geometry \leftarrow Isbell duality \rightarrow algebra

See also topology.

There are many entries on sheaf, stack, site, locale and topos theory including

and pages on various cohomologies, including sheaf cohomology, nonabelian cohomology, differential cohomology, Deligne cohomology, etale cohomology, equivariant cohomology, Bredon cohomology and their cocycle classes including torsors, gerbes, principal 2-bundles as well as the related picture of the descent theory (cf. oriental, descent for simplicial presheaves…). A modern systematic theory of cohomology and descent can be done using the language of (,1)(\infinity,1)-categories and abstract homotopy theory, say via Quillen model categories (e.g. of simplicial presheaves).

duality between algebra and geometry in physics:

Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

Last revised on September 15, 2015 at 14:30:03. See the history of this page for a list of all contributions to it.