DOCSLIB.ORG

Matemaattis-Luonnontieteellinen Matematiikan Ja Tilastotieteen Laitos Joni Leino on Mitchell's Embedding Theorem of Small Abel

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Matemaattis-Luonnontieteellinen Matematiikan Ja Tilastotieteen Laitos Joni Leino on Mitchell's Embedding Theorem of Small Abel HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI Tiedekunta/Osasto — Fakultet/Sektion — Faculty Laitos — Institution — Department Matemaattis-luonnontieteellinen Matematiikan ja tilastotieteen laitos Tekijä — Författare — Author Joni Leino Työn nimi — Arbetets titel — Title On Mitchell’s embedding theorem of small abelian categories and some of its corollaries Oppiaine — Läroämne — Subject Matematiikka Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Pro gradu -tutkielma Kesäkuu 2018 64 s. Tiivistelmä — Referat — Abstract Abelian categories provide an abstract generalization of the category of modules over a unitary ring. An embedding theorem by Mitchell shows that one can, whenever an abelian category is sufficiently small, find a unitary ring such that the given category may be embedded in the category of left modules over this ring. An interesting consequence of this theorem is that one can use it to generalize all diagrammatic lemmas (where the conditions and claims can be formulated by exactness and commutativity) true for all module categories to all abelian categories. The goal of this paper is to prove the embedding theorem, and then derive some of its corollaries. We start from the very basics by defining categories and their properties, and then we start con- structing the theory of abelian categories. After that, we prove several results concerning functors, "homomorphisms" of categories, such as the Yoneda lemma. Finally, we introduce the concept of a Grothendieck category, the properties of which will be used to prove the main theorem. The final chapter contains the tools in generalizing diagrammatic results, a weaker but more general version of the embedding theorem, and a way to assign topological spaces to abelian categories. The reader is assumed to know nothing more than what abelian groups and unitary rings are, except for the final theorem in the proof of which basic homotopy theory is applied. Avainsanat — Nyckelord — Keywords Category theory, abelian categories, homological algebra Säilytyspaikka — Förvaringsställe — Where deposited Kumpulan tiedekirjasto Muita tietoja — Övriga uppgifter — Additional information On Mitchell’s embedding theorem of small abelian categories and some of its corollaries Joni Leino Contents Introduction 2 1 Category theoretical preliminaries 4 2 Abelian categories 10 3 Module categories 26 4 On functors 30 5 On the Grothendieck property and injectivity 42 6 Two embedding theorems 49 7 Some corollaries of the embedding theorems 58 8 References and bibliography 63 1 Introduction On one hand, studying homological algebra involves a lot of category the- oretical constructions. On the other hand, some of its results, namely the ones involving exactness and commutativity in a diagram, can be generalized to categories under certain conditions. As it is not always the case that the objects in these categories have underlying sets, the methods familiar from homological algebra involving the element relation of set theory will not be of use in these generalizations, and this will make proving the results harder inside these categories. Luckily, a way to bypass such difficulties exists. It is easily motivated by a topological analogy. Manifolds, an important category of topological spaces, are approachable using open coverings called atlases. These atlases consist of so called chart neighbourhoods, or open sets homeomorphic to a euclidean space of some dimension. These atlases lead to two pleasant things. The first of these is that, due to the homeomorphism, whatever local properties (for example local compactness or local connectedness) hold in euclidean spaces, they also hold for all manifolds, and the second one is that the mathematician studying a manifold can transfer points from the possibly quite abstract manifold to a space much more familiar and easy to deal with by using these local homeomorphisms. Of course, this comes with the limitation that it need not be possible to consider the whole manifold at once like this, but instead it may be restricted to strict subsets. An interesting thing is that, although seemingly different, this same idea can be used while dealing with abelian categories, or a special type of cate- gories sharing a lot of properties with the category of abelian groups. This is based on the main result of this text, the embedding theorem by Mitchell, and several smaller results. On the way to Mitchell’s theorem, we will prove that any set of objects in an abelian category gives rise to a small abelian category, and this will serve as an analogue for the chart neighborhoods