DOCSLIB.ORG

Spectral Sequences

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectral Sequences J.T. Lyczak, January 2016 Spectral sequences A useful tool in homological algebra is the theory of spectral sequences. The purpose of this text is to introduce the reader to the subject and proofs are generally omitted for clearity. At the end the reader will be hopefully be able to work with spectral sequences as they arise in practice. For a thorough treatise on spectral sequences one can consult at any of the many books on homological algebra. Below the book by [Weibel] is used as a reference for theorems without proofs. 1 Homological spectral sequences Definition 1. A homological spectral sequence (or simply spectral sequences) in an abelian cate- gory (for example modules over a fixed commutative ring) consists of • a non-negative integer a; r r • objects Epq for every r ≥ a indexed by integers p and q. We assume that Epq = 0 for negative p or q; r r r r r • morphisms d : Epq ! Ep−r;q+r−1 for all r ≥ a and all p and q, such that d ◦ d = 0; r+1 r • isomorphisms between Epq and the homology at Epq under the complex given by the morphisms dr. One can think of Er as a family of complexes whose homology groups are the objects of Er+1: • • • • • 0 • • • • • • • • • • 0 • • • • • E0 : • • • • • E1 : 0 • • • • • • • • • • 0 • • • • • • • • • • 0 • • • • • 0 0 0 0 0 0 0 • • • • • 0 0 • • • • • E2 : 0 0 • • • • • 0 0 • • • • • • • • • • 0 0 0 1 1 Homological spectral sequences Note that the objects Er+1 can be recovered from the rth page, but the differentials dr+1 cannot and are part of the given structure. Let us now fix p and q for a moment and look at the objects at position (p; q) for increasing r+1 r r. First note that each Epq is a subquotient of Epq. One also sees that if r > max(p; q + 1), r r r d r d then Epq lives in the complex 0 Epq 0: r ∼ r+1 ∼ r+2 ∼ So we find Epq = Epq = Epq = :::. Hence, for large enough r the objects on position (p; q) 1 stabilize. We will write Epq for this object. The usefulness of spectral sequences lies in convergence, which we will now define. Definition 2. Let Hn be an object for every integer n ≥ 0. We say that a homological spectral a sequence Epq converges to Hp+q, notation a Epq ) Hp+q; if each Hn has a filtration 0 = (Hn)−1 ⊆ (Hn)0 ⊆ (Hn)1 ⊆ ::: ⊆ (Hn)n−1 ⊆ (Hn)n = Hn; and we have isomorphisms 1 ∼ Epq = (Hp+q)p=(Hp+q)p−1: 1 This means that the filtration on Hn allows one to determine the objects of E with p+q = n. ∼ n (Hn)0=(Hn)−1 = (Hn)0 n − 1 (Hn)1=(Hn)0 n − 2 (Hn)2=(Hn)1 . E1 : . .. 1 (Hn)n−1=(Hn)n−2 ∼ q = 0 (Hn)n=(Hn)n−1 = Hn=(Hn)n−1 p = 0 1 2 ::: n − 1 n Convergence of spectral sequences is particular useful when there are a lot of trivial objects on E1. Let us first consider the following extreme case. Theorem 1. Fix an integer n and suppose that we have a converging homological spectral sequence r Epq ) Hp+q: 1 Then Epq = 0 on the diagonal p + q = n if and only if Hn = 0. By far the most applications of spectral sequences occur in a slightly less specific situation named in the following definition. Definition 3. A spectral sequence is said to collapse at Es (s ≥ a) if Es consists of only trivial objects outside of a single row or column. In this case we have Es =∼ E1 (at least for s ≥ 2). For a converging spectral sequence which collapses we have the following relation between the 1-page and the limit. 