# Elementary (ha-ha) Aspects of Topos Theory

Size: px
Start display at page:

Transcription

1 Elementary (ha-ha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces Presheaves on spaces Digression on pointless topology Sheaves on spaces Sheafification Operations on sheaves Sheaves on categories Grothendieck topologies Sheaves on a site Limits and colimits Giraud s Theorem Exercises 8 This is an introductory talk on topos theory from a geometer s perspective 1. From this viewpoint topos theory can be said to be a generalisation of topology - we ll see that we can put a topology not just on a set, but on a category. We ll start with the definition of a sheaf of sets on a topological space, and then extend that definition to get sheaves of sets on categories. Then we ll look at some nice properties of topoi. In this talk, all categories will be locally small. Otherwise, we will mostly ignore any size issues. 1 Sheaves on topological spaces 1.1 Presheaves on spaces Definition Let X be a topological space. A presheaf of sets F on X is the data of: 1 However, I won t really talk about any geometry! 1

2 i) A set F(U) for every open set U X ii) A map ρ UV : F(V ) F(U) for every pair of open sets U V such that: i) ρ UU = id F(U) for all open U ii) If U V W is a triple of open sets, then ρ UV ρ V W = ρ UW We call ρ UV the restriction map from V to U. If s F(V ) then we ll often write s U for ρ UV (s). The two conditions we require are intuitively just some obvious facts about restrictions. We often refer to the elements s F(U) as the sections of F over U. This language is supposed to evoke the idea that the set F(U) lives above U in some sense. Sections of F over X are the global sections. An alternate notation, common in algebraic geometry, is to write Γ(U, F) for F(U). This is often used when U is considered fixed and F is allowed to vary. Example The constant presheaf on X with value A is the presheaf that assigns the set A to every open set U X. The restriction maps are all identity maps. Example If Y is a space, then we can define a presheaf F on X by letting F(U) be the collection of continuous functions from U to Y. The restriction maps ρ UV are the genuine restriction maps f f U. Definition A morphism of presheaves of sets F G on X is an assignment of a map φ U : F(U) G(U) to every open set U X such that the maps φ U are compatible with the restriction maps. Since this is a category theory workshop, we want to try and rephrase the definition of a presheaf in a more categorical manner. This isn t too hard: Definition Given a space X, the category Open(X) of opens in X is the category whose objects are the open sets U X and whose morphisms are the inclusions. Note that a continuous map f:x Y induces a functor Open(Y ) Open(X) by sending an open set U to its preimage under f. Proposition Let X be a space. Then a presheaf of sets on X is the same thing as a functor Open(X) op Set. A morphism of presheaves is a natural transformation between such functors. Definition The category of presheaves on X is the category Psh(X) := Fun(Open(X) op, Set). Definition A presheaf on X is representable if it is isomorphic to a presheaf of the form h U = Hom(, U) for some open subset U of X. 2

3 1.2 Digression on pointless topology A natural question to ask is: how does the category Open(X) compare to the space X? In particular, can we recover X from Open(X)? Definition A topological space X is sober if every irreducible closed subset V is the closure of exactly one point ζ of X. The point ζ is called the generic point of V. The category Sob of sober spaces has objects the sober spaces and morphisms the continuous maps. Example A Hausdorff space is sober, since the only irreducible subsets are the points and every singleton is closed. A sober space is Kolmogorov (any two points are topologically distinguishable). Proposition If X is sober, then the category Open(X) determines the space X up to homeomorphism. So if we only care about sober spaces X, then we might as well deal with the categories Open(X) up to isomorphism. In fact, Open(X) is a certain kind of lattice called a frame. The above Proposition tells us that we can recover a sober space X from its associated frame. Since the functor assigning a space X to its associated frame is contravariant, we really should consider the opposite category of the category of frames: Definition The category Loc of locales is defined to be the opposite of the category of frames. In particular we can define a presheaf on a locale L as a functor L op Set. Locales are pointless versions of topological spaces, since they just remember the structure of the lattice of open sets: Theorem There is an embedding Sob Loc, defined by sending a sober space to its associated locale. Remark This is a weak form of Stone duality. In fact, there is an adjunction between Top and Loc restricting to an equivalence between the full subcategories Sob and sloc, the category of spatial locales. The unit of this adjunction is the soberification map Top Sob and the counit is the pointification map Loc sloc. 1.3 Sheaves on spaces Presheaves are all well and good, but some presheaves have undesirable properties. For example we d like presheaves to be local : if {U i : i I} is an open cover of an open subset U, then we d like the sections of a presheaf on U to be determined by the sections on the U i. Definition A presheaf F on a space X is separated if, for any open set U of X and any open cover {U i : i I} of U, if s, t are two sections of F(U) with s Ui = t Ui for all i I, then s = t. 3

