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\begin{document}

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\section*{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{geometries_and_admissibility_structure}{Geometries and admissibility structure}\dotfill \pageref*{geometries_and_admissibility_structure} \linebreak
\noindent\hyperlink{structure_sheaves_local_algebras}{Structure sheaves: local $\infty$-algebras}\dotfill \pageref*{structure_sheaves_local_algebras} \linebreak
\noindent\hyperlink{InTermsOfClassifying}{As algebras over geometric $(\infty,1)$-theories}\dotfill \pageref*{InTermsOfClassifying} \linebreak
\noindent\hyperlink{LocalMorphisms}{Local morphisms between structure sheaves}\dotfill \pageref*{LocalMorphisms} \linebreak
\noindent\hyperlink{in_terms_of_admissibility_structures}{In terms of admissibility structures}\dotfill \pageref*{in_terms_of_admissibility_structures} \linebreak
\noindent\hyperlink{InTermsOfClassifyingToposes}{In terms of classifying $(\infty,1)$-toposes}\dotfill \pageref*{InTermsOfClassifyingToposes} \linebreak
\noindent\hyperlink{CatOfStructuredToposes}{The $(\infty,1)$-category of structured $(\infty,1)$-toposes}\dotfill \pageref*{CatOfStructuredToposes} \linebreak
\noindent\hyperlink{the_spectrum_construction}{The spectrum construction}\dotfill \pageref*{the_spectrum_construction} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classes_of_examples}{Classes of examples}\dotfill \pageref*{classes_of_examples} \linebreak
\noindent\hyperlink{CanonicalStructureSheafOnObjectsInBigTopos}{Canonical structure sheaves on objects in the big topos}\dotfill \pageref*{CanonicalStructureSheafOnObjectsInBigTopos} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{StrSheafOfcontFuncts}{Structure sheaves of continuous functions}\dotfill \pageref*{StrSheafOfcontFuncts} \linebreak
\noindent\hyperlink{locally_ringed_spaces}{Locally ringed spaces}\dotfill \pageref*{locally_ringed_spaces} \linebreak
\noindent\hyperlink{ordinary_ringed_spaces}{Ordinary ringed spaces}\dotfill \pageref*{ordinary_ringed_spaces} \linebreak
\noindent\hyperlink{derived_ringed_spaces}{Derived ringed spaces}\dotfill \pageref*{derived_ringed_spaces} \linebreak
\noindent\hyperlink{derived_smooth_manifolds}{Derived smooth manifolds}\dotfill \pageref*{derived_smooth_manifolds} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{LimitsAndColimits}{Limits and colimits}\dotfill \pageref*{LimitsAndColimits} \linebreak
\noindent\hyperlink{EmbeddingIntoTheAmbientBigTopos}{Embedding into the ambient big $(\infty,1)$-topos}\dotfill \pageref*{EmbeddingIntoTheAmbientBigTopos} \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}

The notion of \emph{structured $(\infty,1)$-topos} is a generalization of the notion of a [[locally ringed space]] and locally [[ringed topos]] generalized to [[(∞,1)-topos]]es. This is a way to formalize [[higher geometry]]/[[derived geometry]]-structure on [[little topos|little]] [[(∞,1)-topos]]s.

So a structured $(\infty,1)$-topos is an [[(∞,1)-topos]] $\mathcal{X}$ equipped with an [[(∞,1)-sheaf|(∞,1)-]][[structure sheaf]] $\mathcal{O}$ that we think of as the collection of \emph{functions} on $\mathcal{X}$ that preserve extra geometric structure -- for instance [[topology|continuous structure]] or [[smooth structure]].

Being an $\infty$-function $\infty$-algebra, $\mathcal{O}(\mathcal{U})$ is an algebra over an [[(∞,1)-algebraic theory]] $\mathcal{G}$, called the (pre)[[geometry (for structured (∞,1)-toposes)]], since this encodes the nature of the extra geometric structure on $\mathcal{X}$.

Formally therefore a geometric structure $(\infty,1)$-sheaf of $\mathcal{X}$ is a product/limit-preserving [[(∞,1)-functor]]

\begin{displaymath}
\mathcal{O} : \mathcal{G} \to \mathcal{X}
  \,.
\end{displaymath}
Here we think of $\mathcal{X} = Sh_{(\infty,1)}(C)$ as being the [[(∞,1)-sheaf (∞,1)-topos]] on some [[(∞,1)-site]] $C$ and for any $V \in \mathcal{G}$ we think of

\begin{displaymath}
\mathcal{O}_V \in Sh_{(\infty,1)}(C)
\end{displaymath}
as being the [[(∞,1)-sheaf]] of structure-preserving functions on $C$ with values in $V$.

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

Let $S$ be a [[geometry (for structured (∞,1)-toposes)|geometry]] and let $Sh(S)$ be the [[(∞,1)-topos]] of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaves]] on $S$.

