Smooth sets of fields: A pedagogical introduction

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.

Figures (4)

• Figure 1: A schematic illustration of a smooth set. The sets $X(U)$ denote the sets of plots of shape $U$, and the maps $X(\phi)$ determine how these plots change when we change the domain of reference $\varphi \colon U \to V$.
• Figure 2: A schematic illustration of the sheaf/glueing property. If two plots $\phi_1, \phi_2$ coincide on the intersection of their domains, there must be a plot $\phi$ defined on the union of the domains of $\phi_1$, $\phi_2$ whose restrictions coincide with them.
• Figure 3: A schematic illustration of the limit of a functor. The diagram $\mathsf{I}$ is represented (drawn) in the category $\mathsf{C}$ by the functor $F$ as $F(\mathsf{I})$, then the object $c =: \lim_\mathsf{I} F$ is the best covering of $F(\mathsf{I})$ in $\mathsf{C}$.
• Figure 4: Diagram describing the construction of the tangent functor $\widehat{T}$ in the category $\mathsf{SmoothSet}$ as a left Kan extension.

Theorems & Definitions (7)

• Definition 1.1
• Definition 2.1
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 2.5