# Representable presheaves

Size: px
Start display at page:

Transcription

1 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 C (, X). I d like to explain the formula F = (Hom C (,X) F) C F Hom C (, X), both very useful and tautological, which says that any presheaf is a it of representable presheaves. This comes from the Yoneda lemma. Presheaves Definition. A presheaf of sets on C is a contravariant functor F : C Set. A morphism of presheaves F G is a natural transformation of functors, i.e. a collection of morphisms θ X : F(X) G(X) for every object X Ob(C) such that for every morphism X f Y the diagram below is commutative: Y F(Y) θ Y G(Y) f X F( f ) F(X) θ X G( f ) G(X) Following SGA 4, I will denote by Ĉ the category of presheaves on C. The most basic example is the following. Let X be a topological space and let Open(X) be the category of open subsets of X partially ordered by inclusion: { {U V}, if U V Hom Open(X) (U, V) :=, otherwise. A presheaf on the category Open(X) is the same thing as a presheaf on X in the usual sense: for each V U we have the corresponding restriction map and by functoriality, res UU = id F(U) and res UV : F(U) F(V), res UW = res VW res UV for W V U. 1

2 Size issues There is a problem with the definition of Ĉ. Recall that one says that C is a small category if the objects Ob(C) form a set. For instance, the category Open(X) is small. The majority of interesting categories are not. Even the categories of finite sets, finite dimensional vector spaces, etc. are not small (even one-element sets { } do not form a set, since can be any set here). However, these categories are equivalent to small categories. It is very common to say small category when in fact the category in question is equivalent to some small category. In the definition of a category, one normally assumes that the morphisms Hom C (X, Y) between two fixed objects X, Y Ob(C) form a set. Some authors say in this case that the category is locally small. In general, natural transformations F G between two functors F, G : C D do not form a category, unless C is small. In particular, if C is small, then the presheaves on C form a category in the usual sense. To resolve the arising problems, Grothendieck in the appendix to exposé I of SGA 4 developed the theory of universes. Curiously, the text says Nous reproduisons ici, avec son accord, des papiers secrets de N. Bourbaki, but most likely it was written by Grothendieck himself, and it has never appeared in Bourbaki s books. You can check exposé I of SGA 4 to see the statements of some basic results (as the Yoneda lemma) with universes, but I won t go into details related to set theory. Here are a couple of links: Aise Johan de Jong (the maintainer of the Stacks Project) about set theory: (check the comments!) Zhen Lin Low, Universes for category theory : for those who are interested in logic. Limits and its Let s revise some definitions and basic properties, mainly to fix the notation and terminology. Let I be a (small) category and let F : I C be a functor. A limit of F is an object lim I F Ob(C) together with a collection of morphisms {lim I F F(i)} such that every morphism i j in I induces a commutative diagram lim I F F(i) F(j) Moreover, we ask for the following universal property: for each other object X Ob(C) with a collection of morphisms {X F(i)} that commute with the morphisms in I in the above sense (one says that X is a cone with respect to F) there exists a unique morphism X lim I F that gives commutative diagrams lim I F X! F(i) F(j) 2

3 A it of F is an object I F Ob(C) together with a collection of morphisms {F(i) I F} that give for each morphism i j in I a commutative diagram F(i) F(j) I F And one asks that I F satisfies the universal property of its F(i) F(j) I F When a limit or it exists, it is unique up to isomorphism. The terminology varies: X! limit inverse limit projective limit it direct limit inductive limite Actually, the terms inverse limit and direct limit, as well as the common notation lim and lim, should be reserved for the case where the indexing category enjoys some special properties (when it is a directed set, or in general a filtered category). That s why I prefer writing lim and. Instead of lim I F and I F one often writes lim F(i) and F(i), and I will use this notation, even though it s slightly misleading: limits and its are indexed by both objects and morphisms in I. Of course, the basic examples of limits and its are the following: A terminal object in C is a limit over the empty category I = and an initial object is a it over I =. Products X i, fibered products X Z Y (pullbacks) and equalizers Eq(X Y) are particular cases of limits. Coproducts X i, fibered coproducts X Z Y (pushouts) and coequalizers Coeq(X Y) are particular cases of its. Many important categories are complete and cocomplete, meaning that limits and its exist over any small category I. Among those is the category of sets Set, and many categories of sets with additional structure: topological spaces Top, groups Grp, rings Ring, modules R-Mod, and so on. Limits and its in these cases have an explicit description: lim I I F = {(x i ) F(i) F( f )(x i ) = x j for each f : i j}. F = F(i) /, 3

