Model Structures on Diagram Categories

Abstract

Model categories have been an important tool in algebraic topology since rst de ned by Quillen. Given a category and a class of morphisms called weak equivalences one can study the homotopy \category" in which the weak equivalences are turned into isomorphisms by formally giving them inverses. However, the resulting structure might not be a category, and even when it is understanding it can be very di cult. A model structure on a category ensures that formally inverting the weak equivalence does result in a category. It also makes the study of the homotopy category easier by providing two weak factorisation systems on the model category which can be used to understand the homotopy category. We explore the basic consequences of weak factorisation systems and show how one can be co brantly generated from a set of morphisms. We then de ne model categories and discuss some fundamental results about them, including de ning their homotopy categories, and proving a recognition theorem. Having done this we show there is a co brantly generated model structure on the category of compactly generated, weakly Hausdor , topological spaces, T . We take a look at the category of simplicial sets, sSet, which can be considered a generalisation of inductively constructed topological spaces. We later describe a co brantly generated model structure on them and a Quillen adjunction between T and sSet. In stable homotopy theory the important objects of study are categories of D- spectra and the stable model structures on them. We de ne a level model structure on D-spectra for chosen categories D, explain why it is not satisfactory for stable homotopy theory, and then describe the stable model structure on spectra. Finally, we describe the Reedy model structure on diagram categories MC where M is a model category and C is a Reedy category. A recent result classifying those functors between Reedy categories which induce a Quillen functor between diagram categories for any choice of model category by Hirschhorn and Volic is shown using a dual argument to the one in their paper.