DOCSLIB.ORG

Recursive Domains, Indexed Category Theory and Polymorphism

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Recursive Domains, Indexed Category Theory and Polymorphism Recursive Domains, Indexed Category Theory and Polymorphism Paul Taylor 1983{7 ii A dissertation submitted for the degree of Doctor of Philosophy at the University of Cambridge also for the Summer 1986 Prize Fellowship Competition at Trinity College, Cambridge c copyright Paul Taylor, August 1986 I hereby declare that my thesis entitled Recursive Domains, Indexed Category Theory and Polymorphism is not substantially the same as any that I have submitted for a degree or diploma or other qualification at any other University. I further state that no part of thesis has already been (or is being) concurrently submitted for any such degree, diploma or other qualification. Further, I declare that the results presented herein are the product of my own original research where claimed as such. This qualification will be explained in the Introduction. During the course of my research I have been supported financially by the Science and Engineering Research Council, Trinity College and the Department of Pure Mathematics and Mathematical Statistics. This dissertation (in particular Chapter V) has been revised somewhat since submission. iii This work is dedicated with love to the guardians of my sanity James Annett Mark Dawson Ellie Mayger (n´ee Clarke) Mike Mayger Fiona Miller Shane Voss Ian White without whose support it would never have happened. iv Introduction Let me begin by confessing that this is a very eccentric document. It aims to do two things. First, of course, it sets out some original constructions, primarily of a \type of types" in certain categories whose objects have come to be known (rather unimaginatively) as domains, but also of various other things in such categories. Second, however, which is a departure from custom as far as Ph.D. or fellowship dissertations are concerned, it attempts to give an introduction to a substantial but largely undocumented body of knowledge among (certain) Mathematicians, namely Categorical Logic, which has very recently been found to be of considerable importance in Theoretical Computer Science. Category Theory, I like to say, is a tourist subject. It started in Group Theory, but was only taken seriously when Frank Adams needed it to discuss constructions in Topology; after that it visited Linear Algebra, Universal Algebra, Automata Theory and Algebraic Geometry. At the last of these it met Logic and turned into Topos Theory. The journey to that stage was undertaken (almost) entirely in the company of mathematicians, who could perhaps be expected to understand the significance of the souvenirs, and in particular the geometrical language. Now that it has arrived in Computer Science, however, I feel that some basic discussion of what it's all about is called for. This is what I have been trying to do. I have a strong conviction that when one has begun to understand something which is un- derstood by few other people then it is one's duty to formulate on paper that understanding (in whatever terms one has found useful) for the benefit of others trying to do likewise. Moreover I have an habitual urge to do so. This is my defence for the length and verbosity of this work. This policy also requires me to apologise to the originators of the results, and other practitioners of the subject far more expert than myself, for tiring them with the necessity of reading my misrepresentations. This I freely do. I have also included a lot of standard Domain Theory, starting with the Lambda Calculus. This is not a thesis about the Lambda Calculus! This is because I disagreed with some of the existing terminology and because I wished to push the results as far as I could. Also, in Chapter V I wanted to be able to assume that a category of domains was of the form Retr(Λ); this explains my preference for continuous rather than algebraic domains. Since much of the bulk of the work is therefore \standard" material, a notation is called for to indicate what is original. This is done by means of a \+" before the number of the section. The other sections I claim for the most part to be an original account of their subject, though occasionally I have marked sections \ " by way of confession of having forgotten the proof of the result in question. In some cases originality− means improvement (or in a few cases rediscovery) of an existing result; here credit has been given in apparent contradiction to the mark on the section. Certain results in this work have been designated as Facts. (Specifically in 1.1.13, 1.2.11, 2.2.11, 2.5.8.) This indicates an original result which I have not had time to writexx up. The most substantial of these is a technique for dealing with large filtered diagrams in a small category (where we just say \cardinality" in a poset), the proof of which would have taken ten pages. [Postscript: This result is a factorisation system which is similar to that of any functor into a final functor followed by a discrete fibration [Street & Walters, 1973?]