of points. We will then use the claim of the embedding theorem: for every small abelian category, one can always find a unitary ring such that the cat- egory can be embedded into the category of left modules over this ring, and these categories will be the "euclidean spaces" in this context. Applying the metatheorems, according to which diagrammatic lemmas hold in abelian categories if and only if they hold in the categories of abelian groups or these module categories, one obtains an analogue to the local inheritance of compactness and connectedness. Small categories will mean that the class of objects in the given category is a set instead of a proper class. Without spending time in set theoretic considerations, we will use the von Neumann-Bernays-Gödel axioms, so we 2 can speak of the category of sets, groups, topological spaces, and so on, but not of the category of all categories. Due to the simplicity of the examples in this text, it will be sufficient to divide classes into two cases, namely the ones with and the ones without a cardinality, and these will be enough to function as sets and proper classes, respectively. The requirement of smallness will act as a limiting factor, just like some manifolds need to be considered piecewise instead of being homeomorphic to euclidean spaces as whole. Mitchell’s theorem does not allow embedding an arbitrary abelian category at once, but "locally" using subcategories de- termined by choosing a set of objects. It is also possible to generalize the result, but this will also weaken it. In the final section, we will consider abelian categories equivalent to small abelian categories, and induce equiv- alences between the so called essentially small category and a subcategory of a module category. A related example will show that this indeed does generalize Mitchell’s theorem, but does not cover all abelian categories. The text, which is heavily based on Freyd’s book on abelian categories, will be divided into seven sections. The first two will construct the basic theory of general categories and then abelian categories. The third section will be used to prove that the category of left modules over a unitary ring is indeed an abelian category. In the fourth section, we will derive basic facts regarding functors, as these will then serve as tools needed later, and we will prove a sufficient condition for the desired kind of embeddings to exist. In the fifth section, we will develop more necessary tools to be used in the sixth section where the embedding theorem is finally proved. The seventh section will be on several consequences of the theorem by Mitchell, such as the generalization of diagrammatic lemmas, or the existence of a classifying space unique up to weak homotopy equivalence for suitable abelian categories. Finally, I would like to thank Marja Kankaanrinta and Pekka Pankka for their valuable feedback and help. 3 1 Category theoretical preliminaries The goal of this section is to lay down the basic category theoretical con- cepts we will be using later. We begin by defining categories, subcategories and opposite categories, and then we define several commonly encountered properties. Definition 1.1. A category C consists of (1) A class of objects obj C; (2) For any objects A; B 2 obj C, a class HomC(A; B) of morphisms A ! B; (3) A binary operation ◦ for morphisms such that: (i) g◦f 2 HomC(A; C) is defined iff f 2 HomC(A; B) and g 2 HomC(B; C) for some A; B; C 2 obj C; (ii) Whenever h◦(g◦f) is defined, so is (h◦g)◦f, and h◦(g◦f) = (h◦g)◦f; (iii) For every A; B 2 obj C, there are morphisms 1A; 1B called the iden- tities of A and B, respectively, such that 1B ◦ f = f = f ◦ 1A for every f 2 HomC(A; B). Lemma 1.2. For any category C, the identities are unique. 0 Proof. Let A be an object in a category C. If now 1A; 1A are identities of A, then 0 0 1A = 1A ◦ 1A = 1A. Definition 1.3. A subcategory C’ of a category C consists of (1) a subclass obj C’ of obj C; (2) for all A; B 2 obj C’, a subclass HomC’(A; B) of HomC(A; B); (3) the composition ◦ of C such that: (i) if A 2 obj C’, then 1A 2 HomC’(A; A); (ii) if f 2 HomC’(A; B) and g 2 HomC’(B; C), then g◦f 2 HomC’(A; C). If HomC’(A; B) = HomC(A; B) for all A; B 2 obj C’, then C’ is a full subcategory of C. We will give a short list of examples of categories now. The fifth example will be of great importance later when we discuss the abelian categories, an example of which it is. (i) The category Sets with sets as its objects, functions as its morphisms and the ordinary composition of functions as its composition. 4 (ii) The category Top with topological spaces as its objects, continuous maps as its morphisms and the ordinary composition. (iii) The category Haus with Hausdorff spaces as its objects is a subcat- egory of Top. (iv) The category Grp of groups with groups as its objects, group homo- morphisms as its morphisms and the ordinary composition of functions.
