# A QUICK NOTE ON ÉTALE STACKS

Size: px
Start display at page:

Transcription

1 A QUICK NOTE ON ÉTALE STACKS DAVID CARCHEDI Abstract. These notes start by closely following a talk I gave at the Higher Structures Along the Lower Rhine workshop in Bonn, in January. I then give a taste of some applications to foliation theory. Finally, I give a brief account on the general theory of étale stacks. 1. Étale Differentiable Stacks Disclaimer: Throughout these notes, manifolds will not be assumed paracompact, 2 nd -countable, or even Hausdorff. Idea: Étale stacks are like manifolds whose points posses intrinsic (discrete) automorphism groups. Examples: 1) G a discrete group, M a smooth manifold, G M a smooth action M//G - the stacky quotient There is a (surjective) projection π : M M//G making M into a principal G-bundle over M//G, and given a point x M, Aut (π (x)) = G x - the stabilizer subgroup. 2) X locally of the form M//G i, for G i finite groups X is an orbifold. 3) (M, F) a foliated manifold, L : M//F a leaf has Étale stacks form a bicategory: M//F -the stacky leaf space. Aut (L) = Hol (L) - holonomy group. Rough Idea: The points of an étale stack X form not just a set, but a groupoid. If ϕ M X, with M a manifold and X an étale stack, then for all points x M, ψ α α (x) : ϕ (x) ψ (x) 1

2 2 DAVID CARCHEDI is an arrow in this groupoid. In particular, such an arrow must be invertible Hom (M, X ) is a groupoid. So given an étale stack X, we get a functor Mfd op X Gpd M Hom (M, X ). This functor completely characterizes X. Instead of describing what an étale stack is geometrically, we can characterize those functors of the form X. This is the functor of points approach. What properties must such a functor satisfy? Need continuity: If U = (U α ) is an open cover of M, morphisms together with coherent isomorphisms should be the same as morphisms i.e. X is a stack. ϕ α : U α X ϕ α Uα U β = ϕβ Uα U β, M X, To see what other properties we need to characterize examples: X, lets turn back to our 1) There is an action groupoid G M : objects: M arrows: G M, where a pair (g, x) : x g x. This is a groupoid object in Mfd, and the source and target maps (s, t) are local diffeomorphisms. G M is an étale Lie groupoid, and M//G M// (G M) 1. 3) There exists a Lie groupoid Hol (M, F) M, whose arrows are given by holonomy classes of leaf-wise paths. Hol (M, F) is Morita equivalent to an étale Lie groupoid and M//F M// Hol (M, F) 1. 3) If G M with G finite, then G M is étale and proper (Meaning (s, t) : G M M M is proper) and this is a local property. Conversely, if G 1 G 0 is étale and proper, G is locally of the form G i M i. So orbifolds are stacky quotients of the form G 0 //G 1 for étale and proper Lie groupoids. Idea: Étale stacks are quotients of the form G 0//G 1 for an étale Lie groupoid G. More precisely:

3 A QUICK NOTE ON ÉTALE STACKS 3 G is a Lie groupoid ỹ (G) : Mfd op Gpd In general ỹ (G) is not a stack. But the inclusion M Hom (M, G 1 ) Hom (M, G 0 ). St (Mfd) Gpd Mfdop of stacks into general presheaves of groupoids has left-adjoint a which canonically associates to a given presheaf of groupoids its stackification. Definition. Let G 0 //G 1 := a (ỹ (G)). Stacks of this form are called differentiable stacks. A smooth functor G H is a Morita equivalence G 0 //G 1 H 0 //H 1 an equivalence. G and H are Morita equivalent if and only if there is a diagram of Morita equivalences K G H. Maps G 0 //G 1 L 0 //L 1 are the same as diagrams G K with K G a Morita equivalence. They may also be described by principal L bundles over G (Hilsum-Skandalis maps.) Definition. X St (Mfd) is an étale stack X G 0 //G 1, with G Morita equivalent to an étale Lie groupoid Sheaf Theory. Recall: If M is a manifold, and F Sh (M) a sheaf, there exists an étalé space L (F ) L L(F ) M, which is a local diffeomorphism such that sections of L (F ) over an open subset U M 1:1 elements of F (U) ( L (F ) = ) F x, and we have an adjoint equivalence x M Sh (M) Γ L Et/M, with L Γ, where Et/M is the category of local diffeomorphisms over M. We will generalize this for étale stacks:

