A homotopy theory of diffeological spaces
|
|
- Ezra Black
- 5 years ago
- Views:
Transcription
1 A homotopy theory of diffeological spaces Dan Christensen and Enxin Wu MIT Jan. 5, 2012
2 Motivation Smooth manifolds contain a lot of geometric information: tangent spaces, differential forms, de Rham cohomology, etc. These can be put to great use in proving results. But the category of smooth manifolds is not closed under many useful constructions, such as subspaces, quotients and function spaces. The category of (weak Hausdorff compactly generated) topological spaces is closed under these operations, but the geometric information is missing. Can we have the best of both worlds?
3 Diffeological spaces Definition (J. Souriau, 1980). A diffeological space is a set X together with a specified family of maps U X (called plots) for each open U R n and each n, such that for every open U, V R n : (1) Every constant map U X is a plot; (2) If U X is a plot and V U is smooth, then V U X is a plot; (3) If U = i U i is an open cover and U X is a map such that every restriction U i X is a plot, then U X is a plot. Definition. For diffeological spaces X and Y, a function X Y is smooth if for every plot U X in X, U X Y is a plot in Y. Definition. The category of diffeological spaces with smooth maps is denoted by Diff.
4 Properties of Diff The category of smooth manifolds is a full subcategory of Diff: Each smooth manifold M is a diffeological space: a map U M is declared to be a plot iff it is smooth in the usual sense. A function M N between smooth manifolds is smooth in the usual sense iff it is smooth as a smooth map of diffeological spaces. Remark. Kriegl-Michor s result says this is also true for smooth manifolds with boundary.
5 Properties of Diff, II Diff is closed under limits and colimits. For example, if Y is a subset of a diffeological space X, we can declare a function U Y to be a plot if the composition U Y X is a plot. In this case, Y is called a sub-diffeological space of X. If Y is a quotient of a diffeological space X, we can declare a function U Y to be a plot if locally it is of the form U i X Y, with the first map a plot of X. In this case, Y is called a quotient diffeological space of X. If X and Y are diffeological spaces, say that U X Y is a plot if each of U X and U Y is.
6 Properties of Diff, III Diff is cartesian closed: write Diff(X, Y ) for the set of smooth maps from X to Y. This is a diffeological space with a function U Diff(X, Y ) defined to be a plot if the natural map U X Y is smooth. One can show that is a diffeomorphism. Diff(W X, Y ) Diff(W,Diff(X, Y )) Given a diffeological space, we still have geometric information: dimension, differential forms, de Rham cohomology, tangent space at a point, tangent bundles, etc (Laubinger, Iglesias-Zemmour, etc), which are generalizations for smooth manifolds. Diff is locally presentable, but Top is not.
7 Examples of diffeological spaces For any diffeological space X, the loop space Diff(S 1, X ) and the path space Diff(I, X ) are diffeological spaces. Such examples are a big motivation for generalizing manifolds. A cross in R 2 with the following diffeologies: (1) the gluing diffeology. Then any smooth curve making a turn must stop for a finite amount of time. We call this a border crossing; (2) the sub-diffeology. Then there exists a curve which makes a turn but only halts for an instant. We call this a stop sign.
8 Naive homotopy theory of diffeological spaces In his thesis, Iglesias-Zemmour 85 defined the smooth homotopy groups π D n (X, x 0 ) of a pointed diffeological space (X, x 0 ) inductively, using loop spaces. As usual, π D 0 is a set, πd 1 is a group, and πd n is abelian for n 2. For a smooth manifold, the smooth and topological homotopy groups agree by smooth approximation. Every diffeological space has a natural topology (called the D-topology), but we will see that in general the smooth and topological homotopy groups do not agree.
9 Diffeological bundles [Iglesias-Zemmour 85] Definition. A diffeological bundle is a smooth surjective map X Y such that the pullback along any plot of Y is locally trivial. Theorem. If X Y is a diffeological bundle with fibre F, then there is a long exact sequence πn D (F ) πn D (X ) πn D (Y ) πn 1 D (F ) πd 0 (Y ). Theorem. If H is any subgroup of a diffeological group G, then G G/H is a diffeological bundle with fibre H. Example. Let H be a line of irrational slope in the torus T 2 and let A = T 2 /H. Then T 2 A is a diffeological bundle. From the long exact sequence, π D 1 (A) = Z Z. But π 1(A) = 0. All of this and much more is detailed in Iglesias-Zemmour s book in progress, which is freely available online.
