DOCSLIB.ORG

Quillen K-Theory: Lecture Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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 . 77 3.5 Frequently Used Lemmms . 83 3.5.1 A Lemma on Bisimplicial Sets . 84 4 The Quillen-K-Theory 89 4.1 The Classifying spaces of a category . 89 4.1.1 Properties of the Classifying Space . 91 4.1.2 Directed and Filtering Limit . 92 4.1.3 Theorem A: Sufficient condition for a functor to be homotopy equiv- alence . 96 4.1.4 Theorem B: The Exact Homotopy Sequence . 101 4.2 The K-Groups of Exact Categories . 110 4.2.1 Quillen’s Q-Construction . 110 4.2.2 The Higher K-groups of an Exact Category . 115 4.2.3 Higher K-Groups . 120 4.3 Characteristic Exact Sequences and Filtrations . 123 4.3.1 Additivity Theorem ......................... 125 4.3.2 The Prime Examples . 127 4.4 Reduction by Resolution . 130 4.4.1 Extension closed and Resolving Subcategories . 132 4.5 Dévissage and Localization in Abelian Categories . 143 4.5.1 Semisimple Objects . 145 4.5.2 Quillen’s Localization Theorem . 146 4.6 K-Theory Spaces and Reformulations . 149 4.7 Negative K-Theory . 149 4.7.1 Spectra and Negative Homotopy Groups . 150 CONTENTS 5 4.8 Suspension of an Exact Category . 151 4.8.1 Cofinality and Idempotent Completion . 154 4.8.2 The K-theory Spectra . 156 A Not Used 157 A.0.3 Some Jargon from [H] (SKIP)..................... 157 A.0.4 On SSet Realization . 160 A.0.5 Inclusion of Geometric Realizations . 160 6 CONTENTS Chapter 1 Background from Category Theory Among the sources I used in this chapter are [HS, Sv, Wc1] and most importantly [HS]. Importance of Category Theory, as a learning tool, is underrated. To me, Category Theory is a way to learn how analogy works in mathematics, from one area to others. This is a good way to understand different areas of mathematics, just by learning one single area. First, I recall the basic definitions from [HS]. 1.1 Preliminary Definitions Definition 1.1.1. A Category C consists of the following: 1. A class of objects, denoted by Obj(C ). 2. For any two objected A; B in Obj(C ), a set of morphisms, from A to B, denoted by Mor(A; B), more precisely by MorC (A; B).(Depending on the context, several other notations are used, for Mor(A; B). One of them is Hom(A; B). Morphisms are also called "arrows", "maps" and, other things appropriate to the specific context.) 3. To each triple A; B; C of objects, a law of composition Mor(A; B) × Mor(B; C) −! Mor(A; C) sending (f; g) 7! gf: The above data satisfy the following Axioms: A 1 Mor(A; B) are mutually disjoint. A 2 For composeable morphisms f; g; h, the associativity holds: h(gf) = (hg)f. 7 8 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY A 3 For each object A, teethe is a morphism 1A : A −! A such that, for composable f; g: 1Af = f and g1A = g. Given a category C , the following are some obvious definitions: 1. A morphism f : X −! Y in C , is said to be an isomorphism, if there is a morphism g : Y −! X such that fg = 1Y and gf = 1X . In this case, g would be called the inverse of f. 2. A morphism f : X −! Y in C is called a monomorphism or injective, if for g1f = g2f =) g1 = g2, for all pairs of morphisms g1; g2 : A −! X. A morphism f : X −! Y in C is called a epimorphism or surjective, if for fg1 = fg2 =) g1 = g2, for all pairs of morphisms g1; g2 : Y −! B. 3. An object L in C would be called an initial object, if Mor(L; X) is a singleton, for all X in Obj(C . An object R in C would be called a final object (orterminal object), if Mor(X; F ) is a singleton, for all X in Obj(C . An object 0 in C would be called zero object, if it is both initial and final object. 4. Injective objects and projective objects (will skip for now). 