Weakly Exact Categories and the Snake Lemma 3
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Homological Algebra
Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of -
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. -
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience. -
Derived Functors and Homological Dimension (Pdf)
DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic. -
Lecture 15. De Rham Cohomology
Lecture 15. de Rham cohomology In this lecture we will show how differential forms can be used to define topo- logical invariants of manifolds. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and Cechˇ cohomology. 15.1 Cocycles and coboundaries Let us first note some applications of Stokes’ theorem: Let ω be a k-form on a differentiable manifold M.For any oriented k-dimensional compact sub- manifold Σ of M, this gives us a real number by integration: " ω : Σ → ω. Σ (Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i =0, 1. Then d(F∗ω)isa (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], and Stokes’ theorem gives " " " d(F∗ω)= ω − ω. Σ×[0,1] Σ1 Σ1 In particular, if dω =0,then d(F∗ω)=F∗(dω)=0, and we deduce that ω = ω. Σ1 Σ0 This says that k-forms with exterior derivative zero give a well-defined functional on homotopy classes of compact oriented k-dimensional submani- folds of M. We know some examples of k-forms with exterior derivative zero, namely those of the form ω = dη for some (k − 1)-form η. But Stokes’ theorem then gives that Σ ω = Σ dη =0,sointhese cases the functional we defined on homotopy classes of submanifolds is trivial. -
On the Cohomology of the Finite Special Linear Groups, I
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 54, 216-238 (1978) On the Cohomology of the Finite Special Linear Groups, I GREGORY W. BELL* Fort Lewis College, Durango, Colorado 81301 Communicated by Graham Higman Received April 1, 1977 Let K be a finite field. We are interested in determining the cohomology groups of degree 0, 1, and 2 of various KSL,+,(K) modules. Our work is directed to the goal of determining the second degree cohomology of S&+,(K) acting on the exterior powers of its standard I + l-dimensional module. However, our solution of this problem will involve the study of many cohomology groups of degree 0 and 1 as well. These groups are of independent interest and they are useful in many other cohomological calculations involving SL,+#) and other Chevalley groups [ 11. Various aspects of this problem or similar problems have been considered by many authors, including [l, 5, 7-12, 14-19, 211. There have been many techni- ques used in studing this problem. Some are ad hoc and rely upon particular information concerning the groups in question. Others are more general, but still place restrictions, such as restrictions on the characteristic of the field, the order of the field, or the order of the Galois group of the field. Our approach is general and shows how all of these problems may be considered in a unified. way. In fact, the same method may be applied quite generally to other Chevalley groups acting on certain of their modules [l]. -
A Few Points in Topos Theory
A few points in topos theory Sam Zoghaib∗ Abstract This paper deals with two problems in topos theory; the construction of finite pseudo-limits and pseudo-colimits in appropriate sub-2-categories of the 2-category of toposes, and the definition and construction of the fundamental groupoid of a topos, in the context of the Galois theory of coverings; we will take results on the fundamental group of étale coverings in [1] as a starting example for the latter. We work in the more general context of bounded toposes over Set (instead of starting with an effec- tive descent morphism of schemes). Questions regarding the existence of limits and colimits of diagram of toposes arise while studying this prob- lem, but their general relevance makes it worth to study them separately. We expose mainly known constructions, but give some new insight on the assumptions and work out an explicit description of a functor in a coequalizer diagram which was as far as the author is aware unknown, which we believe can be generalised. This is essentially an overview of study and research conducted at dpmms, University of Cambridge, Great Britain, between March and Au- gust 2006, under the supervision of Martin Hyland. Contents 1 Introduction 2 2 General knowledge 3 3 On (co)limits of toposes 6 3.1 The construction of finite limits in BTop/S ............ 7 3.2 The construction of finite colimits in BTop/S ........... 9 4 The fundamental groupoid of a topos 12 4.1 The fundamental group of an atomic topos with a point . 13 4.2 The fundamental groupoid of an unpointed locally connected topos 15 5 Conclusion and future work 17 References 17 ∗e-mail: [email protected] 1 1 Introduction Toposes were first conceived ([2]) as kinds of “generalised spaces” which could serve as frameworks for cohomology theories; that is, mapping topological or geometrical invariants with an algebraic structure to topological spaces. -
