Augmented Homotopical Algebraic Geometry

In this thesis we are interested in extending the theory of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. We introduce the concept of augmentation categories, which are a class of generalised Reedy categories. An augmentation category is a category which has enough structure that we can mirror the simplicial constructions which make up the theory of homotopical algebraic geometry. In particular, we construct a Quillen model structure on their presheaf categories, and introduce the concept of augmented hypercovers to define a local model structure on augmented presheaves. As an application, we show that a crossed simplicial group is an example of an augmentation category. The resulting augmented geometric theory can be thought of as being equivariant. Using this, we define equivariant cohomology theories as special mapping spaces in the category of equivariant stacks. We also define the SO(2)-equivariant derived stack of local systems by using a twisted nerve construction. Moreover, we prove that the category of planar rooted trees appearing in the theory of dendroidal sets is also an augmentation category. The augmented geometry over this setting should be thought of as being stable in the spectral sense of the word. Finally, we show that we can combine the two main examples presented using a categorical amalgamation construction.