Parametrized seminar

In one approach to parametrized stable homotopy theory, discussed in [ABGHR] and [ABG], a parametrized spectrum over a space B is an infinity-functor from the infinity groupoid represented by B to an infinity-category of spectra: to each point of B, the functor associates a spectrum; to each path in B, a map of spectra; to each 2-simplex in B, a homotopy relating the composite map along two of the faces to the map associated with the third; and so on. The purpose of this seminar/study circle is to study infinity-category theory with the goal of being able to understand parametrized homotopy theory from this infinity-categorical point of view.

What are we doing?

For the time being, we are taking turns presenting material from [HTT] needed in Appendix B of [ABGHR]. This appendix explains how to construct an infinity-category of parametrized spectra starting with a simplicial model category of spectra. The following table summarizes the references to [HTT] made in the appendix. (Thank you, Richard!)

ReferenceSummarySection
1.1.5.5The simplicial nerve1.1.5: Comparing infinity-categories and simplicial categories
1.1.5.10The simplicial nerve of a simplicial category with Kan morphisms is a quasicategory.1.1.5: Comparing infinity-categories and simplicial categories
1.2.13Limits and colimits1.2.13: Limits and colimits.
4.3Kan extensions4.3: Kan extensions
4.2.3.14Every simplicial set admits cofinal map from the nerve of a category4.2.3: Decomposition of diagrams
4.1.1.8Cofinal maps K->K’ and colimits over K and K’4.1.1: Cofinality
4.2.4.1Homotopy colimits in simplicial categories and colimits in the corresponding infinity categories.4.2.4: Homotopy colimits
4.3.3.7Kan extensions and adjunctions between functor categories4.3.3: Kan extensions along general functors
5.2.2.8Adjunctions and isomorphisms of mor-sets.5.2.2: Adjunctions

Meetings

We are meeting on Fridays 13:15 in room 04.4.01. Here is a list of past and upcoming meetings:

  • April 1: Anssi Lahtinen
    Adjunctions II
  • March 25: (no seminar)
  • March 18: Richard Hepworth,
    Adjunctions I
  • March 4: Anssi Lahtinen,
    Examples of quasicategorical colimits
  • February 25: Richard Hepworth,
    Colimits of colimits
  • February 18: Anssi Lahtinen,
    More homotopy colimits (cont)
  • February 11: Anssi Lahtinen,
    More homotopy colimits
  • February 4: Samik Basu,
    Homotopy colimits
  • January 28: Alexander Berglund,
    Cofinality II
  • January 21 at 14:15: Alexander Berglund,
    Cofinality
  • January 14: Oscar Randal-Williams,
    Orientations and Thom spectra II
  • January 7, 2011: Oscar Randal-Williams,
    Orientations and Thom spectra
  • December 15: Anssi Lahtinen,
    Straightening and Unstraightening III
  • December 3: Anssi Lahtinen,
    Straightening and Unstraightening II
  • November 26: Anssi Lahtinen,
    Straightening and Unstraightening
  • November 19: Richard Hepworth,
    Kan complexes and infinity-groupoids
  • November 16: Richard Hepworth,
    More Left and Right Fibrations
  • November 2: Richard Hepworth,
    Left Fibrations
  • October 26: Alexander Berglund,
    Quasi-categories vs simplicial categories (cont)
  • October 19: Alexander Berglund,
    Quasi-categories vs simplicial categories
  • October 12: Oscar Randal-Williams,
    Comparing infinity-categories and simplicial categories
  • October 5: Anssi Lahtinen,
    Limits and colimits in quasicategories
  • September 29, 2010: Anssi Lahtinen,
    Generalities about infinity-categories and quasicategories.

References

[ABGHR]
Matthew Ando, Andrew J. Blumberg, David J. Gepner, Michael J. Hopkins, and Charles Rezk: Units of ring spectra and Thom spectra. arXiv:0810.4535v3 [math.AT]
[ABG]
Matthew Ando, Andrew J. Blumberg, and David Gepner: Twists of K-theory and TMF. arXiv:1002.3004v2 [math.AT]
[HTT]
Jacob Lurie: Higher Topos Theory. Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009. Available on Jacob Lurie’s homepage.