5. For a category C , define the opposite category C opp. Where Obj(E ) = Obj(C opp), and for two objects in C , MorC opp (A; B) = MorC (A; B). Definition 1.1.2. A covariant functor F : C −! D from a category C to another category D is defined by the following data: 1. To each object A 2 C , it associates an object FA 2 D. 2. For objects A; B, there is a set theoretic map F : MorE (A; B) −! MorD (F A; F B) 3. Further, F (fg) = F (f)F (g) for any two composeable morphisms f; g. A contravariant functor F : C −! D is a covariant functor C opp −! D. Definition 1.1.3. Given two covariant functors F; G : C −! D, a natural transformation τ : F −! G, associates, to each object X in C , a morphism τX : F (X) −! G(X), such that for any morphism f 2 Mor(X; Y ) the diagram τX F (X) / G(X) F (f) G(f) commutes: F (Y ) / G(Y ) τY 1.1. PRELIMINARY DEFINITIONS 9 If τX is an isomorphism, for all objects X, the τ would be called a natural equivalence. Likewise, define natural transformation and natural equivalence between two contravari- ant functors. Example 1.1.4. Let C be a category and A 2 Obj(C ). Then X 7! MorC (A; X) is a covariant functor C −! Sets: and X 7! MorC (X; A) is a contravariant functor C −! Sets: They are called representable functors. These are, perhaps, the most significant examples of functors. Some special functors: Definition 1.1.5. Let F : C −! D be a functor. 1. F is said to be full, if 8 X; Y 2 Obj(C ) the map F : MorC (X; Y ) −! MorD (F X; F Y ) is surjective: 2. F is called faithful, if 8 X; Y 2 Obj(C ) the map F : MorC (X; Y ) −! MorD (F X; F Y ) is injective: 3. F is called fully faithful, if 8 X; Y 2 Obj(C ) the map F : MorC (X; Y ) −! MorD (F X; F Y ) is bijective: 4. In particular, if F : Obj(C ) −! Obj(D) is an "inclusion", then C is called a sub- category. So, the concept a full subcategory is to be distinguished from a non-full subcategory. 1.1.1 Classical and Standard Examples We recall a some of the classical and Standard examples. Examples 1.1.6. We recall a few examples and establish notations. 1. The category of sets will be denoted by Set. The objects of Set consists of the class of all sets. For two sets X; Y , a morphism f : X −! Y is a set theoretic map. 10 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY 2. The category of Topological spaces will be denoted by Top. The objects of Top con- sists of the class of all topological spaces. For two topological spaces X; Y , Mor(X; Y ) is the set of all continuous maps f : X −! Y . Note the empty set φ is the initial object and singleton sets fxg are the terminal objects in Top. In Topology, there are two more important categories: (a) The category of Pointed Topological spaces will be denoted by Top•. The objects of Top• consists of the class of all pairs (X; x), where X is a topological space and x 2 X is a point, to be called the base point. For two objects (X; x); (Y; y), Mor((X; x); (Y; y)) is the set of all continuous maps f : X −! Y , with f(x) = y. Note, Top• has a zero object, namely (fxg; x) , where fxg is any singleton set. (b) The category of Topological Pairs will be denoted by Top − Pairs. The objects of Top − Pairs consists of the class of all pairs (X; U), where X is a topological space and U ⊆ X is a topological subspace. For two objects (X; U); (Y; V ), Mor((X; x); (Y; y)) is the set of all continuous maps f : X −! Y , with f(U) ⊆ V .
Load more

Recommended publications
  • De Rham Cohomology
    De Rham Cohomology 1. Definition of De Rham Cohomology Let X be an open subset of the plane. If we denote by C0(X) the set of smooth (i. e. infinitely differentiable functions) on X and C1(X), the smooth 1-forms on X (i. e. expressions of the form fdx + gdy where f; g 2 C0(X)), we have natural differntiation map d : C0(X) !C1(X) given by @f @f f 7! dx + dy; @x @y usually denoted by df. The kernel for this map (i. e. set of f with df = 0) is called the zeroth De Rham Cohomology of X and denoted by H0(X). It is clear that these are precisely the set of locally constant functions on X and it is a vector space over R, whose dimension is precisley the number of connected components of X. The image of d is called the set of exact forms on X. The set of pdx + qdy 2 C1(X) @p @q such that @y = @x are called closed forms. It is clear that exact forms and closed forms are vector spaces and any exact form is a closed form. The quotient vector space of closed forms modulo exact forms is called the first De Rham Cohomology and denoted by H1(X). A path for this discussion would mean piecewise smooth. That is, if γ : I ! X is a path (a continuous map), there exists a subdivision, 0 = t0 < t1 < ··· < tn = 1 and γ(t) is continuously differentiable in the open intervals (ti; ti+1) for all i.
