Singular superspaces

TL;DR

This paper introduces a broad category of superspaces called locally finitely generated superspaces, which extends supermanifolds and retains strong categorical properties, enabling new constructions like relative supermanifolds and Weil functors.

Contribution

It defines locally finitely generated superspaces, expanding the framework of supergeometry with enhanced categorical stability and new tools such as inner homs and relative supermanifolds.

Findings

The category is closed under finite fibre products and thickenings.

Morphisms into supermanifolds are described via coordinates.

Inner homs generalize Weil functors for supergeometry.

Abstract

We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in this category, morphisms with values in a supermanifold are still given in terms of coordinates. This framework gives a natural notion of relative supermanifolds over a locally finitely generated base. Moreover, the existence of inner homs, whose source is the spectrum of a Weil superalgebra, is established; they are generalisations of the Weil functors defined for smooth manifolds.