4 4 DAVID CARCHEDI Definition. A morphism G 0 //G 1 H 0 //H 1 with G and H étale is a local diffeomorphism if and only if it corresponds to a diagram K Mor. ϕ G H with ϕ 0 a local diffeomorphism. Given an étale Lie groupoid, the canonical projection G 0 G 0 //G 1 is always a local diffeomorphism. Define a category Site (G) : objects O (G 0 ) open subsets. arrows U α f V. G 0 G 0 G 0 //G 1 Sh (Site (G)) BG - the category of equivariant sheaves on G 0. (The classifying topos of G.) If G and H are Morita equivalent, BG BH, so if X G 0 //G 1, one may define its category of sheaves to be sheaves over Site (G), and this is well defined. Even more, one may define the 2-category of stacks of groupoids over X, by St (X ) St (Site (G)). Theorem (D.C). For an étale differentiable stack X G 0 //G 1, there is an adjoint equivalence St (X ) Γ L Et/X, where Et/X is the bicategory of local diffeomorphisms Y X, with Y an étale differentiable stack. Moreover, for each Z St (X ), and U G 0 open, Z (U) is equivalent to the groupoid of sections L (Z ) σ α L(Z ) U G 0 X. L (Z ) is called the étalé realization of Z. Notice that even when X = M is a manifold, this gives something new, as it says any stack of groupoids on a manifold comes from sections of a local diffeomorphism Y M from an étale differentiable stack.

5 A QUICK NOTE ON ÉTALE STACKS Haefliger Groupoids and Effectivity. Definition. Let M be a manifold, H (M) is the étale Lie groupoid, called the Haefliger groupoid of M, whose objects are M and whose arrows are germs of locally defined diffeomorphisms. Given G étale and x g y in G, choose a neighborhood U g over which s and t are injective t (s U ) 1 is a local diffeomorphism x y G H (G 0 ) (which is a local diffeomorphism). Definition. G is effective G H (G 0 ) is faithful. In fact, we can construct an étale Lie groupoid out of these germs to produce Eff (G), the effective part of G, and G is effective G Eff (G) is an isomorphism. Effectivity is Morita invariant, therefore we can make the following definition: Definition. An étale differentiable stack X is effective if X G 0 //G 1 for G an effective étale Lie groupoid. Geometric meaning: For an étale stack X G 0 //G 1, with π : G 0 X the canonical quotient map, and a point x G 0, the automorphism group Aut (π (x)) acts on the germ of G 0 around x. X is effective if and only if each of these actions are faithful. Proposition. For M and N connected manifolds, H (M) Mor. H (N) dim (M) = dim (N). Definition. Let ( ) ( ) H := (R n //H (R n ) 1 ) R n //H R n. n=0 H is the universal étale stack. Given an étale stack X G 0 //G 1, with dim (G 0 ) = n, we have the composite X G 0 //G 1 G 0 //H (G 0 ) 1 R n //H (R n ) 1 H, denoted by ef X. It is a local diffeomorphism, hence corresponds to a stack of groupoids Ef X over H. Theorem (D.C). X is effective Ef X is in fact a sheaf (of sets) over H. Theorem (D.C). H is a terminal object in the bicategory EtSt et of étale differentiable stacks and local diffeomorphisms, and hence n=0 EtSt et EtSt et /H Et/H St (H). Theorem (D.C). Let Mfd et denote the category of smooth manifolds and local diffeomorphisms. Then there is a canonical equivalence ω : St (H) St ( Mfd et). Moreover, the induced equivalence EtSt et St ( Mfd et) sends a manifold M to its representable sheaf. n=0 1

6 6 DAVID CARCHEDI What this last theorem basically says is, if you have any stack Z St ( Mfd et), that is some moduli problem which is functorial with respect to local diffeomorphisms, then there exists a unique étale differentiable stack Z, such that for a given manifold M, the groupoid Z (M) is equivalent to the groupoid of local diffeomorphisms from M to Z. Conversely, given any étale differentiable stack X, it determines a stack on manifolds and local diffeomorphisms by assigning a manifold M the groupoid of local diffeomorphisms from M to X. These operations are inverse to each other. As an example: Let R : ( Mfd et) op Set be the functor which assigns a manifold M its set of Riemannian metrics. This is not even a functor on Mfd, but it is functorial with respect to local diffeomorphisms, and in fact is a sheaf. So there exists an étale differentiable stack R, such that local diffeomorphisms M R are the same as Riemannian metrics on M. We call R the classifying stack for Riemannian metrics (In fact, the proof of the equivalence tells you not only the existence of such an R, but also an explicit étale Lie groupoid model.) Corollary. There is a canonical equivalence EtSt et eff Sh ( Mfd et), between effective étale differentiable stacks and local diffeomorphisms, and sheaves on Mfd et. Given any stack Z of groupoids, one can always take isomorphism classes objectwise to get a presheaf of sets, and then sheafify the result to get a sheaf, denoted by π 0 (Z ). This produces a left adjoint π 0 to the inclusion of sheaves (of sets) into stacks (of groupoids). Theorem (D.C). Under the equivalences and EtSt et St ( Mfd et) EtSt et eff Sh ( Mfd et), the left adjoint π 0 : St ( Mfd et) Sh (Mfd) et corresponds to the functor sending an étale stack X to its effective part Eff (X ). Let j : Mfd et Mfd be the canonical inclusion. Given any stack Y St (Mfd), one may restrict it to Mfd et to get a stack j Z, hence producing a restriction functor This functor actually has a left adjoint, called the prolongation functor. j : St (Mfd) St ( Mfd et). j! : St ( Mfd et) St (Mfd) Theorem (D.C). A stack X St (Mfd) is an étale differentiable stack if and only if it is in the essential image of j!.

7 A QUICK NOTE ON ÉTALE STACKS Applications to Foliation Theory. Étale differentiable stacks are also enjoy an intimate connection with foliation theory. For example, after recasting into the language of étale stacks, one has the following theorem: Theorem. (Haefliger, Moerdijk, Kock) For M a smooth manifold, equivalence classes of submersions M H are in bijection with regular foliations on M. (If it factors through R q //H (R q ) 1, then it is a q-codimensional foliation.) Fact: Given any submersion X Y between étale stacks, if factors uniquely as X Y Y with X Y a submersion with connected fibers, and Y Y and local diffeomorphism which encodes a sheaf. (Not all local diffeomorphisms encode a sheaf, as the sections could form a groupoid rather than a set.) Moreover, if F : M H is a submersion classifying a foliation on M, the factorization is given by M M// Hol (M, F) 1 H, where Hol (M, F) is the holonomy groupoid of the foliation, and the map M// Hol (M, F) 1 H is the unique local diffeomorphism (since H is terminal). We denote M//F := M// Hol (M, F) and call it the stacky leaf space. When the leaf space happens to be a manifold, it agrees. Example 1. Let R be the classifying stack for Riemannian metrics. Suppose that M R is a submersion. Then it factors uniquely as M R R, with M R a submersion with connected fibers, and R R a sheaf. However, one can also consider the composite M R H by the unique local diffeomorphism R H. This is a submersion F : M H so it classifies a foliation. Moreover, it can be factored as the submersion with connected fibers M R follows by the sheaf R R H (this uses that Riemannian metrics form a sheaf not just a stack). By uniqueness, one has that R = M//F, so that one gets a local diffeomorphism M//F R between the stacky leaf space and the classifying stack for Riemannian metrics. So this corresponds to a Riemannian metric on the stacky leaf space. One can show in fact that such Riemannian metrics on M//F are the same as transverse metrics on M with respect to the foliation F. (I had Camilo Angulo prove this while coadvising his master class thesis.) Hence, one has that submersions into the classifying stack for Riemannian metrics, classify Riemannian foliations.

9 A QUICK NOTE ON ÉTALE STACKS 9 of affine schemes. However, one can still show e.g. that Deligne-Mumford stacks are characterized by prolongations of stacks on the site of affine schemes and étale maps (with some appropriate size-theoretic conditions). All this seems to be new, and is the subject of a paper I am currently working on.