10 A model category candidate for Diff [Work in progress with my supervisor, Dan Christensen. Many variations are being considered as well.] Let A n = {(x 0,, x n ) R n+1 x i = 1}, with the sub-diffeology. Then A n = R n. These A n s behave formally like the standard simplices n, and they form a cosimplicial object. For formal reasons, we get an adjoint pair : sset Diff: S called diffeological realization and smooth singular functors. We have n = A n, that preserves colimits and that S(X ) n =Diff(A n, X ).
11 A model category candidate for Diff, II We define a map X Y in Diff to be a weak equivalence (fibration) if S(X ) S(Y ) is a weak equivalence (fibration) in sset. A map X Y is a cofibration if it has the left lifting property with respect to all trivial fibrations. We haven t yet completed the proof that these form a model category, but we have some partial results.
12 Fibrant objects An important first step is to study the fibrant objects. By definition, these are the diffeological spaces such that S(X ) is a fibrant simplicial set, i.e. a Kan complex. Proposition (Christensen-W.) If X is fibrant, then π D (X ) = π (SX ). Corollary If X and Y are fibrant, then a map X Y is a weak equivalence iff π D (X ) π D (Y ) are isomorphisms for all basepoints. Unfortunately, not all objects are fibrant. For example, the cross in R 2 with the gluing diffeology and the sub-diffeology are both not fibrant.
13 Theorem (Christensen-W.) Every homogeneous space is fibrant. Corollary (Christensen-W.) Every smooth manifold without boundary is fibrant.
14 Theorem (Christensen-W.) Every homogeneous space is fibrant. Proof This follows from the following results: Every diffeological group is fibrant (in Diff), since every simplicial group is fibrant (in sset). Lemma (Christensen-W.) A diffeological bundle with fibrant fibre is a fibration. Lemma (Christensen-W.) If X Y is a diffeological bundle with X fibrant, then Y is fibrant. Corollary (Christensen-W.) Every smooth manifold without boundary is fibrant.
15 Theorem (Christensen-W.) Every homogeneous space is fibrant. Proof This follows from the following results: Every diffeological group is fibrant (in Diff), since every simplicial group is fibrant (in sset). Lemma (Christensen-W.) A diffeological bundle with fibrant fibre is a fibration. Lemma (Christensen-W.) If X Y is a diffeological bundle with X fibrant, then Y is fibrant. Corollary (Christensen-W.) Every smooth manifold without boundary is fibrant. Proof This follows from: (1) every connected smooth manifold without boundary is a homogeneous space. (2) Λ n k is connected under D-topology.
Patrick Iglesias-Zemmour
Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries
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 informationDIFFEOLOGICAL VECTOR SPACES
DIFFEOLOGICAL VECTOR SPACES J. DANIEL CHRISTENSEN AND ENXIN WU Abstract. We study the relationship between many natural conditions that one can put on a diffeological vector space, including being fine
More informationCAHIERS DE TOPOLOGIE ET Vol. LVII-1 (2016) GEOMETRIE DIFFERENTIELLE CATEGORIQUES TANGENT SPACES AND TANGENT BUNDLES FOR DIFFEOLOGICAL SPACES by J. Dan
CAHIERS DE TOPOLOGIE ET Vol. LVII-1 (2016) GEOMETRIE DIFFERENTIELLE CATEGORIQUES TANGENT SPACES AND TANGENT BUNDLES FOR DIFFEOLOGICAL SPACES by J. Daniel CHRISTENSEN and ENXIN WU Résumé. Nous étudions
More informationGroupoids and Orbifold Cohomology, Part 2
Groupoids and Orbifold Cohomology, Part 2 Dorette Pronk (with Laura Scull) Dalhousie University (and Fort Lewis College) Groupoidfest 2011, University of Nevada Reno, January 22, 2012 Motivation Orbifolds:
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 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 information1. Classifying Spaces. Classifying Spaces
Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.
