Tensor Topology

TL;DR

This paper explores how subunits in monoidal categories provide a topological intuition, enabling concepts like restriction and localization, and develops universal constructions to enhance their algebraic structure.

Contribution

It introduces methods to complete monoidal categories so that their subunits form a lattice, preframe, or frame, enriching their topological and algebraic properties.

Findings

Subunits in monoidal categories can be used to model topological notions.

Universal constructions can turn subunits into a lattice, preframe, or frame.

These structures facilitate restriction, localization, and support concepts in categorical settings.

Abstract

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.