サイトマップお問合せヘルプ
シラバス
  • 授業一覧から探す
    • 教養教育院
    • 文学部・人文学研究科
    • 文学研究科
    • 国際言語文化研究科
    • 教育学部・教育発達科学研究科
    • 法学部・法学研究科
    • 経済学部・経済学研究科
    • 情報学部/情報学研究科
    • 情報文化学部
    • 情報科学研究科
    • 理学部・理学研究科
    • 医学部・医学系研究科
    • 工学部・工学研究科
    • 農学部・生命農学研究科
    • 国際開発研究科
    • 多元数理科学研究科
    • 環境学研究科
    • 創薬科学研究科
    • 国際教育交流センター
    • 国際言語センター
    • ※平成29年度学生募集停止
    • オープンキャンパス
    • 名大の研究指導
    • G30 for everyone
    • 名古屋大学ラジオ公開講座
    • 退職記念講義アーカイブ
    • 退職記念講義2016
    • nuocwをフォローしましょう
    • 過去の特集ページ
    • 教員の方へ
    • NU OCW Podcast
      RSS を iTunes の "Podcast" にドラッグ&ドロップすると、ポッドキャストが登録されます。
      (iTunesは最新版をお使いください)
  1. ホーム >
  2. 多元数理科学研究科 >
  3. トポロジー特論 I >
  4. シラバス
授業ホームシラバス講義資料

授業の目標

To every category C, we associate a topological space BC. The space BC is called the classifying space of C. A functor f: C to D gives rise to a continuous map Bf: BC to BD. Moreover, a natural transformation from the functor f: C to D to the functor g: C to D gives rise to a homotopy Balpha: BC times [0,1] to BD from the map Bf to the map Bg. In short, the classifying space construction gives rise to a 2-functor from the 2-category of categories to the 2-category of topological spaces. In this way, properties of categories are reflected in the homotopy type of their classifying spaces.

The classifying space is constructed by gluing together simplices
Definition of Delta of n
The general recipe for constructing a topological space by gluing together simplices is called a simplicial set. The resulting topological space is called the geometric realization of the simplicial set. The first part of the course will focus on simplicial sets and their geometric realization along with the basic category theoretical notions of limits and colimits and adjoints functors which are needed to develop this theory.

The next part of the course focuses on homotopy theory. We introduce homotopy groups and define a continuous map between topological spaces to be a weak equivalence if it induces an isomorphism of the associated homotopy groups. The homotopy category of topological spaces to be the category obtained from the category of topological spaces and continuous maps by formally introducing an inverse map for every weak equivalence. The main techniques for studying the homotopy category are centered around two classes of maps called the fibrations and the cofibrations. The category of topological spaces together with the three classes of maps given by the weak equivalences, the fibrations, and the cofibrations form a model category. In homotopy theory, theorems live in the homotopy category, but their proofs live in the model category.

The final part of the course uses the techniques we have developed to define algebraic K-theory. We prove the so-called additivity theorem from which many of the basic properties of algebraic K-theory are readily derived.

キーワード

Homotopy theory, model categories, algebraic K-theory.

履修に必要な知識

An introductory course in algebraic topology including the fundamental group and covering spaces.

教科書・参考書

The course lecture notes. The following texts are also useful:

  • Mark Hovey, Model Categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society.
  • Daniel G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, New York.
  • Friedhelm Waldhausen, Algebraic K-theory of spaces, Lecture Notes in Mathematics, vol. 1126, Springer-Verlag, New York.

参考資料

The course lecture notes.

Mark Hovey, Model Categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society.

Daniel G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, New York.

Friedhelm Waldhausen, Algebraic K-theory of spaces, Lecture Notes in Mathematics, vol. 1126, Springer-Verlag, New York.

課題

スケジュール

講義内容
1 The classifying space of a category.
2 The geometric realization of a simplicial set.
3 Limits and colimit; filtered colimits and finite limits commute.
4 Adjoint functors, limits and colimits.
5 Characterizing the geometric realization by maps from it.
6 Geometric realization preserves finite products; k-spaces.
7 Homotopy groups; mapping fiber; long-exact sequence.
8 The classifying space of a finite group. (I recommend to skip this lecture.)
9 Weak equivalences, Serre fibrations, Serre cofibrations; Quillen model categories.
10 The homotopy category; Quillen functors.
11 Reedy model structure on the category of simplicial spaces; geometric realization is a left Quillen functor.
12 Bi-simplicial sets and their geometric realization; geometric realization and weak equivalences; Quillen's Theorem A and B.
13 Algebraic K-theory.
14 The additivity theorem.
15 Hochschild homology of a ringoid: An introduction.

成績評価の方法

Occasional exercises reviewed by the teacher.

最終更新日:2009年05月28日
最終更新日の時点の講義内容で公開しております。
現在、この講義は開講されていません。

ページトップへ

http://ocw.nagoya-u.jp/