4 We d also like to be able to glue sections together. If {U i : i I} is a cover of U then we ll often write U ij for U i U j. Definition A separated presheaf F on a space X is a sheaf if, for any open set U of X and any open cover {U i : i I} of U, for any collection s i F(U i ) of sections on the U i such that s i Uij = s j Uij, there exists a section s of F over U, called the gluing of the s i, such that s Ui = s i. Since F is separated this gluing will be unique. Remark It s worth commenting on the sections of a sheaf F over the empty set. Note that is an open subset of X that is covered by the empty covering. Hence the collection of sections is an element of the empty product in Set, which is a final object of Set, a.k.a. a singleton. So F( ) is always a one-element set. The condition that a presheaf is a sheaf can be phrased simply as compatible sections can be glued together uniquely. A sheaf can hence be used to track locally defined data on a space. Example Let f : Y X be a continuous map. The sheaf of sections of f is the sheaf Γ(Y/X) whose value at U is the set of maps s : U Y with f s = id U. The restriction maps are just restriction of functions. This is the prototypical example of a sheaf. Example If X is a smooth manifold then it comes equipped with many sheaves: the sheaf of n-times differentiable functions OX n, sheaf of smooth functions O X (also called the structure sheaf), and the cotangent sheaves Ω p X of differential p-forms on X. If X is a complex manifold then it also comes equipped with a sheaf of holomorphic functions. Example Consider the presheaf F on R sending an open subset U to the collection of bounded real-valued functions on U. Then F is a separated presheaf but not a sheaf: locally bounded functions do not necessarily glue together to give a globally bounded function. Example Let X be a space, x X a point, and A a set. The skyscraper sheaf skyx A is the sheaf whose value on an open set U is { skyx A A if x U (U) = else Again, we d like a categorical characterisation of sheaves: Proposition Let F Psh(X). Then F is a sheaf if and only if, for any open set U of X and any open cover {U i : i I} of U, the diagram F(U) where ρ = i ρ U iu, α = i,j ρ U iju i ρ F(U i ) i α β F(U ij ) i,j and β = i,j ρ U iju j, is an equaliser. A morphism of sheaves is a morphism of presheaves. The collection of sheaves on a space X forms a category Sh(X). 4

5 1.4 Sheafification We d like a method for turning a presheaf into a sheaf. Luckily this is always possible: Proposition The forgetful functor Sh(X) Psh(X) admits a left adjoint, which we call the sheafification. There are many ways of proving this. One way is to use the éspace étalé construction. Example The constant sheaf with value A is the sheafification of the constant presheaf. Proposition Sheafification preserves finite limits. Remark Since it is a left adjoint, sheafification preserves all colimits. 1.5 Operations on sheaves We can push and pull sheaves around by maps: Definition Let f : X Y be a map and let F be a sheaf on X. The direct image sheaf f F is the sheaf on Y with f F(U) = F(f 1 (U)). This operation is functorial. Inverse image sheaves are more complicated, since the image of an open set under a continuous function need not be open. Definition Let f : X Y be a map and let F be a sheaf on Y. The inverse image sheaf f 1 F is the sheaf on X defined as the sheafification of the presheaf U colim V f(u) F(V ). This operation is also functorial. 2 Sheaves on categories 2.1 Grothendieck topologies We have a categorical description of what it means to be a presheaf on a category: Definition Let C be any category. The category Psh(C) of presheaves on C is the functor category Fun(C op, Set). However, the notion of a sheaf required us to have the concept of a topology. We re going to need to put a topology on a category. Grothendieck s realisation was that the correct thing to axiomatise is the notion of covering family: Definition Let C be a category. A Grothendieck topology on C is an assignment of, to every U C, a collection of sets of maps {U α U}, called the covering families of U, such that the following conditions are satisfied: 5