### D-manifolds and derived differential geometry

D-manifolds and derived differential geometry Dominic Joyce, Oxford University September 2014 Based on survey paper: arxiv:1206.4207, 44 pages and preliminary version of book which may be downloaded from

### Categorical Properties of Topological and Dierentiable Stacks

Categorical Properties of Topological and Dierentiable Stacks ISBN 978-90-5335-441-4 Categorical Properties of Topological and Dierentiable Stacks Categorische Eigenschappen van Topologische en Dierentieerbare

### Stacks in Representation Theory.

What is a representation of an algebraic group? Joseph Bernstein Tel Aviv University May 22, 2014 0. Representations as geometric objects In my talk I would like to introduce a new approach to (or rather

### Groupoids 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:

### Three 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

### What are stacks and why should you care?

What are stacks and why should you care? Milan Lopuhaä October 12, 2017 Todays goal is twofold: I want to tell you why you would want to study stacks in the first place, and I want to define what a stack

### BASIC MODULI THEORY YURI J. F. SULYMA

BASIC MODULI THEORY YURI J. F. SULYMA Slogan 0.1. Groupoids + Sites = Stacks 1. Groupoids Definition 1.1. Let G be a discrete group acting on a set. Let /G be the category with objects the elements of

### THE 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

### Elementary (ha-ha) Aspects of Topos Theory

Elementary (ha-ha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces 1 1.1 Presheaves on spaces......................... 1 1.2 Digression on pointless topology..................

### Coarse Moduli Spaces of Stacks over Manifolds

Coarse Moduli Spaces of Stacks over Manifolds (joint work with Seth Wolbert) Jordan Watts (CMS Summer Meeting 2014) University of Illinois at Urbana-Champaign June 8, 2014 Introduction Let G be a Lie group,

### The d-orbifold programme. Lecture 3 of 5: D-orbifold structures on moduli spaces. D-orbifolds as representable 2-functors

The d-orbifold programme. Lecture 3 of 5: D-orbifold structures on moduli spaces. D-orbifolds as representable 2-functors Dominic Joyce, Oxford University May 2014 For the first part of the talk, see preliminary

### PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES

PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What

### Derived Differential Geometry

Derived Differential Geometry Lecture 1 of 3: Dominic Joyce, Oxford University Derived Algebraic Geometry and Interactions, Toulouse, June 2017 For references, see http://people.maths.ox.ac.uk/ joyce/dmanifolds.html,

### Mini-Course on Moduli Spaces

Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional

### Lecture 2 Sheaves and Functors

Lecture 2 Sheaves and Functors In this lecture we will introduce the basic concept of sheaf and we also will recall some of category theory. 1 Sheaves and locally ringed spaces The definition of sheaf

### Differentiable Stacks, Gerbes, and Twisted K-Theory. Ping Xu, Pennsylvania State University

Differentiable Stacks, Gerbes, and Twisted K-Theory Ping Xu, Pennsylvania State University 4 septembre 2017 2 Table des matières 1 Lie Groupoids and Differentiable Stacks 5 1.1 Groupoids.....................................

### The maximal atlas of a foliation. 1 Maximal atlas, isonomy, and holonomy

The maximal atlas of a foliation Talk at 62. PSSL, Utrecht Oct. 1996 Anders Kock We shall describe such maximal atlas and provide it with an algebraic structure that brings along the holonomy groupoid

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

### h M (T ). The natural isomorphism η : M h M determines an element U = η 1

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli

### Homology 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

### VECTOR FIELDS AND FLOWS ON DIFFERENTIABLE STACKS

Theory and Applications of Categories, Vol. 22, No. 21, 2009, pp. 542 587. VECTOR FIELDS AND FLOWS ON DIFFERENTIABLE STACKS RICHARD HEPWORTH Abstract. This paper introduces the notions of vector field

### Translation Groupoids and Orbifold Cohomology

Canad. J. Math. Vol. 62 (3), 2010 pp. 614 645 doi:10.4153/cjm-2010-024-1 c Canadian Mathematical Society 2010 Translation Groupoids and Orbifold Cohomology Dorette Pronk and Laura Scull Abstract. We show

### Some remarks on Frobenius and Lefschetz in étale cohomology

Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)

### Sheaves. S. Encinas. January 22, 2005 U V. F(U) F(V ) s s V. = s j Ui Uj there exists a unique section s F(U) such that s Ui = s i.