2 2 Construction of homological spectral sequences Theorem 2. Suppose that a converging spectral sequence r Epq ) Hp+q s s collapses at E (s ≥ 2). Then Hn is isomorphic to the unique non-zero Epq with p + q = n. In other situations there is still something to say, but we do not get isomorphisms between Hn and objects of the spectral sequence any more. We will just be able to say something on Hn up to extension, in the sense made clear by the following exercises. Exercise 1 ([Weibel], Ex 5.2.1 (Corrected)). Suppose that a spectral sequence sequence converg- 2 ing to H∗ has Epq = 0 unless p = 0; 1. Show that there are exact sequences 2 2 0 E0n Hn E1;n−1 0: Generalize to two non-zero columns at p = k; l ≥ 0. Exercise 2 ([Weibel], Ex 5.2.2). Suppose that a spectral sequence sequence converging to H∗ has 2 Epq = 0 unless q = 0; 1. Show that there is a long exact sequence 2 2 d 2 2 d 2 ::: Hp+1 Ep+1;0 Ep−1;1 Hp Ep0 Ep−2;1 Hp−1 :::: Generalize to two non-zero rows at q = k; l ≥ 0. Exercise 3 (Five term sequence). Consider a converging spectral sequence 2 Epq ) Hp+q: Then we have an exact sequence of a the first terms of low degree 2 2 d 2 2 H2 E20 E01 H1 E10 0: Exercise 4 (Edge morphisms). Show that for a spectral sequences and all r larger than 2 (and a), we have natural maps r 1 1 r E0n E0n and En0 ⊆ En0: If the spectral sequence converges to Hn we get maps r 1 1 r E0n E0n ⊆ Hn and Hn En0 ⊆ En0: These maps are called the edge morphisms. 2 Construction of homological spectral sequences We have seen some terminology and theory concering spectral sequences. Now we will give a few ways to produce spectral sequences. The first one being the most general. 2.1 Spectral sequence of a filtered complex Let C• be a complex concentrated in non-negative degrees and let (C)n be a filtration of the complex which is canonically bounded, i.e. 0 = (Cn)−1 ⊆ (Cn)0 ⊆ ::: ⊆ (Cn)n−1 = (Cn)n = Cn for all n. We can depict this as 3 2 Construction of homological spectral sequences d3 d2 d1 C : ::: C3 C2 C1 C0 0 . (C)2 : ::: (C3)2 (C2)2 C1 C0 0 (C)1 : ::: (C3)1 (C2)1 (C1)1 C0 0 (C)0 : ::: (C3)0 (C2)0 (C1)0 (C0)0 0 (C)−1 : ::: 0 0 0 0 0 in which all the horizontal maps are restriction of the differenatials in the original complex. So all rows are complexes too. In each column beneath Cn we have exactly n + 1 interesting quotients (Cn)p=(Cn)p−1. Let us put these on the diagonal in a grid as we did for the filtration of the family H• above. 4 (C4)0 3 (C3)0 (C4)1=(C4)0 0 E : 2 (C2)0 (C3)1=(C3)0 (C4)2=(C4)1 1 (C1)0 (C2)1=(C2)0 (C3)2=(C3)1 (C4)3=(C4)2 q = 0 C0 C1=(C1)0 C2=(C2)1 C3=(C3)2 C4=(C4)3 p = 0 1 2 3 4 Here the horizontal maps for a fixed p come from the complex (C)p. We can extend this zeroth page into a spectral sequence. The construction, like the construction of the connecting homomorphisms or the snake map, is by direct definition and not very enlightening. We will just state the existence of the differentials on subsequent pages and its convergence properties. Theorem 3. Let (C)k be a canonically bounded filtration on a complex C•. There exists a natural 0 spectral sequence starting with Epq = (Cp+q)p=(Cp+q)p−1 such that it converges to the homology of C•: 0 Epq ) Hp+q(C): Exercise 5. Verify the above theorem for the filtration (C)n = C• for all n ≥ 0. Do the same for the filtration ( 0 for l < k; (Ck)l = Ck for l ≥ k: 4 2 Construction of homological spectral sequences Now change (C1)0 into C1. What happens? Exercise 6. Consider the complex ·2 ·2 ·2 ::: Z=4Z Z=4Z Z=4Z 0 of abelian groups. Define the filtration ·2 ·2 ::: R2 R1 R0 0 . ·2 ·2 ::: R22 R12 R02 0 ·2 ·2 ::: 2R2 R11 R01 0 ·2 ::: 0 2R1 R00 0 ::: 0 0 0 where each Rn and Rpq is a copy of Z=4Z. r The indices are simply to keep track of our objects. Every object Epq is a subquotient of objects in the filtration. So make sure every object you compute below is written as either Rpq, 2Rp, Rpq=2Rp or 0. a) Determine E0 and note that the differentials come directly from the diagram above. Com- pute E1 while keeping track of your indices. b) Keeping your indices in mind, remark that there is only one obvious choice for the morphisms d1. Use this to compute E2. c) Determine the morphisms d2 (again, there is only one natural choice). Now compute E3 and show that this is actually E1. Conclude that you just defined a spectral sequence which converges to the homology of the complex above. 