. The difference is that the first part of my factorisation is also regular epi whilst the second is characterised by a condition I called cosignposted; the latter places a bound on the size of the intermediate category in terms of that of the codomain, whilst the former retains the property of final functors that v vi they yield equivalent filtered colimits. The final-fibration factorisation for functors which preserve connected limits is the same as that giving Diers' spectrum and Berry's trace.] I also have another result whose proof is of a similar length but which is not mentioned in the text. This is that an interpretation by means of partial maps can be given to s simple imperative language (including a while construct) in a category with certain finitary exactness conditions. Specifically, we use equalisers of a form in which one of the maps is a mono to interpret the while; the important stage is the proof (in an elementary topos, though the category itself does not need higher order logic) that a certain graph has the diamond property and hence that the map from the equaliser to the coequaliser of a pair of maps one of which is mono is mono. There is no mention of anything of an infinitary nature in this proof: it uses the adjoint functor theorem in the way in which 4.4 tries to indicate that it can be used to replace infinitary arguments. The history ofx Computer Science from Babbage onwards is one of generalisation and abstrac- tion. Foremost among the ideas are those of subroutines and high-level (i.e. machine-independent) languages, enabling the essence (algorithm) of some process to be expressed without reference to either its application or implementation. As programs become ever more complicated, it becomes necessary to provide formal rather than ad hoc techniques for understanding them and prov- ing them correct. This is the r^oleof Mathematics, specifically Category Theory, in theoretical Computer Science. The major languages which were available for programming until the end of the 1960s were imperative, which is to say they set out precisely how | and in particular in what order | the operations were to be performed. The exception to this was LISP. This was nominally based on the λ-calculus, a mathematical tool for the study of recursion with its origins in symbolic logic. LISP and modern theoretical languages are called functional because they express only the behaviour of the program as a function from inputs to outputs, abstracting away details of sequencing just as earlier high level languages had hidden space allocation. This change shifts the emphasis as regards the meaning of the program from the operational to the mathematical viewpoint. It also places functions on the same footing as data items. Also, whereas the imperative style provided recursion (inductive definition) by means of program loops or jumps, the functional style expresses the function as a solution of an equation. This is always of a specific form, a fixed point, so for instance the factorial function is the fixed point of 1 if n = 0 f n 7! 7! n f(n 1)otherwise × − which is a function N N. Since we have placed no restriction on the nature of this function (and cannot, because of the!Halting Problem), the domains on which it is defined cannot be ordinary sets. Another feature of some of the modern languages is polymorphism, which is the abstraction of the function away from the types of its arguments; for instance the same code may be used to calculate determinants of matrices with either real or complex entries. Stronger forms of polymorphism may also be contemplated, as far as asking for a type of all types. It is clear that concepts such as these cannot be expressed straightforwardly in terms of discrete sets. We can express them in the same way as is customary in existing imperative programming practice, namely operationally. However to do this rigorously we more or less have to invent or describe a particular machine, and this immediately loses the abstraction which we have tried to gain. In the absence of recursion there are mathematical interpretations of functional languages and even polymorphism naturally provided by category theory. The former is clearly described by a cartesian closed category, which has an exponential or function space operation. The determinant example above, which is uniform over the ring definition, is reminiscent of the naturality of the double dual in Linear Algebra which led Eilenberg and Mac Lane to category theory. In fact this example can be used to motivate the idea of a classifying topos.