4 where the equivalence relation is generated by F(i) x i F( f )(x i ) F(j) for each f : i j. One can make the (obvious) adjustments for the case of sets with additional structure. For instance, in the case of topological spaces, F(i) / should be the space with the quotient topology. Here is an important fact that may be verified from the definitions: Observation. The contravariant Hom C (, X) converts its to limits: Hom C ( F(i), X) = lim Hom C (F(i), X). Limits and its are used in some canonical constructions, so the following is also important: Another observation. (1) If for a fixed small category I and a category C all limits and its exist, then one can choose their particular representatives (in general, limits and its are defined only up to an isomorphism), and one has functors lim I, I : Fun(I, C) C. Indeed, for any natural transformation η : F G, the limit lim I F gives a cone for G, and hence a canonical morphism lim I F lim I G. Similarly, the it I G gives a cocone for F and a canonical morphism I F I G. F(i) X F(j) G(i) η i F(i) G(j) η j F(j) η i G(i) η j G(j) X (2) Every functor between indexing categories φ : J I induces canonical morphisms φ : lim I F lim J F φ, φ : F φ F, J I and these constructions are functorial in the sense that (φ ψ) = φ ψ and (φ ψ) = ψ φ. Limits and its of presheaves For every small category C the category of presheaves Ĉ is complete and cocomplete. That is, for every functor I Ĉ, i F i 4

5 on a small category I, there exist its limit and it lim F i and F i. That s because we can see our functor as I C Set, i.e. calculate the limit and it pointwise : (lim F i )(X) = lim F i (X) and ( and the category Set is complete and cocomplete. In particular, F i )(X) = F i (X), For products of presheaves For coproducts of presheaves (F G)(X) = F(X) G(X). (F G)(X) = F(X) G(X). The category of presheaves has a terminal object, which is the presheaf F such that F(X) = { } is a one-element set a terminal object in the category of sets. The category of presheaves has an initial object, which is the presheaf F such that F(X) = for each X, where is an initial object in the category of sets. Very often (for instance, to develop sheaf cohomology) one considers not presheaves of sets but presheaves of, say, abelian groups. But we can use the fact that the products and coproducts are calculated pointwise and say that a presheaf of abelian groups F : C Ab is an abelian group object in the category of presheaves of sets Ĉ. The words abelian group object mean that for a presheaf F we specify morphisms + : F F F (addition), : F F (subtraction), and 0: { } F (zero) that fit into commutative diagrams that express that + is associative and commutative (i.e. x + (y + z) = x + (y + z) and x + y = y + x), that 0 is the neutral element with respect to + (i.e. x + 0 = x), and that is inverse to + (i.e. x + ( x) = 0). For instance, the associativity is expressed by F F F + id F F and the commutativity is expressed by id + F F + F + F F + F where τ is the canonical morphism defined by τ F F + p 1 τ = p 2 and p 2 τ = p 1, and p 1, p 2 : A A A are the canonical projections. 5

6 In general, we have that A presheaf of groups abelian groups rings modules..... is a group abelian group ring module..... object in the category Ĉ. So all the constructions for Ĉ may be generalized to presheaves with any algebraic structure defined by commutative diagrams involving terminal and initial objects, products and coproducts, and so on. One should be careful though: this does not work, for instance, for presheaves of topological spaces F : C Top, because one can t describe topologies via such diagrams. Yoneda lemma: the importance of representable presheaves The most basic and important example of presheaves is the presheaf represented by a fixed object X Ob(C). Namely, it is the functor Hom C (, X) : C Set. A morphism Y f Z induces a natural morphism Hom C (Z, X) Hom C (Y, X) which maps Z g X to Y g f X. Now the famous Yoneda lemma says that For every X Ob(C) and every presheaf F : C Set there is a natural bijection HomĈ(Hom C (, X), F) = F(X). The construction goes as follows. For a morphism of presheaves θ Y : Hom C (Y, X) F(Y) the only obvious element of F(X) that one can produce is θ X (id X ). In the other direction, starting from an element x F(X), one can define a morphism of presheaves θy x : Hom C(Y, X) F(Y) by sending Y f X to F( f )(x) F(Y), which is again the only obvious possibility. It only remains to check that this gives us a natural bijection. In particular, for two representable presheaves Hom C (, X) and Hom C (, Y) the above bijection is given by HomĈ(Hom C (, X), Hom C (, Y)) = Hom C (X, Y). This means that we can think of C as of a full subcategory of Ĉ: C Ĉ, X Hom C (, X). This is known as the Yoneda embedding. By abuse of notation, in certain diagrams in SGA 4, X in fact denotes the corresponding presheaf Hom C (, X). 6

