Homotopical Algebra Yuri Berest, Sasha Patotski Fall 2015 These are lecture notes of a graduate course MATH 7400 currently taught by Yuri Berest at Cornell University. The notes are taken by Sasha Patotski; the updates will appear every week. If you are reading the notes, please send us corrections; your questions and comments are also very much welcome. Emails:
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[email protected]. ii Contents 1 Homotopy theory of categories 1 1 Basics of simplicial sets . .1 1.1 Definitions . .1 1.2 Motivation . .3 1.3 Examples of (co-)simplicial sets . .5 1.4 Remarks on basepoints and reduced simplicial sets . .7 1.5 The nerve of a category . .8 1.6 Examples of nerves . .9 1.7 One example: the nerve of a groupoid . 11 2 Geometric realization . 12 2.1 General remarks on spaces . 12 2.2 Definition of geometric realization . 14 2.3 Two generalizations of geometric realization . 16 2.4 Kan extensions . 17 2.5 Comma categories . 20 2.6 Kan extensions using comma categories . 22 3 Homotopy theory of categories . 24 3.1 The classifying space of a small category . 24 3.2 Homotopy-theoretic properties of the classifying spaces . 26 3.3 Connected components . 28 3.4 Coverings of categories . 28 3.5 Explicit presentations of fundamental groups of categories . 30 3.6 Homology of small categories . 31 3.7 Quillen's Theorem A . 34 3.8 Fibred and cofibred functors . 37 3.9 Quillen's Theorem B . 39 2 Algebraic K-theory 41 1 Introduction and overview .