Beijing Institute of Mathematical Sciences and Applications

In this course we will discuss topological and simplicial sides of homotopy theory. The course will start with a necessary background of the notion of homotopy in the category of topological spaces and foundations of categorical approach to homotopy theory. This will serve as a basis for introducing the classical definition of model category structure on a given category as a convenient framework for doing homotopy. We will discuss various meaningful choices of such model structures on Top and sSet and alternative frameworks such as categories of fractions and simplicial categories. The rest of the course will be dedicated to studying classical and modern objects of interest in Top and sSet.