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\begin{document}

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\section*{geometry (for structured (infinity,1)-toposes)}

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

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

[[!include higher geometry - contents]]

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

[[!include (infinity,1)-topos - 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{Geometry}{Geometry}\dotfill \pageref*{Geometry} \linebreak
\noindent\hyperlink{Pregeometry}{Pregeometry}\dotfill \pageref*{Pregeometry} \linebreak
\noindent\hyperlink{smooth_morphisms}{Smooth morphisms}\dotfill \pageref*{smooth_morphisms} \linebreak
\noindent\hyperlink{StructTop}{Structured $(\infty,1)$-topos}\dotfill \pageref*{StructTop} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{tale_geometry}{\'E{}tale geometry}\dotfill \pageref*{tale_geometry} \linebreak
\noindent\hyperlink{smooth_geometry}{Smooth geometry}\dotfill \pageref*{smooth_geometry} \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}

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

\begin{enumerate}%
\item the structure of an [[(∞,1)-site]];


\item the structure of an [[essentially algebraic (∞,1)-theory]].



\end{enumerate}
The [[object]]s of $\mathcal{G}$ are to be thought of as test-[[space]]s with certain [[higher geometry]] structure and the [[morphism]]s as [[homomorphism]]s preserving that geometric structure.

These two structures gives rise to

\begin{enumerate}%
\item The [[big topos|big]] [[(∞,1)-topos]] $Sh(\mathcal{G})$ of [[(∞,1)-sheaves]] on $\mathcal{G}$. Its objects are generalized [[space]]s given by rules

$X : \mathcal{G}^{op} \to$ [[∞Grpd]]

for how to map test spaces into them.


\item The [[∞-algebra over an (∞,1)-algebraic theory|(∞,1)-algebras]] over $\mathcal{G}$ in some [[little topos]] $\mathcal{X}$, given by rules

\begin{displaymath}
\mathcal{O} : \mathcal{G} \to \mathcal{X}
\end{displaymath}
that send each obect $U\in \mathcal{G}$ to a $U$-valued [[structure sheaf]].



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

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

\hypertarget{Geometry}{}\subsubsection*{{Geometry}}\label{Geometry}

\begin{uremark}
A \textbf{geometry} on an [[(∞,1)-category]] $\mathcal{G}$ is a [[Grothendieck topology]] on $\mathcal{G}$ together with

\begin{itemize}%
\item the extra structure given the information of which covering morphisms are to be thought of as [[local homeomorphism]]s


\item the extra property that it has all finite [[limit]]s.



\end{itemize}
If only all finite [[product]]s exist we speak of a pre-geometry. Every pregeometry $\mathcal{T}$ extends uniquely $\mathcal{T} \hookrightarrow \mathcal{G}$ to an \emph{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 space]]s. This distinction is used to encode [[smooth map|smoothness]] of maps between test spaces: a morphism in $\mathcal{G}$ is smooth if it locally factors through admissible maps between objects in $\mathcal{T}$.

\end{uremark}
\begin{udefn}
An \textbf{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 \textbf{admissible morphisms} have the following properties

\begin{itemize}%
\item admissible morphisms are stable under [[(∞,1)-pullback]];


\item admissible morphisms satisfy ``left cancellability'', meaning that whenever in

\begin{displaymath}
\itexarray{
    && Y 
    \\
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    X &&\stackrel{h}{\to}&& Z
  }
\end{displaymath}
$g$ and $h$ are admissible, then so is $f$.


\item admissible morphisms are closed under [[retracts]].



\end{itemize}
Equivalently, this is a [[Grothendieck topology]] on $\mathcal{G}$ which is \emph{generated} from admissible morphisms.

\end{udefn}
\begin{remark}
\label{}\hypertarget{}{}
As will become clear when looking at examples, the notion of admissible morphisms models the idea of maps between test spaces that behave like \emph{open injections} or, more generally, as \emph{local homeomorphisms} .

\end{remark}
\begin{udefn}
A \textbf{geometry (for $(\infty,1)$-toposes)} is

\begin{itemize}%
\item An [[(∞,1)-category]] $\mathcal{G}$ that

\begin{itemize}%
\item is [[essentially small category|essentially small]];


\item has all finite [[limit]]s


\item is [[idempotent complete category|idempotent complete]].



\end{itemize}

\item equipped with an admissibility structure.



\end{itemize}
\end{udefn}
\begin{udefn}
The \textbf{discrete geometry} $\mathcal{G}^0$ on $\mathcal{G}$ is given by

\begin{itemize}%
\item the admissible morphisms in $\mathcal{G}$ are precisely the equivalences


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



\end{itemize}
Every [[small (∞,1)-category]] $C$ becomes a geometry by regarding it as a discrete geometry in the above way.

\end{udefn}
\hypertarget{Pregeometry}{}\subsubsection*{{Pregeometry}}\label{Pregeometry}

\begin{udefn}
A \textbf{pregeometry} (for structured [[(∞,1)-topos]]es) is

\begin{itemize}%
\item an [[(∞,1)-category]] $\mathcal{T}$;


\item equipped with an admissibility structure (homotopical topology)



\end{itemize}
such that

\begin{itemize}%
\item $\mathcal{T}$ has all [[product]]s.

