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!)

Reference | Summary | Section |
---|---|---|

1.1.5.5 | The simplicial nerve | 1.1.5: Comparing infinity-categories and simplicial categories |

1.1.5.10 | The simplicial nerve of a simplicial category with Kan morphisms is a quasicategory. | 1.1.5: Comparing infinity-categories and simplicial categories |

1.2.13 | Limits and colimits | 1.2.13: Limits and colimits. |

4.3 | Kan extensions | 4.3: Kan extensions |

4.2.3.14 | Every simplicial set admits cofinal map from the nerve of a category | 4.2.3: Decomposition of diagrams |

4.1.1.8 | Cofinal maps K->K’ and colimits over K and K’ | 4.1.1: Cofinality |

4.2.4.1 | Homotopy colimits in simplicial categories and colimits in the corresponding infinity categories. | 4.2.4: Homotopy colimits |

4.3.3.7 | Kan extensions and adjunctions between functor categories | 4.3.3: Kan extensions along general functors |

5.2.2.8 | Adjunctions 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.