group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
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 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.
shape modality$\dashv$ flat modality $\dashv$ sharp modality
reduction modality$\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality
Last revised on November 4, 2013 at 22:30:24. See the history of this page for a list of all contributions to it.