Notice that if $S = Op(X)$ is the nerve of the [[category of open subsets]] of some [[topological space]] $X$, then $Sh(X) \coloneqq Sh(S)$ is the [[(∞,1)-category of (∞,1)-sheaves]] on $X$, as in the above motivating introduction.

We want to define a \emph{structure sheaf} on $S$ (for instance on $Op(X)$) of quantities modeled on some $(\infty,1)$-category $V$ to be an [[(∞,1)-functor]]

\begin{displaymath}
O_X : V \to Sh(S)
\end{displaymath}
to be thought of as the assignment to each \emph{value space} $v \in V$ of an $(\infty,1)$-sheaf of $v$-valued functions on $S$ (on $X$ if $S = Op(X)$).

But since we are taking care of the sheaf condition on $S$, we also want to allow a similar kind of co-sheaf condition on $V$. In order to do so, $V$ is taken to be equipped with extra structure encoding covers in $V$, and $O_X$ is then required to respect this structure suitably.

\hypertarget{geometries_and_admissibility_structure}{}\subsubsection*{{Geometries and admissibility structure}}\label{geometries_and_admissibility_structure}

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{admissiblility structure} on an $(\infty,1)$-category $V$ is

\begin{itemize}%
\item a choice of [[sub-quasi-category|sub (∞,1)-category]] $V^{ad} \hookrightarrow V$, whose morphisms are to be called the \textbf{admissible morphisms}, such that

\begin{itemize}%
\item for every admissible morphism $U \to X$ and any morphism $X' \to X$ there is a diagram

\begin{displaymath}
\itexarray{
    U' &\to& U
    \\
    \downarrow && \downarrow
    \\
    X' &\to& X 
  }
\end{displaymath}
with $U' \to X'$ admissible;


\item for every diagram of the form

\begin{displaymath}
\itexarray{
  && Y
  \\
  & \nearrow && \searrow
  \\
  X &&\to&& Z
  }
\end{displaymath}
with $X \to Z$ and $Y \to Z$ admissible, also $X \to Y$ is admissible.



\end{itemize}

\item a [[Grothendieck topology]] on $V$ which has the property that it is generated from a [[coverage]] consisting of admissible morphisms.



\end{itemize}
\end{defn}
This is [[Structured Spaces|StrSh, def 1.2.1]] in view of remark 1.2.4 below that.

\begin{defn}
\label{}\hypertarget{}{}
An $(\infty,1)$-category $V$ equipped with an admissiblility structure is a \textbf{geometry} if it is essentially small, admits finite limits and is [[idempotent complete (infinity,1)-category|idempotent complete]].

\end{defn}
The admissible morphisms in an admissibility structure are roughly to be thought of as those morphisms that behave as \emph{open immersions} ,

\hypertarget{structure_sheaves_local_algebras}{}\subsubsection*{{Structure sheaves: local $\infty$-algebras}}\label{structure_sheaves_local_algebras}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(structure sheaf)}

Let $\mathcal{G}$ be a geometry and $\mathcal{X}$ an $(\infty,1)$-topos An [[(∞,1)-functor]]

\begin{displaymath}
O_X : \mathcal{G} \to \mathcal{X}
\end{displaymath}
is a $\mathcal{G}$-\textbf{structure} on $\mathcal{X}$ or $\mathcal{G}$-\textbf{[[structure sheaf]]} on $\mathcal{X}$ if

\begin{itemize}%
\item it is a left [[exact (∞,1)-functor]];


\item it respects gluing in $\mathcal{G}$ in that for $\{U_i \to V\}_i$ a covering [[sieve]] consisting of admissible morphism, the induced morphism

\begin{displaymath}
\coprod_i O_X(U_i) \to O_X(V)
\end{displaymath}
is an [[effective epimorphism]] in $\mathcal{X}$.



\end{itemize}
\end{defn}
Write $Str_{\mathcal{G}}(\mathcal{X}) \subset Func(\mathcal{G},\mathcal{X})$ for the full subcategory of such morphisms of the [[(∞,1)-category of (∞,1)-functors]].

\begin{remark}
\label{}\hypertarget{}{}
Without the condition on preservations of covers, the above defined the [[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]] over the [[essentially algebraic (∞,1)-theory]] $\mathcal{G}$. The preservation of covers encodes the \textbf{local} $\mathcal{G}$-algebras.

Therefore we shall equivalently write

\begin{displaymath}
\mathcal{G}Alg_{loc}(\mathcal{X}) \simeq Str_{\mathcal{G}}(\mathcal{X})
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
$\mathcal{G}Alg_{loc}(\mathcal{X})$ is the $(\infty,1)$-category of algebras over an $(\infty,1)$-[[geometric theory]].

\end{remark}
This is discussed \hyperlink{InTermsOfClassifying}{below}.