Load more

Recommended publications
  • Diagram Chasing in Abelian Categories
    Diagram Chasing in Abelian Categories Daniel Murfet October 5, 2006 In applications of the theory of homological algebra, results such as the Five Lemma are crucial. For abelian groups this result is proved by diagram chasing, a procedure not immediately available in a general abelian category. However, we can still prove the desired results by embedding our abelian category in the category of abelian groups. All of this material is taken from Mitchell’s book on category theory [Mit65]. Contents 1 Introduction 1 1.1 Desired results ...................................... 1 2 Walks in Abelian Categories 3 2.1 Diagram chasing ..................................... 6 1 Introduction For our conventions regarding categories the reader is directed to our Abelian Categories (AC) notes. In particular recall that an embedding is a faithful functor which takes distinct objects to distinct objects. Theorem 1. Any small abelian category A has an exact embedding into the category of abelian groups. Proof. See [Mit65] Chapter 4, Theorem 2.6. Lemma 2. Let A be an abelian category and S ⊆ A a nonempty set of objects. There is a full small abelian subcategory B of A containing S. Proof. See [Mit65] Chapter 4, Lemma 2.7. Combining results II 6.7 and II 7.1 of [Mit65] we have Lemma 3. Let A be an abelian category, T : A −→ Ab an exact embedding. Then T preserves and reflects monomorphisms, epimorphisms, commutative diagrams, limits and colimits of finite diagrams, and exact sequences. 1.1 Desired results In the category of abelian groups, diagram chasing arguments are usually used either to establish a property (such as surjectivity) of a certain morphism, or to construct a new morphism between known objects.
    [Show full text]
  • The Freyd-Mitchell Embedding Theorem States the Existence of a Ring R and an Exact Full Embedding a Ñ R-Mod, R-Mod Being the Category of Left Modules Over R
    The Freyd-Mitchell Embedding Theorem Arnold Tan Junhan Michaelmas 2018 Mini Projects: Homological Algebra arXiv:1901.08591v1 [math.CT] 23 Jan 2019 University of Oxford MFoCS Homological Algebra Contents 1 Abstract 1 2 Basics on abelian categories 1 3 Additives and representables 6 4 A special case of Freyd-Mitchell 10 5 Functor categories 12 6 Injective Envelopes 14 7 The Embedding Theorem 18 1 Abstract Given a small abelian category A, the Freyd-Mitchell embedding theorem states the existence of a ring R and an exact full embedding A Ñ R-Mod, R-Mod being the category of left modules over R. This theorem is useful as it allows one to prove general results about abelian categories within the context of R-modules. The goal of this report is to flesh out the proof of the embedding theorem. We shall follow closely the material and approach presented in Freyd (1964). This means we will encounter such concepts as projective generators, injective cogenerators, the Yoneda embedding, injective envelopes, Grothendieck categories, subcategories of mono objects and subcategories of absolutely pure objects. This approach is summarised as follows: • the functor category rA, Abs is abelian and has injective envelopes. • in fact, the same holds for the full subcategory LpAq of left-exact functors. • LpAqop has some nice properties: it is cocomplete and has a projective generator. • such a category embeds into R-Mod for some ring R. • in turn, A embeds into such a category. 2 Basics on abelian categories Fix some category C. Let us say that a monic A Ñ B is contained in another monic A1 Ñ B if there is a map A Ñ A1 making the diagram A B commute.
    [Show full text]
  • Diagram Chasing in Abelian Categories
    Diagram Chasing in Abelian Categories Toni Annala Contents 1 Overview 2 2 Abelian Categories 3 2.1 Denition and basic properties . .3 2.2 Subobjects and quotient objects . .6 2.3 The image and inverse image functors . 11 2.4 Exact sequences and diagram chasing . 16 1 Chapter 1 Overview This is a short note, intended only for personal use, where I x diagram chasing in general abelian categories. I didn't want to take the Freyd-Mitchell embedding theorem for granted, and I didn't like the style of the Freyd's book on the topic. Therefore I had to do something else. As this was intended only for personal use, and as I decided to include this to the application quite late, I haven't touched anything in chapter 2. Some vague references to Freyd's book are made in the passing, they mean the book Abelian Categories by Peter Freyd. How diagram chasing is xed then? The main idea is to chase subobjects instead of elements. The sections 2.1 and 2.2 contain many standard statements about abelian categories, proved perhaps in a nonstandard way. In section 2.3 we dene the image and inverse image functors, which let us transfer subobjects via a morphism of objects. The most important theorem in this section is probably 2.3.11, which states that for a subobject U of X, and a morphism f : X ! Y , we have ff −1U = U \ imf. Some other results are useful as well, for example 2.3.2, which says that the image functor associated to a monic morphism is injective.
    [Show full text]
  • Snake Lemma - Wikipedia, the Free Encyclopedia
    Snake lemma - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Snake_lemma Snake lemma From Wikipedia, the free encyclopedia The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms. Contents 1 Statement 2 Explanation of the name 3 Construction of the maps 4 Naturality 5 In popular culture 6 See also 7 References 8 External links Statement In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram: where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernels and cokernels of a, b, and c: Furthermore, if the morphism f is a monomorphism, then so is the morphism ker a → ker b, and if g' is an epimorphism, then so is coker b → coker c. Explanation of the name To see where the snake lemma gets its name, expand the diagram above as follows: 1 of 4 28/11/2012 01:58 Snake lemma - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Snake_lemma and then note that the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering snake. Construction of the maps The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity.
    [Show full text]
  • Weakly Exact Categories and the Snake Lemma 3
    WEAKLY EXACT CATEGORIES AND THE SNAKE LEMMA AMIR JAFARI Abstract. We generalize the notion of an exact category and introduce weakly exact categories. A proof of the snake lemma in this general setting is given. Some applications are given to illustrate how one can do homological algebra in a weakly exact category. Contents 1. Introduction 1 2. Weakly Exact Category 3 3. The Snake Lemma 4 4. The3 × 3 Lemma 6 5. Examples 8 6. Applications 10 7. Additive Weakly Exact Categories 12 References 14 1. Introduction Short and long exact sequences are among the most fundamental concepts in mathematics. The definition of an exact sequence is based on kernel and cokernel, which can be defined for any category with a zero object 0, i.e. an object that is both initial and final. In such a category, kernel and cokernel are equalizer and coequalizer of: f A // B . 0 arXiv:0901.2372v1 [math.CT] 16 Jan 2009 Kernel and cokernel do not need to exist in general, but by definition if they exist, they are unique up to a unique isomorphism. As is the case for any equalizer and coequalizer, kernel is monomorphism and cokernel is epimorphism. In an abelian category the converse is also true: any monomorphism is a kernel and any epimor- phism is a cokernel. In fact any morphism A → B factors as A → I → B with A → I a cokernel and I → B a kernel. This factorization property for a cate- gory with a zero object is so important that together with the existence of finite products, imply that the category is abelian (see 1.597 of [FS]).
    [Show full text]
  • A Primer on Homological Algebra
    A Primer on Homological Algebra Henry Y. Chan July 12, 2013 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section... Before telling you what a module is, you probably should know what a ring is... Definition 1.1. A ring is a set R with two operations + and ∗ and two identities 0 and 1 such that 1. (R; +; 0) is an abelian group. 2. (Associativity) (x ∗ y) ∗ z = x ∗ (y ∗ z), for all x; y; z 2 R. 3. (Multiplicative Identity) x ∗ 1 = 1 ∗ x = x, for all x 2 R. 4. (Left Distributivity) x ∗ (y + z) = x ∗ y + x ∗ z, for all x; y; z 2 R. 5. (Right Distributivity) (x + y) ∗ z = x ∗ z + y ∗ z, for all x; y; z 2 R. A ring is commutative if ∗ is commutative. Note that multiplicative inverses do not have to exist! Example 1.2. 1. Z; Q; R; C with the standard addition, the standard multiplication, 0, and 1. 2. Z=nZ with addition and multiplication modulo n, 0, and 1. 3. R [x], the set of all polynomials with coefficients in R, where R is a ring, with the standard polynomial addition and multiplication. 4. Mn×n, the set of all n-by-n matrices, with matrix addition and multiplication, 0n, and In. For convenience, from now on we only consider commutative rings. Definition 1.3. Assume (R; +R; ∗R; 0R; 1R) is a commutative ring. A R-module is an abelian group (M; +M ; 0M ) with an operation · : R × M ! M such that 1 1.