6 i) (Base change) If {U α U} is a covering family and V U is any arrow, then the fibre products U α U V exist, and the collection of projections {U α U V V } is a covering family. ii) (Locality) If {U α U} is a covering family, and for each index α we have a covering family {U αβ U α }, then the composite family {U αβ U} is a covering. Informally, this states that a cover of a cover is again a cover. iii) (Isomorphisms) If V U is an isomorphism, then the one-element family {V U} is a covering family. A category together with a choice of Grothendieck topology is called a site. Strictly this is the notion of a Grothendieck pretopology; in this talk we can do without the more general notion of a Grothendieck topology. Grothendieck pretopologies require C to have certain fibre products, whereas Grothendieck topologies do not. Moreover, different pretopologies may give the same topology. Example If X is a topological space, then Open(X) admits the structure of a site where the covering families of U are the open covers of U. Example The category Top admits a Grothendieck topology, the global classical topology, where a covering of U is a jointly surjective collection of open injections {U α U}. Top also admits another Grothendieck topology (the global étale topology) where we require the covers to be local homeomorphisms, not just open injections. Example Any category admits the indiscrete topology, where the covering families are the isomorphisms. 2.2 Sheaves on a site If U i U and U j U are members of a covering family of U, then we ll denote the fibre product U i U U j by U ij. This agrees with our earlier notation for topological spaces. We can now easily state what it means to be a sheaf: Definition Let C be a site. Then a presheaf F on C is a sheaf if and only if, for every object U of C and every covering family {U i U}, the diagram F(U) F(U i ) i F(U ij ) is an equaliser. A morphism of sheaves is a morphism of presheaves. If C is a site, then Sh(C) is the category of sheaves of sets on C. Definition A topos 2 is a category equivalent to the category of sheaves of sets on a site. 2 Every time I say topos in this talk I will mean Grothendieck topos. i,j 6

7 Example The category Set is a topos; it can be identified with the category of sheaves of sets on a point. Proposition Let C be a category. The forgetful functor Sh(C) Psh(C) admits a left adjoint, which we call the sheafification. Remark This is harder to prove than in the case of sheaves of sets on spaces. Definition The canonical topology on a category is the largest topology for which all representable functors are sheaves. A subcanonical topology is any subtopology of the canonical topology. We can also characterise subcanonical topologies as the topologies for which all representable functors are sheaves. 2.3 Limits and colimits If C is any category, then Psh(C) is both complete and cocomplete. Moreover, there is an easy way to compute limits and colimits: Proposition [Limits and colimits are computed pointwise] Let i F i be a diagram of presheaves in a category C. Let X be an object of C. Then we have isomorphisms (lim F i )(X) = lim(f i (X)) (colim F i )(X) = colim(f i (X)) Moreover, if every F i is a sheaf, then so is lim F i. Remark The assertion that the limit of sheaves is a sheaf is easy to see, since an equaliser is a limit and limits commute with limits. Colimits of sheaves are not computed pointwise! We have to sheafify, since a colimit of sheaves is not necessarily a sheaf. Recall that the Yoneda Lemma tells us that we have a fully faithful embedding (the Yoneda embedding) C Psh(C) defined by the assignment U h U. Since Psh(C) is a cocomplete category, we can regard the Yoneda embedding as a cocompletion. In fact, it is the free cocompletion: Proposition The Yoneda embedding C Psh(C) is universal among functors from C into cocomplete categories, i.e. any functor F : C D to a cocomplete catgeory factors through Psh(C), and the map Psh(C) D preserves colimits. We have also seen the co-yoneda lemma, which tells us that every presheaf is a canonical colimit of representables. 2.4 Giraud s Theorem We d like an intrinsic definition of what it means to be a topos. First we need to set up some terminology. 7

