nLab
higher geometry

higher geometry $\leftarrow$ Isbell duality $\to$ higher algebra

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Idea

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 $(\mathrm{\infty },1)$-topos enocodes itself a space. Axioms for equipping little $(\mathrm{\infty },1)$-toposes with geometric structure have been given in (Lurie) in terms of the notion of structured (∞,1)-toposes.

• A big $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-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.

Axiomatizations

Structured little $(\mathrm{\infty },1)$-toposes

The notion of geometry is formalized in the form of an (∞,1)-category $𝒢$ whose objects play the role of test-spaces on which all other spaces are modeled, in a hierarchy of generalized objects:

• test spaces $\hookrightarrow$ spaces locally equivalent to test spaces $\hookrightarrow$ concrete spaces with structure sheaves taking values in test spaces $\hookrightarrow$ spaces probeable by test spaces.

technically modeled by:

• geometry (for structured (∞,1)-toposes) $𝒢$ $\hookrightarrow$ generalized schemes $\hookrightarrow$ formal duals to $𝒢$-structured (∞,1)-toposes $\hookrightarrow$ (∞,1)-topos of ∞-stacks on $𝒢$.

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 $A_{\mathrm{\infty }}$-categories are presentations for stable (∞,1)-categories and by the stable Giraud theorem presentable stable $(\mathrm{\infty },1)$-categories play a very similar role to (unstable) ∞-stack (∞,1)-toposes. In particular they may be obtained from the latter by stabilization.

Structured big $(\mathrm{\infty },1)$-toposes

(…)

Examples

• derived algebraic geometry

  • étale (∞,1)-site, dg-geometry, Hochschild cohomology of dg-algebras

  • schematic homotopy type

• derived noncommutative geometry

  • noncommutative geometry

• higher differential geometry

  • motivation for higher differential geometry

  • differential geometry, differential topology

  • higher prequantum geometry

  • derived smooth manifold

  • smooth ∞-groupoid, ∞-Lie algebroid

• higher symplectic geometry

• higher Klein geometry

• higher Cartan geometry

Related concepts

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

For relation to physics see

• higher category theory and physics

• geometry of physics

References

An axiomatization of higher geometry of little (∞,1)-toposes is proposed in

• Jacob Lurie, Structured Spaces .

In

• Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

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

• cohesive (∞,1)-topos.