7 A little bit of history The Yoneda lemma and Yoneda embedding are named after the Japanese mathematician Nobuo Yoneda ( ). The Yoneda lemma was stated and popularized by Saunders Mac Lane after he met Yoneda. In texts like SGA, the Yoneda lemma is used extensively without ever mentioning Yoneda. The origin of the lemma is an article on homological algebra Nobuo Yoneda. On the Homology Theory of Modules. Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics Vol. 7 No. 2, p MR In this article Yoneda introduced what is known now as the Yoneda Ext, which is a construction of Ext n (M, N) that does not use derived functors and coincides with the usual definition Ext n (M, N) := R n Hom(M, )(N) when the base category has enough injective objects. Then Yoneda mentions something which corresponds to the following curious result. Let F : A Ab be an additive functor defined on an abelian category A (Yoneda uses the term A-module ). Then for each X Ob(A) one has the functor Hom A (X, ) : A Ab represented by X and a natural isomorphism of abelian groups Similarly, for representable contravariant functors Nat(Hom A (X, ), F) = F(X). (L 0 ) Nat(Hom A (, X), F) = F(X). (R 0 ) This is an abelian version of what we know nowadays as Yoneda lemma. Under the assumption that A has enough injective/projective objects, this generalizes to isomorphisms and Nat(Ext n A (X, ), F) = L n F(X) (L n ) Nat(Ext n A (, X), F) = R n F(X). (R n ) 7

8 This is an interesting interpretation of derived functors! One possible modern reference for this result is A Course in Homological Algebra by Hilton and Stammbach. I took some time to track Yoneda s paper in the library. On pages he mentions the formulas (L 0 ) and (R 0 ), and then uses (L n ) and (R n ) as the definitions for L n F and R n X. If I understand correctly, then Yoneda shows that these definitions coincide with the usual ones. Here s the relevant part of his text (Yoneda uses the term satelite, and for a right exact functor F its satellite S n F is naturally isomorphic to the left derived functor L n F, while for a left exact functor F its satellite S n F is naturally isomorphic to R n F). Every presheaf is a it of representable presheaves Observation. 1) Every presheaf may be seen in a canonical way as a it of the representable presheaves. Namely, we have F = (Hom C (,X) F) C F Hom C (, X), where C F is the category where the objects are presheaf morphisms Hom C(, X) F, and morphisms in C F are morphisms f : X Y in C that induce commutative diagrams of 8

9 presheaves Hom C (, X) f Hom C (, Y). F 2) In particular, for every pair of presheaves F and G there is a natural bijection HomĈ(F, G) = lim (Hom C (,X) F) G(X). C F First of all, we see that part 2) is a consequence of 1), the fact that HomĈ(, G) converts its to limits, and the Yoneda lemma: HomĈ(F, G) = HomĈ( (Hom C (,X) F) Hom C (, X), G) C F = lim (Hom C (,X) F) C F HomĈ(Hom C (, X), G) = lim (Hom C (,X) F) C F G(X). Part 1) seems to be more interesting, and in fact it s a very important principle, but it turns out to be something tautological as well, and it may be easily seen from the construction of the category C F. By the Yoneda lemma, each morphism of presheaves Hom C (, X) F corresponds to an element x F(X), and the category C F has the following equivalent description: the objects are pairs (X Ob(C), x F(X)), the morphisms are f : X Y in C such that F( f )(y) = x. Because of that the category C F is also known as the category of elements of F. Then, by the definition of C F, the presheaf F gives a cocone with respect to (Hom C (,X) F) C F Hom C(, X), i.e. there are commutative diagrams Hom C (, X) f Hom C (, Y). F Then, for any other presheaf G that gives a cocone Hom C (, X) f Hom C (, Y) F G 9