8 Definition An equivalence relation R on an object X of a category C is a map R X X such that for all objects Y, the image of the map Hom(Y, R) Hom(Y, X) Hom(Y, X) is an equivalence relation on the set Hom(Y, X). Example Let X Y be a morphism. The limit, if it exists, of the diagram X Y X is an equivalence relation. This limit is the kernel pair of the morphism X Y. Definition Let R be an equivalence relation on an object X. Let X/R be the coequaliser of the diagram R X X, called the quotient of the relation. Say that R is effective if the induced map R X X/R X is an isomorphism. Example Let G be a group and H a normal subgroup. Then the quotient group G/H is isomorphic to the quotient of the relation G H G G, where the map sends (g, h) to (g, gh). Definition The following statements about a category C are referred to as Giraud s axioms: i) C has a set of generators. ii) C has all colimits, and colimits commute with fibre products. iii) Sums are disjoint, i.e. X X Y Y is the initial object of C for all objects X, Y of C. iv) Every equivalence relation in C is effective. Example A topos satisfies Giraud s axioms. In fact more can be said: Theorem (Giraud). Let C be a category. The following are equivalent: i) C is a topos. ii) C satisfies Giraud s axioms. iii) There is a small category D and an inclusion C Psh(D) that admits a left adjoint which preserves finite limits. 3 Exercises Examples of sheaves i) Give an example of a presheaf that is not separated. ii) Give an example of a separated presheaf that is not a sheaf. iii) ([3]) Let U be a subset of C and let F(U) be the set of homolomorphic functions f on U such that df dz = 1/z. Show that F is a sheaf under restriction of functions. 8

9 Sheaves on some small spaces i) Let X be the space {0, 1} where we declare the subsets, {0}, and {0, 1} to be open (this space is known as the Sierpiński space). Describe explicitly the presheaves on X. Describe explicitly the sheaves on X. ii) Same question, but where X now has the discrete topology. iii) Same question, but where X now is any set with the indiscrete topology. Stalks The stalk of a sheaf F on X at the point x X is defined to be the set F x := lim F(U), where the direct limit is indexed over all open sets U containing U x. i) Let x be a point of X and let j : {x} X be the inclusion map. Show that the stalk F x is the same as the inverse image sheaf j 1 F. ii) Show that a morphism of sheaves is an epimorphism (resp. monomorphism, isomorphism) if and only if the induced map on the stalks is an epimorphism (resp. monomorphism, isomorphism). iii) Compute the stalks of the sheaf from Examples of sheaves, iii) iv) Compute the stalks of a skyscraper sheaf. Limits and colimits i) Prove Proposition ii) Give an example of a topological space X and a direct system of sheaves F i on X such that the limit presheaf lim i F i is not a sheaf. References [1] Saunders MacLane, Ieke Moerdijk, Sheaves in geometry and logic: A first introduction to topos theory, Springer, 1994 [2] Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory, available at [3] George Kempf, Algebraic Varieties, Cambridge University Press,

### 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

### LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain

### A Grothendieck site is a small category C equipped with a Grothendieck topology T. A Grothendieck topology T consists of a collection of subfunctors

Contents 5 Grothendieck topologies 1 6 Exactness properties 10 7 Geometric morphisms 17 8 Points and Boolean localization 22 5 Grothendieck topologies A Grothendieck site is a small category C equipped

### SOME OPERATIONS ON SHEAVES

SOME OPERATIONS ON SHEAVES R. VIRK Contents 1. Pushforward 1 2. Pullback 3 3. The adjunction (f 1, f ) 4 4. Support of a sheaf 5 5. Extension by zero 5 6. The adjunction (j!, j ) 6 7. Sections with support

### which is a group homomorphism, such that if W V U, then

4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV

### Modules over a Ringed Space

Modules over a Ringed Space Daniel Murfet October 5, 2006 In these notes we collect some useful facts about sheaves of modules on a ringed space that are either left as exercises in [Har77] or omitted

### 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

### 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 3

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 3 RAVI VAKIL CONTENTS 1. Kernels, cokernels, and exact sequences: A brief introduction to abelian categories 1 2. Sheaves 7 3. Motivating example: The sheaf of differentiable

### 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

### 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

### Some glances at topos theory. Francis Borceux

Some glances at topos theory Francis Borceux Como, 2018 2 Francis Borceux francis.borceux@uclouvain.be Contents 1 Localic toposes 7 1.1 Sheaves on a topological space.................... 7 1.2 Sheaves

### 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

### 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,

### Direct Limits. Mathematics 683, Fall 2013

Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday

### 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

### Category theory and set theory: algebraic set theory as an example of their interaction

Category theory and set theory: algebraic set theory as an example of their interaction Brice Halimi May 30, 2014 My talk will be devoted to an example of positive interaction between (ZFC-style) set theory

### University of Oxford, Michaelis November 16, Categorical Semantics and Topos Theory Homotopy type theor

Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011 November 16, 2011 References Johnstone, P.T.: Sketches of an Elephant. A Topos-Theory Compendium.