\hypertarget{InTermsOfClassifying}{}\subsubsection*{{As algebras over geometric $(\infty,1)$-theories}}\label{InTermsOfClassifying}

By the [[(∞,1)-Yoneda lemma]], a cover-preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ [[Yoneda extension|Yoneda extends]] equivalently to a [[(∞,1)-colimit]]-preserving [[(∞,1)-functor]]

\begin{displaymath}
\mathcal{O} : Sh_{(\infty,1)}(\mathcal{G}) \to \mathcal{X}
  \,.
\end{displaymath}
By the [[adjoint (∞,1)-functor theorem]] this has a [[right adjoint|right]] [[adjoint (∞,1)-functor]] and if $\mathcal{O}$ preserves finite [[(∞,1)-limits]] then so does its extension. Therefore local $\mathcal{G}$-[[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]] in $\mathcal{X}$ are equivalent to [[(∞,1)-geometric morphism]]s

\begin{displaymath}
\mathcal{X} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to}
  Sh_{(\infty,1)}(\mathcal{G})
  \,.
\end{displaymath}
This means that structure sheaves $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ are equivalently encoded in [[geometric morphism]]s to the [[(∞,1)-category of (∞,1)-sheaves]] on the geometry.

Formally we have:

\begin{prop}
\label{}\hypertarget{}{}
For $\mathcal{G}$ a geometry, precomposition of the [[inverse image]] functor with the [[(∞,1)-Yoneda embedding]] $y : \mathcal{G} \to Sh_{(\infty,1)}(\mathcal{G})$ induces an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G}))
  \stackrel{\simeq}{\to}
  \mathcal{G}Alg_{loc}(\mathcal{X})
\end{displaymath}
between the [[(∞,1)-category of (∞,1)-functors|(∞,1)-category of (∞,1)-geometric morphisms]] from $\mathcal{X}$ to $Sh_{(\infty,1)}(\mathcal{G})$ and the $(\infty,1)$-category of local $\mathcal{G}$-[[∞-algebra over an (∞,1)-algebraic theory|∞-algebra]]s in $\mathcal{X}$.

\end{prop}
This is (\hyperlink{Lurie}{StrSp, prop 1.42}).

\begin{proof}
This follows from the general fact, discussed in the section  at [[(∞,1)-Yoneda lemma]] that the essential image of the $(\infty,1)$-functor

\begin{displaymath}
Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G}))
  \stackrel{L}{\to}
  Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G}))
  \stackrel{y}{\to}
  Func(\mathcal{G}, \mathcal{X})
\end{displaymath}
is spanned by the left exact and cover preserving functors.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
We may think of this as saying that $Sh_{(\infty,1)}(\mathcal{G})$ is the $(\infty,1)$-[[classifying topos]] for the $(\infty,1)$-[[geometric theory]] of \textbf{local [[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]]} over the [[essentially algebraic (∞,1)-theory]] $\mathcal{G}$.

\end{remark}
\hypertarget{LocalMorphisms}{}\subsubsection*{{Local morphisms between structure sheaves}}\label{LocalMorphisms}

\begin{remark}
\label{}\hypertarget{}{}
The $(\infty,1)$-category $Str_{\mathcal{G}}(\mathcal{X})$ of $\mathcal{G}$-structure sheaves on an $(\infty,1)$-topos $\mathcal{X}$ does \emph{not} depend on the admissibility structure of $\mathcal{G}$, but only on the [[Grothendieck topology]] induced by it.

\end{remark}
(See [[Structured Spaces|StrSp, remark below prop. 1.4.2]]).

The admissibility structure does serve to allow the following definition of \emph{local} morphisms of structure sheaves.

\hypertarget{in_terms_of_admissibility_structures}{}\paragraph*{{In terms of admissibility structures}}\label{in_terms_of_admissibility_structures}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(local morphism of structure sheaves)}

A [[natural transformation]] $\eta : \mathcal{O} \to \mathcal{O}' : \mathcal{G} \to \mathcal{X}$ of structure sheaves is \textbf{local} if for every admissible morphism $U \to X$ in $\mathcal{G}$ the naturality diagram

\begin{displaymath}
\itexarray{
    \mathcal{O}(U) &\stackrel{\eta(U)}{\to}& \mathcal{O}'(U)
    \\
    \downarrow && \downarrow
    \\
    \mathcal{O}(X) &\stackrel{\eta(X)}{\to}& \mathcal{O}'(X)
  }
\end{displaymath}
is a [[limit in a quasi-category|pullback square]] in $\mathcal{X}$.

Write

\begin{displaymath}
Str^{loc}_{\mathcal{G}}(\mathcal{X}) \subset Str_{\mathcal{G}}(\mathcal{X})
\end{displaymath}
for the [[sub-quasi-category|sub-(∞,1)-category]] of $\mathcal{G}$-structures on $\mathcal{X}$ spanned by local transformations between them.