2 2 If you did the above exercise, then you probably tried to compute the differential d : E20 ! 2 E01. For this you just had to notice that the image of the morphism R22 ! R12 even lies in 2R1 ,! R12. This is exactly what happens in the construction of this spectral sequence for general filtered complexes, which uses the intermediate objects r Ap;n−p := fx 2 (Cn)p j dn(x) lies in (Cn−1)p−r ⊆ (Cn−1)pg : Since (Cn)p is part of Cn and (Cn−1)p−r is smaller than (Cn−1)p, we can think of (Cn)p becoming r closer to Cn for large p and (Cn−1)p−r going to 0.
Load more

Recommended publications
  • Categorical Semantics of Constructive Set Theory
    Categorical semantics of constructive set theory Beim Fachbereich Mathematik der Technischen Universit¨atDarmstadt eingereichte Habilitationsschrift von Benno van den Berg, PhD aus Emmen, die Niederlande 2 Contents 1 Introduction to the thesis 7 1.1 Logic and metamathematics . 7 1.2 Historical intermezzo . 8 1.3 Constructivity . 9 1.4 Constructive set theory . 11 1.5 Algebraic set theory . 15 1.6 Contents . 17 1.7 Warning concerning terminology . 18 1.8 Acknowledgements . 19 2 A unified approach to algebraic set theory 21 2.1 Introduction . 21 2.2 Constructive set theories . 24 2.3 Categories with small maps . 25 2.3.1 Axioms . 25 2.3.2 Consequences . 29 2.3.3 Strengthenings . 31 2.3.4 Relation to other settings . 32 2.4 Models of set theory . 33 2.5 Examples . 35 2.6 Predicative sheaf theory . 36 2.7 Predicative realizability . 37 3 Exact completion 41 3.1 Introduction . 41 3 4 CONTENTS 3.2 Categories with small maps . 45 3.2.1 Classes of small maps . 46 3.2.2 Classes of display maps . 51 3.3 Axioms for classes of small maps . 55 3.3.1 Representability . 55 3.3.2 Separation . 55 3.3.3 Power types . 55 3.3.4 Function types . 57 3.3.5 Inductive types . 58 3.3.6 Infinity . 60 3.3.7 Fullness . 61 3.4 Exactness and its applications . 63 3.5 Exact completion . 66 3.6 Stability properties of axioms for small maps . 73 3.6.1 Representability . 74 3.6.2 Separation . 74 3.6.3 Power types .
    [Show full text]
  • Notes Towards Homology in Characteristic One
    Notes Towards Homology in Characteristic One David Jaz Myers Abstract In their paper Homological Algebra in Characteristic One, Connes and Consani develop techniques of Grandis to do homological calculations with boolean modules { commutative idempotent monoids. This is a writeup of notes taken during a course given by Consani in Fall of 2017 explaining some of these ideas. This note is in flux, use at your own peril. Though the integers Z are the initial ring, they are rather complex. Their Krull dimension is 1, suggesting that they are the functions on a curve; but each of the epimorphisms out of them has a different characteristic, so they are not a curve over any single field. This basic observation, and many more interesting and less basic observations, leads one to wonder about other possible bases for the algebraic geometry of the integers. Could there be an \absolute base", a field over which all other fields lie? What could such a “field” be? As a first approach, let's consider the characteristic of this would-be absolute base field. Since it must lie under the finite fields, it should also be finite, and therefore its characteristic must divide the order of all finite fields. There is only one number which divides all prime powers (and also divides 0): the number 1 itself. Therefore, we are looking for a “field of characteristic one". Now, the axioms of a ring are too restrictive to allow for such an object. Let's free ourselves up by considering semirings, also known as rigs { rings without negatives.