Load more

Recommended publications
  • 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]
  • 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]
  • Part II Topological Dualities
    Part II Top ological dualities Chapter Top ology and armative predicates In the rst part of this monograph we considered predicates to be subsets of an abstract set of states If we think of the states as the denotations of results of computations of programs then predicates b ecome computationally mean ingful in the sense that we can use partial information ab out a computation to tell whether or not a predicate holds for that computation A predicate for which only nite information ab out a computation is needed to arm whether it holds is called an armative predicate The set of armative predicates is closed under nite intersections and ar bitrary unions Hence armative predicates can be identied with the op en sets of a top ological space The idea that op en sets are observable predi cates was prop osed by Smyth in although it is also brie y mentioned in Smyth interprets op en sets as semidecidable prop erties in some eectively given top ological space More generally op en sets can be inter preted as nitely observable predicates Alp ern and Schneider and Kwiatkowska use op en sets as nite liveness predicates and closed sets as safety predicates to formalize the informal characterization of liveness and safety prop erties of Lamp ort The name armative predicates has b een intro duced by Vickers for denoting the abstract op en sets of a frame Armative predicates are also called veriable predicates by Rewitzky who uses the term observable for predicates which are b oth armative and refutative Bonsangue In this chapter we intro
    [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]
  • Spatio-Temporal Domains: an Overview David Janin
    Spatio-temporal domains: an overview David Janin To cite this version: David Janin. Spatio-temporal domains: an overview. 2018. hal-01634897v2 HAL Id: hal-01634897 https://hal.archives-ouvertes.fr/hal-01634897v2 Preprint submitted on 17 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spatio-temporal domains: an overview David Janin? UMR LaBRI, Bordeaux INP Université de Bordeaux [email protected] Abstract. We consider the possibility of defining a general mathemat- ical framework for the homogeneous modeling and analysis of hetero- geneous spatio-temporal computations as they occur more and more in modern computerized systems of systems. It appears that certain fibra- tions of posets into posets, called here spatio-temporal domains, eventu- ally provide a fully featured category that extends to space and time the category of cpos and continuous functions, aka Scott Domains, used in classical denotational semantics. 1 Introduction Research context. Program semantics is classically divided between two com- plementary approaches : denotational semantics and operational semantics. De- notational semantics generally refers to what the partial functions encoded by programs are : what is the relationship between (models of) their input val- ues (or input memory state) and their output values (or output memory state).
    [Show full text]
  • On the Categorical Semantics of Elementary Linear Logic
    Theory and Applications of Categories, Vol. 22, No. 10, 2009, pp. 269{301. ON THE CATEGORICAL SEMANTICS OF ELEMENTARY LINEAR LOGIC OLIVIER LAURENT Abstract. We introduce the notion of elementary Seely category as a notion of cate- gorical model of Elementary Linear Logic (ELL) inspired from Seely's de¯nition of models of Linear Logic (LL). In order to deal with additive connectives in ELL, we use the ap- proach of Danos and Joinet [DJ03]. From the categorical point of view, this requires us to go outside the usual interpretation of connectives by functors. The ! connective is decomposed into a pre-connective ] which is interpreted by a whole family of functors (generated by id, ­ and &). As an application, we prove the strati¯ed coherent model and the obsessional coherent model to be elementary Seely categories and thus models of ELL. Introduction The goal of implicit computational complexity is to give characterizations of complexity classes which rely neither on a particular computation model nor on explicit bounds. In linear logic (LL) [Gir87], the introduction of the exponential connectives gives a precise status to duplication and erasure of formulas (the qualitative analysis). It has been shown that putting constraints on the use of exponentials permits one to give a quantitative analysis of the cut elimination procedure of LL and to de¯ne light sub-systems of LL characterizing complexity classes (for example BLL [GSS92], LLL [Gir98] or SLL [Laf04] for polynomial time and ELL [Gir98, DJ03] for elementary time). In order to have a better understanding of the mathematical structures underlying these systems, various proposals have been made in the last years with the common goal of de¯ning denotational models of light systems [MO00, Bai04, DLH05, LTdF06, Red07].
    [Show full text]
  • Compact Topologies on Locally Presentable Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 38, No 3 (1997), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES PANAGIS KARAZERIS Compact topologies on locally presentable categories Cahiers de topologie et géométrie différentielle catégoriques, tome 38, no 3 (1997), p. 227-255 <http://www.numdam.org/item?id=CTGDC_1997__38_3_227_0> © Andrée C. Ehresmann et les auteurs, 1997, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIER DE TOPOLOGIE ET J Illume XXXVIII-3 (1997) GEOMETRIE DIFFERENTIELLE CATEGORIQUES COMPACT TOPOLOGIES ON LOCALLY PRESENTABLE CATEGORIES by Panagis KARAZERIS RESUME. Les topologies sur les categories localement pr6sentables g6n6rallsent les notions famill6res suivantes, d’une part les topologies de Grothendieck sur des petites categories, d’autre part les topologies de Gabriel sur des categories abéliennes à generateurs. Dans cet article on introduit une condition, qui peut être v6rifi6e pour les topologies pr6c6dentes, appel6e "compacit6". Dans le cas des topologies de Grothendieck, cette condition signifie qu’un recouvrement quelconque a un sous-recouvrement fini. Les topologies compactes correspondantes ont des localisations ferm6es dans la cat6gorie donnee pour des colimites filtrantes monomorphiques. On examine aussi la fermeture des objets s6par6s et des faisceaux pour les colimites filtrantes. Les topologies compactes sur une cat6gorie localement de presentation finie forment un locale. Si cette cat6gorie est un topos coherent, alors le locale est compact et localement compact.