\end{defn}
\hypertarget{InTermsOfClassifyingToposes}{}\paragraph*{{In terms of classifying $(\infty,1)$-toposes}}\label{InTermsOfClassifyingToposes}

Alternatively, the local transformations can be characterized as follows

it turns out the local transformations are the right half of a [[factorization system]] on $Str_{\mathcal{G}}(\mathcal{X})$, and that this factorization system depends functorially on $\mathcal{X}$, in that for every geometric morphism $\mathcal{X} \to \mathcal{Y}$ the induced $Str_{\mathcal{G}}(\mathcal{X}) \to Str_{\mathcal{G}}(\mathcal{Y})$ respects these factorization systems. (theorem 1.3.1)

This one can turn around, to characterize local transformations (and hence admissibility structures on $\mathcal{G}$) in terms of functorial factorization systems on classifying $(\infty,1)$-toposes (def. 1.4.3):

For $\mathcal{K}$ an $(\infty,1)$-topos, declare that a \emph{geometric structure} on $\mathcal{K}$ is a choice of factorization systems on $Topos_{geom}(\mathcal{X}, \mathcal{K})^{op}$ that is functorial in $\mathcal{X}$ . Given such we have another way of saying ``local transformation'': this is the non-full subcategory $Str^{loc}_{\mathcal{K}}(\mathcal{X})$ of $Topos_{geom}(\mathcal{X}, \mathcal{K})^{op}$ on all objects and on the right part of the factorization system.

And this is indeed the same kind of datum as an admissibility structure on a geometry (example 1.4.4): in the case that $\mathcal{K} = Sh(\mathcal{G})$ is the classifying topos for the geometry $\mathcal{G}$, the defining equivalence $Topos_{geom}(\mathcal{X}, Sh(\mathcal{G}))^{op} \stackrel{\simeq}{\to} Str_{\mathcal{G}}(\mathcal{X})$ identifies the two sub-categories of local transformations, $Str^{loc}_{\mathcal{G}}(\mathcal{X})$ and $Str^{loc}_{Sh(\mathcal{G})}(\mathcal{X})$.

\hypertarget{CatOfStructuredToposes}{}\subsubsection*{{The $(\infty,1)$-category of structured $(\infty,1)$-toposes}}\label{CatOfStructuredToposes}

Let $(\infty,1)Toposes \subset$ [[(∞,1)Cat]] be the [[sub-quasi-category|sub (∞,1)-category]] of [[(∞,1)-topos]]es: objects are [[(∞,1)-topos]]es, morphisms are [[geometric morphism]]s.

Write $LTop \coloneqq (\infty,1)Toposes^{op}$.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{($(\infty,1)$-category of $\mathcal{G}$-structured $(\infty,1)$-toposes)}

For $\mathcal{G}$ a geometry, the $(\infty,1)$-category of $\mathcal{G}$-structured $(\infty,1)$-toposes

\begin{displaymath}
LTop(\mathcal{G})
\end{displaymath}
is defined as follows.

It is the [[sub-quasi-category|sub (∞,1)-category]]

\begin{displaymath}
LTop(\mathcal{G})
  \subset
  Func(\mathcal{G}, E LTop) \times_{Func(\mathcal{G}, LTop)} LTop
  \,,
\end{displaymath}
where $E LTop \to LTop$ is the [[coCartesian fibration]] associated by the [[(∞,1)-Grothendieck construction]] to the inclusion functor $LTop \hookrightarrow (\infty,1)Cat$, spanned by the following objects and morphisms:

\begin{itemize}%
\item objects are $\mathcal{G}$-structures $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ on some $(\infty,1)$-topos $\mathcal{X}$:

an object in $Func(\mathcal{G}, E LTop) \times_{Func(\mathcal{G}, LTop)} LTop$ is an object $\mathcal{X} \on LTop$ together with a functor $\mathcal{G} \to E LTop|_{\mathcal{X}}$ into the fiber of $E Top$ over that object; but that fiber is $\mathcal{X}$ itself, so an object in the fiber product is a functor $\mathcal{G} \to \mathcal{X}$ and this is in $LTop(\mathcal{G})$ if it is a $\mathcal{G}$-structure on $\mathcal{X}$;


\item morphisms $\alpha : \mathcal{O} \to \mathcal{O}'$ are \emph{local} morphisms of $\mathcal{G}$-structures:

for $f^* : \mathcal{X} \to \mathcal{Y}$ the image of $\alpha$ in $LTop$, $\alpha$ is in $LTop(\mathcal{G})$ precisely if for every admissible morphism $U \to X$ in $\mathcal{G}$ the square

\begin{displaymath}
\itexarray{
   f^* \mathcal{O}(U) &\to& f^*\mathcal{O}(X)
   \\
   \downarrow^{} && \downarrow^{}
   \\
   \mathcal{O}'(U) &\to& \mathcal{O}'(X)
  }
\end{displaymath}
is a pullback square in $\mathcal{Y}$.



