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\begin{document}

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\section*{locally representable structured (infinity,1)-topos}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{affine_schemes}{Affine $\mathcal{G}$-schemes}\dotfill \pageref*{affine_schemes} \linebreak
\noindent\hyperlink{schemes}{$\mathcal{G}$-Schemes}\dotfill \pageref*{schemes} \linebreak
\noindent\hyperlink{smooth_schemes}{Smooth $\mathcal{G}$-schemes}\dotfill \pageref*{smooth_schemes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ordinary_schemes}{Ordinary schemes}\dotfill \pageref*{ordinary_schemes} \linebreak
\noindent\hyperlink{ordinary_delignemumford_stacks}{Ordinary Deligne-Mumford stacks}\dotfill \pageref*{ordinary_delignemumford_stacks} \linebreak
\noindent\hyperlink{derived_schemes}{Derived schemes}\dotfill \pageref*{derived_schemes} \linebreak
\noindent\hyperlink{derived_smooth_manifolds}{Derived smooth manifolds}\dotfill \pageref*{derived_smooth_manifolds} \linebreak
\noindent\hyperlink{derived_delignemumford_stacks}{Derived Deligne-Mumford stacks}\dotfill \pageref*{derived_delignemumford_stacks} \linebreak
\noindent\hyperlink{derived_schemes_with_ring_valued_structure_sheaves}{Derived schemes with $E_\infty$-ring valued structure sheaves}\dotfill \pageref*{derived_schemes_with_ring_valued_structure_sheaves} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $\mathcal{G}$ a [[geometry (for structured (∞,1)-toposes)]] a $\mathcal{G}$-[[structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is \emph{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 \emph{locally finite presented} .

This generalizes

\begin{itemize}%
\item the notion of [[smooth manifold]] from [[differential geometry]];


\item the notion of [[scheme]] from [[algebraic geometry]].


\item etc.



\end{itemize}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $\mathcal{G}$ be a [[geometry (for structured (∞,1)-toposes)]]. Write $\mathcal{G}_0$ for the underlying discrete geometry. The identity functor

\begin{displaymath}
p : \mathcal{G}_0 \to \mathcal{G}
\end{displaymath}
is then a morphism of geometries.

Recall the notation $LTop(\mathcal{G})$ for the [[(∞,1)-category]] of $\mathcal{G}$-[[structured (∞,1)-topos]]es and [[geometric morphism]]s between them.

\hypertarget{affine_schemes}{}\subsubsection*{{Affine $\mathcal{G}$-schemes}}\label{affine_schemes}

\begin{theorem}
\label{}\hypertarget{}{}
There is a pair of [[adjoint (∞,1)-functor]]s

\begin{displaymath}
p^* : LTop(\mathcal{G}) \stackrel{\leftarrow}{\to}
  LTop(\mathcal{G}_0)
  : 
  \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}
\end{displaymath}
with $\mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}$ left adjoint to the canonical functor $p^*$ given by precomposition with $p$.

\end{theorem}
\begin{remark}
\label{}\hypertarget{}{}
There is a canonical morphism

\begin{displaymath}
can :
  Pro(\mathcal{G})^{op}
  \to
  LTop(\mathcal{G}_0)
\end{displaymath}
\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
Write $\mathbf{Spec}^{\mathcal{G}}$ for the [[(∞,1)-functor]]

\begin{displaymath}
\mathbf{Spec}^{\mathcal{G}} :
  Pro(\mathcal{G})^{op}
  \stackrel{can}{\to}
  LTop(\mathcal{G}_0)
  \stackrel{
    \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}
  }{\to}
  LTop(\mathcal{G})
  \,.
\end{displaymath}
A $\mathcal{G}$-[[structured (∞,1)-topos]] in the image of this functor is an \textbf{affine $\mathcal{G}$-scheme}.

\end{defn}
\hypertarget{schemes}{}\subsubsection*{{$\mathcal{G}$-Schemes}}\label{schemes}

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{G}$ be a [[geometry (for structured (∞,1)-toposes)]].

A $\mathcal{G}$-[[structured (∞,1)-topos]] $(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is a \textbf{$\mathcal{G}$-scheme} if

\begin{itemize}%
\item there exists a collection $\{U_i \in \mathcal{X}\}$

\end{itemize}
such that

\begin{itemize}%
\item 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]];


\item for every $U_i$ there exists an equivalence

\begin{displaymath}
(\mathcal{X}/{U_i}, \mathcal{O}_{\mathcal{X}}|_{U_i})
 \simeq \mathbf{Spec}^{\mathcal{G}} A_i
\end{displaymath}
of structured $(\infty,1)$-toposes for some $A_i \in Pro(\mathcal{G})$ (in the [[(∞,1)-category]] of [[pro-object]]s of $\mathcal{G}$).



\end{itemize}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{T}$ a pregeometry, a $\mathcal{T}$-[[structured (infinity,1)-topos]] $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a \textbf{$\mathcal{T}$-scheme} if it is a $\mathcal{G}$-scheme for [[generalized the|the]] geometric envelope $\mathcal{G}$ of $\mathcal{T}$.

This means that for $f : \mathcal{T} \to \mathcal{G}$ [[generalized the|the]] geometric envelope and for $\mathcal{O}'_{\mathcal{X}}$ [[generalized the|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.

\end{defn}
\hypertarget{smooth_schemes}{}\subsubsection*{{Smooth $\mathcal{G}$-schemes}}\label{smooth_schemes}

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 \emph{smooth} test spaces -- for instance the [[cartesian product]]s 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 \emph{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 $\mathcal{Tau}$.

\begin{defn}
\label{}\hypertarget{}{}
With an envelope $\Tau \hookrightarrow \mathcal{G}$ fixed, a $\mathcal{G}$-scheme is called \textbf{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 $\tau \hookrightarrow \mathcal{G}$.

\end{defn}
\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\hypertarget{ordinary_schemes}{}\paragraph*{{Ordinary schemes}}\label{ordinary_schemes}

See the discussion at [[derived scheme]] for how ordinary [[scheme]]s are special cases of [[generalized scheme]]s.

\hypertarget{ordinary_delignemumford_stacks}{}\paragraph*{{Ordinary Deligne-Mumford stacks}}\label{ordinary_delignemumford_stacks}

See the discussion at [[derived Deligne-Mumford stack]] for how ordinary [[Deligne-Mumford stack]]s are special cases of [[derived Deligne-Mumford stack]]s.

\hypertarget{derived_schemes}{}\paragraph*{{Derived schemes}}\label{derived_schemes}

\begin{defn}
\label{}\hypertarget{}{}
Let $k$ be a commutative ring. Recall the pregoemtry $\mathcal{T}_{Zar}(k)$.

A \textbf{[[derived scheme]]} over $k$ is a $\mathcal{T}_{Zar}(k)$-scheme.

\end{defn}
\hypertarget{derived_smooth_manifolds}{}\paragraph*{{Derived smooth manifolds}}\label{derived_smooth_manifolds}

\begin{itemize}%
\item [[derived smooth manifold]].

\end{itemize}
\hypertarget{derived_delignemumford_stacks}{}\paragraph*{{Derived Deligne-Mumford stacks}}\label{derived_delignemumford_stacks}

\begin{defn}
\label{}\hypertarget{}{}
Let $k$ be a commutative ring. Recall the pregeometry $\mathcal{T}_{et}(k)$

A \textbf{[[derived Deligne-Mumford stack]]} over $k$ is a $\mathcal{T}_{et}(k)$-scheme.

\end{defn}
\hypertarget{derived_schemes_with_ring_valued_structure_sheaves}{}\paragraph*{{Derived schemes with $E_\infty$-ring valued structure sheaves}}\label{derived_schemes_with_ring_valued_structure_sheaves}

The above [[derived scheme]]s have structure sheaves with values in [[simplicial object|simplicial]] commutative rings. There is also a notion of derived scheme whose structure sheaf takes values in [[E-infinity ring]]s. The theory of these is to be described in full detail in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Spectral Schemes]]} .

\end{itemize}
An indication of some details is in

\begin{itemize}%
\item [[Paul Goerss]], \emph{[[Topological Algebraic Geometry - A Workshop]]}

\end{itemize}
See at \emph{[[E-∞ scheme]]} and \emph{[[E-∞ geometry]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[geometry (for structured (∞,1)-toposes)]]


\item [[structured (∞,1)-topos]]


\item \textbf{locally representable structured (∞,1)-topos}



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Generalized schemes are definition 2.3.9 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Structured Spaces|Derived Algebraic Geometry V - Structured Spaces]]}

\end{itemize}
The definition of affine $\mathcal{G}$-schemes (absolute spectra) is in section 2.2.

[[!redirects locally representable structured (∞,1)-topos]]

[[!redirects locally representable structured (∞,1)-toposes]] [[!redirects locally representable structured (infinity,1)-toposes]]



\end{document}