    [Show full text]
  • Quillens Q-Construction
    Topics in Algebraic topolgy Talk: Quillens Q-construction Julie Zangenberg Rasmusen 02-01-2019 Contents 1 The Q-construction 1 1.1 Quillens Q-construction . .1 1.2 An 1-categorical Q-construction . .3 2 Higher algebraic K-theory 6 2.1 Introduction . .6 2.2 The Devissage theorem . .8 Denition 0.1. Let F : C!D be a functor and d 2 obD a xed object. Then we dene a new category F=d which consist of pairs (c; u) where u : F (c) ! d with c 2 obC, in which morphisms (c; u) ! (c0; u0) is a map v : c ! c0 such that the square F (c) u / d F (v) F (c0) / d u0 commutes. Theorem 0.2 (Theorem A). Let F : C!D be a functor and d 2 obD a xed object. Then if the category F=d is contractible for every object d 2 obD, then the functor F is a homotopy equivalence. 1 The Q-construction 1.1 Quillens Q-construction Assume that C is an exact category. First of all we wish to dene a new category QC called Quillens Q-construction. QC has the same objects as C, i.e obQC = obC and we dene the morphisms in the following way: 1 1.1 Quillens Q-construction 1 THE Q-CONSTRUCTION Let c0; c1 2 obC and consider all diagrams of the form p r c0 o o c01 / / c1 (1) in C with p an admissible epimorphism and r an admissible monomorphism. We will say that p r p 0 r c0 o o c01 / / c1 ∼ c0 o o c01 / / c1 if and only if there exist an isomorphism 0 which makes the following diagram γ : c01 ! c01 commute: p c o o c / r / c 0 a a 01 = 1 γ p0 = r0 0 c01 A morphism f : c0 ! c1 in QC is all diagrams (1) up to the above equivalence.
    [Show full text]
  • Introduction to Coherent Spaces
    Introduction to coherent spaces Arnold Neumaier Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria email: [email protected] WWW: http://www.mat.univie.ac.at/~neum arXiv:1804.01402 September 28, 2018 Abstract. The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept. Coherent spaces provide a setting for the study of geometry in a different direction than traditional metric, topological, and differential geometry. Just as it pays to study the properties of manifolds independently of their embedding into a Euclidean space, so it appears fruitful to study the properties of coherent spaces independent of their embedding into a Hilbert space. Coherent spaces have close relations to reproducing kernel Hilbert spaces, Fock spaces, and unitary group representations, and to many other fields of mathematics, statistics, and physics. This paper is the first of a series of papers and defines concepts and basic theorems about coherent spaces, associated vector spaces, and their topology. Later papers in the series dis- cuss symmetries of coherent spaces, relations to homogeneous spaces, the theory of group representations, C∗-algebras, hypergroups, finite geometry, and applications to quantum physics. While the applications to quantum physics were the main motiviation for develop- ing the theory, many more applications exist in complex analysis, group theory, probability theory, statistics, physics, and engineering. For the discussion of questions concerning coherent spaces, please use the discussion forum https://www.physicsoverflow.org.