    [Show full text]
  • Harish-Chandra 1923–1983
    NATIONAL ACADEMY OF SCIENCES HARISH- C HANDRA 1 9 2 3 – 1 9 8 3 A Biographical Memoir by R O G E R H O W E Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 2011 NATIONAL ACADEMY OF SCIENCES WASHINGTON, D.C. Photo by Herman Landshoff; Courtesy Archives of the Institute for Advanced Study. HARISH-CHANDRA October 11, 1923–October 12, 1983 BY ROGER HOWE He taught them the Kshatria code of honor: that a warrior may never refuse a challenge…. The Five Sons of Pandu, The Story of the Mahabharata Retold by Elizabeth Seeger ARISH-CHANDRA WAS, if not the exclusive architect, cer- Htainly the chief engineer of harmonic analysis on semisimple Lie groups. This subject, with roots deep in mathematical physics and analysis, is a synthesis of Fou- rier analysis, special functions and invariant theory, and it has become a basic tool in analytic number theory, via the theory of automorphic forms. It essentially did not ex- ist before World War II, but in very large part because of the labors of Harish-Chandra, it became one of the major mathematical edifices of the second half of the twentieth century. Harish-Chandra was born in 1923 in Uttar Pradesh, in northern India. His family belonged to the Kshatria (war- rior) caste. Kshatria traditionally were rulers, landowners, and military leaders, and more recently have commonly been businessmen or civil servants. Harish-Chandra’s father, Chandrakishore, was a civil engineer who monitored and maintained the dikes and irrigation canals that sustain agri- 3 B IOGRA P HICAL MEMOIRS culture on the North Indian plains.
    [Show full text]
  • Arxiv:1305.5974V1 [Math-Ph]
    INTRODUCTION TO SPORADIC GROUPS for physicists Luis J. Boya∗ Departamento de F´ısica Te´orica Universidad de Zaragoza E-50009 Zaragoza, SPAIN MSC: 20D08, 20D05, 11F22 PACS numbers: 02.20.a, 02.20.Bb, 11.24.Yb Key words: Finite simple groups, sporadic groups, the Monster group. Juan SANCHO GUIMERA´ In Memoriam Abstract We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups Zp, and the alternating groups Altn>4. After a quick revision of finite fields Fq, q = pf , with p prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 extra “sporadic” groups, which gather in three interconnected “generations” (with 5+7+8 groups) plus the Pariah groups (6). We point out a couple of physical applications, in- cluding constructing the biggest sporadic group, the “Monster” group, with close to 1054 elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory. ∗[email protected] arXiv:1305.5974v1 [math-ph] 25 May 2013 1 Contents 1 Introduction 3 1.1 Generaldescriptionofthework . 3 1.2 Initialmathematics............................ 7 2 Generalities about groups 14 2.1 Elementarynotions............................ 14 2.2 Theframeworkorbox .......................... 16 2.3 Subgroups................................. 18 2.4 Morphisms ................................ 22 2.5 Extensions................................. 23 2.6 Familiesoffinitegroups ......................... 24 2.7 Abeliangroups .............................. 27 2.8 Symmetricgroup ............................. 28 3 More advanced group theory 30 3.1 Groupsoperationginspaces. 30 3.2 Representations.............................. 32 3.3 Characters.Fourierseries . 35 3.4 Homological algebra and extension theory . 37 3.5 Groupsuptoorder16..........................
