nLab
geometry (for structured (infinity,1)-toposes)

Idea

The spaces of relevance in many applications – notably in those related to physics – crucially carry more structure than plain topological space, they carry certain geometric structure, specified by local model spaces.

Geometries

This may be formalized by fixing an (∞,1)-category $𝒢$ whose objects we think of as loci – test spaces with which all spaces with $𝒢$-geometry structure may be probed – and whose morphisms we think of maps between loci that respect the geometric structure in question.

Since the specification of $𝒢$ encodes what we want to mean by geometric structure, $𝒢$ is called a geometry.

By the general abstract nonsense of space and quantity, the most general notion of space modeled on the test objects in $𝒢$ is an ∞-stack on $𝒢$. We write

$$H:={\mathrm{Sh}}_{\mathrm{\infty }}(\mathrm{Pro}(C))$$\mathbf{H} := Sh_\infty(Pro(C))

for a choice of (∞,1)-category of (∞,1)-sheaves on pro-objects in $𝒢$: the gros (∞,1)-topos of $𝒢$-geometric spaces. The choice of $H$ on top of the choice of $𝒢$ encodes the notion locality of spaces modeled on $𝒢$.

The Yoneda embedding $𝒢\hookrightarrow {\mathrm{Sh}}_{\mathrm{\infty }}(𝒢)$ ensures that every test space in $𝒢$ may canonically be regarded as a general space modeled on $𝒢$. When studying geometry it is of interest to refine this inclusion of very simple into very general spaces through a hierarchy of types of spaces of decreasing rigid geometric structure, for instance:

$$\begin{array}{ccccc}𝒢& \stackrel{{\mathrm{Spec}}^{𝒢}}{\hookrightarrow }& \mathrm{Sch}(𝒢)& \hookrightarrow & {\mathrm{Sh}}_{\mathrm{\infty }}(\mathrm{Pro}(𝒢))\\ & & \downarrow & & \downarrow \\ & & \mathrm{Str}(𝒢)& \hookrightarrow & {\mathrm{PSh}}_{\mathrm{\infty }}(\mathrm{Pro}(𝒢))\end{array}$$\array{ \mathcal{G} &\stackrel{Spec^{\mathcal{G}}}{\hookrightarrow}& Sch(\mathcal{G}) &\hookrightarrow& Sh_\infty(Pro(\mathcal{G})) \\ && \downarrow && \downarrow \\ && Str(\mathcal{G}) &\hookrightarrow& PSh_\infty(Pro(\mathcal{G})) }

where

• $\mathrm{Str}(𝒢)$ are the $𝒢$-structured (∞,1)-toposes: those $𝒢$-probeable spaces that have something like an underlying topological space in the generalized form of an underlying petit (∞,1)-topos which is equipped with structure sheaves of function quantities with values in objects of $𝒢$;

• $\mathrm{Sch}(𝒢)$ are the $𝒢$-generalized schemes: those $𝒢$-structured spaces that are not only probeable by object of $𝒢$, but are locally isomorphic to objects in $𝒢$.

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

A structured (∞,1)-topos $T\simeq {\mathrm{Sh}}_{\mathrm{\infty }}(S)$ is an (∞,1)-topos equipped with a generalized quantity : a $C$-valued co-presheaf

$$C(-):G\to T$$C(-) : G \to T

on some (∞,1)-category $G$.

This is to be interpreted as

• assigning to each object $V\in G$ – which we think of a co-test space

• the (∞,1)-sheaf of structure preserving maps from $S$ to $V$,

  • i.e. the sheaf that assigns to each test space $U\in S$ the collection of structure preserving maps from $U$ to $V$

Here the structure that is “preserved” is to be thought of as defined (implicitly) by the choice of $C(-)$, following the general idea of space and quantity.

In standard motivating example (in the context of petit toposes), that of derived smooth manifolds, we have

• $S=\mathrm{Op}(X)$ is the category of open subsets of a topological space $X$

• $G=\mathrm{CartSp}$ is the full subcategory of Diff on Cartesian manifolds (manifolds of the form ${ℝ}^n$)

• the choice $C(-):G\to {\mathrm{Sh}}_{\mathrm{\infty }}(\mathrm{Op}(X))$ is to be thought of as specifying a differential geometry on $X$ in that it amounts to specifying for each open (topological!) subset $U\subset X$ the collection of maps $U\to {ℝ}^n$ that are to count as smooth .

It is in this sense that one thinks of $G$ as encoding geometry over topology: $G$ is to be thought of as a category of co-test spaces that are topological spaces equipped with extra geometrical structure.

Definition

Geometry

Summary

A geometry on an (∞,1)-category $𝒢$ is a Grothendieck topology on $𝒢$ together with

• the extra structure given the information of which covering morphisms are to be thought of as local homeomorphisms

• the extra property that it has all finite limits.