    [Show full text]
  • Homological Algebra Lecture 3
    Homological Algebra Lecture 3 Richard Crew Summer 2021 Richard Crew Homological Algebra Lecture 3 Summer 2021 1 / 21 Exactness in Abelian Categories Suppose A is an abelian category. We now try to formulate what it means for a sequence g X −!f Y −! Z (1) to be exact. First of all we require that gf = 0. When it is, f factors through the kernel of g: X ! Ker(g) ! Y But the composite Ker(f ) ! X ! Ker(g) ! Y is 0 and Ker(g) ! Y is a monomorphism, so Ker(f ) ! X ! Ker(g) is 0. Therefore f factors Richard Crew Homological Algebra Lecture 3 Summer 2021 2 / 21 X ! Im(f ) ! Ker(g) ! Y : Since Im(f ) ! Y is a monomorphism, Im(f ) ! Ker(g) is a monomorphism as well. Definition A sequence g X −!f Y −! Z is exact if gf = 0 and the canonical monomorphism Im(f ) ! Ker(g) is an isomorphism. Richard Crew Homological Algebra Lecture 3 Summer 2021 3 / 21 We need the following lemma for the next proposition: Lemma Suppose f : X ! Z and g : Y ! Z are morphisms in a category C which has fibered products, and suppose i : Z ! Z 0 is a monomorphism. Set f 0 = if : X ! Z 0 and g 0 = ig : Y ! Z 0. The canonical morphism X ×Z Y ! X ×Z 0 Y is an isomorphism. Proof: The canonical morphism comes from applying the universal property of the fibered product to the diagram p2 X ×Z Y / Y p1 g f g 0 X / Z i f 0 , Z 0 Richard Crew Homological Algebra Lecture 3 Summer 2021 4 / 21 The universal property of X ×Z Y is that the set of morphisms T ! X ×Z Y is in a functorial bijection with the set of pairs of morphisms a : T ! X and b : T ! Y such that fa = gb.
    [Show full text]
  • The Snail Lemma
    Theory and Applications of Categories, Vol. 31, No. 19, 2016, pp. 484{501. THE SNAIL LEMMA ENRICO M. VITALE Resum´ e.´ The classical snake lemma produces a six terms exact sequence starting from a commutative square with one of the edge being a regular epimorphism. We establish a new diagram lemma, that we call snail lemma, removing such a condition. We also show that the snail lemma subsumes the snake lemma and we give an interpretation of the snail lemma in terms of strong homotopy kernels. Our results hold in any pointed regular protomodular category. 1. Introduction One of the basic diagram lemmas in homological algebra is the snake lemma (also called kernel-cokernel lemma). In an abelian category, it can be stated in the following way : from the commutative diagram kf f Ker(f) / A / B (1) K(α) α β Ker(f0) / A0 / B0 kf0 f0 and under the assumption that f is an epimorphism, it is possible to get an exact sequence Ker(K(α)) / Ker(α) / Ker(β) / Cok(K(α)) / Cok(α) / Cok(β) If we replace \epimorphism" with \regular epimorphism" and if α; β and K(α) are proper morphisms (Definition 2.2) the snake lemma holds also in several important non-abelian categories, like groups, crossed modules, Lie algebras, not necessarily unitary rings. Despite its very clear formulation, the snake lemma is somehow asymmetric because of the hypothesis on the morphism f: The aim of this paper is to study what happens if we remove this condition. What we prove is that we can get a six-term exact sequence (the snail sequence) starting from any commutative diagram like f A / B (2) α β A0 / B0 f0 Received by the editors 2014-07-05 and, in final form, 2016-06-07.
    [Show full text]
  • Lectures on Homological Algebra
    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 of Sciences, Beijing 100049, China Email: [email protected] Contents 1 Categories and functors 1 1.1 Categories . 1 1.2 Functors . 3 1.3 Universal constructions . 7 1.4 Adjunction . 11 1.5 Additive categories . 16 1.6 Abelian categories . 21 1.7 Projective and injective objects . 30 1.8 Projective and injective modules . 32 2 Derived categories and derived functors 41 2.1 Complexes . 41 2.2 Homotopy category, triangulated categories . 47 2.3 Localization of categories . 56 2.4 Derived categories . 61 2.5 Extensions . 70 2.6 Derived functors . 78 2.7 Double complexes, derived Hom ..................... 83 2.8 Flat modules, derived tensor product . 88 2.9 Homology and cohomology of groups . 98 2.10 Spectral objects and spectral sequences . 101 Summary of properties of rings and modules 105 iii iv CONTENTS Chapter 1 Categories and functors Very rough historical sketch Homological algebra studies derived functors between • categories of modules (since the 1940s, culminating in the 1956 book by Cartan and Eilenberg [CE]); • abelian categories (Grothendieck’s 1957 T¯ohokuarticle [G]); and • derived categories (Verdier’s 1963 notes [V1] and 1967 thesis of doctorat d’État [V2] following ideas of Grothendieck). 1.1 Categories Definition 1.1.1. A category C consists of a set of objects Ob(C), a set of morphisms Hom(X, Y ) for every pair of objects (X, Y ) of C, and a composition law, namely a map Hom(X, Y ) × Hom(Y, Z) → Hom(X, Z), denoted by (f, g) 7→ gf (or g ◦ f), for every triple of objects (X, Y, Z) of C.