Sheaves. S. Encinas January 22, 2005 Definition 1. Let X be a topological space. A presheaf over X is a functor F : Op(X) op Sets, such that F( ) = { }. Where Sets is the category of sets, { } denotes

### Derived Differential Geometry

Different kinds of spaces in algebraic geometry What is derived geometry? The definition of schemes Some basics of category theory Moduli spaces and moduli functors Algebraic spaces and (higher) stacks

### Groupoid Representation Theory

Groupoid Representation Theory Jeffrey C. Morton University of Western Ontario Seminar on Stacks and Groupoids February 12, 2010 Jeffrey C. Morton (U.W.O.) Groupoid Representation Theory UWO Feb 10 1 /

### 1 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

### The Differential Structure of an Orbifold

The Differential Structure of an Orbifold AMS Sectional Meeting 2015 University of Memphis Jordan Watts University of Colorado at Boulder October 18, 2015 Introduction Let G 1 G 0 be a Lie groupoid. Consider

### SJÄLVSTÄNDIGA ARBETEN I MATEMATIK

SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Equivariant Sheaves on Topological Categories av Johan Lindberg 2015 - No 7 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET,

### in 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

### The Fundamental Gerbe of a Compact Lie Group

The Fundamental Gerbe of a Compact Lie Group Christoph Schweigert Department of Mathematics, University of Hamburg and Center for Mathematical Physics, Hamburg Joint work with Thomas Nikolaus Sophus Lie

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### via Topos Theory Olivia Caramello University of Cambridge The unification of Mathematics via Topos Theory Olivia Caramello

in University of Cambridge 2 / 23 in in In this lecture, whenever I use the word topos, I really mean Grothendieck topos. Recall that a Grothendieck topos can be seen as: a generalized space a mathematical

### Cyclic homology of deformation quantizations over orbifolds

Cyclic homology of deformation quantizations over orbifolds Markus Pflaum Johann Wolfgang Goethe-Universität Frankfurt/Main CMS Winter 2006 Meeting December 9-11, 2006 References N. Neumaier, M. Pflaum,

### BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

### D-manifolds and d-orbifolds: a theory of derived differential geometry. III. Dominic Joyce, Oxford UK-Japan Mathematical Forum, July 2012.

D-manifolds and d-orbifolds: a theory of derived differential geometry. III. Dominic Joyce, Oxford UK-Japan Mathematical Forum, July 2012. Based on survey paper: arxiv:1206.4207, 44 pages and preliminary

### THE LUSTERNIK-SCHNIRELMANN CATEGORY FOR A DIFFERENTIABLE STACK

THE LUSTERNIK-SCHNIRELMANN CATEGORY FOR A DIFFERENTIABLE STACK SAMIRAH ALSULAMI, HELLEN COLMAN, AND FRANK NEUMANN Abstract. We introduce the notion of Lusternik-Schnirelmann category for differentiable

### The transverse index problem for Riemannian foliations

The transverse index problem for Riemannian foliations John Lott UC-Berkeley http://math.berkeley.edu/ lott May 27, 2013 The transverse index problem for Riemannian foliations Introduction Riemannian foliations

### We 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.

### Lecture 9: Sheaves. February 11, 2018

Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with

### STACKY HOMOTOPY THEORY

STACKY HOMOTOPY THEORY GABE ANGEINI-KNO AND EVA BEMONT 1. A stack by any other name... argely due to the influence of Mike Hopkins and collaborators, stable homotopy theory has become closely tied to moduli

### Matrix factorizations over projective schemes

Jesse Burke (joint with Mark E. Walker) Department of Mathematics University of California, Los Angeles January 11, 2013 Matrix factorizations Let Q be a commutative ring and f an element of Q. Matrix

### Introduction to derived algebraic geometry

Introduction to derived algebraic geometry Bertrand Toën Our main goal throughout these lectures will be the explicate the notion of a derived Artin stack. We will avoid homotopy theory wherever possible.