\end{itemize}
\end{defn}
This is [[Structured Spaces|StrSp, def 1.4.8]]

\hypertarget{the_spectrum_construction}{}\subsubsection*{{The spectrum construction}}\label{the_spectrum_construction}

For $f : \mathcal{G} \to \mathcal{G}'$ a morphism of geometries, let

\begin{displaymath}
\mathcal{O}_{\mathcal{G}}^{\mathcal{G}'}
  : 
  Topos(\mathcal{G}') \to Topos(\mathcal{G})
\end{displaymath}
be the induced functor on categories of structured toposes.

\begin{theorem}
\label{}\hypertarget{}{}
This functor is a left [[adjoint (∞,1)-functor]]

\begin{displaymath}
(
    \mathcal{O}_{\mathcal{G}}^{\mathcal{G}'}
      \dashv
    Spec_{\mathcal{G}^{\mathcal{G}'}})
  )
  : 
  Topos(\mathcal{G})
  \stackrel{\overset{\mathcal{O}_{\mathcal{G}}^{\mathcal{G}'}
}{\leftarrow}}{\underset{Spec_{\mathcal{G}}^{\mathcal{G}'}
}{\to}}
  Topos(\mathcal{G}')
  \,.
\end{displaymath}
\end{theorem}
This is (\hyperlink{Lurie}{Lurie, theorem 2.1.1}).

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{G}$ a geometry, let $\mathcal{G}_0$ be the corresponding discrete geometry. We have a canonical morphism $\mathcal{G}_0 \to \mathcal{G}$.

Write

\begin{displaymath}
Spec^{\mathcal{G}} 
    : 
  Pro(\mathcal{G})
   \to 
  Topos(\mathcal{G}_0)
   \stackrel{Spec_{\mathcal{G}_0}^{\math}}{\to}
  Topos(\mathcal{G})
\end{displaymath}
for the composite.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
This fits into an adjunction

\begin{displaymath}
(\mathcal{O} \dashv Spec) : 
  Pro \mathcal{G} \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}
  Top(\mathcal{G})
  \,.
\end{displaymath}
\end{theorem}
This is (\hyperlink{Lurie}{Lurie, theorem xyz}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{classes_of_examples}{}\subsubsection*{{Classes of examples}}\label{classes_of_examples}

\hypertarget{CanonicalStructureSheafOnObjectsInBigTopos}{}\paragraph*{{Canonical structure sheaves on objects in the big topos}}\label{CanonicalStructureSheafOnObjectsInBigTopos}

For $\mathcal{G}$ a [[geometry (for structured (infinity,1)-toposes)|geometry]] let

\begin{displaymath}
\mathbf{H} \coloneqq Sh_\infty(\mathcal{G})
\end{displaymath}
be the [[(∞,1)-category of (∞,1)-sheaves]] on $\mathcal{G}$. This is the [[big topos]] of [[higher geometry]] modeled on $\mathcal{G}$. By the \hyperlink{InTermsOfClassifying}{above discussion} it is also the classifying topos of $\mathcal{G}$-[[structure sheaves]] on toposes:

a $\mathcal{G}$-valued structure sheaf $\mathcal{O}_{\mathcal{X}} : \mathcal{G} \to \mathcal{X}$ on an [[(∞,1)-topos]] $\mathcal{X}$ is equivalently an [[(∞,1)-geometric morphism]]

\begin{displaymath}
(p^* \dashv p_*) 
   :  
  \mathcal{X} \stackrel{\leftarrow}{\to} \mathbf{H}
  \stackrel{j}{\leftarrow}
  \mathcal{G}
\end{displaymath}
in that $\mathcal{O}_{\mathcal{X}} = p^* j$, where $j$ is the [[(∞,1)-Yoneda embedding]].

Notice that for every object $X \in \mathcal{H}$ its [[little topos]]-incarnation is the [[over-(∞,1)-topos]] $\mathbf{H}/X$. This canonically sits over $\mathbf{H}$ by its [[etale geometric morphism]]

\begin{displaymath}
\mathcal{X} \coloneqq \mathbf{H}/X
  \stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}
  \mathbf{H}
  \,.
\end{displaymath}
So we have

\begin{lemma}
\label{}\hypertarget{}{}
The [[little topos]] $\mathcal{X} \coloneqq \mathbf{H}/X$ of every object $X$ in the big topos $\mathbf{H}$ over $\mathcal{G}$ is canonically equipped with a $\mathcal{G}$-structure sheaf

\begin{displaymath}
\mathcal{O}_X : \mathcal{G} \stackrel{j}{\to} \mathbf{H}
  \stackrel{X^*}{\to}
  \mathbf{H}/X
  \,.
\end{displaymath}
\end{lemma}
We want to show that for $V \in \mathcal{G}$ the [[(∞,1)-sheaf]] $\mathcal{O}_X(V)$ may indeed be thought of as the ``sheaf of $V$-valued functions on $X$''.

