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

Idea

A geometry $\mathcal{G}$ is an (∞,1)-category equipped in a compatible way with

1. the structure of an (∞,1)-site;

2. the structure of an essentially algebraic (∞,1)-theory.

The objects of $\mathcal{G}$ are to be thought of as test-spaces with certain higher geometry structure and the morphisms as homomorphisms preserving that geometric structure.

These two structures gives rise to

1. The big (∞,1)-topos $Sh(\mathcal{G})$ of (∞,1)-sheaves on $\mathcal{G}$. Its objects are generalized spaces given by rules

   $X : \mathcal{G}^{op} \to$ ∞Grpd

   for how to map test spaces into them.

2. The (∞,1)-algebras over $\mathcal{G}$ in some little topos $\mathcal{X}$, given by rules

   $$\mathcal{O} : \mathcal{G} \to \mathcal{X}$$

   that send each obect $U\in \mathcal{G}$ to a $U$-valued structure sheaf.

Using the additional structure of a site on $\mathcal{G}$ allows to identify those structure sheaves $\mathcal{O}$ that are local in that they respect coverings. This constitutes a generalized notion of locally ringed toposes called $\mathcal{G}$-structured (∞,1)-toposes. Equivalently these local structure sheaves are given by (∞,1)-geometric morphisms $\mathcal{X} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} \mathbf{H} = Sh(\mathcal{G})$ to the big topos over $\mathcal{G}$.

Definition

Geometry

Summary

A geometry on an (∞,1)-category $\mathcal{G}$ is a Grothendieck topology on $\mathcal{G}$ 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-geometry. Every pregeometry $\mathcal{T}$ extends uniquely $\mathcal{T} \hookrightarrow \mathcal{G}$ to an enveloping geometry $\mathcal{G}$.

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

Definition (admissibility structure, StSp 1.2.1)

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

• admissible morphisms are stable under (∞,1)-pullback;

• admissible morphisms satisfy “left cancellability”, meaning that whenever in

  $$\array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z }$$

  $g$ and $h$ are admissible, then so is $f$.

• admissible morphisms are closed under retracts.

Equivalently, this is a Grothendieck topology on $\mathcal{G}$ 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 $(\infty,1)$-toposes) is

• An (∞,1)-category $\mathcal{G}$ that

  • is essentially small;

  • has all finite limits

  • is idempotent complete.

• equipped with an admissibility structure.

Definition (discrete geometry, StSp, 1.2.10)

The discrete geometry $\mathcal{G}^0$ on $\mathcal{G}$ is given by

• the admissible morphisms in $\mathcal{G}$ are precisely the equivalences

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

Every small (∞,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 $\mathcal{T}$;

• equipped with an admissibility structure (homotopical topology)

such that

• $\mathcal{T}$ 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 $\mathcal{T}$ is called smooth if it is locally stably admissible 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

$$\array{ U_i &\stackrel{u_i}{\to}& X \\ \downarrow && \downarrow \\ S \times V_i &\stackrel{p_1}{\to}& S } \,.$$

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 $\mathcal{T}$-structures $\mathcal{O} : \mathcal{T} \to \mathcal{X}$ preserve pullbacks of smooth morphisms.

Structured $(\infty,1)$-topos

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

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

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

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 admissible morphisms in $G$, the induced morphism

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

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

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

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

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

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

• $\mathcal{O}$ preserves finite products.

• $\mathcal{O}$ preserves pullbacks of admissible morphism in that for every pullback diagram

  $$\array{ U' &\to& U \\ \downarrow && \downarrow^f \\ X' &\to& X }$$

  in $\mathcal{T}$ with $f$ admissible, the image

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

  is again a pullback.

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

Remark

The first clause says that $\mathcal{O} : \mathcal{T} \to \mathcal{X}$ is in particular an $\infty$-algebra over the (multi-sorted) (∞,1)-algebraic theory $\mathcal{T}$.

The other two clauses encode that this $\infty$-algebra $\mathcal{O}$ indeed behaves like a function algebra .

Definition (geometric envelope, StrSp 3.4.1)

…the universal geometry extending a pregeometry…

Proposition (StSp 3.4.5)

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

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

Examples

Étale geometry

If we regard the ordinary étale site as a pregeometry $\mathcal{T}_{et}$, then its geometric envelope $\mathcal{G}_{et}$ is the étale (∞,1)-site. See derived étale geometry for the precise statement.

Proposition

The 1-localic $\mathcal{G}_{et}$-generalized schemes are precisely Deligne-Mumford stacks (without the separation axiom).

See Deligne-Mumford stack for details.

Smooth geometry

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

Related concepts

Analogous structures in the axiomatic context of differential cohesion are discussed in differential cohesion – structure sheaves.

References

The general theory is developed in

• Jacob Lurie, Structured Spaces

The definition of a geometry $\mathcal{G}$ is def. 1.2.5.

A $\mathcal{G}$-structure on an (∞,1)-topos is in def. 1.2.8.

The notion of $\mathcal{G}$-spectrum – which are (∞,1)-toposes – is the subject of section 2.1 .

The inclusion

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

is definition 2.1.2.

The definition of $\mathcal{G}$-generalized scheme is definition 2.3.9, page 51.

The inclusion

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

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.