10 there is a unique morphism of presheaves F G making the diagram commute. Indeed, by the equivalent description of C F, the commutative diagrams as above mean that for each X Ob(C) and x F(X) one has x G(X), such that for every morphism f : X Y in C F( f )(y) = x and G( f )(y ) = x. (*) The required morphism of presheaves is given by τ X : F(X) G(X), x x, and its naturality in X is the condition (*). This shows the formula 1). Here are some observations: The Yoneda embedding C Ĉ preserves limits: for any functor F : I C we have Hom C (, lim F(i)) = lim Hom C (, F(i)) (in a sense, this is dual to the fact that Hom C ( F(i), ) = lim Hom C (F(i), )). So a limit of representable presheaves is also a representable presheaf. The above observation is about its. If F = Hom C (, X) is a representable presheaf, then C F has a terminal object, which is the identity morphism Hom C (, X) Hom C (, X): Hom C (, Y)! Hom C (, X) Hom C (, X) id and because of that C Hom C(,X) Hom C(, Y) is isomorphic to Hom C (, X). Let X be a topological space and let Open(X) be the corresponding category of open subsets. A representable presheaf in this case is something boring, as Hom Open(X) (U, V) is either a one element set or an empty set. The category Open(X) F is formed by pairs (V, x) where V X is an open subset and x F(V). A morphism (V, x) (W, y) is an inclusion V W such that res WV (y) = x. The it (V,x) Hom Open(X) (U, V) Open(X) F is given by a disjoint union / / V X x F(V) Hom Open(X) (U, V) = V X x F(V) U V { } = V X U V F(V) where in the las expression identifies x F(V) with y F(W) if res WV (y) = x. So the it formula gives in this case F(U) = F(V), V X U V which is something obvious. /, 10

11 Example: direct and inverse image of presheaves (SGA 4) Let s conclude with one particular example of the formula F = (X,x) C F Hom C(, X): the inverse image of a presheaf. A functor between two small categories u : C D induces the inverse image functor u : D Ĉ that sends any presheaf G on D to the presheaf u G := G u on C: (u G)(X) := G(u(X)). In fact, there is another, less obvious functor u! that goes in the opposite direction: u! : Ĉ D, that is left adjoint to u, meaning that there is a natural bijection Hom D (u!f, G) = HomĈ(F, u G) for every presheaf F on C and G on D. In order to discover this functor, we may use the following common trick: first we see what happens if F is a representable presheaf, and then generalize the construction to an arbitrary presheaf using the wonderful it formula. If F = Hom C (, X) for some X Ob(C), then by the Yoneda lemma, the natural bijection as above corresponds to Hom D (u! Hom C (, X), G) = G(u(X)). Again by the Yoneda lemma, we see that u! Hom C (, X) should be isomorphic to the representable presheaf Hom D (, u(x)). Now any presheaf F is naturally isomorphic to F = and the following construction comes to mind. (X,x) Hom C (, X), C F Definition. Let u : C D be a functor between two small categories. Then the direct image of a presheaf F on C is the presheaf on D given by u! F := (X,x) Hom D (, u(x)). C F We see that this defines a covariant functor u! : Ĉ D. With this definition, we get natural isomorphisms Hom D (u!f, G) = Hom D ( (X,x) Hom D (, u(x)), G) C F = lim (X,x) C F Hom D (Hom D(, u(x)), G) = lim (X,x) C F G(u(X)) = lim (X,x) C F HomĈ(Hom C (, X), u G) = HomĈ( (X,x) C F Hom C (, X), u G) = HomĈ(F, u G), 11

12 just as required. Here s a trivial example. Let be the category with one object and one arrow id and let C be another small category. A functor u : C is defined by one object X Ob(C) such that X, so specifying a presheaf on is equivalent to specifying a set S. Let s see what u! S is. We have to consider the category S with objects being the elements of S and no arrows (except for the identity for each object). Such a category without nontrivial arrows is called a discrete category, and a it over a discrete category is isomorphic to the coproduct indexed by its objects. Therefore, u! S = Hom C (, X). S To give a more interesting example, every continuous map between topological spaces induces a functor which goes in the opposite direction: f : X Y u := f 1 : Open(Y) Open(X). Now if F is a presheaf on X, then the presheaf u F is called the direct image of F and it is denoted by f F: ( f F)(U) := (u F)(X) = F( f 1 (U)). For a presheaf G on Y, the presheaf u! G on X is called the inverse image and it is denoted by f G. This presheaf is given by the it over the category Open(Y) G which boils down to ( f G)(U) = V Y open U f 1 (V) G(V) = V Y open V f (U) G(V). This is the well-known formula for the inverse image. This is simply G( f (U)) if f (U) is an open subset, but in general f (U) is not open, so one has to take the it. 12