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 informationLecture 4: Stabilization
Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3
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 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 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 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 informationOverview of Atiyah-Singer Index Theory
Overview of Atiyah-Singer Index Theory Nikolai Nowaczyk December 4, 2014 Abstract. The aim of this text is to give an overview of the Index Theorems by Atiyah and Singer. Our primary motivation is to understand
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 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 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 informationConvenient Categories of Smooth Spaces
Convenient Categories of Smooth Spaces John C. Baez and Alexander E. Hoffnung Department of Mathematics, University of California Riverside, California 92521 USA email: baez@math.ucr.edu, alex@math.ucr.edu
More informationBEN KNUDSEN. Conf k (f) Conf k (Y )
CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X Y is an injective
More informationAlgebraic models of homotopy type theory
Algebraic models of homotopy type theory Nicola Gambino School of Mathematics University of Leeds CT2016 Halifax, August 9th 1 Theme: property vs structure Fundamental distinction: satisfaction of a property
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
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 informationNonabelian Poincare Duality (Lecture 8)
Nonabelian Poincare Duality (Lecture 8) February 19, 2014 Let M be a compact oriented manifold of dimension n. Then Poincare duality asserts the existence of an isomorphism H (M; A) H n (M; A) for any
More informationSIMPLICIAL COCHAIN ALGEBRAS FOR DIFFEOLOGICAL SPACES
SIMPLICIAL COCHAIN ALGEBRAS FOR DIFFEOLOGICAL SPACES KATSUHIKO KURIBAYASHI Abstract. We introduce a de Rham complex endowed with an integration map to the singular cochain complex which gives the de Rham
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the
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 informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationFINITE SPECTRA CARY MALKIEWICH
FINITE SPECTRA CARY MALKIEWICH These notes were written in 2014-2015 to help me understand how the different notions of finiteness for spectra are related. I am usually surprised that the basics are not
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 informationTopological Field Theories
Topological Field Theories RTG Graduate Summer School Geometry of Quantum Fields and Strings University of Pennsylvania, Philadelphia June 15-19, 2009 Lectures by John Francis Notes by Alberto Garcia-Raboso
More informationDerived Algebraic Geometry IX: Closed Immersions
Derived Algebraic Geometry I: Closed Immersions November 5, 2011 Contents 1 Unramified Pregeometries and Closed Immersions 4 2 Resolutions of T-Structures 7 3 The Proof of Proposition 1.0.10 14 4 Closed
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 informationLecture 6: Classifying spaces
Lecture 6: Classifying spaces A vector bundle E M is a family of vector spaces parametrized by a smooth manifold M. We ask: Is there a universal such family? In other words, is there a vector bundle E
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 information10. The subgroup subalgebra correspondence. Homogeneous spaces.
10. The subgroup subalgebra correspondence. Homogeneous spaces. 10.1. The concept of a Lie subgroup of a Lie group. We have seen that if G is a Lie group and H G a subgroup which is at the same time a
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 informationAn Introduction to the Stolz-Teichner Program
Intro to STP 1/ 48 Field An Introduction to the Stolz-Teichner Program Australian National University October 20, 2012 Outline of Talk Field Smooth and Intro to STP 2/ 48 Field Field Motivating Principles
More informationAlgebraic Topology I Homework Spring 2014
Algebraic Topology I Homework Spring 2014 Homework solutions will be available http://faculty.tcu.edu/gfriedman/algtop/algtop-hw-solns.pdf Due 5/1 A Do Hatcher 2.2.4 B Do Hatcher 2.2.9b (Find a cell structure)
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 informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
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 information7. Homotopy and the Fundamental Group
7. Homotopy and the Fundamental Group The group G will be called the fundamental group of the manifold V. J. Henri Poincaré, 895 The properties of a topological space that we have developed so far have
More informationMotivic integration on Artin n-stacks
Motivic integration on Artin n-stacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of
More informationProfinite Homotopy Theory
Documenta Math. 585 Profinite Homotopy Theory Gereon Quick Received: April 4, 2008 Revised: October 28. 2008 Communicated by Stefan Schwede Abstract. We construct a model structure on simplicial profinite
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 informationMULTIPLE DISJUNCTION FOR SPACES OF POINCARÉ EMBEDDINGS
MULTIPLE DISJUNCTION FOR SPACES OF POINCARÉ EMBEDDINGS THOMAS G. GOODWILLIE AND JOHN R. KLEIN Abstract. Still at it. Contents 1. Introduction 1 2. Some Language 6 3. Getting the ambient space to be connected
More information32 Proof of the orientation theorem
88 CHAPTER 3. COHOMOLOGY AND DUALITY 32 Proof of the orientation theorem We are studying the way in which local homological information gives rise to global information, especially on an n-manifold M.
