Bøger af Peter Gabriel

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The main purpose of the present work is to present to the reader a particularly nice category for the study of homotopy, namely the homo- topic category (IV). This category is, in fact, - according to Chapter VII and a well-known theorem of J. H. C. WHITEHEAD - equivalent to the category of CW-complexes modulo homotopy, i.e. the category whose objects are spaces of the homotopy type of a CW-complex and whose morphisms are homotopy classes of continuous mappings between such spaces. It is also equivalent (I, 1.3) to a category of fractions of the category of topological spaces modulo homotopy, and to the category of Kan complexes modulo homotopy (IV). In order to define our homotopic category, it appears useful to follow as closely as possible methods which have proved efficacious in homo- logical algebra. Our category is thus the" topological" analogue of the derived category of an abelian category (VERDIER). The algebraic machinery upon which this work is essentially based includes the usual grounding in category theory - summarized in the Dictionary - and the theory of categories of fractions which forms the subject of the first chapter of the book. The merely topological machinery reduces to a few properties of Kelley spaces (Chapters I and III). The starting point of our study is the category,10 Iff of simplicial sets (C.S.S. complexes or semi-simplicial sets in a former terminology).

• Are you at that point in life where you are beginning to wonder and ask yourself'Why am I here? Am I suppose to be doing what I am doing?• Are you frustrated that you are not making headway in life?• Have you come to a crossroad in your life and tired of going round and round the circle of life?• Do you think you can do more than what you are doing right now?Your answer to some of these questions might be no or yes, but let me tell you; we develop a detachment from who we really are, to who we become over the years of our development, we grow out of our true identity through the everyday stress of life. The Truth About You, goes through the stages of how to identify your roadmap to locating your reason for existence. In this enlightening book, you will find out how other successful people have used these principles and guidelines to navigate their way and how you can use them as a guide to locating your purpose and make a great impact on your generation.

The main purpose of the present work is to present to the reader a particularly nice category for the study of homotopy, namely the homo topic category (IV). It is also equivalent (I, 1.3) to a category of fractions of the category of topological spaces modulo homotopy, and to the category of Kan complexes modulo homotopy (IV).

Der heutige Hochschulunterricht für Mathematiker gründet meist auf Abstraktion und führt vom Allgemeinen zum Speziellen. Die Methode hat Vorteile, sie stärkt das Denkvermögen und meidet lästige Wiederholungen. Doch sie "stellt den Pflug vor die Ochsen", weil Abstraktion auf Spezialfälle baut, die dem Lernenden oft fremd sind. So bleibt der Erfolg den Glücklichen vorbehalten, die den Weg von der Abstraktion zu den Beispielen finden. Dieses Lehrbuch führt von zwei Spezialfällen zur Allgemeinheit und gründet nicht auf Abstraktion. Die Beweise der abstrakten Algebra werden zuerst am konkreten Beispiel der Matrizen vorgeführt. Zur Schärfung der Anschauung wird dann die Begriffswelt der Elementargeometrie durchleuchtet. Die Auseinandersetzung mit dem Lehrstoff der Schule dient der Vorbereitung auf die geometrisch gefärbte Sprache der linearen Algebra, die am Ende des Buches erläutert wird. Dem Text sind Anwendungsbeispiele und zahlreiche historische Kommentare beigefügt.

[Gabriel and Roiter] are pioneers in this subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature.