(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
For $\mathcal{G}$ a geometry (for structured (∞,1)-toposes) a $\mathcal{G}$-structured (∞,1)-topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is locally representable if it is locally equivalent to $Spec U$ for $U \in Pro(\mathcal{G})$ (the pro-objects in an (∞,1)-category), or $U \in \mathcal{G}$ itself if it is locally finite presented .
This generalizes
the notion of smooth manifold from differential geometry;
the notion of scheme from algebraic geometry.
etc.
Let $\mathcal{G}$ be a geometry (for structured (∞,1)-toposes). Write $\mathcal{G}_0$ for the underlying discrete geometry. The identity functor
is then a morphism of geometries.
Recall the notation $LTop(\mathcal{G})$ for the (∞,1)-category of $\mathcal{G}$-structured (∞,1)-toposes and geometric morphisms between them.
There is a pair of adjoint (∞,1)-functors
with $\mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}$ left adjoint to the canonical functor $p^*$ given by precomposition with $p$.
There is a canonical morphism
Write $\mathbf{Spec}^{\mathcal{G}}$ for the (∞,1)-functor
A $\mathcal{G}$-structured (∞,1)-topos in the image of this functor is an affine $\mathcal{G}$-scheme.
Let $\mathcal{G}$ be a geometry (for structured (∞,1)-toposes).
A $\mathcal{G}$-structured (∞,1)-topos $(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is a $\mathcal{G}$-scheme if
such that
the $\{U_i\}$ cover $\mathcal{X}$ in that the canonical morphism $\coprod_i U_i \to {*}$ (with ${*}$ the terminal object of $\mathcal{X}$) is an effective epimorphism;
for every $U_i$ there exists an equivalence
of structured $(\infty,1)$-toposes for some $A_i \in Pro(\mathcal{G})$ (in the (∞,1)-category of pro-objects of $\mathcal{G}$).
For $\mathcal{T}$ a pregeometry, a $\mathcal{T}$-structured (infinity,1)-topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a $\mathcal{T}$-scheme if it is a $\mathcal{G}$-scheme for the geometric envelope $\mathcal{G}$ of $\mathcal{T}$.
This means that for $f : \mathcal{T} \to \mathcal{G}$ the geometric envelope and for $\mathcal{O}'_{\mathcal{X}}$ the $\mathcal{G}$-structure on $\mathcal{X}$ such that $\mathcal{O}_{\mathcal{X}} \simeq \mathcal{O}'_{\mathcal{X}} \circ f$, we have that $(\mathcal{X}, \mathcal{O}'_{\mathcal{X}})$ is a $\mathcal{G}$-scheme.
Let $\Tau$ be a pregeometry (for structured (∞,1)-toposes) and let $\Tau \hookrightarrow \mathcal{G}$ be an inclusion into an enveloping geometry (for structured (∞,1)-toposes).
We think of the objects of $\Tau$ as the smooth test spaces – for instance the cartesian products of some affine line $R$ with itsef – and of the objects of $\mathcal{G}$ as affine test spaces that may have singular points where they are not smooth.
The idea is that a smooth $\mathcal{G}$-scheme is a $\mathcal{G}$-structured space that is locally not only equivalent to objects in $\mathcal{G}$, but even to the very nice – “smooth” – objects in $\mathcal{Tau}$.
With an envelope $\Tau \hookrightarrow \mathcal{G}$ fixed, a $\mathcal{G}$-scheme is called smooth if there the affine schemes $\mathbf{Spec}^{\mathcal{G}} A_i$ appearing in its definition may be chosen with $A_i$ in the image of the includion $\tau \hookrightarrow \mathcal{G}$.
See the discussion at derived scheme for how ordinary schemes are special cases of generalized schemes.
See the discussion at derived Deligne-Mumford stack for how ordinary Deligne-Mumford stacks are special cases of derived Deligne-Mumford stacks.
Let $k$ be a commutative ring. Recall the pregoemtry $\mathcal{T}_{Zar}(k)$.
A derived scheme over $k$ is a $\mathcal{T}_{Zar}(k)$-scheme.
Let $k$ be a commutative ring. Recall the pregeometry $\mathcal{T}_{et}(k)$
A derived Deligne-Mumford stack over $k$ is a $\mathcal{T}_{et}(k)$-scheme.
The above derived schemes have structure sheaves with values in simplicial commutative rings. There is also a notion of derived scheme whose structure sheaf takes values in E-infinity rings. The theory of these is to be described in full detail in
An indication of some details is in
See at E-∞ scheme and E-∞ geometry.
locally representable structured (∞,1)-topos
Generalized schemes are definition 2.3.9 of
The definition of affine $\mathcal{G}$-schemes (absolute spectra) is in section 2.2.