More informationTHE H-PRINCIPLE, LECTURE 14: HAEFLIGER S THEOREM CLASSIFYING FOLIATIONS ON OPEN MANIFOLDS
THE H-PRINCIPLE, LECTURE 14: HAELIGER S THEOREM CLASSIYING OLIATIONS ON OPEN MANIOLDS J. RANCIS, NOTES BY M. HOYOIS In this lecture we prove the following theorem: Theorem 0.1 (Haefliger). If M is an open
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 informationTopological K-theory, Lecture 3
Topological K-theory, Lecture 3 Matan Prasma March 2, 2015 1 Applications of the classification theorem continued Let us see how the classification theorem can further be used. Example 1. The bundle γ
More informationSupplementary Notes October 13, On the definition of induced representations
On the definition of induced representations 18.758 Supplementary Notes October 13, 2011 1. Introduction. Darij Grinberg asked in class Wednesday about why the definition of induced representation took
More informationInverse Galois Problem for C(t)
Inverse Galois Problem for C(t) Padmavathi Srinivasan PuMaGraSS March 2, 2012 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem
More informationMATH8808: ALGEBRAIC TOPOLOGY
MATH8808: ALGEBRAIC TOPOLOGY DAWEI CHEN Contents 1. Underlying Geometric Notions 2 1.1. Homotopy 2 1.2. Cell Complexes 3 1.3. Operations on Cell Complexes 3 1.4. Criteria for Homotopy Equivalence 4 1.5.
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 informationHomotopy and homology of fibred spaces
Homotopy and homology of fibred spaces Hans-Joachim Baues and Davide L. Ferrario Abstract We study fibred spaces with fibres in a structure category F and we show that cellular approximation, Blakers Massey
More informationEXCISION ESTIMATES FOR SPACES OF DIFFEOMORPHISMS. Thomas G. Goodwillie. Brown University
EXCISION ESTIMATES FOR SPACES OF DIFFEOMORPHISMS Thomas G. Goodwillie Brown University Abstract. [This is not finished. The surgery section needs to be replaced by a reference to known results. The obvious
More informationarxiv: v1 [math.ct] 10 Jul 2016
ON THE FIBREWISE EFFECTIVE BURNSIDE -CATEGORY arxiv:1607.02786v1 [math.ct] 10 Jul 2016 CLARK BARWICK AND SAUL GLASMAN Abstract. Effective Burnside -categories, introduced in [1], are the centerpiece of
More informationarxiv:math/ v1 [math.at] 5 Oct 1999
arxiv:math/990026v [math.at] 5 Oct 999 REPRESENTATIONS OF THE HOMOTOPY SURFACE CATEGORY OF A SIMPLY CONNECTED SPACE MARK BRIGHTWELL AND PAUL TURNER. Introduction At the heart of the axiomatic formulation
More informationBredon, Introduction to compact transformation groups, Academic Press
1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions
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 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 information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationSpectra and the Stable Homotopy Category
Peter Bonventre Graduate Seminar Talk - September 26, 2014 Abstract: An introduction to the history and application of (topological) spectra, the stable homotopy category, and their relation. 1 Introduction
More informationSheaf models of type theory
Thierry Coquand Oxford, 8 September 2017 Goal of the talk Sheaf models of higher order logic have been fundamental for establishing consistency of logical principles E.g. consistency of Brouwer s fan theorem
More informationHOMOTOPY THEORY OF POSETS
Homology, Homotopy and Applications, vol. 12(2), 2010, pp.211 230 HOMOTOPY THEORY OF POSETS GEORGE RAPTIS (communicated by J. Daniel Christensen) Abstract This paper studies the category of posets Pos
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationTHE STEENROD ALGEBRA. The goal of these notes is to show how to use the Steenrod algebra and the Serre spectral sequence to calculate things.
THE STEENROD ALGEBRA CARY MALKIEWICH The goal of these notes is to show how to use the Steenrod algebra and the Serre spectral sequence to calculate things. 1. Brown Representability (as motivation) Let
More information9. The Lie group Lie algebra correspondence
9. The Lie group Lie algebra correspondence 9.1. The functor Lie. The fundamental theorems of Lie concern the correspondence G Lie(G). The work of Lie was essentially local and led to the following fundamental
More informationWe then have an analogous theorem. Theorem 1.2.
1. K-Theory of Topological Stacks, Ryan Grady, Notre Dame Throughout, G is sufficiently nice: simple, maybe π 1 is free, or perhaps it s even simply connected. Anyway, there are some assumptions lurking.