### Lecture 4 Super Lie groups

Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is

### On 2-Representations and 2-Vector Bundles

On 2-Representations and 2-Vector Bundles Urs April 19, 2007 Contents 1 Introduction. 1 1.1 Fibers for 2-Vector Bundles...................... 2 1.2 The canonical 2-representation.................... 3

### ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up

### Lecture 6: Etale Fundamental Group

Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and

### ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.

ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)

### SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY

SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes

### HOLONOMY GROUPOIDS OF SINGULAR FOLIATIONS

j. differential geometry 58 (2001) 467-500 HOLONOMY GROUPOIDS OF SINGULAR FOLIATIONS CLAIRE DEBORD Abstract We give a new construction of Lie groupoids which is particularly well adapted to the generalization

### AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

### ON THE HOMOTOPY TYPE OF LIE GROUPOIDS

ON THE HOMOTOPY TYPE OF LIE GROUPOIDS HELLEN COLMAN Abstract. We propose a notion of groupoid homotopy for generalized maps. This notion of groupoid homotopy generalizes the notions of natural transformation

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### RTG Mini-Course Perspectives in Geometry Series

RTG Mini-Course Perspectives in Geometry Series Jacob Lurie Lecture IV: Applications and Examples (1/29/2009) Let Σ be a Riemann surface of genus g, then we can consider BDiff(Σ), the classifying space

### Derived 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

### Infinite root stacks of logarithmic schemes

Infinite root stacks of logarithmic schemes Angelo Vistoli Scuola Normale Superiore, Pisa Joint work with Mattia Talpo, Max Planck Institute Brown University, May 2, 2014 1 Let X be a smooth projective

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

### PART IV.2. FORMAL MODULI

PART IV.2. FORMAL MODULI Contents Introduction 1 1. Formal moduli problems 2 1.1. Formal moduli problems over a prestack 2 1.2. Situation over an affine scheme 2 1.3. Formal moduli problems under a prestack

### -Chern-Simons functionals

-Chern-Simons functionals Talk at Higher Structures 2011, Göttingen Urs Schreiber November 29, 2011 With Domenico Fiorenza Chris Rogers Hisham Sati Jim Stasheff Details and references at http://ncatlab.org/schreiber/show/differential+

### The moduli stack of vector bundles on a curve

The moduli stack of vector bundles on a curve Norbert Hoffmann norbert.hoffmann@fu-berlin.de Abstract This expository text tries to explain briefly and not too technically the notions of stack and algebraic

### AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.

### PICARD GROUPS OF MODULI PROBLEMS II

PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

### 3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

### Chern 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

### DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground

### LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

### Realizing Families of Landweber Exact Theories

Realizing Families of Landweber Exact Theories Paul Goerss Department of Mathematics Northwestern University Summary The purpose of this talk is to give a precise statement of 1 The Hopkins-Miller Theorem

### Critical point theory for foliations

Critical point theory for foliations Steven Hurder University of Illinois at Chicago www.math.uic.edu/ hurder TTFPP 2007 Bedlewo, Poland Steven Hurder (UIC) Critical point theory for foliations July 27,

### QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when

### Olivia Caramello. University of Insubria - Como. Deductive systems and. Grothendieck topologies. Olivia Caramello. Introduction.

duality University of Insubria - Como 2 / 27 duality Aim of the talk purpose of this talk is to illustrate the relevance of the notion of topology. I will show that the classical proof system of geometric

### A Survey of Lurie s A Survey of Elliptic Cohomology

A Survey of Lurie s A Survey of Elliptic Cohomology Aaron Mazel-Gee Abstract These rather skeletal notes are meant for readers who have some idea of the general story of elliptic cohomology. More than

### Integrability and Associativity

Department of Mathematics University of Illinois at Urbana-Champaign, USA Aim Theorem (Lie III) Every finite dimensional Lie algebra is the Lie algebra of a Lie group Aim Theorem (Lie III) Every finite

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### Rigid Geometry and Applications II. Kazuhiro Fujiwara & Fumiharu Kato

Rigid Geometry and Applications II Kazuhiro Fujiwara & Fumiharu Kato Birational Geometry from Zariski s viewpoint S U D S : coherent (= quasi-compact and quasi-separated) (analog. compact Hausdorff) U

### Noncommutative geometry and quantum field theory

Noncommutative geometry and quantum field theory Graeme Segal The beginning of noncommutative geometry is the observation that there is a rough equivalence contravariant between the category of topological

### Unipotent groups and some

Unipotent groups and some A 1 -contractible smooth schemes math.ag/0703137 Aravind Asok (joint w/ B.Doran) July 14, 2008 Outline 1. History/Historical Motivation 2. The A 1 -homotopy black box 3. Main

### Higher Descent. 1. Descent for Sheaves. 2. Cosimplicial Groups. 3. Back to Sheaves. Amnon Yekutieli. 4. Higher Descent: Stacks. 5.