### 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

### 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Some constructions using universal properties, old and new 1 2. Adjoint functors 3 3. Sheaves 6 Last day: What is algebraic geometry? Crash

### 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

### 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

### 3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection

3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given

### TOPICS IN ALGEBRA COURSE NOTES AUTUMN Contents. Preface Notations and Conventions

TOPICS IN ALGEBRA COURSE NOTES AUTUMN 2003 ROBERT E. KOTTWITZ WRITTEN UP BY BRIAN D. SMITHLING Preface Notations and Conventions Contents ii ii 1. Grothendieck Topologies and Sheaves 1 1.1. A Motivating

### Exercises from Vakil

Exercises from Vakil Ian Coley February 6, 2015 2 Sheaves 2.1 Motivating example: The sheaf of differentiable functions Exercise 2.1.A. Show that the only maximal ideal of the germ of differentiable functions

### 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

### Section Higher Direct Images of Sheaves

Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will

### Constructible Derived Category

Constructible Derived Category Dongkwan Kim September 29, 2015 1 Category of Sheaves In this talk we mainly deal with sheaves of C-vector spaces. For a topological space X, we denote by Sh(X) the abelian

### Topos-theoretic background

opos-theoretic background Olivia Caramello IHÉS September 22, 2014 Contents 1 Introduction 2 2 erminology and notation 3 3 Grothendieck toposes 3 3.1 he notion of site............................ 3 3.2

### Category Theory. Travis Dirle. December 12, 2017

Category Theory 2 Category Theory Travis Dirle December 12, 2017 2 Contents 1 Categories 1 2 Construction on Categories 7 3 Universals and Limits 11 4 Adjoints 23 5 Limits 31 6 Generators and Projectives

### An introduction to locally finitely presentable categories

An introduction to locally finitely presentable categories MARU SARAZOLA A document born out of my attempt to understand the notion of locally finitely presentable category, and my annoyance at constantly

### 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

### A QUICK NOTE ON ÉTALE STACKS

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

### Review of category theory

Review of category theory Proseminar on stable homotopy theory, University of Pittsburgh Friday 17 th January 2014 Friday 24 th January 2014 Clive Newstead Abstract This talk will be a review of the fundamentals

### C2.7: CATEGORY THEORY

C2.7: CATEGORY THEORY PAVEL SAFRONOV WITH MINOR UPDATES 2019 BY FRANCES KIRWAN Contents Introduction 2 Literature 3 1. Basic definitions 3 1.1. Categories 3 1.2. Set-theoretic issues 4 1.3. Functors 5

### 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

### 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)

### IndCoh Seminar: Ind-coherent sheaves I

IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### Topological aspects of restriction categories

Calgary 2006, Topological aspects of restriction categories, June 1, 2006 p. 1/22 Topological aspects of restriction categories Robin Cockett robin@cpsc.ucalgary.ca University of Calgary Calgary 2006,

### Manifolds, sheaves, and cohomology

Manifolds, sheaves, and cohomology Torsten Wedhorn These are the lecture notes of my 3rd year Bachelor lecture in the winter semester 2013/14 in Paderborn. This manuscript differs from the lecture: It

### 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.

### arxiv: v2 [math.ag] 28 Jul 2016

HIGHER ANALYTIC STACKS AND GAGA THEOREMS MAURO PORTA AND TONY YUE YU arxiv:1412.5166v2 [math.ag] 28 Jul 2016 Abstract. We develop the foundations of higher geometric stacks in complex analytic geometry

### Assume the left square is a pushout. Then the right square is a pushout if and only if the big rectangle is.

COMMUTATIVE ALGERA LECTURE 2: MORE CATEGORY THEORY VIVEK SHENDE Last time we learned about Yoneda s lemma, and various universal constructions initial and final objects, products and coproducts (which

### PART I. Abstract algebraic categories

PART I Abstract algebraic categories It should be observed first that the whole concept of category is essentially an auxiliary one; our basic concepts are those of a functor and a natural transformation.

