Level {\epsilon}

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.