Approaches to abstract homotopy theory fall roughly into two types: classical and higher categorical. Classical models of homotopy theories are some structured categories equipped with weak equivalences, e.g. model categories or (co)fibration categories. From the perspective of higher category theory homotopy theories are the same as (infinity,1)-categories, e.g. quasicategories or complete Segal spaces. The higher categorical point of view allows us to consider the homotopy theory of homotopy theories and to use homotopy theoretic methods to compare various notions of homotopy theory. Most of the known notions of (infinity,1)-categories are equivalent to each other. This raises a question: are the classical approaches equivalent to the higher categorical ones? I will provide a positive answer by constructing the homotopy theory of cofibration categories and explaining how it is equivalent to the homotopy theory of (finitely) cocomplete quasicategories. This is achieved by encoding both these homotopy theories as fibration categories and exhibiting an explicit equivalence between them.

In this talk we will discuss some results concerning the zero sets of real analytic functions on open sets in $\mathbb{R}^n$. We will consider the related notion of analytic uniqueness sequences and, as an application, we will show that the zero set of every non-constant real analytic function on a domain has always empty interior with respect to the fine topology (which strictly contains the Euclidean one). Further, we will see that for a certain category of sets $E$ (containing the finely open sets), a function is real analytic on some open neighbourhood of $E$ if and only if it is real analytic ''at each point'' of $E$. (Joint work with Andre Boivin and Paul Gauthier.)

We will introduce the Univalence Axiom and discuss a few of its immediate consequences such as existence of types that are not sets, function extensionality or preservation of n-types by dependent products.