Augmented Homotopical Algebraic Geometry

TL;DR

This paper extends homotopical algebraic geometry by introducing augmentation categories, establishing model structures, and providing new examples, thereby broadening the framework for studying geometric structures in a homotopical context.

Contribution

It defines augmentation categories, proves the existence of compatible model structures, and introduces a method to generate new examples, advancing the theoretical foundation of augmented homotopical algebraic geometry.

Findings

Existence of a closed Quillen model structure on presheaf categories.

Identification of crossed simplicial groups and planar rooted tree category as augmentation categories.

A categorical pushout construction method for generating new augmentation examples.

Abstract

We develop the framework for augmented homotopical algebraic geometry. This is an extension of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. To do so, we define the notion of augmentation categories, which are a special class of generalised Reedy categories. For an augmentation category, we prove the existence of a closed Quillen model structure on the presheaf category which is compatible with the Kan-Quillen model structure on simplicial sets. Moreover, we use the concept of augmented hypercovers to define a local model structure on the category of augmented presheaves. We prove that crossed simplicial groups, and the planar rooted tree category are examples of augmentation categories. Finally, we introduce a method for generating new examples from old via a categorical pushout construction.