Model categories have been an important tool in algebraic topology since first defined 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 difficult. 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 get a handle on the homotopy category.
I will talk about weak factorisation systems and the 'small object argument' for building them out of collections of maps. Then we will move on to what that has to do with model categories and talk about some important examples and basic results.
Please note that this seminar will be held in our brand new building! Hopefully everything will be in order, and the seminar room (on the ground floor) will actually be usable.
The PhD seminar is a weekly seminar held on Tuesday mornings, at 11am, for PhD students to talk about their thesis work or other projects.