Notice that for any $V \in \mathbf{H}$ we have that $X^*(F) = (p_2 : V \times X \to X)$.

Now assume first that $X$ is itself representable. Then by the discussion at [[over-(∞,1)-topos]] we have that $\mathbf{H}/X$ is a lucalization of $PSh_\infty(\mathcal{G})/X \simeq PSh_\infty(\mathcal{G}/X)$, where $\mathcal{G}/X$ is the [[big site]] of $X$. Under this equivalence (more details on this at [[over-topos]]) we have that $(V \times X \to X)$ identifies with the presheaf given by

\begin{displaymath}
(U \to X) \mapsto \mathcal{G}(U,V)
  \,.
\end{displaymath}
This is the ``sheaf of $V$-valued functions on $X$''.

(\ldots{})

\hypertarget{specific_examples}{}\subsubsection*{{Specific examples}}\label{specific_examples}

\hypertarget{StrSheafOfcontFuncts}{}\paragraph*{{Structure sheaves of continuous functions}}\label{StrSheafOfcontFuncts}

Consider $X$ an ordinary [[topological space]] and $Sh(X)$ the ordinary [[category of sheaves]] on its [[category of open subsets]]. Let $\mathcal{G} = Top$ be some [[small category|small]] version of [[Top]] with its usual [[Grothendieck topology]] with admissible [[covering]] families being [[open cover]]s. Consider the functor

\begin{displaymath}
O_X : Top \to Sh(X)
\end{displaymath}
that sends a topological space $V$ to the sheaf of [[continuous function]]s with values in $V$:

\begin{displaymath}
O_X(V) : U \mapsto C(U,V) = Hom_{Top}(U,V)
     \,.
\end{displaymath}
By general properties of the [[hom-functor]], this respects [[limit]]s. The gluing condition says that for $V_1, V_2 \subset V$ an open cover of $V$ by two patches, the morphism of sheaves

\begin{displaymath}
O_X(V_1) \coprod O_X(V_2) \to O_X(V)
\end{displaymath}
is an [[epimorphism]] of sheaves. This means that for each point $x \in X$ the map of [[stalk]]s

\begin{displaymath}
O_X(V_1)_x \coprod O_X(V_2)_x \to O_X(V)_x
\end{displaymath}
is an epimorphism of sets. But this just says that given any function $f : U_x \to V$ on a neighbourhood $U_x$ of $x$, there is a smaller neighbourhood $W_x \subset U_x$ such that the restriction $f|_{W_x}$ factors either through $V_1$ or through $V_1$. This is clearly the case by the fact that $V_1,V_2$ form an open cover. (A neighbourhood of $f(x) \in V$ exists which is contained in $V_1$ or in $V_2$, so take its preimage under $f$ as $U_x$).

\hypertarget{locally_ringed_spaces}{}\paragraph*{{Locally ringed spaces}}\label{locally_ringed_spaces}

\href{http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0459v1.pdf#page=71}{StrSh, remark 2.5.11}

Let $X$ be a topological space as before, but consider now the geometry $\mathcal{G} = CRing^{op}$ to be the opposite category of commutative [[ring]]s, where a covering family of $Spec R \in CRing^{op}$ is a family of maps of the form $R \to R[\frac{1}{r_i}]$ with $\{r_i \in R\}_i$ generating the unit ideal in $R$. So we think of $Spec R[\frac{1}{r_i}]$ as an the open subset of $Spec R$ on wich the function $r_i$ does not vanish, and a covering family we think of as an open cover by such open subsets, in direct analogy to the above example.

Now given a sheaf of rings

\begin{displaymath}
\bar O_X \in Sh(X,CRing)
\end{displaymath}
on $X$ (making $X$ a [[ringed space]]), which we may regard as the functor

\begin{displaymath}
O_X : CRing^{op} \to Sh(X)
\end{displaymath}
that it [[representable functor|represents]]

\begin{displaymath}
O_X(Spec R) : U \mapsto  
                   Hom_{CRing^{op}}(\bar O_X(U), Spec R)
                           = Hom_{CRing}(R, \bar O_X(U))
\end{displaymath}
we check what conditions this has to satisfy to qualify as a structure sheaf in the above sense.

The condition that

\begin{displaymath}
\coprod_i O_X(Spec R[\frac{1}{r_i}])
     \to Spec R
\end{displaymath}
is an epimorphism of sheaves again means that it is [[stalk]]wise an epimorphism of sets. Now, a ring homomorphism $R[\frac{1}{r_i}] \to \bar O_X(U)$ is given by a ring homomorphism $f : R \to O_X(U)$ such that $f(r_i)$ is invertible in $O_X(U)$. (We think of this as the pullback of functions on $Spec R$ to functions on $U$ by a map $U \to Spec R$ that lands only in the open subset where the functoin $r_i$ is non-vanishing).