If only all finite products exist we speak of a pre-geopmetry. Every pregeometry $𝒯$ extends uniquely $𝒯\hookrightarrow 𝒢$ to an enveloping geometry $𝒢$.

When the objects of the geopmetry $𝒢$ are thought of as test spaces (affine schemes), the objects of the pregeometry $𝒯\hookrightarrow 𝒢$ are to be thought of as the affine spaces. This distinction is used to encode smoothness? of maps between test spaces: a morphism in $𝒢$ is smooth if it locally factors through admissible maps between objects in $𝒯$.

Definition (admissibility structure, StSp )

An admissibility structure on an (∞,1)-category $𝒢$ is a Grothendieck topology on $𝒢$ that is generated from its intersection with a subcategory ${𝒢}^{\mathrm{ad}}\subset 𝒢$ whose morphisms – called the admissible morphisms have the following properties

• admissible morphisms are stable under pullback;

• admissible morphism satisfy 2-out-of-3.

Equivalently, this is a Grothendieck topology on $𝒢$ which is generated from admissible morphisms.

Admissibility

As will become clear when looking at examples, the notion of admissible morphisms models the idea of maps between test spaces that behave like open injections or, more generally, as local homeomorphisms .

Definition ( Structured Spaces 1.2.5 )

A geometry (for $(\mathrm{\infty }\mathrm{,1})$-toposes) is

• An (∞,1)-category $𝒢$ that

  • is essentially small;

  • has all finite limits

  • is idempotent complete.

• equipped with a homotopical topology .

Definition (discrete geometry, StSp, 1.2.10)

The disscrete geometry ${𝒢}^0$ on $𝒢$ is given by

• the admissible morphisms in $𝒢$ are precisely the equivalences

• the Grothendieck topology on $𝒢$ is trivial: a sieve is covering only if it is maximal.

Every (essentially small) (infinity,1)-category $C$ becomes a geometry by regarding it as a discrete geometry in the above way.

Pregeometry

Definition (pregeometry, StSp, 3.1.1)

A pregeometry (for structured (∞,1)-toposes) is

• an (∞,1)-category $𝒯$;

• equipped with an admissibility structure (homotopical topology)

such that

• $𝒯$ has all products.

Remark

So a geometry differs from a pregeometry in that it is idempotent complete and closed not only under products but under all finite limits.

Various concepts for geometries have immediate analogues for pregeometries.

Smooth morphisms

Definition (smooth morphism, Structured Spaces, 3.1.7)

A morphism $f:X\to S$ in a pregeometry $𝒯$ is called smooth if it is locally stably admissaible in that there exists a cover $\{u_i:U_i\to X\}$ (meaning: generators of a covering sieve) of $X$ by admissible morphisms, such that on $U_i$ the morphism $f$ factors admissibly through some $S\times V_i$ in that there is a commuting diagram

$$\begin{array}{ccc}{U}_i& \stackrel{u_i}{\to }& X\\ \downarrow & & \downarrow \\ S\times V_i& \stackrel{p_1}{\to }& S\end{array}.$$\array{ U_i &\stackrel{u_i}{\to}& X \\ \downarrow && \downarrow \\ S \times V_i &\stackrel{p_1}{\to}& S } \,.

Remark

To interpret this, recall that we think of admissible morphisms as injections of open subsets.

Proposition (Structured Spaces, 3.1.8)

• Smooth morphisms are stable under pullback.

• pregeometric $𝒯$-structures $𝒪:𝒯\to 𝒳$ preserve pullbacks of smooth morphisms.

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

Definition (geometry structure on an $(\mathrm{\infty }\mathrm{,1})$-topos, StSp, 1.2.8)

For $𝒢$ a geometry, and $T\simeq {\mathrm{Sh}}_{\mathrm{\infty }}(S)$ an (∞,1)-topos, a $𝒢$-structure on the $(\mathrm{\infty }\mathrm{,1})$-topos $T$ making it a structured (∞,1)-topos is a (∞,1)-functor

$$C(-):𝒢\to T$$C(-) : \mathcal{G} \to T

such that

• $C(-)$ is left exact (preserves finite limits);

• $C(-)$ satisfies codescent (the dual notion of descent): for $\pi :(V={\coprod }_i{V}_i)\to W$ any cover by admissable morphisms in $G$, the induced morphism

  $$C(\pi ):C(V)\to C(W)$$C(\pi) : C(V) \to C(W)

  is an effective epimorphism in $T$, i.e. its Čech nerve is a simplicial resolution of $C(W)$:

  $$Čech(C(\pi ))\stackrel{\simeq }{\to }C(W).$$\mathop{Čech}(C(\pi)) \stackrel{\simeq}{\to} C(W) \,.

