Smooth sets of fields: A pedagogical introduction

TL;DR

This paper introduces the concept of smooth sets within a categorical framework, illustrating their properties and applications in physical theories through examples like tangent structures and variational bicomplexes.

Contribution

It provides a pedagogical introduction to smooth sets and explores their properties and relevance in modeling spaces of fields in physics.

Findings

The topos of smooth sets has specific categorical properties.

Smooth sets can incorporate geometrical structures such as tangent functors.

Applications include modeling spaces of fields in physical theories.

Abstract

In order to provide a good categorical setting to the many different spaces of fields arising in the description of physical theories, a pedagogical introduction to the categorical notion of smooth sets is provided and some simple properties of the topos of smooth sets are discussed. The introduction of geometrical structures into such spaces is illustrated via the specific examples of the tangent functor and the variational bicomplex.