A Categorical Formulation of Superalgebra and Supergeometry

TL;DR

This paper presents a categorical reformulation of superalgebra and supergeometry using the functor of points, enabling natural extensions to infinite dimensions and clarifying foundational aspects.

Contribution

It introduces a fully categorical approach to supergeometry, clarifies key concepts, and facilitates the construction of superspaces of morphisms, inspired by Molotkov's ideas.

Findings

Extended supergeometry to infinite-dimensional contexts

Clarified the relation between different supergeometry approaches

Constructed superspaces of morphisms easily

Abstract

We reformulate superalgebra and supergeometry in completely categorical terms by a consequent use of the functor of points. The increased abstraction of this approach is rewarded by a number of great advantages. First, we show that one can extend supergeometry completely naturally to infinite-dimensional contexts. Secondly, some subtle and sometimes obscure-seeming points of supergeometry become clear in light of these results, e.g., the relation between the Berezin-Leites-Kostant and the de Witt-Rogers approaches, and the precise geometric meaning of odd parameters in supergeometry. In addition, this method allows us to construct in an easy manner superspaces of morphisms between superobjects, i.e., the inner Hom objects associated with the sets of such morphisms. The results of our work rely heavily and were inspired by the ideas of V. Molotkov, who first outlined the approach extensively expounded here.