    [Show full text]
  • Objects, Interference, and the Yoneda Embedding
    Syracuse University SURFACE College of Engineering and Computer Science - Former Departments, Centers, Institutes and College of Engineering and Computer Science Projects 1995 Objects, Interference, and the Yoneda Embedding Peter W. O'Hearn Syracuse University Uday S. Reddy University of Illinois at Urbana-Champaign Follow this and additional works at: https://surface.syr.edu/lcsmith_other Part of the Programming Languages and Compilers Commons Recommended Citation O'Hearn, Peter W. and Reddy, Uday S., "Objects, Interference, and the Yoneda Embedding" (1995). College of Engineering and Computer Science - Former Departments, Centers, Institutes and Projects. 14. https://surface.syr.edu/lcsmith_other/14 This Article is brought to you for free and open access by the College of Engineering and Computer Science at SURFACE. It has been accepted for inclusion in College of Engineering and Computer Science - Former Departments, Centers, Institutes and Projects by an authorized administrator of SURFACE. For more information, please contact [email protected]. Electronic Notes in Theoretical Computer Science to app ear Ob jects Interference and the Yoneda Emb edding Peter W OHearn Syracuse University Uday S Reddy University of Il linois at UrbanaChampaign Dedicated to John C Reynolds in honor of his th birthday Abstract We present a new semantics for Algollike languages that combines metho ds from two prior lines of development the ob jectbased approach of where the meaning of an imp erative program is describ ed in terms of sequences of observable
    [Show full text]
  • LINEAR LOGIC, -AUTONOMOUS CATEGORIES and COFREE COALGEBRAS1 R.A.G. Seely Girard 1987]
    LINEAR LOGIC AUTONOMOUS CATEGORIES 1 AND COFREE COALGEBRAS RAG Seely ABSTRACT A brief outline of the categorical characterisation of Girards linear logic is given analagous to the relationship b etween cartesian closed cat egories and typed calculus The linear structure amounts to a autonomous category a closed symmetric monoidal category G with nite pro ducts and a closed involution Girards exp onential op erator is a cotriple on G which carries the canonical comonoid structure on A with resp ect to cartesian pro duct to a comonoid structure on A with resp ect to tensor pro duct This makes the Kleisli category for cartesian closed INTRODUCTION In Linear logic JeanYves Girard introduced a logical system he describ ed as a logic b ehind logic Linear logic was a consequence of his analysis of the structure of qualitative domains Girard he noticed that the interpretation of the usual conditional could b e decomp osed into two more primitive notions a linear conditional and a unary op erator called of course which is formally rather like an interior op erator X Y X Y The purp ose of this note is to answer two questions and p erhaps p ose some others First if linear category means the structure making valid the prop ortion linear logic linear category typed calculus cartesian closed category then what is a linear category This question is quite easy and in true categorical spirit one nds that it was answered long b efore b eing put namely by Barr Our intent here is mainly to supply a few details to make the matter more precise
    [Show full text]
  • (1) Weibel, C. Intro to Algebraic K-Theory, In-Progress, but Available Online; (2) Quillen, D
    ALGEBRAIC K-THEORY BERTRAND GUILLOU 1. The Q construction 1.1. Introduction References: (1) Weibel, C. Intro to Algebraic K-theory, in-progress, but available online; (2) Quillen, D. "Higher algebraic K-theory: I" in Springer LNM v.341, 1973; (3) Grason, D. \Higher algebraic K-theory: II" in Springer LNM v.551, 1976; (4) Srinivas, V. Algebraic K-Theory, Second Edition, Birkhauser, 1996. We have seen a definition of the higher K-groups of a ring as the homotopy groups of a space: + Ki(R) = πi(BGl(R) K0(R)): × Quillen was able to calculate the K-theory of finite fields with this definition, and Borel calculated the ranks of the rational K-groups of the ring of integers in a number field (=finite extension of Q). On the other hand, one would like to extend some of the fundamental structure theorems from classical K-theory to the higher K-groups, and so we need a more general construction. 1.2. Exact Categories Definition 1. An exact category is an additive category C with a family of sequences E j 0 B i C D 0 ! −! −! ! such that there is an embedding of C as a full subcategory of some abelian category satisfying A (1) A sequence of the above form in C is in if and only if the sequence is a short exact sequence in . E A (2) C is closed under extensions in (i.e. if we have a short exact sequence 0 B A ! ! C D 0 in with B; D C then C C , up to isomorphism).
    [Show full text]