    [Show full text]
  • The Fundamental Group
    The Fundamental Group Tyrone Cutler July 9, 2020 Contents 1 Where do Homotopy Groups Come From? 1 2 The Fundamental Group 3 3 Methods of Computation 8 3.1 Covering Spaces . 8 3.2 The Seifert-van Kampen Theorem . 10 1 Where do Homotopy Groups Come From? 0 Working in the based category T op∗, a `point' of a space X is a map S ! X. Unfortunately, 0 the set T op∗(S ;X) of points of X determines no topological information about the space. The same is true in the homotopy category. The set of `points' of X in this case is the set 0 π0X = [S ;X] = [∗;X]0 (1.1) of its path components. As expected, this pointed set is a very coarse invariant of the pointed homotopy type of X. How might we squeeze out some more useful information from it? 0 One approach is to back up a step and return to the set T op∗(S ;X) before quotienting out the homotopy relation. As we saw in the first lecture, there is extra information in this set in the form of track homotopies which is discarded upon passage to [S0;X]. Recall our slogan: it matters not only that a map is null homotopic, but also the manner in which it becomes so. So, taking a cue from algebraic geometry, let us try to understand the automorphism group of the zero map S0 ! ∗ ! X with regards to this extra structure. If we vary the basepoint of X across all its points, maybe it could be possible to detect information not visible on the level of π0.
    [Show full text]
  • 3. Closed Sets, Closures, and Density
    3. Closed sets, closures, and density 1 Motivation Up to this point, all we have done is define what topologies are, define a way of comparing two topologies, define a method for more easily specifying a topology (as a collection of sets generated by a basis), and investigated some simple properties of bases. At this point, we will start introducing some more interesting definitions and phenomena one might encounter in a topological space, starting with the notions of closed sets and closures. Thinking back to some of the motivational concepts from the first lecture, this section will start us on the road to exploring what it means for two sets to be \close" to one another, or what it means for a point to be \close" to a set. We will draw heavily on our intuition about n convergent sequences in R when discussing the basic definitions in this section, and so we begin by recalling that definition from calculus/analysis. 1 n Definition 1.1. A sequence fxngn=1 is said to converge to a point x 2 R if for every > 0 there is a number N 2 N such that xn 2 B(x) for all n > N. 1 Remark 1.2. It is common to refer to the portion of a sequence fxngn=1 after some index 1 N|that is, the sequence fxngn=N+1|as a tail of the sequence. In this language, one would phrase the above definition as \for every > 0 there is a tail of the sequence inside B(x)." n Given what we have established about the topological space Rusual and its standard basis of -balls, we can see that this is equivalent to saying that there is a tail of the sequence inside any open set containing x; this is because the collection of -balls forms a basis for the usual topology, and thus given any open set U containing x there is an such that x 2 B(x) ⊆ U.
    [Show full text]
  • 1. CW Complex 1.1. Cellular Complex. Given a Space X, We Call N-Cell a Subspace That Is Home- Omorphic to the Interior of an N-Disk
    1. CW Complex 1.1. Cellular Complex. Given a space X, we call n-cell a subspace that is home- omorphic to the interior of an n-disk. Definition 1.1. Given a Hausdorff space X, we call cellular structure on X a n collection of disjoint cells of X, say ffeαgα2An gn2N (where the An are indexing sets), together with a collection of continuous maps called characteristic maps, say n n n k ffΦα : Dα ! Xgα2An gn2N, such that, if X := feαj0 ≤ k ≤ ng (for all n), then : S S n (1) X = ( eα), n2N α2An n n n (2)Φ α Int(Dn) is a homeomorphism of Int(D ) onto eα, n n F k (3) and Φα(@D ) ⊂ eα. k n−1 eα2X We shall call X together with a cellular structure a cellular complex. n n n Remark 1.2. We call X ⊂ feαg the n-skeleton of X. Also, we shall identify X S k n n with eα and vice versa for simplicity of notations. So (3) becomes Φα(@Dα) ⊂ k n eα⊂X S Xn−1. k k n Since X is a Hausdorff space (in the definition above), we have that eα = Φα(D ) k n k (where the closure is taken in X). Indeed, we easily have that Φα(D ) ⊂ eα by n n continuity of Φα. Moreover, since D is compact, we have that its image under k k n k Φα also is, and since X is Hausdorff, Φα(D ) is a closed subset containing eα. Hence we have the other inclusion (Remember that the closure of a subset is the k intersection of all closed subset containing it).