More informationThe formal theory of adjunctions, monads, algebras, and descent
The formal theory of adjunctions, monads, algebras, and descent Emily Riehl Harvard University http://www.math.harvard.edu/~eriehl Reimagining the Foundations of Algebraic Topology Mathematical Sciences
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 informationNOTES ON FIBER BUNDLES
NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement
More informationChern Classes and the Chern Character
Chern Classes and the Chern Character German Stefanich Chern Classes In this talk, all our topological spaces will be paracompact Hausdorff, and our vector bundles will be complex. Let Bun GLn(C) be the
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 informationHomotopical Algebra. Yuri Berest, Sasha Patotski
Homotopical Algebra Yuri Berest, Sasha Patotski Fall 2015 These are lecture notes of a graduate course MATH 7400 currently taught by Yuri Berest at Cornell University. The notes are taken by Sasha Patotski;
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 informationMath-Net.Ru All Russian mathematical portal
Math-Net.Ru All Russian mathematical portal V. A. Voevodskii, On the Zero Slice of the Sphere Spectrum, Tr. Mat. Inst. Steklova, 2004, Volume 246, 106 115 Use of the all-russian mathematical portal Math-Net.Ru
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 informationREPRESENTABILITY OF DERIVED STACKS
REPRESENTABILITY OF DERIVED STACKS J.P.PRIDHAM Abstract. Lurie s representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack.
More information6. Lecture cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not
6. Lecture 6 6.1. cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not only motivic cohomology, but also to Morel-Voevodsky
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 informationDimension in diffeology
Dimension in diffeology Patrick Iglesias-Zemmour typeset September 11, 2007 Abstract We define the dimension function for diffeological spaces, a simple but new invariant. We show then how it can be applied
More information121B: ALGEBRAIC TOPOLOGY. Contents. 6. Poincaré Duality
121B: ALGEBRAIC TOPOLOGY Contents 6. Poincaré Duality 1 6.1. Manifolds 2 6.2. Orientation 3 6.3. Orientation sheaf 9 6.4. Cap product 11 6.5. Proof for good coverings 15 6.6. Direct limit 18 6.7. Proof
More informationParametrized Homotopy Theory. J. P. May J. Sigurdsson
Parametrized Homotopy Theory J. P. May J. Sigurdsson Department of Mathematics, The University of Chicago, Chicago, IL 60637 Department of Mathematics, The University of Notre Dame, Notre Dame, IN, 46556-4618
More information110:615 algebraic topology I
110:615 algebraic topology I Topology is the newest branch of mathematics. It originated around the turn of the twentieth century in response to Cantor, though its roots go back to Euler; it stands between
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 informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More informationA Taxonomy of 2d TQFTs
1 Section 2 Ordinary TFTs and Extended TFTs A Taxonomy of 2d TQFTs Chris Elliott October 28, 2013 1 Introduction My goal in this talk is to explain several extensions of ordinary TQFTs in two dimensions
More informationBUILDING A MODEL CATEGORY OUT OF MULTIPLIER IDEAL SHEAVES
Theory and Applications of Categories, Vol. 32, No. 13, 2017, pp. 437 487. BUILDING A MODEL CATEGORY OUT OF MULTIPLIER IDEAL SHEAVES SEUNGHUN LEE Abstract. We will construct a Quillen model structure out
More informationGeometry and Topology, Lecture 4 The fundamental group and covering spaces
1 Geometry and Topology, Lecture 4 The fundamental group and covering spaces Text: Andrew Ranicki (Edinburgh) Pictures: Julia Collins (Edinburgh) 8th November, 2007 The method of algebraic topology 2 Algebraic
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More informationThe topology of the space of knots
The topology of the space of knots Felix Wierstra August 22, 2013 Master thesis Supervisor: prof.dr. Sergey Shadrin KdV Instituut voor wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica
More informationp,q H (X), H (Y ) ), where the index p has the same meaning as the
There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore
More informationNOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY
NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY 1. Closed and exact forms Let X be a n-manifold (not necessarily oriented), and let α be a k-form on X. We say that α is closed if dα = 0 and say
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 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 informationPart II. Algebraic Topology. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section II 18I The n-torus is the product of n circles: 5 T n = } S 1. {{.. S } 1. n times For all n 1 and 0
More informationConstructive Set Theory from a Weak Tarski Universe
Università degli Studi di Roma Tor Vergata Facoltà di Scienze Matematiche Fisiche e Naturali Corso di laurea magistrale in Matematica Pura ed Applicata Constructive Set Theory from a Weak Tarski Universe
More informationBERTRAND GUILLOU. s G q+r
STABLE A 1 -HOMOTOPY THEORY BERTRAND GUILLOU 1. Introduction Recall from the previous talk that we have our category pointed A 1 -homotopy category Ho A 1, (k) over a field k. We will often refer to an
More information