nLab higher geometry

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

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Idea

Higher geometry or homotopical geometry is the study of concepts of space and geometry in the context of higher category theory and homotopy theory.

Slogan

higher geometry = geometry + homotopy theory/higher category theory

Higher geometry subsumes notably the theory of orbifolds and geometric stacks, as well as the theory of more general stacks such as moduli stacks, and generalizes all this to ∞-stacks and derived stacks. This way higher geometry includes what is called derived geometry and it subsumes at least parts of (derived) noncommutative geometry. Many other phenomena are naturally part of higher geometry, see the list of Examples below.

In any given instance of higher geometry, one starts with a notion of “local models” for the geometry. An affine space will then be a formal dual of such a local model, and a general space will be formed by “gluing” these affine spaces in some appropriate way. There are two ways of formalizing this idea, coming from Alexander Grothendieck‘s two definitions of scheme in algebraic geometry via locally ringed spaces and functors of points. Both are built on (∞,1)-topos theory: in one direction, a petit (∞,1)-topos (with some additional structure) encodes a space itself; in another direction, a space is an object of a gros (∞,1)-topos of ∞-stacks on some (∞,1)-site. We discuss these axiomatizations below in Formalization.

These approaches do not apply to noncommutative algebraic geometry, which requires a different approach to deal with a more complicated notion of gluing; we discuss this below in Noncommutative algebraic geometry.

Formalizations

We discuss two different (but closely related) formalizations of these ideas.

In

• Gros (∞,1)-topos

we discuss the approach of considering one big (∞,1)-topos $\mathbf{H}$ (with “big”/gros being formalized for instance by cohesion) such that (some of) its objects are to be regarded as higher geometric spaces.

In

• Petit (∞,1)-toposes

we discuss the approach of encoding a would-be higher geometric space $X$ by a structured (∞,1)-topos to be thought of as the petit (∞,1)-topos of (∞,1)-sheaves $(Sh_\infty(X), \mathcal{O}_X)$ of $X$, canonically equipped with a structure sheaf $\mathcal{O}_X$.

Gros (∞,1)-toposes

Let $\mathcal{G}$ be an (∞,1)-site whose objects are to be viewed as “local models” or “test spaces” for a geometry. Within the context of this geometry, we make the following definitions:

An affine space is a formal dual of an object of $\mathcal{G}$, so that the (∞,1)-category of affine spaces is the opposite of $\mathcal{G}$. A stack is an ∞-stack on $\mathcal{G}$, so that the (∞,1)-category of stacks is the gros (∞,1)-sheaf (∞,1)-topos on $\mathcal{G}$. Finally, a space is a stack $X$ that has a cover by a family of affine spaces $(f_i : U_i \to X)_i$, where each $f_i$ belongs to some nice class of morphisms (e.g. open immersions, etale morphisms or smooth morphisms).

When the underlying (∞,1)-category of $\mathcal{G}$ is the (∞,1)-category of commutative algebras in a symmetric monoidal (∞,1)-category, this is known as homotopical algebraic geometry. When it is the (∞,1)-category of algebras over a Lawvere theory, this is discussed at derived geometry.

Following Bill Lawvere, one may ask for a set of axioms on the (∞,1)-sheaf (∞,1)-topos $Sh_\infty(\mathcal{G})$ that ensure that it is appropriate to view (∞,1)-sheaves on $\mathcal{G}$ as generalized geometric spaces. One such set of axioms is cohesion.

Petit (∞,1)-toposes

As above, let $\mathcal{G}$ be an (∞,1)-category whose objects will be viewed as “local models” for the kind of geometry to be developed. Following Jacob Lurie (based on the theory of geometry via ringed toposes by Alexander Grothendieck and Monique Hakim), a $\mathcal{G}$-structured (∞,1)-topos is the data of an (∞,1)-topos together with a structure sheaf valued in $\mathcal{G}$. Given an appropriate choice of $\mathcal{G}$, one gets the following hierarchy of generalized spaces this way:

• 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)$\mathcal{G}$

  $\hookrightarrow$

  generalized schemes

  $\hookrightarrow$ formal duals to $\mathcal{G}$-structured (∞,1)-toposes $\hookrightarrow$ (∞,1)-topos of ∞-stacks on $\mathcal{G}$.

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.

Relation between these approaches

Given a gros cohesive (∞,1)-topos $\mathbf{H}$ and an object $X \in \mathbf{H}$, one may in turn assign to $X$ a petit structured (∞,1)-topos $Sh_{\mathbf{H}}(X)$ of internal sheaves over $X$. See at differential cohesion for how this works. This connects the “gros” perspective back to the “petit” perspective.

Conversely, given a structured (∞,1)-topos one may consider its associated functor of points, which will be an object in the gros (∞,1)-topos.

Examples

List of examples

• homotopical algebraic geometry

  • 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

  • derived smooth manifold

  • smooth ∞-groupoid, ∞-Lie algebroid

• higher symplectic geometry

• higher complex analytic geometry

• higher prequantum geometry

• higher Klein geometry

• higher Cartan geometry

Noncommutative algebraic geometry

The above frameworks for higher geometry are not suitable for describing noncommutative algebraic geometry, because of the more complicated notions of localization, gluing and descent in the latter setting. Indeed, noncommutative spaces are supposed to be obtained from affine ones (formal duals of associative algebras or dg-algebras) by gluing along bimodules. A good setting for such gluing is that of pretriangulated dg-categories (or stable (∞,1)-categories). Thus in derived noncommutative algebraic geometry, a noncommutative space is defined to be a stable (∞,1)-category.

Connes-style noncommutative geometry

The process of forming groupoid convolution algebras is a 2-functor from suitable topological and differentiable stacks to C*-algebras with Hilbert bimodules between them. Much of Connes-style noncommutative geometry turns out to deal with objects in the image of this functor, and to the extent that it does, Connes-style noncommutative geometry may be regarded as being a way of speaking about higher geometry, specifically the higher differential geometry of differentiable stacks.

Related concepts

• higher structures

• higher differential geometry

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

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

For relation to physics see

• higher category theory and physics

• geometry of physics

References

Both approaches to higher geometry are described, in the special case of derived algebraic geometry, in

• Jacob Lurie, Derived Algebraic Geometry, Ph.D. thesis.

The gros topos approach is described, in the case of homotopical algebraic geometry, in

• Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry II: geometric stacks and applications, 2004, arXiv:math/0404373.

A general exposition of the petit topos approach 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.

Review:

• Urs Schreiber, Higher Topos Theory in Physics [arXiv:2311.11026]

• Luigi Alfonsi, §1 in: Higher geometry in physics, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.07308]