    [Show full text]
  • An Axiomatic Survey of Diagram Lemmas for Non-Abelian Group-Like Structures
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 370 (2012) 387–401 Contents lists available at SciVerse ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra An axiomatic survey of diagram lemmas for non-abelian group-like structures Zurab Janelidze 1 Mathematics Division, Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa article info abstract Article history: It is well known that diagram lemmas for abelian groups (and Received 20 December 2011 more generally in abelian categories) used in algebraic topology, Availableonline21August2012 can be suitably extended to “non-abelian” structures such as Communicated by Michel Van den Bergh groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five Keywords: Exact sequence lemma this is well known, and in the present paper we establish Snake lemma this for the snake lemma and the 3 × 3lemma,which,when 3 × 3 lemma suitably formulated, turn out to be equivalent to each other for Diagram chasing all (pointed) algebraic structures, and also in general categories Ideal determined variety of a special type. In particular, we show that among varieties Normal category of universal algebras, they are satisfied precisely in the so-called Subtractive category (pointed) ideal determined varieties. Subtraction © 2012 Elsevier Inc. All rights reserved. Introduction In this paper we explore a connection between classical diagram lemmas of homological algebra and properties encountered in categorical and universal algebra, which arise there in the axiomatic study of “group-like” structures such as groups, rings, modules, algebras, topological groups, loops, etc.
    [Show full text]
  • Homological Algebra in Abelian Categories
    Homological Algebra in Abelian Categories Rylee Lyman March 3, 2019 1 Preliminaries Recall that a category is a collection of objects with arrows between them. We use the word \collection" advisedly, making no claim that we are dealing with sets. We require each object A to have an associated arrow A 1A A called the identity. We also require composition of arrows to be defined whenever it makes sense, and we require this composition to be associative. Furthermore, given f : A ! B, f1A = f = 1Bf We follow the treatment in [Mac71] to which the curious reader is referred. In category theory, a dual statement is one in which all the arrows have been reversed. For example, an arrow m is monic if it is \right-cancellative," i.e. mf = mg =) f = g The dual notion is an epi. An isomorphism is an arrow that has a two-sided inverse. Exercise 1. Isomorphisms are both monic and epi The converse does not always hold. A standard example is the inclusion Z ! Q in the category of rings. An object z is initial if, given another object t, there is a unique arrow z A: Dually, an object is terminal if there is a unique arrow to the object from any given object. A zero object is one that is both initial and terminal. Exercise 2. If z and w are both initial, terminal, or zero objects, there exists a unique isomorphism z ! w. We will denote our zero objects as 0, as well as any arrow that factors through the zero object.
    [Show full text]
  • Problem Set 1 Due: September 10, 2015
    Math 7260 Fall 2015 Homological Algebra P. Achar Problem Set 1 Due: September 10, 2015 1. (Do not hand in) Weibel, Exercises 1.1.1, 1.1.2, 1.1.5. 2. (Do not hand in) Show that any isomorphism of chain complexes is a quasi-isomorphism. 3. (Do not hand in) Weibel, Exercise 1.1.3. Also, give an example of a quasi-isomorphism that is not an isomorphism. 4. Weibel, Exercise 1.1.4. 5. Let R = C[t]=(t2). Regarding R as a module over itself, consider the R-module homomorphism f : R ! R given by f(r) = tr. Next, let N be the R-module R=(t) =∼ C[t]=(t). Consider the following two chain complexes (with nonzero terms only in degrees 0 and −1): f C = (···! 0 ! R ! R ! 0 !··· ); D = (···! 0 ! N !0 N ! 0 !··· ): ∼ Prove that Hn(C) = Hn(D) for all n, but that there is no quasi-isomorphism u : C ! D or u : D ! C. 6.A filtered abelian group is an abelian group M together with a fixed collection of subgroups labelled by integers: · · · ⊂ F−1M ⊂ F0M ⊂ F1M ⊂ · · · ⊂ M: A filtered homomorphism f : F•M ! F•N is a homomorphism of abelian groups f : M ! N such that f(FnM) ⊂ FnN for all n. (a) (Do not hand in) Show that the category FAb of filtered abelian groups is additive. (b) Show that FAb has kernels and cokernels. Also, describe in concrete terms the monic and epic morphisms in this category. (c) Show that FAb is not abelian. (Hint: Find a morphism that is both epic and monic, but not an isomorphism.) 7.
    [Show full text]