Definition (pregeometry structure on an $(\mathrm{\infty }\mathrm{,1})$-topos, StrSp 3.1.4)

Let $𝒯$ be a pregeometry and $𝒳$ an (∞,1)-topos.

A $𝒯$-structure on $𝒳$ is an (infinity,1)-functor $𝒪:𝒯\to 𝒳$ such that

• $𝒪$ preserves finite products.

• $𝒪$ preserves pullbacks of admissible morphism in that for every pullback diagram

  $$\begin{array}{ccc}U\prime & \to & U\\ \downarrow & & {\downarrow }^f\\ X\prime & \to & X\end{array}$$\array{ U' &\to& U \\ \downarrow && \downarrow^f \\ X' &\to& X }

  in $𝒯$ with $f$ admissable, the image

  $$\begin{array}{ccc}𝒪(U\prime )& \to & 𝒪(U)\\ \downarrow & & {\downarrow }^f\\ 𝒪(X\prime )& \to & 𝒪(X)\end{array}$$\array{ \mathcal{O}(U') &\to& \mathcal{O}(U) \\ \downarrow && \downarrow^f \\ \mathcal{O}(X') &\to& \mathcal{O}(X) }

  is again a pullback.

• $𝒪$ respects covers by admissable morphisms in that for every covering sieve $\{f_i:U_i\to X\}$ in $𝒯$ by admissble $f_i$ the induced map ${\coprod }_i𝒪(U_i)\to 𝒪(X)$ is an effective epimorphism in $𝒳$.

Definition (geometric envelope, StrSp 3.4.1)

…the universal geometry extending a pregeometry…

Proposition (StSp 3.4.5)

Let $𝒯$ be a pregeometry and $f:𝒯\to 𝒢$ a morphism that exhibts the geometry $𝒢$ as a geometric envelope of $𝒯$. Then for every (∞,1)-topos $𝒳$ precomposition with $f$ induces an equivalence of (∞,1)-categories of $𝒯$- and $𝒢$-structures on $𝒳$:

$${\mathrm{Str}}_{𝒢}(𝒳)\stackrel{\simeq }{\to }{\mathrm{Str}}_{𝒯}(𝒳),{\mathrm{Str}}_{𝒢}(𝒳{)}^{\mathrm{loc}}\stackrel{\simeq }{\to }{\mathrm{Str}}_{𝒯}(𝒳{)}^{\mathrm{loc}}$$Str_{\mathcal{G}}(\mathcal{X}) \stackrel{\simeq}{\to} Str_{\mathcal{T}}(\mathcal{X}) \,,\;\;\;\; Str_{\mathcal{G}}(\mathcal{X})^{loc} \stackrel{\simeq}{\to} Str_{\mathcal{T}}(\mathcal{X})^{loc}

Examples

Deligne-Mumford stacks

There is a geometry $𝒢={𝒢}_{\mathrm{et}}(k)$, the etale geometry, such that $𝒢$-generalized schemes that are 1-localic are precisely Deligne-Mumford stacks. See there for more details.

Derived smooth manifolds

There should be a geometry $𝒢$ such that $𝒢$-generalized schemes are precisely derived smooth manifolds.

References

The general theory is developed in

• StrSh Jacob Lurie, Structured Spaces

The definition of a geometry $𝒢$ is def. 1.2.5.

A $𝒢$-stucture on an (infinity,1)-topos is in def. 1.2.8.

The notion of $𝒢$-spectrum – which are (infinity,1)-toposes – is the subject of section 2.1 .

The inclusion

$${\mathrm{Spec}}^{𝒢}:𝒢\hookrightarrow \mathrm{Str}(𝒢)$$Spec^{\mathcal{G}} : \mathcal{G} \hookrightarrow Str(\mathcal{G})

is definition 2.1.2.

The definition of $𝒢$-generalized scheme is definition 2.3.9, page 51.

The inclusion

$$\mathrm{Sch}(𝒢)\hookrightarrow {\mathrm{Sh}}_{\mathrm{\infty }}(\mathrm{Ind}(𝒢))$$Sch(\mathcal{G}) \hookrightarrow Sh_\infty(Ind(\mathcal{G}))

is the topic of section 2.4, theorem 2.4.1

The special case of “smoothly structured spaces” called derived smooth manifold is discussed in

• David Spivak, Derived smooth manifolds PhD thesis (original version pdf arXiv:0810.5174)

Apart from looking at the special case this article also contains useful introduction and details on the general case.

In the version of this that is available on the arXiv (arXiv) the point of view is more on bi-presheaves, a useful discussion to the relation to structured morphisms here is in section 10.1. there.