Augmented homotopical algebraic geometry

Scott Balchin – 13 September 2017

The theory of homotopical algebraic geometry, or oo-stacks, hinges on a certain Quillen model structure on the category of simplicial presheaves over a Grothendieck site. The goal of augmented homotopical algebraic geometry is to study the result of replacing the simplex category in this theory with a suitable, more exotic category.

In the first of what will probably constitute a couple of talks, we will introduce the notion of an augmentation category. The definition of an augmentation category gives sufficient conditions for when we can equip the category of "augmented presheaves" with a Quillen model structure which has a Quillen restriction to the category of oo-stacks. We will show that when our site is the category of derived affine schemes, it is possible to construct n-Deligne-Mumford (or n-Artin) augmented derived stacks by using a modified hypergroupoid construction.

In the sequel(s), we will explore some examples of augmentation categories, and discuss what the "augmentation" is achieving with regards to the geometry over certain sites