Level {\epsilon}

TL;DR

This paper introduces a new concept called level {} in the context of local geometric morphisms, providing foundational properties and concrete examples inspired by algebraic geometry and topos theory.

Contribution

It offers an elementary definition of level {} associated with local geometric morphisms and explores its properties and examples, inspired by Lawvere's observations.

Findings

Defined level {} for local geometric morphisms

Established basic properties of level {}

Provided concrete descriptions for several pre-cohesive morphisms

Abstract

Lawvere has observed that certain 'gros' toposes in algebraic geometry suggest the existence of an 'infinitesimal level', closely related to finite-dimensional local algebras. Motivated by this observation we propose an elementary definition of level {\epsilon} associated to a local geometric morphism, establish some relevant basic properties suggested by geometric intuition, and give concrete descriptions of the level {\epsilon} determined by several pre-cohesive geometric morphisms.