    [Show full text]
  • Math 601 Algebraic Topology Hw 4 Selected Solutions Sketch/Hint
    MATH 601 ALGEBRAIC TOPOLOGY HW 4 SELECTED SOLUTIONS SKETCH/HINT QINGYUN ZENG 1. The Seifert-van Kampen theorem 1.1. A refinement of the Seifert-van Kampen theorem. We are going to make a refinement of the theorem so that we don't have to worry about that openness problem. We first start with a definition. Definition 1.1 (Neighbourhood deformation retract). A subset A ⊆ X is a neighbourhood defor- mation retract if there is an open set A ⊂ U ⊂ X such that A is a strong deformation retract of U, i.e. there exists a retraction r : U ! A and r ' IdU relA. This is something that is true most of the time, in sufficiently sane spaces. Example 1.2. If Y is a subcomplex of a cell complex, then Y is a neighbourhood deformation retract. Theorem 1.3. Let X be a space, A; B ⊆ X closed subspaces. Suppose that A, B and A \ B are path connected, and A \ B is a neighbourhood deformation retract of A and B. Then for any x0 2 A \ B. π1(X; x0) = π1(A; x0) ∗ π1(B; x0): π1(A\B;x0) This is just like Seifert-van Kampen theorem, but usually easier to apply, since we no longer have to \fatten up" our A and B to make them open. If you know some sheaf theory, then what Seifert-van Kampen theorem really says is that the fundamental groupoid Π1(X) is a cosheaf on X. Here Π1(X) is a category with object pints in X and morphisms as homotopy classes of path in X, which can be regard as a global version of π1(X).
    [Show full text]
  • 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]
  • Full Text (PDF Format)
    ASIAN J. MATH. © 1997 International Press Vol. 1, No. 2, pp. 330-417, June 1997 009 K-THEORY FOR TRIANGULATED CATEGORIES 1(A): HOMOLOGICAL FUNCTORS * AMNON NEEMANt 0. Introduction. We should perhaps begin by reminding the reader briefly of Quillen's Q-construction on exact categories. DEFINITION 0.1. Let £ be an exact category. The category Q(£) is defined as follows. 0.1.1. The objects of Q(£) are the objects of £. 0.1.2. The morphisms X •^ X' in Q(£) between X,X' G Ob{Q{£)) = Ob{£) are isomorphism classes of diagrams of morphisms in £ X X' \ S Y where the morphism X —> Y is an admissible mono, while X1 —y Y is an admissible epi. Perhaps a more classical way to say this is that X is a subquotient of X''. 0.1.3. Composition is defined by composing subquotients; X X' X' X" \ ^/ and \ y/ Y Y' compose to give X X' X" \ s \ s Y PO Y' \ S Z where the square marked PO is a pushout square. The category Q{£) can be realised to give a space, which we freely confuse with the category. The Quillen if-theory of the exact category £ was defined, in [9], to be the homotopy of the loop space of Q{£). That is, Ki{£)=ni+l[Q{£)]. Quillen proved many nice functoriality properties for his iT-theory, and the one most relevant to this article is the resolution theorem. The resolution theorem asserts the following. 7 THEOREM 0.2. Let F : £ —> J be a fully faithful, exact inclusion of exact categories.
    [Show full text]
  • Open Sets in Topological Spaces
    International Mathematical Forum, Vol. 14, 2019, no. 1, 41 - 48 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.913 ii – Open Sets in Topological Spaces Amir A. Mohammed and Beyda S. Abdullah Department of Mathematics College of Education University of Mosul, Mosul, Iraq Copyright © 2019 Amir A. Mohammed and Beyda S. Abdullah. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce a new class of open sets in a topological space called 푖푖 − open sets. We study some properties and several characterizations of this class, also we explain the relation of 푖푖 − open sets with many other classes of open sets. Furthermore, we define 푖푤 − closed sets and 푖푖푤 − closed sets and we give some fundamental properties and relations between these classes and other classes such as 푤 − closed and 훼푤 − closed sets. Keywords: 훼 − open set, 푤 − closed set, 푖 − open set 1. Introduction Throughout this paper we introduce and study the concept of 푖푖 − open sets in topological space (푋, 휏). The 푖푖 − open set is defined as follows: A subset 퐴 of a topological space (푋, 휏) is said to be 푖푖 − open if there exist an open set 퐺 in the topology 휏 of X, such that i. 퐺 ≠ ∅,푋 ii. A is contained in the closure of (A∩ 퐺) iii. interior points of A equal G. One of the classes of open sets that produce a topological space is 훼 − open.
