nLab affine modality

Idea

Given a suitable line object $\mathbb{A}^1$ in a suitable ambient (∞,1)-topos, then there exists the cohomology localization at morphisms that induces equivalences in cohomology with coefficients in $\mathbb{A}^1$.

In this case the right adjoint to the reflector typically has the interpretation of producing spaces which are “affine” in that they are entirely characterized by their function ∞-algebra with coefficients in $\mathbb{A}^1$.

Therefore in this case the localization modality deserves to be called the affine modality.

Examples

Examples for this in higher algebraic geometry and synthetic differential geometry are discussed at function algebras on ∞-stacks in the section Localization of the (∞,1)-topos at R-cohomology.

Related concepts

• n-truncation modality, double negation modality

• shape modality$\dashv$ flat modality $\dashv$ sharp modality

• reduction modality$\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality