Augmented Homotopical Algebraic Geometry
Scott Balchin

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
