Fibration structure for Gromov h-principle

TL;DR

This paper explores the structure of flexible sheaves within the context of the Gromov h-principle, revealing a fibrant object interpretation that enhances understanding of partial differential relations.

Contribution

It introduces a fibrant object perspective for flexible sheaves, linking the h-principle to model category theory for the first time.

Findings

Flexible sheaves can be viewed as fibrant objects in a model structure.

Provides a new categorical framework for understanding the h-principle.

Bridges sheaf theory with homotopical algebra in differential topology.

Abstract

The h-principle is a powerful tool for obtaining solutions to partial differential inequalities and partial differential equations. Gromov discovered the h-principle for the general partial differential relations to generalize the results of Hirsch and Smale. In his book, Gromov generalizes his theorem and discusses the sheaf theoretic h-principle, in which an object called a flexible sheaf plays an important role. We show that a flexible sheaf can be interpreted as a fibrant object with respect to a model structure.