Math 231Br: Algebraic Topology
algebraic topology Lectures delivered by Michael Hopkins Notes by Eva Belmont and Akhil Mathew Spring 2011, Harvard fLast updated August 14, 2012g Contents Lecture 1 January 24, 2010 x1 Introduction 6 x2 Homotopy groups. 6 Lecture 2 1/26 x1 Introduction 8 x2 Relative homotopy groups 9 x3 Relative homotopy groups as absolute homotopy groups 10 Lecture 3 1/28 x1 Fibrations 12 x2 Long exact sequence 13 x3 Replacing maps 14 Lecture 4 1/31 x1 Motivation 15 x2 Exact couples 16 x3 Important examples 17 x4 Spectral sequences 19 1 Lecture 5 February 2, 2010 x1 Serre spectral sequence, special case 21 Lecture 6 2/4 x1 More structure in the spectral sequence 23 x2 The cohomology ring of ΩSn+1 24 x3 Complex projective space 25 Lecture 7 January 7, 2010 x1 Application I: Long exact sequence in H∗ through a range for a fibration 27 x2 Application II: Hurewicz Theorem 28 Lecture 8 2/9 x1 The relative Hurewicz theorem 30 x2 Moore and Eilenberg-Maclane spaces 31 x3 Postnikov towers 33 Lecture 9 February 11, 2010 x1 Eilenberg-Maclane Spaces 34 Lecture 10 2/14 x1 Local systems 39 x2 Homology in local systems 41 Lecture 11 February 16, 2010 x1 Applications of the Serre spectral sequence 45 Lecture 12 2/18 x1 Serre classes 50 Lecture 13 February 23, 2011 Lecture 14 2/25 Lecture 15 February 28, 2011 Lecture 16 3/2/2011 Lecture 17 March 4, 2011 Lecture 18 3/7 x1 Localization 74 x2 The homotopy category 75 x3 Morphisms between model categories 77 Lecture 19 March 11, 2011 x1 Model Category on Simplicial Sets 79 Lecture 20 3/11 x1 The Yoneda embedding 81 x2 82 -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
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. -
1 Equalizers and Coequalizers
Travaux Dirigés Equalizers, epi-mono factorization, first-order logic λ-calculs et catégories (7 octobre 2019) 1 Equalizers and coequalizers In this exercise, we study the notion of equalizer (called “égalisateur” in French) and its dual notion of coequalizer. Suppose given a pair of coinitial and cofinal arrows f, g : X Y in a category C . An equalizer of f and g is an arrow m : E X such that f ◦ m = g ◦ m and such that, for every arrow n : F X such that f ◦ n = g ◦ n there exists a unique arrow h : F E such that n = m ◦ h. §1. Show that every pair of functions f, g : X −→ Y has an equalizer m : E −→ X in the category Sets and describe this equalizer. §2. Show that when it exists in a category C , the equalizer m : E −→ X of a pair of arrows f, g : X −→ Y is a mono. §3. Formulate the dual notion of coequalizer e : Y −→ Q of two arrows f, g : X Y in a category C . §4. Show that when it exists in a category C , the coequalizer e : Y −→ Q of two arrows f, g : X −→ Y is an epi. §5. Show that every pair of functions f, g : X −→ Y has a coequalizer e : Y −→ Q in the category Sets and describe this coequalizer. §6. An epi e : Y −→ Q is called regular when there exists a pair of arrows f, g : X −→ Y such that e is a coequalizer of f and g as in the diagram below: f X Y e Q g Show that every surjective function e : A −→ B is a regular epi in the category Sets. -
Introduction to Homology
Introduction to Homology Matthew Lerner-Brecher and Koh Yamakawa March 28, 2019 Contents 1 Homology 1 1.1 Simplices: a Review . .2 1.2 ∆ Simplices: not a Review . .2 1.3 Boundary Operator . .3 1.4 Simplicial Homology: DEF not a Review . .4 1.5 Singular Homology . .5 2 Higher Homotopy Groups and Hurweicz Theorem 5 3 Exact Sequences 5 3.1 Key Definitions . .5 3.2 Recreating Groups From Exact Sequences . .6 4 Long Exact Homology Sequences 7 4.1 Exact Sequences of Chain Complexes . .7 4.2 Relative Homology Groups . .8 4.3 The Excision Theorems . .8 4.4 Mayer-Vietoris Sequence . .9 4.5 Application . .9 1 Homology What is Homology? To put it simply, we use Homology to count the number of n dimensional holes in a topological space! In general, our approach will be to add a structure on a space or object ( and thus a topology ) and figure out what subsets of the space are cycles, then sort through those subsets that are holes. Of course, as many properties we care about in topology, this property is invariant under homotopy equivalence. This is the slightly weaker than homeomorphism which we before said gave us the same fundamental group. 1 Figure 1: Hatcher p.100 Just for reference to you, I will simply define the nth Homology of a topological space X. Hn(X) = ker @n=Im@n−1 which, as we have said before, is the group of n-holes. 1.1 Simplices: a Review k+1 Just for your sake, we review what standard K simplices are, as embedded inside ( or living in ) R ( n ) k X X ∆ = [v0; : : : ; vk] = xivi such that xk = 1 i=0 For example, the 0 simplex is a point, the 1 simplex is a line, the 2 simplex is a triangle, the 3 simplex is a tetrahedron.