Outline Outline Higher Descent Amnon Yekutieli Department of Mathematics Ben Gurion University Notes available at http://www.math.bgu.ac.il/~amyekut/lectures Written 21 Nov 2012 1. Descent for Sheaves

### CHAPTER 1. Étale cohomology

CHAPTER 1 Étale cohomology This chapter summarizes the theory of the étale topology on schemes, culminating in the results on l-adic cohomology that are needed in the construction of Galois representations

### Algebraic 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

### FUNDAMENTAL GROUPS OF TOPOLOGICAL STACKS WITH SLICE PROPERTY

FUNDAMENTAL GROUPS OF TOPOLOGICAL STACKS WITH SLICE PROPERTY BEHRANG NOOHI Abstract. The main result of the paper is a formula for the fundamental group of the coarse moduli space of a topological stack.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26 RAVI VAKIL CONTENTS 1. Proper morphisms 1 Last day: separatedness, definition of variety. Today: proper morphisms. I said a little more about separatedness of

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### 1 Categorical Background

1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,

### Solutions to some of the exercises from Tennison s Sheaf Theory

Solutions to some of the exercises from Tennison s Sheaf Theory Pieter Belmans June 19, 2011 Contents 1 Exercises at the end of Chapter 1 1 2 Exercises in Chapter 2 6 3 Exercises at the end of Chapter

### Foliations, Fractals and Cohomology

Foliations, Fractals and Cohomology Steven Hurder University of Illinois at Chicago www.math.uic.edu/ hurder Colloquium, University of Leicester, 19 February 2009 Steven Hurder (UIC) Foliations, fractals,

### Classification of matchbox manifolds

Classification of matchbox manifolds Steven Hurder University of Illinois at Chicago www.math.uic.edu/ hurder Workshop on Dynamics of Foliations Steven Hurder (UIC) Matchbox Manifolds April 28, 2010 1

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### LIE GROUPOIDS AND LIE ALGEBROIDS LECTURE NOTES, FALL Contents 1. Lie groupoids Definitions 30

LIE GROUPOIDS AND LIE ALGEBROIDS LECTURE NOTES, FALL 2017 ECKHARD MEINRENKEN Abstract. These notes are very much under construction. They contain many errors, and the references are very incomplete. Apologies!

### THE KEEL MORI THEOREM VIA STACKS

THE KEEL MORI THEOREM VIA STACKS BRIAN CONRAD 1. Introduction Let X be an Artin stack (always assumed to have quasi-compact and separated diagonal over Spec Z; cf. [2, 1.3]). A coarse moduli space for

### Quotient Stacks. Jacob Gross. Lincoln College. 8 March 2018

Quotient Stacks Jacob Gross Lincoln College 8 March 2018 Abstract These are notes from a talk on quotient stacks presented at the Reading Group on Algebraic Stacks; meeting weekly in the Quillen Room of

### Representable presheaves

Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last

### 1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim

Reference: [BS] Bhatt, Scholze, The pro-étale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the pro-étale topos, and whose derived categories

### Parabolic sheaves, root stacks and the Kato-Nakayama space

Parabolic sheaves, root stacks and the Kato-Nakayama space Mattia Talpo UBC Vancouver February 2016 Outline Parabolic sheaves as sheaves on stacks of roots, and log geometry. Partly joint with A. Vistoli,

### Gauge Theory and Mirror Symmetry

Gauge Theory and Mirror Symmetry Constantin Teleman UC Berkeley ICM 2014, Seoul C. Teleman (Berkeley) Gauge theory, Mirror symmetry ICM Seoul, 2014 1 / 14 Character space for SO(3) and Toda foliation Support