higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A geometry $\mathcal{G}$ is an (∞,1)-category equipped in a compatible way with
the structure of an (∞,1)-site;
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
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.
The (∞,1)-algebras over $\mathcal{G}$ in some little topos $\mathcal{X}$, given by rules
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 $\mathcal{G}$.
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 $\mathcal{T}$.
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
$g$ 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.
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 .
A geometry (for $(\infty,1)$-toposes) is
An (∞,1)-category $\mathcal{G}$ that
has all finite limits
equipped with an admissibility structure.
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.
A pregeometry (for structured (∞,1)-toposes) is
an (∞,1)-category $\mathcal{T}$;
equipped with an admissibility structure (homotopical topology)
such that
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.
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
To interpret this, recall that we think of admissible morphisms as injections of open subsets.
Smooth morphisms are stable under pullback.
pregeometric $\mathcal{T}$-structures $\mathcal{O} : \mathcal{T} \to \mathcal{X}$ preserve pullbacks of smooth morphisms.
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
such that
$C(-)$ 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
is an effective epimorphism in $T$, i.e. its Čech nerve is a simplicial resolution of $C(W)$:
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
in $\mathcal{T}$ with $f$ admissible, the image
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 $\mathcal{X}$.
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 .
…the universal geometry extending a pregeometry…
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 $\mathcal{X}$:
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
The 1-localic $\mathcal{G}_{et}$-generalized schemes are precisely Deligne-Mumford stacks (without the separation axiom).
See Deligne-Mumford stack for details.
There should be a geometry $\mathcal{G}$ such that $\mathcal{G}$-generalized schemes are precisely derived smooth manifolds.
Analogous structures in the axiomatic context of differential cohesion are discussed in differential cohesion – structure sheaves.
The general theory is developed in
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
is definition 2.1.2.
The definition of $\mathcal{G}$-generalized scheme is definition 2.3.9, page 51.
The inclusion
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
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.