Pre-sheaves of spaces and the Grothendieck construction for infinity-categories
by Daniel Stevenson
Abstract: The notion of pre-stack in algebraic geometry can be formulated either in terms of categories fibered in groupoids, or else as a functor to the category of groupoids with composites only preserved up to a coherent system of natural isomorphisms. The device which lets one shift from one perspective to the other is known as the `Grothendieck construction’ in category theory.
A pre-sheaf in higher geometry is a functor to the infinity-category of infinity-groupoids; in this context the need to keep track of all the coherent natural isomorphisms between composites becomes particularly acute. Fortunately there is an analog of the Grothendieck construction in this context, due to Lurie, which lets one `straighten out’ a presheaf into a certain kind of fibration. In this talk we will give a new perspective on this straightening procedure which allows for a more conceptual proof of Lurie’s straightening theorem.