertrand Toën Laboratoire Emile Picard UMR CNRS 5580 Université Paul Sabatier, Toulouse France Abstract Gabriele Vezzosi
More informationRigidity and Exotic Models for the K-local Stable Homotopy Category
Rigidity and Exotic Models for the K-local Stable Homotopy Category Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität
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 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 informationHigher Categories, Homotopy Theory, and Applications
Higher Categories, Homotopy Theory, and Applications Thomas M. Fiore http://www.math.uchicago.edu/~fiore/ Why Homotopy Theory and Higher Categories? Homotopy Theory solves topological and geometric problems
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 informationMODEL STRUCTURES ON PRO-CATEGORIES
Homology, Homotopy and Applications, vol. 9(1), 2007, pp.367 398 MODEL STRUCTURES ON PRO-CATEGORIES HALVARD FAUSK and DANIEL C. ISAKSEN (communicated by J. Daniel Christensen) Abstract We introduce a notion
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23 6.4. Homotopy uniqueness of projective resolutions. Here I proved that the projective resolution of any R-module (or any object of an abelian category
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 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 informationPostnikov extensions of ring spectra
1785 1829 1785 arxiv version: fonts, pagination and layout may vary from AGT published version Postnikov extensions of ring spectra DANIEL DUGGER BROOKE SHIPLEY We give a functorial construction of k invariants
More informationsset(x, Y ) n = sset(x [n], Y ).
1. Symmetric monoidal categories and enriched categories In practice, categories come in nature with more structure than just sets of morphisms. This extra structure is central to all of category theory,
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 informationMORITA THEORY IN STABLE HOMOTOPY THEORY
MORITA THEORY IN STABLE HOMOTOPY THEORY BROOKE SHIPLEY Abstract. We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This
More informationAn introduction to stable homotopy theory. Abelian groups up to homotopy spectra ()generalized cohomology theories
An introduction to stable homotopy theory Abelian groups up to homotopy spectra ()generalized cohomology theories Examples: 1. Ordinary cohomology: For A any abelian group, H n (X; A) =[X +,K(A, n)]. Eilenberg-Mac
More informationDifferential Graded Algebras and Applications
Differential Graded Algebras and Applications Jenny August, Matt Booth, Juliet Cooke, Tim Weelinck December 2015 Contents 1 Introduction 2 1.1 Differential Graded Objects....................................
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationElmendorf s Theorem via Model Categories
Elmendorf s Theorem via Model Categories Marc Stephan January 29, 2010 1 Introduction In [2], working in the category of compactly generated spaces U, Elmendorf relates the equivariant homotopy theory
More informationarxiv: v2 [math.at] 26 Jan 2012
RATIONAL EQUIVARIANT RIGIDITY DAVID BARNES AND CONSTANZE ROITZHEIM arxiv:1009.4329v2 [math.at] 26 Jan 2012 Abstract. We prove that if G is S 1 or a profinite group, then the all of the homotopical information
More informationModules over Motivic Cohomology
Noname manuscript No. will be inserted by the editor) Oliver Röndigs, Paul Arne Østvær Modules over Motivic Cohomology Oblatum date & date Abstract For fields of characteristic zero, we show the homotopy
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 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 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 informationOn differential graded categories
On differential graded categories Bernhard Keller Abstract. Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review
More informationAlgebraic Topology (topics course) Spring 2010 John E. Harper. Series 6
Algebraic Toology (toics course) Sring 2010 John E. Harer Series 6 Let R be a ring and denote by Ch + R (res. Mod R) the category of non-negative chain comlexes over R (res. the category of left R-modules).