So the condition that the above is an epimorphism on small enough $U$ says that for every ring homomorphism $\phi : R \to \bar O_X(U)$ the value of $\phi$ on at least one of the $r_i$ is invertible element in $O_X(U)$.

Thinking of this dually this is just the same kind of statement as in the first example, really. But now we can say this more algebraically:

by assumption there is a linear combination of the $r_i$ to the identity in $R$

\begin{displaymath}
1 = \sum_i \alpha_i r_i
\end{displaymath}
in $R$ (the partition of unity of functions on $Spec R$) and hence $\sum_i \alpha_i \phi(r_1) = 1$ in $(O_X)_x$ That for this invertible finite sum at least one of the summands is invertible is the condition that $(O_X)_x$ is a \emph{local ring} .

So a [[ringed space]] has a structure sheaf in the above sense if it is a [[locally ringed space]].

\hypertarget{ordinary_ringed_spaces}{}\paragraph*{{Ordinary ringed spaces}}\label{ordinary_ringed_spaces}

It may be worthwhile to retell the motivating example in the ``Idea'' introduction above for the maybe more familiar case of ordinary (1-categorical) structure sheaves with values in (unital) rings.

An ordinary [[topological space]] $X$ with its [[category of open subsets]] $Op(X)$ is a [[ringed space]] or $Op(X)$ is a [[ringed site]] if it is equipped with a [[sheaf]] $O_X : Op(X)^{op} \to Rings$ with values in the category of [[rings]]. For $U \subset X$ one thinks of $O_X(U)$ as the ring of allowed functions on $U$.

If for the moment we ignore the technicality that $O_X$ is supposed to be a [[sheaf]] and just regard it as a [[presheaf]], and if we furthermore invoke the idea of [[space and quantity]] and think of a ring $R$ as a generalized quantity in form of a copresheaf, canonically the [[representable functor|co-representable]] co-presheaf

\begin{displaymath}
R : (Ring^fin)^{op} \to \Set
\end{displaymath}
on finitely generated rings, which sends

\begin{displaymath}
R : R' \mapsto Hom(R,R')
\end{displaymath}
then we find that $O_X$ is in fact a presheaf on $Op(X)$ with values in a co-presheaf on $(Ring^{fin})^{op}$

\begin{displaymath}
O_X : Op(X)^{op} \to [(Ring^{fin})^{op}, Set]
\end{displaymath}
or equivalently a generalized quantity on $(Ring^{fin})^{op}$ with values in presheaves on $X$:

\begin{displaymath}
O_X : (Ring^{fin})^{op} \to [Op(X)^{op}, Set]
  \,.
\end{displaymath}
Since rings can be identified with left-exact functors $(Ring^{fin})^{op}\to Set$, we don't need to impose any admissibility structure in order to recover the notion of a sheaf of rings, since left-exactness is part of the definition of a ``structure sheaf.'' We do, however, need an admissibility structure if we want to recover the notion of a sheaf of \emph{local} rings, as in the previous example above.

\hypertarget{derived_ringed_spaces}{}\paragraph*{{Derived ringed spaces}}\label{derived_ringed_spaces}

Now formulate the previous example according to the above definition:

Let $CRing^{fin}$ be the category of finitely generated commutative rings There is a standard admissibility structure on $(CRing^{fin})^{op}$ that makes it a \emph{geometry} in the above sense.

Then for $X$ a topological space an $(\infty,1)$-functor $(CRing^{fin})^{op} \to Sh(S)$ to $(infty,1)$-sheaves on $X$ is a sheaf of local commutative rings on $X$. (\href{http://arxiv.org/abs/0905.0459}{StrSh, example 1.2.13})

To generalize this to \textbf{derived structure sheaves} we replace the category of rings here with the $(\infty,1)$-category of simplicial rings.

\textbf{Definition} (\href{http://arxiv.org/abs/0905.0459}{StrSh def 4.1.1})

The $(\infty,1)$-category of \textbf{simplicial commutative rings} over an ordinary commutative ring $k$ is

\begin{displaymath}
SCR_k \coloneqq PSh_\Sigma(FreeAlg_k)
\end{displaymath}
the $(\infty,1)$-category of [[(∞,1)-presheaves]] on commutative $k$-algebras of the form $k[x_1, \cdots, x_n]$.

Then\ldots{}

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

(\href{http://arxiv.org/abs/0905.0459}{StrSh, example 4.5.2})

Every ordinary [[smooth manifold]] $X$ becomes canonically a generalized space with structure sheaf as follows:

Let $V \coloneqq Diff$ be some version of the category of smooth manifolds. This becomes a \emph{pregeometry} in the above sense by taking admissible morphisms to be inclusions of open submanifolds.