### 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

### 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

### 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

### COMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY

COMMUTATIVE ALGEBRA LECTURE 1: SOME CATEGORY THEORY VIVEK SHENDE A ring is a set R with two binary operations, an addition + and a multiplication. Always there should be an identity 0 for addition, an

### 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

### 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

### 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..................

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### Category Theory (UMV/TK/07)

P. J. Šafárik University, Faculty of Science, Košice Project 2005/NP1-051 11230100466 Basic information Extent: 2 hrs lecture/1 hrs seminar per week. Assessment: Written tests during the semester, written

### 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

### 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

### 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

### 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

### 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

### EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY

EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1. Categories 1.1. Generalities. I ve tried to be as consistent as possible. In particular, throughout the text below, categories will be denoted by capital

### 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)

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?

### GALOIS CATEGORIES MELISSA LYNN

GALOIS CATEGORIES MELISSA LYNN Abstract. In abstract algebra, we considered finite Galois extensions of fields with their Galois groups. Here, we noticed a correspondence between the intermediate fields

### SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES

SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there

### 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

### 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

### 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

### 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

### Categories and Modules

Categories and odules Takahiro Kato arch 2, 205 BSTRCT odules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract

### MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to

### 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

### 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

### 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

### 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

### 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

### 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

### CATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.

CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists

### 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,

### A TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor

A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it

### 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

The Adjoint Functor Theorem. Kevin Buzzard February 7, 2012 Last modified 17/06/2002. 1 Introduction. The existence of free groups is immediate from the Adjoint Functor Theorem. Whilst I ve long believed

### 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

### Lectures on Homological Algebra. Weizhe Zheng

Lectures on Homological Algebra Weizhe Zheng Morningside Center of Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100190, China University of the Chinese Academy

### Math 762 Spring h Y (Z 1 ) (1) h X (Z 2 ) h X (Z 1 ) Φ Z 1. h Y (Z 2 )

Math 762 Spring 2016 Homework 3 Drew Armstrong Problem 1. Yoneda s Lemma. We have seen that the bifunctor Hom C (, ) : C C Set is analogous to a bilinear form on a K-vector space, : V V K. Recall that

### 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

### arxiv: v2 [math.ra] 14 Sep 2016

ON THE NEGATIVE-ONE SHIFT FUNCTOR FOR FI-MODULES arxiv:1603.07974v2 [math.ra] 14 Sep 2016 WEE LIANG GAN Abstract. We show that the negative-one shift functor S 1 on the category of FI-modules is a left

### Derived 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

### 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

### A NOTE ON ENRICHED CATEGORIES

U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 4, 2010 ISSN 1223-7027 A NOTE ON ENRICHED CATEGORIES Adriana Balan 1 În această lucrare se arată că o categorie simetrică monoidală închisă bicompletă V cu biproduse

### 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

### sset(x, Y ) n = sset(x [n], Y ).

1. Symmetric monoidal categories and enriched categories In practice, categories come in nature with more structure than just sets of morphisms. This extra structure is central to all of category theory,

### 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

### 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

### 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

### 58 CHAPTER 2. COMPUTATIONAL METHODS

58 CHAPTER 2. COMPUTATIONAL METHODS 23 Hom and Lim We will now develop more properties of the tensor product: its relationship to homomorphisms and to direct limits. The tensor product arose in our study

### 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

### Amalgamable diagram shapes

Amalgamable diagram shapes Ruiyuan hen Abstract A category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram

### AN INTRODUCTION TO PERVERSE SHEAVES. Antoine Chambert-Loir

AN INTRODUCTION TO PERVERSE SHEAVES Antoine Chambert-Loir Antoine Chambert-Loir Université Paris-Diderot. E-mail : Antoine.Chambert-Loir@math.univ-paris-diderot.fr Version of February 15, 2018, 10h48 The

### 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

### LECTURE 3: RELATIVE SINGULAR HOMOLOGY

LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following

### 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

### Universal Properties

A categorical look at undergraduate algebra and topology Julia Goedecke Newnham College 24 February 2017, Archimedeans Julia Goedecke (Newnham) 24/02/2017 1 / 30 1 Maths is Abstraction : more abstraction

### 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

### PERVERSE SHEAVES. Contents

PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a

### 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

### 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

### = Spec(Rf ) R p. 2 R m f gives a section a of the stalk bundle over X f as follows. For any [p] 2 X f (f /2 p), let a([p]) = ([p], a) wherea =

