From a categorical point of view, the standard Zermelo-Frankel set theoretic approach to the foundations of mathematics is fundamentally deficient: it is based on the notion of equality of objects in a set. Equalities between objects are not preserved by equivalences of categories, and thus the notion of equality is 'incorrect' in category theory. It should be replaced by the notion of 'isomorphism'.
Moving higher up the categorical ladder, the notion of isomorphism between objects is 'incorrect' from the point of view of 2-category, and should be replaced by the notion of 'equivalence'...
Recently, people have started to take seriously the idea that one should be less dogmatic about working with set-theoretic axiomatisiations of mathematics, and adopt the more fluid point of view that different foundations of mathematics might be better suited to different areas of mathematics. In particular, there are currently serious attempts to develop foundations for mathematics built on homotopy types, or, in another language, ∞-groupoids.
An (∞,1)-topos should admit an internal 'homotopical logic', just as an ordinary (1-)topos admits an internal logic modelling set theory.
It turns out that formalising such a logic is rather closely related to the problem of finding good foundations for 'intensional dependent type theory' in theoretical computer science/logic. This is sometimes referred to as the attempt to construct a 'homotopy lambda calculus'.
It is expected that a homotopy theoretic formalisation of the foundations of mathematics would be of genuine practical significance to the average mathematician!
In this talk we will give an introduction to these ideas, and to the recent work of Vladimir Voevodsky and others in this area.