Homotopy category
Supervisor: Christian Schlichtkrull, email: christian.schlichtkrull math.uib.no
Main content
Prerequisites: MAT242.
Description: In this project we shall study the homotopy category of topological spaces. The homotopy category is what one obtains if, instead of looking at continuous maps between topological spaces, one considers homotopy classes of continuous maps. This category plays a fundamental role in algebraic topology since many questions about topological spaces can be answered by analysing such homotopy classes of maps.
There are two types of continuous maps that play a special role in the study of the homotopy category: the 铿乥rations and the co铿乥rations. For example, any covering map is a 铿乥ration and any 鈥渘ice inclusion鈥� is a co铿乥ration. The first part of the project is to understand these types of maps and to analyse various examples.
The properties of the 铿乥rations and the co铿乥rations are formalised in the general notion of a model category. The second part of the project is to understand the notion of a model category and to analyse how the homotopy category of topological spaces fits into this general setting.
References:
[1] W. G. Dwyer and J. Spalinsky, Homotopy theories and model categories, .
[2] A. Str酶m, The homotopy category is a homotopy category, Arch. Math. (Basel) 23, (1972), 435鈥�441. ().