higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
higher geometry $\leftarrow$ Isbell duality $\to$ higher algebra
Higher geometry studies the notions of space and geometry in higher category theory. Specifically if the spaces on which the geometry is modeled are themselves objects in higher category theory this is also called derived geometry .
Higher geometry is typically built on (∞,1)-topos theory, which (see topos) provides a general context in which to speak of generalizations of topological spaces. The axioms of higher geometry typically impose extra structures on (∞,1)-toposes that encode genuine geometry .
There are two aspects to this, induced from the two aspects of big and little toposes:
A little $(\infty,1)$-topos encodes itself a space. Axioms for equipping little $(\infty,1)$-toposes with geometric structure have been given in (Lurie) in terms of the notion of structured (∞,1)-toposes.
A big $(\infty,1)$-topos is an (∞,1)-category whose objects are generalized spaces. Axioms for characterizing big toposes that encode geometry have been given in (Lawvere). Their generalization to $(\infty,1)$-topos theory is given by the notion of a cohesive (∞,1)-topos.
Under forming groupoid convolution algebras and their higher analog, at least parts of higher geometry translate to noncommutative geometry.
The notion of geometry is formalized in the form of an (∞,1)-category $\mathcal{G}$ whose objects play the role of test-spaces on which all other spaces are modeled, in a hierarchy of generalized objects:
technically modeled by:
A plethora of proposals for formalizations of higher geometry find their home in this pattern, for instance most of the concepts listed at generalized smooth space.
A notable exception to this is possibly the program by Maxim Kontsevich and others where under the term noncommutative geometry and derived noncommutative geometry spaces are modeled as the formal dual to A-∞-categories. But $A_\infty$-categories are presentations for stable (∞,1)-categories and by the stable Giraud theorem presentable stable $(\infty,1)$-categories play a very similar role to (unstable) ∞-stack (∞,1)-toposes. In particular they may be obtained from the latter by stabilization.
duality between algebra and geometry in physics:
For relation to physics see
An axiomatization of higher geometry of little (∞,1)-toposes is proposed in
In
an axiomatization of generalized geometry is proposed in terms of 1-category theory. The evident generalization of this to (∞,1)-category theory provides an axiomatization for higher geometry. This is discussed at