University of Regina Topology Seminar

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.

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.

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.

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

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.

Automorphism groups arise in many contexts. For instance, the automorphism group of the set of n elements is the symmetric group $\Sigma_n$ on n elements. The automorphism group of an n-dimensional vector space is the general linear group $GL(n)$, consisting of invertible $n \times n$ matrices.

In a homotopical context, what should the analogous notion be? Taking the automorphism group in the homotopy category loses too much information. In this talk, I will explain how the hammock localization can be used to construct a space of homotopy automorphisms with good properties.

We have talked about model categories, we have talked about simplicial categories. Now we must tackle the two monsters together. This talk will be about the interactions of the simplicial and the model structure on the category of simplicial objects in a model category.

We give a gentle introduction to homotopy colimits.

We are going to describe and prove a model category structure on the category of nonnegatively graded chain complexes.

I will present some explicit formulae for homotopy limits and colimits.

In this talk I will introduce quasicategories, which are simplicial sets with the weak Kan property and generalize the notion of ordinary categories.

I will talk about applying manifold calculus to study the Lagrangian embeddings in a symplectic manifold. Using Gromov's h-principle for directed embeddings, we will prove that the analytic approximation of the Lagrangian embeddings functor is the totally real embeddings functor.