Then for $Sh(X) \coloneqq Sh(Op(X))$ the $(\infty,1)$-topos of $(\infty,1)$-sheaves on $X$, the obvious $(\infty,1)$-functor

\begin{displaymath}
O_X : V \to Sh(X)
\end{displaymath}
which for every co-test manifold $v$ is the sheaf

\begin{displaymath}
O_X(v) : (U \subset X) \mapsto Hom_{Diff}(U,v)
\end{displaymath}
is a $Diff$-structure sheaf. Notice that this is precisely nothing but the structure sheaf of smooth functions from the introduction above.

The point is that there are other, more fancy structure sheaves

\begin{displaymath}
O_X : V \to Sh(X)
\end{displaymath}
possible. They describe \textbf{[[derived smooth manifolds]]} as described in \href{http://math.berkeley.edu/~dspivak/thesis2.pdf}{DerSmooth}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{LimitsAndColimits}{}\subsubsection*{{Limits and colimits}}\label{LimitsAndColimits}

Let $\mathcal{G}$ be a [[geometry for structured (infinity,1)-toposes]]. Write $F : (\infty,1)Topos(\mathcal{G}) \to$ [[(∞,1)Topos]] for the [[forgetful functor|forgetful]] [[(∞,1)-functor]] from $\mathcal{G}$-structured $(\infty,1)$-toposes to their underlying $(\infty,1)$-topos.

\hypertarget{proposition_2}{}\paragraph*{{Proposition}}\label{proposition_2}

The $(\infty,1)$-category $(\infty,1)Topos(\mathcal{G})$ has all [[filtered (∞,1)-category|cofiltered]] [[(∞,1)-limit]]s and the forgetful functor $F : (\infty,1)Topos(\mathcal{G}) \to (\infty,1)Topos$ preserves these.

This appears as (\hyperlink{Lurie}{Lurie, corl 1.5.4}).

\hypertarget{EmbeddingIntoTheAmbientBigTopos}{}\subsubsection*{{Embedding into the ambient big $(\infty,1)$-topos}}\label{EmbeddingIntoTheAmbientBigTopos}

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

\begin{displaymath}
\hat \mathcal{G} \coloneqq Pro(\mathcal{G})
\end{displaymath}
for its [[(∞,1)-category]] of [[ind-object in an (∞,1)-category|pro-objects]].

Write $\widehat{\infty Grpd}$ for the very large [[(∞,1)-category]] of large [[∞-groupoid]]s and

\begin{displaymath}
\hat Sh(\hat \mathcal{G}, \widehat{\infty Grpd})
\end{displaymath}
for the [[very large (∞,1)-sheaf (∞,1)-topos]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The canonical inclusion

\begin{displaymath}
Scheme(\mathcal{G}) \to 
  \hat PSh(\hat \mathcal{G}, \widehat{\infty Grpd})
\end{displaymath}
of [[locally representable structured (∞,1)-topos]]es by

\begin{displaymath}
(X, \mathcal{O}_X) \mapsto Hom(Spec(-), (X, \mathcal{O}_X))
\end{displaymath}
is a [[full and faithful (∞,1)-functor]].

\end{prop}
This is (\hyperlink{Lurie}{Lurie, theorem, 2.4.1}).

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

\begin{itemize}%
\item [[ringed space]], [[locally ringed space]]


\item [[ringed site]], [[locally ringed site]]


\item [[ringed topos]], [[locally ringed topos]]


\item [[locally algebra-ed topos]]


\item \textbf{locally algebra-ed (∞,1)-topos}


\item [[geometry (for structured (∞,1)-toposes)]], \textbf{structured $(\infty,1)$-topos} , [[locally representable structured (∞,1)-topos]]



\end{itemize}
Analogous structures in the axiomatic context of [[differential cohesion]] are discussed in \emph{\href{cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructureSheaves}{differential cohesion -- Structure sheaves}}.

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

The notion of structured $(\infty,1)$-toposes was introduced in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Structured Spaces]]}

\end{itemize}
Analogous precursor discussion in 1-category theory, hence for [[ringed toposes]] is in

\begin{itemize}%
\item [[Monique Hakim]], \emph{Topos annel\'e{}s et sch\'e{}mas relatifs}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).

\end{itemize}
The special case of ``smoothly structured spaces'' ([[derived smooth manifolds]]) is discussed in

\begin{itemize}%
\item [[David Spivak]], \emph{Derived smooth manifolds} PhD thesis (\href{http://www.uoregon.edu/~dspivak/files/thesis1.pdf}{pdf})

\end{itemize}
[[!redirects structured (∞,1)-topos]]

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

[[!redirects structured generalized spaces]] [[!redirects structured generalized space]] [[!redirects ringed (∞,1)-topos]] [[!redirects ringed generalized spaces]] [[!redirects ringed generalized space]]

[[!redirects locally algebra-ed (∞,1)-topos]] [[!redirects locally algebra-ed (infinity,1)-topos]] [[!redirects locally algebra-ed (∞,1)-toposes]] [[!redirects locally algebra-ed (infinity,1)-toposes]]



\end{document}