Diagram Chasing in Abelian Categories
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The Left and Right Homotopy Relations We Recall That a Coproduct of Two
The left and right homotopy relations We recall that a coproduct of two objects A and B in a category C is an object A q B together with two maps in1 : A → A q B and in2 : B → A q B such that, for every pair of maps f : A → C and g : B → C, there exists a unique map f + g : A q B → C 0 such that f = (f + g) ◦ in1 and g = (f + g) ◦ in2. If both A q B and A q B 0 0 0 are coproducts of A and B, then the maps in1 + in2 : A q B → A q B and 0 in1 + in2 : A q B → A q B are isomorphisms and each others inverses. The map ∇ = id + id: A q A → A is called the fold map. Dually, a product of two objects A and B in a category C is an object A × B together with two maps pr1 : A × B → A and pr2 : A × B → B such that, for every pair of maps f : C → A and g : C → B, there exists a unique map (f, g): C → A × B 0 such that f = pr1 ◦(f, g) and g = pr2 ◦(f, g). If both A × B and A × B 0 are products of A and B, then the maps (pr1, pr2): A × B → A × B and 0 0 0 (pr1, pr2): A × B → A × B are isomorphisms and each others inverses. The map ∆ = (id, id): A → A × A is called the diagonal map. Definition Let C be a model category, and let f : A → B and g : A → B be two maps. -
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0. -
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. -
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. -
Math 395: Category Theory Northwestern University, Lecture Notes
Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere. -
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. -
Homotopy Commutative Diagrams and Their Realizations *
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 57 (1989) 5-24 5 North-Holland HOMOTOPY COMMUTATIVE DIAGRAMS AND THEIR REALIZATIONS * W.G. DWYER Department of Mathematics, University of Notre Dame, Notre Dame, IN465.56, U.S.A D.M. KAN Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. J.H. SMITH Department of Mathematics, Johns Hopkins University, Baltimore, MD21218, lJ.S.A Communicated by J.D. Stasheff Received 11 September 1986 Revised 21 September 1987 In this paper we describe an obstruction theory for the problem of taking a commutative diagram in the homotopy category of topological spaces and lifting it to an actual commutative diagram of spaces. This directly generalizes the work of G. Cooke on extending a homotopy action of a group G to a topological action of G. 1. Introduction 1.1. Summary. In [2], Cooke asked when a homotopy action of a group G on a CW-complex X is equivalent, in an appropriate sense, to a topological action of G on some homotopically equivalent space. He converted this problem into a lifting problem, which then gave rise to a sequence of obstructions, whose vanishing insured the existence of the desired topological action. These obstructions were ele- ments of the cohomology of G with local coefficients in the homotopy groups of the function space Xx. As a group is just a category with one object in which all maps are invertible, one can consider the corresponding problem for homotopy actions of an arbitrary small topological category D on a set {X,} of CW-complexes, indexed by the objects DE D. -
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]). -
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. -
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. -
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. -
14. Limits One of the More Interesting Notions of Category Theory, Is the Theory of Limits
14. Limits One of the more interesting notions of category theory, is the theory of limits. Definition 14.1. Let I be a category and let F : I −! C be a func- tor. A prelimit for F is an object L of C, together with morphisms fI : L −! F (I), for every object I of I, which are compatible in the fol- lowing sense: Given a morphism f : I −! J in I, the following diagram commutes f L I- F (I) F (f) - fJ ? F (J): The limit of F , denoted L = lim F is a prelimit L, which is uni- I versal amongst all prelimits in the following sense: Given any prelimit L0 there is a unique morphism g : L0 −! L, such that for every object I in I, the following diagram commutes g L0 - L fI f 0 - I ? F (I): Informally, then, if we think of a prelimit as being to the left of every object F (I), then the limit is the furthest prelimit to the right. Note that limits, if they exist at all, are unique, up to unique isomorphism, by the standard argument. Note also that there is a dual notion, the notion of colimits. In this case, F is a contravariant functor and all the arrows go the other way (informally, then, a prelimit is to the right of every object F (I) and a limit is any prelimit which is furthest to the left). Let us look at some special cases. First suppose we take for I the category with one object and one morphism.