geometry $\leftarrow$ Isbell duality $\rightarrow$ algebra
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
See also topology.
differential geometry, differential topology, Diff, cobordism
differential form, tangent space, tangent bundle, cotangent bundle, cotangent complex
symplectic geometry, symplectic manifold, Poisson manifold, Lagrangian submanifold
multisymplectic geometry, n-symplectic manifold, foliation, integrable distribution, G-structure
fibre bundle, principal bundle, noncommutative principal bundle, vector bundle
connection (and links therein), connection on a bundle
Morse function, Morse lemma, Morse theory, perfect Morse function
Casson invariant, Donaldson-Thomas invariant, Kähler manifold, mirror symmetry
metric space, convex set, Riemannian manifold, geodesic flow, geodesic convexity, star-shaped
algebraic geometry, analytic geometry, arithmetic geometry, GAGA, book entry EGA
scheme, quasicompact, noetherian scheme, reduced scheme, integral scheme
formal scheme, formal group scheme, formal group law, algebraic group
noncommutative geometry, derived noncommutative algebraic geometry
noncommutative algebraic geometry, noncommutative scheme, noncommutative thin scheme
rational map, rational variety, unirational variety, birational map, birational geometry, image of a rational map
smooth scheme, smooth morphism of schemes, etale morphism, formally smooth morphism
D-module, local system, regular differential operator, holonomic D-module
flag variety, geometric quantization, coherent state, orbit, coadjoint orbit
Lie group, Lie groupoid, Lie algebroid, Courant algebroid, Atiyah Lie groupoid
sheaf of ideals, defining sheaf, conormal sheaf, conormal bundle, subscheme of an Abelian category
synthetic differential geometry, infinitesimal object, smooth topos, Kock-Lawvere axiom, infinitesimal singular simplicial complex, differential forms in synthetic differential geometry
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 $(\infinity,1)$-categories and abstract homotopy theory, say via Quillen model categories (e.g. of simplicial presheaves).
duality between algebra and geometry in physics: