# University of Regina Topology Seminar

## Fall 2019

### Topic: Models for homotopy theories.

Time: Wednesdays 4:30 - 6:00 PM, unless otherwise noted.
Location: CL 251

In this learning seminar, we will study different models in which one can do homotopy theory, starting from Quillen's seminal work on model categories. We will discuss examples and applications, such as computing homotopy (co)limits.

The lecture series will be interspersed with guest research talks.

Date Speaker Title and abstract References
September 11 Martin Frankland Introduction to model categories

In classical homotopy theory, one can look at continuous maps up to homotopy and consider two topological spaces as essentially the same if they are homotopy equivalent. Model categories, introduced by Quillen, allow us to make similar homotopical constructions in a much broader context. I will present the definition and some basic properties. I will then focus on the examples of topological spaces and chain complexes.

W.G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology (1995).
September 18 Martin Frankland Model categories, part 2

I will describe the homotopy category of a model category and how it forms a localization. I will then say more about the model structures on spaces, and come to the example of chain complexes.

September 25 Martin Frankland Model categories, part 3

I will say more about the model structures on topological spaces and on chain complexes. We will see that homological algebra is a form of homotopical algebra.

October 2 Larry So The equivalence between simplicial sets and topological spaces

I will talk about the model structure on simplicial sets and the equivalence between the homotopy theory of simplicial sets and topological spaces.

P.G. Goerss and J.F. Jardine, Simplicial homotopy theory (1999), Chapter I.
October 9 Paul Songhafouo Tsopméné Simplicial categories and the hammock localization

Given a category C and a class W of "weak equivalences" in C, one can construct a new category $C[W^{-1}]$ which as the same objects as C and is obtained from C by formally inverting the maps of W. As shown by Dwyer-Kan, the category $C[W^{-1}]$ reflects just one aspect of a much richer object, the simplicial localization LC, which is a simplicial category. Because it is difficult to get a hold on the homotopy type of the simplicial set LC(X, Y), for X, Y in C, Dwyer-Kan considered a homotopy variation of LC, the hammock localization $L^H C$. In this talk I will recall the notion of simplicial category, and explain the construction of the hammock localization. I will also go over some properties of this localization.

W.G. Dwyer and D.M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980).
October 16   No seminar.
October 23 Martin Frankland Classification complexes and obstruction theory

October 30 Paul Songhafouo Tsopméné Simplicial model categories

November 6   Fall Break. No seminar.
November 13 Don Stanley TBA

November 20 TBA

November 27 TBA

December 4 TBA