### Chapter 1 Sheaf theory

This version: 28/02/2014 Chapter 1 Sheaf theory The theory of sheaves has come to play a central rôle in the theories of several complex variables and holomorphic differential geometry. The theory is also

### SMA. Grothendieck topologies and schemes

SMA Grothendieck topologies and schemes Rafael GUGLIELMETTI Semester project Supervised by Prof. Eva BAYER FLUCKIGER Assistant: Valéry MAHÉ April 27, 2012 2 CONTENTS 3 Contents 1 Prerequisites 5 1.1 Fibred

### 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

### Algebraic Geometry

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

### Modules over a Scheme

Modules over a Scheme Daniel Murfet October 5, 2006 In these notes we collect various facts about quasi-coherent sheaves on a scheme. Nearly all of the material is trivial or can be found in [Gro60]. These

### 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

### Categories and functors

Lecture 1 Categories and functors Definition 1.1 A category A consists of a collection ob(a) (whose elements are called the objects of A) for each A, B ob(a), a collection A(A, B) (whose elements are called

### Compactness in Toposes

Algant Master Thesis Compactness in Toposes Candidate: Mauro Mantegazza Advisor: Dr. Jaap van Oosten Coadvisors: Prof. Sandra Mantovani Prof. Ronald van Luijk Università degli Studi di Milano Universiteit

### 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

### DESCENT THEORY (JOE RABINOFF S EXPOSITION)

DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,

### 1. Introduction. Let C be a Waldhausen category (the precise definition

K-THEORY OF WLDHUSEN CTEGORY S SYMMETRIC SPECTRUM MITY BOYRCHENKO bstract. If C is a Waldhausen category (i.e., a category with cofibrations and weak equivalences ), it is known that one can define its

### A Giraud-type Theorem for Model Topoi

Algant Master Thesis A Giraud-type Theorem for Model Topoi Candidate: Marco Vergura Advisor: Dr. Matan Prasma Coadvisor: Prof. Bas Edixhoven Università degli Studi di Padova Universiteit Leiden Academic

### 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

### Derived categories, perverse sheaves and intermediate extension functor

Derived categories, perverse sheaves and intermediate extension functor Riccardo Grandi July 26, 2013 Contents 1 Derived categories 1 2 The category of sheaves 5 3 t-structures 7 4 Perverse sheaves 8 1

### Fuzzy sets and presheaves

Fuzzy sets and presheaves J.F. Jardine Department of Mathematics University of Western Ontario London, Ontario, Canada jardine@uwo.ca November 27, 2018 Abstract This note presents a presheaf theoretic

### Generalized Topological Covering Systems on. Quantum Events Structures

Generalized Topological Covering Systems on Quantum Events Structures ELIAS ZAFIRIS University of Athens Department of Mathematics Panepistimiopolis, 15784 Athens Greece Abstract Homologous operational

### A CATEGORICAL INTRODUCTION TO SHEAVES

A CATEGORICAL INTRODUCTION TO SHEAVES DAPING WENG Abstract. Sheaf is a very useful notion when defining and computing many different cohomology theories over topological spaces. There are several ways

Adjunctions! Everywhere! Carnegie Mellon University Thursday 19 th September 2013 Clive Newstead Abstract What do free groups, existential quantifiers and Stone-Čech compactifications all have in common?

### On the characterization of geometric logic

Utrecht University Faculty of Beta Sciences Mathematical institute Master of Science Thesis On the characterization of geometric logic by Ralph Langendam (3117618) Supervisor: dr. J. van Oosten Utrecht,

### Algebraic Geometry I Lectures 14 and 15

Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal

### 0.1 Spec of a monoid

These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.

### Lectures on algebraic stacks

Rend. Mat. Appl. (7). Volume 38, (2017), 1 169 RENDICONTI DI MATEMATICA E DELLE SUE APPLICAZIONI Lectures on algebraic stacks Alberto Canonaco Abstract. These lectures give a detailed and almost self-contained

### Basic Modern Algebraic Geometry

Version of December 13, 1999. Final version for M 321. Audun Holme Basic Modern Algebraic Geometry Introduction to Grothendieck s Theory of Schemes Basic Modern Algebraic Geometry Introduction to Grothendieck

### 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,

### 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

### Math 256A: Algebraic Geometry

Math 256A: Algebraic Geometry Martin Olsson Contents 1 Introduction 1 2 Category theory 3 3 Presheaves and sheaves 5 4 Schemes 10 4.1 he underlying set of a scheme.......................................