    [Show full text]
  • Ordered Tensor Categories and Representations of the Mackey Lie Algebra of Infinite Matrices
    Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices Alexandru Chirvasitu, Ivan Penkov To Jean-Louis Koszul on his 95th birthday Abstract We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices M M gl (V; V∗). Here gl (V; V∗) is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces V∗ ⊗ V ! K, where K is the base field. Tensor M representations of gl (V; V∗) are defined as arbitrary subquotients of finite direct sums of tensor ∗ ⊗m ⊗n ⊗p ∗ products (V ) ⊗ (V∗) ⊗ V where V denotes the algebraic dual of V . The category 3 M which they comprise, extends a category M previously studied in [5, 15, Tgl (V;V∗) Tgl (V;V∗) 3 20]. Our main result is that M is a finite-length, Koszul self-dual, tensor category Tgl (V;V∗) with a certain universal property that makes it into a “categorified algebra" defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the 3 category M established in [15]. Finally, we discuss the extension of M obtained Tgl (V;V∗) Tgl (V;V∗) ∗ by adjoining the algebraic dual (V∗) of V∗. Key words: Mackey Lie algebra, tensor category, Koszulity MSC 2010: 17B65, 17B10, 16T15 Contents 1 Background 3 2 Ordered Grothendieck categories5 2.1 Definition and characterization of indecomposable injectives..............5 2.2 Bounds on non-vanishing ext functors..........................7 2.3 C as a comodule category.................................8 2.4 C as a highest weight category...............................8 2.5 Universal properties for C and CFIN ............................9 2.6 The monoidal case....................................
    [Show full text]
  • Between Model Theory and Combinatorics
    INFINITE LIMITS OF FINITE-DIMENSIONAL PERMUTATION STRUCTURES, AND THEIR AUTOMORPHISM GROUPS: BETWEEN MODEL THEORY AND COMBINATORICS BY SAMUEL WALKER BRAUNFELD A dissertation submitted to the School of Graduate Studies Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Gregory Cherlin And approved by New Brunswick, New Jersey May, 2018 ABSTRACT OF THE DISSERTATION Infinite Limits of Finite-Dimensional Permutation Structures, and their Automorphism Groups: Between Model Theory and Combinatorics by SAMUEL WALKER BRAUNFELD Dissertation Director: Gregory Cherlin In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders [7], and this structural viewpoint is taken up here. The majority of this thesis is concerned with Cameron's problem of classifying the homogeneous structures in a language of finitely many linear orders, which we call finite-dimensional permutation structures. Towards this problem, we present a construction that we conjecture produces all such structures. Some evidence for this conjecture is given, including the classification of the homogeneous 3-dimensional permutation structures. We next consider the topological dynamics, in the style of Kechris, Pestov, and Todorˇcevi´c,of the automorphism groups of the homogeneous finite-dimensional per- mutation structures we have constructed, which requires proving a structural Ramsey theorem for all the associated amalgamation classes. Because the ;-definable equiv- alence relations in these homogeneous finite-dimensional permutation structures may form arbitrary finite distributive lattices, the model-theoretic algebraic closure opera- tion may become quite complex, and so we require the framework recently introduced ii by Hubiˇcka and Neˇsetril [16].
    [Show full text]
  • ON the UNIVERSAL EXTENSIONS in TANNAKIAN CATEGORIES Contents 1. Introduction 1 2. Notation 4 3. Universal Extensions 5 4. Variat
    ON THE UNIVERSAL EXTENSIONS IN TANNAKIAN CATEGORIES MARCO D'ADDEZIO AND HEL´ ENE` ESNAULT Abstract. We use the notion of universal extension in a linear abelian category to study extensions of variations of mixed Hodge structure and convergent and overconvergent isocrystals. The results we obtain apply, for example, to prove the exactness of some homotopy sequences for these categories and to study F -able isocrystals. Contents 1. Introduction 1 2. Notation 4 3. Universal extensions 5 4. Variations of mixed Hodge structure 7 5. Isocrystals 10 Appendix A. Observable functors 14 References 17 1. Introduction In this note we define and construct universal extensions (Definition 3.1) of objects belonging to a linear abelian category. This construction is convenient to study all the possible extensions of two objects of the category at the same time. On the one hand, a universal extension enjoys more properties than single extensions. On the other hand, every non-trivial extension between two objects embeds into a universal extension (Proposition 3.4). Some examples of universal extensions appeared already in the literature. For example, in [EH06, x4], universal extensions are used to study the category of flat connections over a smooth variety in characteristic 0. In our article, we use universal extensions to solve some open problems on variations of mixed Hodge structure and convergent and overconvergent isocrystals. Let us present our first application. Let X be a smooth connected complex variety. Write MHS for the category of graded-polarisable finite-dimensional mixed Q-Hodge structures, LS(X) for the category of finite-rank Q-local systems over X, and M(X) for the category of (graded-polarisable) admissible variations of mixed Q-Hodge structure.