LECTURES ON ALGEBRAIC GEOMETRY MATH 202A 41 5. Affine schemes The definition of an a ne scheme is very abstract. We will bring it down to Earth. However, we will concentrate on the definitions. Properties

### arxiv:math/ v1 [math.at] 6 Oct 2004

arxiv:math/0410162v1 [math.at] 6 Oct 2004 EQUIVARIANT UNIVERSAL COEFFICIENT AND KÜNNETH SPECTRAL SEQUENCES L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL Abstract. We construct hyper-homology spectral sequences

### arxiv: v1 [math.ct] 28 Oct 2017

BARELY LOCALLY PRESENTABLE CATEGORIES arxiv:1710.10476v1 [math.ct] 28 Oct 2017 L. POSITSELSKI AND J. ROSICKÝ Abstract. We introduce a new class of categories generalizing locally presentable ones. The

### 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

### 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

### MTH 428/528. Introduction to Topology II. Elements of Algebraic Topology. Bernard Badzioch

MTH 428/528 Introduction to Topology II Elements of Algebraic Topology Bernard Badzioch 2016.12.12 Contents 1. Some Motivation.......................................................... 3 2. Categories

### Lectures on Galois Theory. Some steps of generalizations

= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame

### 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

### Adjoints, naturality, exactness, small Yoneda lemma. 1. Hom(X, ) is left exact

(April 8, 2010) Adjoints, naturality, exactness, small Yoneda lemma Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ The best way to understand or remember left-exactness or right-exactness

### 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

### SOME EXERCISES. This is not an assignment, though some exercises on this list might become part of an assignment. Class 2

SOME EXERCISES This is not an assignment, though some exercises on this list might become part of an assignment. Class 2 (1) Let C be a category and let X C. Prove that the assignment Y C(Y, X) is a functor

### CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

### Cartesian Closed Topological Categories and Tensor Products

Cartesian Closed Topological Categories and Tensor Products Gavin J. Seal October 21, 2003 Abstract The projective tensor product in a category of topological R-modules (where R is a topological ring)

### Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0

1. Show that if B, C are flat and Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0 is exact, then A is flat as well. Show that the same holds for projectivity, but not for injectivity.

### 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

### An extension of Dwyer s and Palmieri s proof of Ohkawa s theorem on Bousfield classes

An extension of Dwyer s and Palmieri s proof of Ohkawa s theorem on Bousfield classes Greg Stevenson Abstract We give a proof that in any compactly generated triangulated category with a biexact coproduct

### Boolean Algebras, Boolean Rings and Stone s Representation Theorem

Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to

### Relative Affine Schemes

Relative Affine Schemes Daniel Murfet October 5, 2006 The fundamental Spec( ) construction associates an affine scheme to any ring. In this note we study the relative version of this construction, which

### 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

### Exact Categories in Functional Analysis

Script Exact Categories in Functional Analysis Leonhard Frerick and Dennis Sieg June 22, 2010 ii To Susanne Dierolf. iii iv Contents 1 Basic Notions 1 1.1 Categories............................. 1 1.2

### Recall: a mapping f : A B C (where A, B, C are R-modules) is called R-bilinear if f is R-linear in each coordinate, i.e.,

23 Hom and We will do homological algebra over a fixed commutative ring R. There are several good reasons to take a commutative ring: Left R-modules are the same as right R-modules. [In general a right

### IND-COHERENT SHEAVES AND SERRE DUALITY II. 1. Introduction

IND-COHERENT SHEAVES AND SERRE DUALITY II 1. Introduction Let X be a smooth projective variety over a field k of dimension n. Let V be a vector bundle on X. In this case, we have an isomorphism H i (X,

### 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,

### Derived Categories. Mistuo Hoshino

Derived Categories Mistuo Hoshino Contents 01. Cochain complexes 02. Mapping cones 03. Homotopy categories 04. Quasi-isomorphisms 05. Mapping cylinders 06. Triangulated categories 07. Épaisse subcategories

### LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

### Elements of Category Theory

Elements of Category Theory Robin Cockett Department of Computer Science University of Calgary Alberta, Canada robin@cpsc.ucalgary.ca Estonia, Feb. 2010 Functors and natural transformations Adjoints and

### INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES

INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES 1. Why correspondences? This part introduces one of the two main innovations in this book the (, 2)-category of correspondences as a way to encode

### Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)

http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product

### 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