étale morphism (submersion, immersion)

generally in differential cohesion

reduction modality $\dashv$ infinitesimal shape modality

formally étale (formally smooth, formally unramified)

étale topos

étale homotopy

étale cohomology

étale ∞-groupoid

in topology

local homeomorphism, étale space

étale groupoid

in differential geometry

local diffeomorphism, (submersion, immersion of smooth manifolds)

étale Lie groupoid

in algebraic geometry

étale morphism of schemes, formally étale morphism of schemes

étale site, pro-étale site, étale (∞,1)-site

Deligne-Mumford stack

ℓ-adic cohomology

There are several concepts of an immersion, depending on the category of spaces. The most basic ones are