    [Show full text]
  • An Induction Theorem for the Unit Groups of Burnside Rings of 2-Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE Journal of Algebra 289 (2005) 105–127 provided by Elsevier - Publisher Connector www.elsevier.com/locate/jalgebra An induction theorem for the unit groups of Burnside rings of 2-groups Ergün Yalçın Department of Mathematics, Bilkent University, Ankara 06800, Turkey Received 11 May 2004 Communicated by Michel Broué Abstract × Let G be a 2-group and B(G) denote the group of units of the Burnside ring of G. For each × × subquotient H/K of G, there is a generalized induction map from B(H/K) to B(G) defined as the composition of inflation and multiplicative induction maps. We prove that the product of × × generalized induction maps B(H/K) → B(G) is surjective when the product is taken over the set of all subquotients that are isomorphic to the trivial group or a dihedral 2-group of order 2n with n 4. As an application, we give an algebraic proof for a theorem by Tornehave [The unit group for the Burnside ring of a 2-group, Aarhus Universitet Preprint series 1983/84 41, May 1984] × which states that tom Dieck’s exponential map from the real representation ring of G to B(G) is surjective. We also give a sufficient condition for the surjectivity of the exponential map from the × Burnside ring of G to B(G) . 2005 Elsevier Inc. All rights reserved. Keywords: Units of Burnside ring; Real representation ring 1. Introduction The Burnside ring of a finite group G is defined to be the Grothendieck ring of the semi-ring generated by isomorphism classes of finite (left) G-sets where the addition and E-mail address: [email protected].
    [Show full text]
  • Graph Minor from Wikipedia, the Free Encyclopedia Contents
    Graph minor From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Profile ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 Abelian group 5 2.1 Definition ............................................... 5 2.2 Facts ................................................. 5 2.2.1 Notation ........................................... 5 2.2.2 Multiplication table ...................................... 6 2.3 Examples ............................................... 6 2.4 Historical remarks .......................................... 6 2.5 Properties ............................................... 6 2.6 Finite abelian groups ......................................... 7 2.6.1 Classification ......................................... 7 2.6.2 Automorphisms ....................................... 7 2.7 Infinite abelian groups ........................................ 8 2.7.1 Torsion groups ........................................ 9 2.7.2 Torsion-free and mixed groups ................................ 9 2.7.3 Invariants and classification .................................. 9 2.7.4 Additive groups of rings ................................... 9 2.8 Relation to other mathematical topics ................................. 10 2.9 A note on the typography ......................................