More informationTwo results from Morita theory of stable model categories
Two results from Morita theory of stable model categories Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität
More informationPOSTNIKOV EXTENSIONS OF RING SPECTRA
POSTNIKOV EXTENSIONS OF RING SPECTRA DANIEL DUGGER AND BROOKE SHIPLEY Abstract. We give a functorial construction of k-invariants for ring spectra, and use these to classify extensions in the Postnikov
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 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 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 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 informationOn Obstructions to Realizing Diagrams of Π-algebras
On Obstructions to Realizing Diagrams of Π-algebras Mark W. Johnson mwj3@psu.edu March 16, 2008. On Obstructions to Realizing Diagrams of Π-algebras 1/13 Overview Collaboration with David Blanc and Jim
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 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 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 informationRational homotopy theory
Rational homotopy theory Alexander Berglund November 12, 2012 Abstract These are lecture notes for a course on rational homotopy theory given at the University of Copenhagen in the fall of 2012. Contents
More informationProperties of Triangular Matrix and Gorenstein Differential Graded Algebras
Properties of Triangular Matrix and Gorenstein Differential Graded Algebras Daniel Maycock Thesis submitted for the degree of Doctor of Philosophy chool of Mathematics & tatistics Newcastle University
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 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 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 informationTopic Proposal Algebraic K-Theory and the Additivity Theorem
Topic Proposal Algebraic K-Theory and the Additivity Theorem Mona Merling Discussed with Prof. Peter May and Angelica Osorno Winter 2010 1 Introduction Algebraic K-theory is the meeting ground for various
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 informationAlgebraic models for higher categories
Algebraic models for higher categories Thomas Nikolaus Fachbereich Mathematik, Universität Hamburg Schwerpunkt Algebra und Zahlentheorie Bundesstraße 55, D 20 146 Hamburg Abstract We introduce the notion
More informationAn introduction to derived and triangulated categories. Jon Woolf
An introduction to derived and triangulated categories Jon Woolf PSSL, Glasgow, 6 7th May 2006 Abelian categories and complexes Derived categories and functors arise because 1. we want to work with complexes
More informationLecture 007 (April 13, 2011) Suppose that A is a small abelian category, and let B be a full subcategory such that in every exact sequence
Lecture 007 (April 13, 2011) 16 Abelian category localization Suppose that A is a small abelian category, and let B be a full subcategory such that in every exact sequence 0 a a a 0 in A, a is an object
More informationTOWARD A RING SPECTRUM MAP
TOWARD A RING SPECTRUM MAP FROM K(ku) TO E 2 BY EIVIND E. DAHL THESIS FOR THE DEGREE OF MASTER OF SCIENCE (MASTER I MATEMATIKK) DEPARTMENT OF MATHEMATICS FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY
More informationModel categories in equivariant rational homotopy theory
Model categories in equivariant rational homotopy theory SPUR Final Paper, Summer 2017 Luke Sciarappa Mentor: Robert Burklund Project suggested by: Robert Burklund August 2, 2017 Abstract Model categories
More informationInduced maps on Grothendieck groups
Niels uit de Bos Induced maps on Grothendieck groups Master s thesis, August, 2014 Supervisor: Lenny Taelman Mathematisch Instituut, Universiteit Leiden CONTENTS 2 Contents 1 Introduction 4 1.1 Motivation
More informationDiagram spaces, diagram spectra and spectra of units
msp Algebraic & Geometric Topology 13 (2013) 1857 1935 Diagram spaces, diagram spectra and spectra of units JOHN ALIND This article compares the infinite loop spaces associated to symmetric spectra, orthogonal
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 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 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 informationMODEL CATEGORY STRUCTURES ON CHAIN COMPLEXES OF SHEAVES
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 6, Pages 2441 2457 S 0002-9947(01)02721-0 Article electronically published on January 3, 2001 MODEL CATEGORY STRUCTURES ON CHAIN COMPLEXES
More informationON THE DERIVED CATEGORY OF AN ALGEBRA OVER AN OPERAD. Dedicated to Mamuka Jibladze on the occasion of his 50th birthday
ON THE DERIVED CATEGORY OF AN ALGEBRA OVER AN OPERAD CLEMENS BERGER AND IEKE MOERDIJK Dedicated to Mamuka Jibladze on the occasion of his 50th birthday Abstract. We present a general construction of the
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationGeometric symmetric powers in the homotopy categories of schemes over a field
Geometric symmetric powers in the homotopy categories of schemes over a field Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy
More informationarxiv: v2 [math.dg] 28 Jul 2017
arxiv:1707.01145v2 [math.dg] 28 Jul 2017 Quasi-coherent sheaves in differential geometry Dennis Borisov University of Goettingen, Germany Bunsenstr. 3-4, 37073 Göttingen dennis.borisov@gmail.com July 31,
More informationAlgebraic model structures
Algebraic model structures Emily Riehl University of Chicago http://www.math.uchicago.edu/~eriehl 22 June, 2010 International Category Theory Conference Università di Genova Emily Riehl (University of
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 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 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 informationFUNCTORS BETWEEN REEDY MODEL CATEGORIES OF DIAGRAMS
FUNCTORS BETWEEN REEDY MODEL CATEGORIES OF DIAGRAMS PHILIP S. HIRSCHHORN AND ISMAR VOLIĆ Abstract. If D is a Reedy category and M is a model category, the category M D of D-diagrams in M is a model category
More informationAlgebraic models for higher categories
Algebraic models for higher categories Thomas Nikolaus Organisationseinheit Mathematik, Universität Hamburg Schwerpunkt Algebra und Zahlentheorie Bundesstraße 55, D 20 146 Hamburg Abstract We establish
More informationRealization problems in algebraic topology
Realization problems in algebraic topology Martin Frankland Universität Osnabrück Adam Mickiewicz University in Poznań Geometry and Topology Seminar June 2, 2017 Martin Frankland (Osnabrück) Realization
More informationCorrection to: The p-order of topological triangulated categories. Journal of Topology 6 (2013), Stefan Schwede
Correction to: The p-order of topological triangulated categories Journal of Topology 6 (2013), 868 914 Stefan Schwede Zhi-Wei Li has pointed out a gap in the proof of Proposition A.4 and a missing argument
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 information