Homological and homotopic algebra (NMAG562) - information about the course in winter semester 2014/2015.

## Basic information

The contents of the course and other basic information is available in the Student Information System.

The schedule: Tuesday 2.00-3.30pm in the semiar room of the Department of Algebra (to be found also in the Student Information System).

The course will focus on the theory built upon a simple idea of identifying algebraic or geometric objects which intuitively "are the same." Typical instances are the identification of homotopically equivalent spaces or the indentification of chain complexes with isomorphic homology. Several powerful tools have been born from tackling this problem: a calculus of non-commutative fractions, Quillen model categories, Joyal's/Lurie's ∞-categories or Grothendieck derivators. The aim is to introduce the students to the field and the tools, preferably so that they do not immediately drown in the mire of formalism.

Currently, the lecture is focused on three particular goals:

1. explaining techniques for working with localized categories (non-commutative fractions, model categories),
2. explaining the relation of localization of categories to homological algebra (derived categories and functors),
3. if time permits, discussing how to axiomatize homotopy invariant constructions.

The course is related to the course of the same name from summer semester 2009/10. The contents has been, however, substantially revised.

## What has been lectured

A brief overview of what has been taught in individual lectures will be updated below.

October 7, 2014
Motivation and goals, localization of categories and the calculus of left fractions ([Kr], sec. 3; [GZ], chapter I).
October 14, 2014
The calculus of left fractions finished, homotopy categories of complexes ([Kr], sec. 3; [GZ], chapter I; [Ha], sec. I.3; [Ver], sec. I.2).
October 21, 2014
Basic properties of homotopies of maps between complexes and of contractible complexes ([KS], chapter 11).
November 4, 2014
Suspension of a complex, mapping cones, analogies with topology. A proof that the quasi-isomorphisms in a homotopy category of complexes form a multiplicative system ([Wei], chapter 1).
November 18, 2014
Derived functors, the mapping cone as a left derived functor of cokernel ([Gr], chapter 3).
November 25, 2014
The mapping cone as a left derived functor of cokernel (proof finished), localization of categories and natural transformations, the homotopy category of topological spaces ([Gr], chapter 3; [Sw], chapter 2).
December 2, 2014
Weak homotopy equivalences of topological spaces, the long line, CW complexes and Whitehead's theorem, lifting and extending homotopies, weak factorization systems ([Sw], chapters 3 and 5, Theorem 6.32 and Definitions 4.2 and 6.3; [Jo], sec. 15.3; [May], chapter 10; [St], sec. 4; [Fox]).
December 9, 2014
Weak factorization systems continued, λ-sequences and their compositions, relative cell complexes, Quillen's small object argument ([St], sec. 4; [Hov], sec. 2.1; [Hir], sec. 10.5).
December 16, 2014
Model categories, abstract homotopies, basic properties of the homotopy category of a model category ([Hov], sec. 1.1, 1.2, 2.3, 2.4; [Hir], sec. 8.3).
January 6, 2015
The existence and a description of derived functors using model categories, complete cotorsion pairs of complexes and the relation to weak factorization systems, abelian model structures ([Hir], sec. 8.4, 8.5; [St], sec. 5.4, 6.2; [Hov2]; [Hov3]).

## Literature

The lectures are going to be compiled from several sources. I will keep adding references here during the semester, along with updating the what-has-been-lectured list. The literature list for the course from 2010 may also serve as an illustration, but the literature for the present course will certainly differ from that.

 [Fox] R. H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945), 429-432. [GZ] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Springer-Verlag New York, Inc., New York, 1967. [Gr] M. Groth, Selected topics in topology: Derivators, lecture notes under construction, 2014. [PDF] [Ha] D. Happel, Triangulated Categories in the Representation Theory of Finite-dimensional Algebras, LMS Lecture Note Series 119, Cambridge Univ. Press, 1988. [Hir] P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, AMS, Providence, RI, 2003. [Hov] M. Hovey, Model categories, Mathematical Surveys and Monographs 63, AMS, Providence, RI, 1999. [Hov2] M. Hovey, Cotorsion pairs and model categories, Interactions between homotopy theory and algebra, 277-296, Contemp. Math. 436, AMS, Providence, RI, 2007. [Hov3] M. Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), 553-592. [Jo] K. D. Joshi, Introduction to General Topology, John Wiley & Sons, Inc., New York, 1983. [KS] M. Kashiwara, P. Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer-Verlag, Berlin, 2006. [Kr] H. Krause, Localization theory for triangulated categories, in Triangulated Categories, LMS Lecture Note Series 375, Cambridge Univ. Press, 2010. [PDF - preprint] [May] J. P. May, A Concise Course in Algebraic Topology, on-line lecture notes, 2007. [PDF] [Mil] D. Miličić, Lectures on Derived Categories, on-line lecture notes, 2010. [PDF] [St] J. Šťovíček, Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves, in Advances in Representation Theory of Algebras, EMS Series of Congress Reports, 2014. [PDF - preprint] [Sw] R. M. Switzer, Algebraic topology - homotopy and homology, Springer-Verlag, New York-Heidelberg, 1975. [Ver] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque No. 239 (1996). [Wei] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge, 1994.