\end{itemize}
\end{udefn}
\begin{uremark}
So a geometry differs from a pregeometry in that it is [[idempotent complete category|idempotent complete]] and closed not only under [[products]] but under all finite [[limit]]s.

Various concepts for geometries have immediate analogues for pregeometries.

\end{uremark}
\hypertarget{smooth_morphisms}{}\paragraph*{{Smooth morphisms}}\label{smooth_morphisms}

\begin{udefn}
A morphism $f : X \to S$ in a pregeometry $\mathcal{T}$ is called \textbf{smooth} if it is \emph{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

\begin{displaymath}
\itexarray{
    U_i &\stackrel{u_i}{\to}& X
    \\
    \downarrow && \downarrow
    \\
    S \times V_i &\stackrel{p_1}{\to}& S
  }
  \,.
\end{displaymath}
\end{udefn}
\begin{uremark}
To interpret this, recall that we think of admissible morphisms as injections of open subsets.

\end{uremark}
\begin{uprop}
\begin{itemize}%
\item Smooth morphisms are stable under [[pullback]].


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



\end{itemize}
\end{uprop}
\hypertarget{StructTop}{}\subsubsection*{{Structured $(\infty,1)$-topos}}\label{StructTop}

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

\begin{displaymath}
C(-) : \mathcal{G} \to T
\end{displaymath}
such that

\begin{itemize}%
\item $C(-)$ is left [[exact functor|exact]] (preserves finite [[limit]]s);


\item $C(-)$ 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

\begin{displaymath}
C(\pi) : C(V) \to C(W)
\end{displaymath}
is an [[effective epimorphism]] in $T$, i.e. its [[?ech nerve]] is a [[simplicial resolution]] of $C(W)$:

\begin{displaymath}
\operatorname{&#268;ech}(C(\pi)) \stackrel{\simeq}{\to} C(W)
  \,.
\end{displaymath}


\end{itemize}
\end{udefn}
\begin{udefn}
Let $\mathcal{T}$ be a pregeometry and $\mathcal{X}$ an [[(∞,1)-topos]].

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

\begin{itemize}%
\item $\mathcal{O}$ preserves finite [[product]]s.


\item $\mathcal{O}$ preserves [[pullback]]s of admissible morphism in that for every [[pullback]] diagram

\begin{displaymath}
\itexarray{
    U' &\to& U
    \\
    \downarrow && \downarrow^f
    \\
    X' &\to& X
  }
\end{displaymath}
in $\mathcal{T}$ with $f$ admissible, the image

\begin{displaymath}
\itexarray{
    \mathcal{O}(U') &\to& \mathcal{O}(U)
    \\
    \downarrow && \downarrow^f
    \\
    \mathcal{O}(X') &\to& \mathcal{O}(X)
  }
\end{displaymath}
is again a [[pullback]].


\item $\mathcal{O}$ 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 $\mathcal{X}$.



\end{itemize}
\end{udefn}
\begin{uremark}
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 \emph{function algebra} .

\end{uremark}
\begin{udefn}
\ldots{}the universal geometry extending a pregeometry\ldots{}

\end{udefn}
\begin{uprop}
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 [[(infinity,1)-category|(∞,1)-categories]] of $\mathcal{T}$- and $\mathcal{G}$-structures on $\mathcal{X}$:

\begin{displaymath}
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}
\end{displaymath}
\end{uprop}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{tale_geometry}{}\subsubsection*{{\'E{}tale geometry}}\label{tale_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  for the precise statement

\begin{uprop}
The [[n-localic (∞,1)-topos|1-localic]] $\mathcal{G}_{et}$-[[generalized scheme]]s are precisely [[Deligne-Mumford stack]]s (without the [[separated (infinity,1)-presheaf|separation axiom]]).

\end{uprop}
See [[Deligne-Mumford stack]] for details.

\hypertarget{smooth_geometry}{}\subsubsection*{{Smooth geometry}}\label{smooth_geometry}

There should be a geometry $\mathcal{G}$ such that $\mathcal{G}$-[[generalized scheme]]s are precisely [[derived smooth manifold]]s.

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

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 general theory is developed in

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

\end{itemize}
The definition of a \textbf{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)-topos]]es -- is the subject of section 2.1 .

The inclusion

\begin{displaymath}
Spec^{\mathcal{G}} : 
  \mathcal{G} \hookrightarrow Str(\mathcal{G})
\end{displaymath}
is definition 2.1.2.

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

The inclusion

\begin{displaymath}
Sch(\mathcal{G}) \hookrightarrow Sh_\infty(Ind(\mathcal{G}))
\end{displaymath}
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

\begin{itemize}%
\item [[David Spivak]], \emph{Derived smooth manifolds} PhD thesis (\href{http://www.uoregon.edu/~dspivak/files/thesis1.pdf}{original version pdf} \href{http://arxiv.org/abs/0810.5174}{arXiv:0810.5174})

\end{itemize}
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 (\href{http://arxiv.org/abs/0810.5174}{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.

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

[[!redirects geometry (for structured ((∞,1)-toposes)]]



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