Messages

Following Mandell-May (2002) and Schwede we will consider orthogonal G-spectra and the associated stable equivariant homotopy category.

I will talk about the stable model structures on sequential and symmetric motivic T-spectra. Here are my notes.

Morel-Voevodsky (1999), Jardine (2000), Hovey (2001).

I will discuss motivic homotopy, motivic cohomology, and motivic sequential T-spectra.

Motivic spaces are defined as Nisnevich- and A^1-localizations of simplicial presheaves on Sm/S.  We study the projective, flasque and injective model structures underlying their homotopy theory.

I will discuss the stable model structures on symmetric and orthogonal spectra, and the lifted model structures on ring and module spectra.

I will discuss Bousfield localization of model structures, and the stable (projective) model structure on sequential spectra.

I will discuss the projective model structure for orthogonal spectra. The main references are Mandell-May-Schwede-Shipley (2001), Hovey (2001) and Hirschhorn (2003).

Sabrina Pauli (sabrinp@student.matnat.uio.no) is the student representative for MAT9580 this term.

I will discuss the projective model structures on sequential and symmetric spectra.  The main references are Hovey-Shipley-Smith (2000), and Hovey (2001).

After discussing closed symmetric monoidal categories, I will talk about monoidal model structures, and prove the Schwede-Shipley theorem about model structures on monoids, modules or algebras in a monoidal model category.  The main reference is Schwede-Shipley (2000).

After explaining Quillen's small object argument, I will discuss cofibrantly generated model structures, including the examples of topological spaces and simplicial sets.

I will talk about Quillen adunction and Quillen equivalences between model categories, and the induced adjunctions and equivalences of their homotopy categories.  Thereafter I will talk about ordinals, cardinals and small objects, leading up to Quillen's small object argument.  References include the books of Hovey (1999), Hirschhorn (2003) and May-Ponto (2012).

Note the modified time and place.

I will say a little more about the Dwyer-Kan simplicial localization, then turn to model structures.  Reference include Dwyer-Spalinski (1995), and Hovey (1999), in additional to the classics by Quillen (1967/69) and Str?m (1972).

I will start with a discussion of some of the problems that model category theory are intended to solve.  This involves the homotopy categories of topological spaces, localization and group completion, simplicial methods and simplicial localizations.  References include Gabriel-Zisman (1967), Dwyer-Kan (JPAA 1980) and Fiedorowicz (1984), and are available in the documents folder for MAT9580.  Here are my rough notes.

Possible sources:

•     Hovey, Mark    Model categories.    Mathematical Surveys and Monographs, 63. American Mathematical Society, Providence, RI, 1999. xii+209 pp.
•     Schwede, Stefan; Shipley, Brooke E.    Algebras and modules in monoidal model categories.    Proc. London Math. Soc. (3) 80 (2000), no. 2, 491-511.
•     Hovey, Mark; Shipley, Brooke; Smith, Jeff    Symmetric spectra.    J. Amer. Math. Soc. 13 (2000), no. 1, 149-208.
•    Jardine, J. F.     Motivic symmetric spectra.     Doc. Math. 5 (2000), 445-5 52.
•     Hovey, Mark    Spectra and symmetric spectra in general model categories.    J. Pure Appl. Algebra 165 (2001), no. 1, 63-127.
•     Mandell, M. A.; May, J. P.; Schwede...