Vorlesung: Ausgewählte Themen aus der Topologie (Master): Model categories
Birgit Richter, email: richter at math.uni-hamburg.de
Outline: Model categories are everywhere. Whenever you have a situation where you want to construct an associated homotopy category to your favorite category, model structures are the right tool for doing so. The definition of model categories goes back to Quillen, and by now they are widely used tool in very different areas of mathematics.
I'll start by presenting the basic concept of model category structures and the construction of the associated homotopy category. After that we will cover some standard examples (chain complexes, topological spaces,...). In these examples and others we will see the need for additional tools and concepts that allow us to transfer model structures, to prove additional properties, to establish equivalences of the associated homotopy categories and so on.
My aim is to describe some more advanced results and examples during the lecture course, leading up to current open problems in research.
  • William G. Dwyer, Jan Spalinski, Homotopy theories and model categories, Handbook of algebraic topology, 73-126, North-Holland, Amsterdam, 1995.
  • Mark Hovey, Model categories, Mathematical Surveys and Monographs, 63. American Mathematical Society, Providence, RI, 1999. xii+209 pp.
  • William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs, 113. American Mathematical Society, Providence, RI, 2004. viii+181 pp.
  • Paul G. Goerss, John F. Jardine, Simplicial homotopy theory, Reprint of the 1999 edition. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2009. xvi+510 pp.
  • Paul G. Goerss, Kristen Schemmerhorn, Model categories and simplicial methods Interactions between homotopy theory and algebra, 3-49, Contemp. Math., 436, Amer. Math. Soc., Providence, RI, 2007.
  • Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, 99. American Mathematical Society, Providence, RI, 2003. xvi+457 pp.
  • J. Peter May, Kate Ponto, More concise algebraic topology, Localization, completion, and model categories, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2012. xxviii+514 pp.
  • Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43 Springer-Verlag, Berlin-New York 1967 iv+156 pp.
Exam: The final exam for this course is an oral exam at the end of term. In order to qualify for the exam, you should take an active part in the exercise sessions.
When and where: Tu, Fr 12-14h, H4. Exercise class: Tu 14-16h, 432.
  1. Show that retracts of isomorphisms are isomorphisms.
  2. Prove that in any model structure fibrations are precisely the maps with the RLP with respect to acyclic cofibrations.
  3. Show that every category has a model structure with the isomorphisms being the weak equivalences.
  4. Verify that the opposite category of a model category is a model category.
  5. Is the inclusion functor from a subcategory to its ambient category always faithful?
  6. Construct a cylinder object in the model category of non-negatively graded chain complexes such that left homotopies in the model category sense correspond to chain homotopies.
  7. Prove the exponential law in chain complexes, i.e. show that the category of chain complexes is closed symmetric monoidal.
  8. Show that for any associative ring R, retracts of monos (epis) of R-module maps are monos (epis).
  9. Identify the R-module of chain maps from \(\mathbb{S}^n\) to a chain complex \(C_*\) with the n-cycles of \(C_*\).
  10. What does the R-module of chain maps from the n-disk to \(C_*\) correspond to? Cover also the case of the n-disk on an R-module \(M\), \(\mathbb{D}^n(M)\).
  11. Show that the fibrations in the (projective) model structure on non-negatively graded chain complexes are precisely the maps with the RLP wrt the set of maps \(0 \rightarrow \mathbb{D}^n\) for \(n \geq 1\). Similarly, prove that the acyclic fibrations are exactly the morphisms with the RLP wrt the inclusions of spheres into disks.
  12. Let \(T \subset X\) be a cofinal subset of a totally ordered set \(X\) and let \(F\colon X \rightarrow \mathcal{C}\) be a functor with \(\mathcal{C}\) cocomplete. Prove that the natural map \(\mathrm{colim}_T F \rightarrow \mathrm{colim}_X F\) is an iso.
  13. Convince yourself that the statements in 8.11 are actually true.
  14. What is the cardinality of \(\Delta([m],[n])\)?
  15. Show that \(\Delta_n\) is not a Kan complex for positive \(n\).
  16. Prove that every simplicial group is a Kan complex.
  17. Assume that \(X\) is a Kan complex. Is then \(\pi_0(X)\) bijective to \(\pi_0(|X|)\)?
  18. Prove that the category of simplicial sets is closed symmetric monoidal.
  19. Show that there is a homotopy from the identity on \(\Delta_n\) to the constant map at \(n\) but not vice versa.
  20. Assume that \(\mathcal{M}\) is a category satisfying the assumptions of 11.5. What is the simplicial structure on the category of cosimplicial objects in \(\mathcal{M}\)?
  21. Are the model categories from Theorem 12.3. automatically cofibrantly generated?
  22. Show that the Moore complex is isomorphic to the direct sum of the normalized chains and the subcomplex of degenerate elements.
  23. On Jan 21 we'll talk about the paper: Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories, Algebraic & Geometric Topology 3 (2003) 287--334.
  24. Make sure that you understand the free functors from chain complexes to dgas and dgcas.
  25. Show that the category of rational differential (non-negatively) graded commutative algebras has a model structure that is created by the forgetful functor.