### PART II.1. IND-COHERENT SHEAVES ON SCHEMES

PART II.1. IND-COHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Ind-coherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. t-structure 3 2. The direct image functor 4 2.1. Direct image

### Journal 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

### Category Theory. Categories. Definition.

Category Theory Category theory is a general mathematical theory of structures, systems of structures and relationships between systems of structures. It provides a unifying and economic mathematical modeling

### 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

### Adjunctions, the Stone-Čech compactification, the compact-open topology, the theorems of Ascoli and Arzela

Adjunctions, the Stone-Čech compactification, the compact-open topology, the theorems of Ascoli and Arzela John Terilla Fall 2014 Contents 1 Adjoint functors 2 2 Example: Product-Hom adjunction in Set

### 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

### 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

### Locally G-ringed spaces and rigid spaces

18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Rigid analytic spaces (at last!) We are now ready to talk about rigid analytic spaces in earnest. I ll give the

### DERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisse-étale and the flat-fppf sites

DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisse-étale and the flat-fppf sites 1 4. Derived categories of quasi-coherent modules 5

### APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP

APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP In this appendix we review some basic facts about étale cohomology, give the definition of the (cohomological) Brauer group, and discuss

### Topos Theory. Lectures 21 and 22: Classifying toposes. Olivia Caramello. Topos Theory. Olivia Caramello. The notion of classifying topos

Lectures 21 and 22: toposes of 2 / 30 Toposes as mathematical universes of Recall that every Grothendieck topos E is an elementary topos. Thus, given the fact that arbitrary colimits exist in E, we can

### Convenient 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

### Geometry 2: Manifolds and sheaves

Rules:Exam problems would be similar to ones marked with! sign. It is recommended to solve all unmarked and!-problems or to find the solution online. It s better to do it in order starting from the beginning,

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### 1 Motivation. If X is a topological space and x X a point, then the fundamental group is defined as. the set of (pointed) morphisms from the circle

References are: [Szamuely] Galois Groups and Fundamental Groups [SGA1] Grothendieck, et al. Revêtements étales et groupe fondamental [Stacks project] The Stacks Project, https://stacks.math.columbia. edu/

### Etale cohomology of fields by Johan M. Commelin, December 5, 2013

Etale cohomology of fields by Johan M. Commelin, December 5, 2013 Etale cohomology The canonical topology on a Grothendieck topos Let E be a Grothendieck topos. The canonical topology T on E is given in

### Postulated colimits and left exactness of Kan-extensions

Postulated colimits and left exactness of Kan-extensions Anders Kock If A is a small category and E a Grothendieck topos, the Kan extension LanF of a flat functor F : A E along any functor A D preserves

### Math 535a Homework 5

Math 535a Homework 5 Due Monday, March 20, 2017 by 5 pm Please remember to write down your name on your assignment. 1. Let (E, π E ) and (F, π F ) be (smooth) vector bundles over a common base M. A vector

### Algebra and Topology

Algebra and Topology Course at Paris VI University, 2007/2008 1 Pierre Schapira http://www.math.jussieu.fr/ schapira/lectnotes schapira@math.jussieu.fr 1/9/2011, v2 1 To the students: the material covered

### An Introduction to Spectral Sequences

An Introduction to Spectral Sequences Matt Booth December 4, 2016 This is the second half of a joint talk with Tim Weelinck. Tim introduced the concept of spectral sequences, and did some informal computations,

### Exercises on chapter 0

Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that

### 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,

### PART II.2. THE!-PULLBACK AND BASE CHANGE

PART II.2. THE!-PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2.

### Topos Theory. Lectures 17-20: The interpretation of logic in categories. Olivia Caramello. Topos Theory. Olivia Caramello.

logic s Lectures 17-20: logic in 2 / 40 logic s Interpreting first-order logic in In Logic, first-order s are a wide class of formal s used for talking about structures of any kind (where the restriction

### 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