    [Show full text]
  • Quillen K-Theory: Lecture Notes
    Quillen K-Theory: Lecture Notes Satya Mandal, U. of Kansas 2 April 2017 27 August 2017 6, 10, 13, 20, 25 September 2017 2, 4, 9, 11, 16, 18, 23, 25, 30 October 2017 1, 6, 8, 13, 15, 20, 27, 29 November 2017 December 4, 6 2 Dedicated to my mother! Contents 1 Background from Category Theory 7 1.1 Preliminary Definitions . .7 1.1.1 Classical and Standard Examples . .9 1.1.2 Pullback, Pushforward, kernel and cokernel . 12 1.1.3 Adjoint Functors and Equivalence of Categories . 16 1.2 Additive and Abelian Categories . 17 1.2.1 Abelian Categories . 18 1.3 Exact Category . 20 1.4 Localization and Quotient Categories . 23 1.4.1 Quotient of Exact Categories . 27 1.5 Frequently Used Lemmas . 28 1.5.1 Pullback and Pushforward Lemmas . 29 1.5.2 The Snake Lemma . 30 1.5.3 Limits and coLimits . 36 2 Homotopy Theory 37 2.1 Elements of Topological Spaces . 37 2.1.1 Frequently used Topologies . 41 2.2 Homotopy . 42 2.3 Fibrations . 47 2.4 Construction of Fibrations . 49 2.5 CW Complexes . 53 2.5.1 Product of CW Complexes . 57 2.5.2 Frequently Used Results . 59 3 4 CONTENTS 3 Simplicial and coSimplicial Sets 61 3.1 Introduction . 61 3.2 Simplicial Sets . 61 3.3 Geometric Realization . 65 3.3.1 The CW Structure on jKj ....................... 68 3.4 The Homotopy Groups of Simplicial Sets . 73 3.4.1 The Combinatorial Definition . 73 3.4.2 The Definitions of Homotopy Groups .
    [Show full text]
  • Awodey-Butz-Simpson
    Annals of Pure and Applied Logic 165 (2014) 428–502 Contents lists available at SciVerse ScienceDirect Annals of Pure and Applied Logic www.elsevier.com/locate/apal Relating first-order set theories, toposes and categories of classes ∗ Steve Awodey a,CarstenButzb,1, Alex Simpson c, ,2, Thomas Streicher d a Department of Philosophy, Carnegie Mellon University, Pittsburgh, USA b IT University of Copenhagen, Denmark c LFCS, School of Informatics, University of Edinburgh, UK d Fachbereich Mathematik, Technische Universität Darmstadt, Germany article info abstract Article history: This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first- Received 22 December 2010 order set theory extending the internal logic of elementary toposes. Given an elementary Received in revised form 12 December 2012 topos, together with the extra structure of a directed structural system of inclusions (dssi) Accepted 12 May 2013 on the topos, a forcing-style interpretation of the language of first-order set theory in the Available online 8 July 2013 topos is given, which conservatively extends the internal logic of the topos. This forcing MSC: interpretation applies to an arbitrary elementary topos, since any such is equivalent to one 03E70 carrying a dssi. We prove that the set theory BIST+Coll (where Coll is the strong Collection 03F50 axiom) is sound and complete relative to forcing interpretations in toposes with natural 03G30 numbers object (nno). Furthermore, in the case that the structural system of inclusions is 18B05 superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can (up to equivalence) be endowed with such superdirected systems Keywords: of inclusions.
    [Show full text]
  • Group Structures on Families of Subsets of a Group
    Group Structures on Families of Subsets of a Group Mario G´omez, Sergio R. L´opez-Permouth , Fernando Mazariegos, Alejandro Jos´eVargas De Le´on and Rigoberto Zelada Cifuentes October 18, 2018 Abstract A binary operation on any set induces a binary operation on its sub- sets. We explore families of subsets of a group that become a group under the induced operation and refer to such families as power groups of the given group. Our results serve to characterize groups in terms of their power groups. In particular, we consider when the only power groups of a group are the factor groups of its subgroups and when that is the case up to isomorphism. We prove that the former are precisely those groups for which every element has finite order and provide examples to illustrate that the latter is not always the case. In the process we consider sev- eral natural questions such as whether the identity element of the group must belong to the identity element of a power group or the inverse of an element in a power group must consist of the inverses of its elements. 1 Introduction Traditionally, group theory has been built upon understanding the subgroups and the factor groups of a given group. In other words, our understanding of groups has largely rested on understanding their subquotients (factors of subgroups or subgroups of factors; it’s the same.) We propose the power groups of a group as a larger collection of groups that might give us a better insight into arXiv:1604.01119v2 [math.GR] 26 Aug 2016 the structure of the original group.
    [Show full text]