# Thomas Blom: Profinite homotopy theory

**Time: **
Tue 2019-02-12 13.15 - 15.00

**Lecturer: **
Thomas Blom

**Location: ** Room 3418, Lindstedtsvägen 25, 4th floor, Department of Mathematics KTH

Abstract: In his article "Profinite homotopy theory", Gereon Quick constructs a model structure on the category of simplicial profinite sets, which can serve as a setting for doing profinite homotopy theory. This model structure is used in the article "Profinite completion of operads and the Grothendieck-Teichmüller group" by Geoffroy Horel, in which he proves that the homotopy automorphism group of the profinite completion of the little 2-discs operad is the profinite Grothendieck-Teichmüller group. In this proof, Horel also uses a model structure on the category of profinite groupoids. In my thesis, I studied these two model categories, filling in some gaps in the proofs by Gereon Quick and Geoffroy Horel. In this talk, I will give an overview of the history of profinite homotopy, and introduce pro-categories along the way. We will then look at profinite groupoids and simplicial profinite sets, and see how the above-mentioned model structures defined by Horel and Quick are constructed.