Power to the people

Dominic Verity – 22 February 2017, 1 March 2017

In the category of simplicial sets every map is exponentiable (or powerful), in the sense that pullback along such maps admits a right adjoint, sometimes called "push-forward" or "fibred product". Restricting our attention to the theory of quasi-categories, it is natural to study the conditions under which this push-forward functor is quasi-categorically well behaved (in some sense). Two results of this kind are absolutely indispensable in many applications of quasi-categories. These show that any cocartesian fibration of quasi-categories is powerful, in the sense that push-forward along it preserves isofibrations and cartesian fibrations of quasi-categories.

These results were originally established by Lurie, and he frames them as model category theoretic results. In this talk we seek to re-frame those results in more elementary terms. Our hope is that by doing so we might throw further conceptual light on some important themes and link them back more clearly to corresponding approaches in the categorical literature.