    [Show full text]
  • An Introduction to Spectra
    An introduction to spectra Aaron Mazel-Gee In this talk I’ll introduce spectra and show how to reframe a good deal of classical algebraic topology in their language (homology and cohomology, long exact sequences, the integration pairing, cohomology operations, stable homotopy groups). I’ll continue on to say a bit about extraordinary cohomology theories too. Once the right machinery is in place, constructing all sorts of products in (co)homology you may never have even known existed (cup product, cap product, cross product (?!), slant products (??!?)) is as easy as falling off a log! 0 Introduction n Here is a table of some homotopy groups of spheres πn+k(S ): n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 n πn(S ) Z Z Z Z Z Z Z Z Z Z n πn+1(S ) 0 Z Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 n πn+2(S ) 0 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 n πn+3(S ) 0 Z2 Z12 Z ⊕ Z12 Z24 Z24 Z24 Z24 Z24 Z24 n πn+4(S ) 0 Z12 Z2 Z2 ⊕ Z2 Z2 0 0 0 0 0 n πn+5(S ) 0 Z2 Z2 Z2 ⊕ Z2 Z2 Z 0 0 0 0 n πn+6(S ) 0 Z2 Z3 Z24 ⊕ Z3 Z2 Z2 Z2 Z2 Z2 Z2 n πn+7(S ) 0 Z3 Z15 Z15 Z30 Z60 Z120 Z ⊕ Z120 Z240 Z240 k There are many patterns here. The most important one for us is that the values πn+k(S ) eventually stabilize.
    [Show full text]
  • Van Kampen Colimits As Bicolimits in Span*
    Van Kampen colimits as bicolimits in Span? Tobias Heindel1 and Pawe lSoboci´nski2 1 Abt. f¨urInformatik und angewandte kw, Universit¨atDuisburg-Essen, Germany 2 ECS, University of Southampton, United Kingdom Abstract. The exactness properties of coproducts in extensive categories and pushouts along monos in adhesive categories have found various applications in theoretical computer science, e.g. in program semantics, data type theory and rewriting. We show that these properties can be understood as a single universal property in the associated bicategory of spans. To this end, we first provide a general notion of Van Kampen cocone that specialises to the above colimits. The main result states that Van Kampen cocones can be characterised as exactly those diagrams in that induce bicolimit diagrams in the bicategory of spans Span , C C provided that C has pullbacks and enough colimits. Introduction The interplay between limits and colimits is a research topic with several applica- tions in theoretical computer science, including the solution of recursive domain equations, using the coincidence of limits and colimits. Research on this general topic has identified several classes of categories in which limits and colimits relate to each other in useful ways; extensive categories [5] and adhesive categories [21] are two examples of such classes. Extensive categories [5] have coproducts that are “well-behaved” with respect to pullbacks; more concretely, they are disjoint and universal. Extensivity has been used by mathematicians [4] and computer scientists [25] alike. In the presence of products, extensive categories are distributive [5] and thus can be used, for instance, to model circuits [28] or to give models of specifications [11].
    [Show full text]
  • LOOP SPACES for the Q-CONSTRUCTION Charles H
    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 52 (1988) l-30 North-Holland LOOP SPACES FOR THE Q-CONSTRUCTION Charles H. GIFFEN* Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S. A Communicated by C.A. Weibel Received 4 November 1985 The algebraic K-groups of an exact category Ml are defined by Quillen as KJ.4 = TT,+~(Qkd), i 2 0, where QM is a category known as the Q-construction on M. For a ring R, K,P, = K,R (the usual algebraic K-groups of R) where P, is the category of finitely generated projective right R-modules. Previous study of K,R has required not only the Q-construction, but also a model for the loop space of QP,, known as the Y’S-construction. Unfortunately, the S-‘S-construction does not yield a loop space for QfLQ when M is arbitrary. In this paper, two useful models of a loop space for Qu, with no restriction on the exact category L&, are described. Moreover, these constructions are shown to be directly related to the Y’S- construction. The simpler of the two constructions fails to have a certain symmetry property with respect to dualization of the exact category mm. This deficiency is eliminated in the second construction, which is somewhat more complicated. Applications are given to the relative algebraic K-theory of an exact functor of exact categories, with special attention given to the case when the exact functor is cofinal.
    [Show full text]
  • DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
    DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V .
    [Show full text]