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\begin{document}

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\section*{space}

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

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

[[!include higher geometry - contents]]

\textbf{space} $\leftarrow$ [[Isbell duality]] $\rightarrow$ [[algebra]]


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{NotionsOfSpace}{Notions of Space}\dotfill \pageref*{NotionsOfSpace} \linebreak
\noindent\hyperlink{spaces_probeable_by_model_spaces_stacks}{spaces probeable by model spaces: $\infty$-stacks}\dotfill \pageref*{spaces_probeable_by_model_spaces_stacks} \linebreak
\noindent\hyperlink{concrete_spaces_coprobeable_by_model_spaces_structured_toposes}{concrete spaces co-probeable by model spaces: structured $(\infty,1)$-toposes}\dotfill \pageref*{concrete_spaces_coprobeable_by_model_spaces_structured_toposes} \linebreak
\noindent\hyperlink{spaces_locally_like_model_spaces_generalized_schemes}{Spaces locally like model spaces: generalized schemes}\dotfill \pageref*{spaces_locally_like_model_spaces_generalized_schemes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The term `space' is quite generic and can mean many different things depending on context. Often we will link it because we want to include many of these contexts at once!

Probably the default spaces are [[topological spaces]] (as defined by [[Bourbaki]]), with [[continuous maps]] between them. There are many variants, from [[convergence spaces]] (general) to [[metric space|metric (metrisable) spaces]] (specific), all of which fall under the topic of [[topology]]. An important variation gives [[locales]]; many of the theorems from topology that require the [[axiom of choice]] for Bourbaki spaces become [[constructive mathematics|constructively]] valid for locales.

Often one restricts to [[nice topological spaces]] or to a [[nice category of spaces]]; these are not always the same kind of restriction! There are many examples under those two headings; [[m-cofibrant spaces]] in particular allow one to identify [[homotopy equivalence]] with [[weak homotopy equivalence]]. More generally in [[homotopy theory]], one uses the [[homotopy hypothesis]] to identify spaces with $\infty$-[[infinity-groupoid|groupoids]], of which there are several models. Since one common model is [[simplicial sets]], some homotopy theorists use ``space'' to mean ``simplicial set.''

Often one is also interested in spaces with additional structure. For instance, smooth [[manifolds]] and [[generalized smooth spaces]] are spaces with smooth structure, and [[algebraic varieties]] and [[schemes]] are spaces with algebraic structure. One can also speak more generally of spaces ``modeled'' on any suitable starting category; in this way one often comes to consider [[sheaves]] on a suitable [[site]] (and variations such as [[presheaves]], [[copresheaf|copresheaves]], or objects of an [[Isbell envelope]]) as spaces. Generalized smooth spaces and schemes are two commonly encountered examples of spaces ``modeled'' on a starting category in this way.

One can also [[vertical categorification|categorify]] the concept of space. As [[space and quantity|space is dual to quantity]], space and quantity can be [[categorification|categorified]] together; see $\infty$-[[infinity-space|space]] and compare $\infty$-[[infinity-quantity|quantity]]. Other types of categorified spaces include [[Grothendieck toposes]] (which are categorified locales) and [[stacks]] (which are categorified [[sheaves]]) and even their $\infty$-versions ($(\infty,1)$-[[(infinity,1)-topos|toposes]] and $\infty$-[[infinity-stack|stacks]]).

\hypertarget{NotionsOfSpace}{}\subsection*{{Notions of Space}}\label{NotionsOfSpace}

In [[Structured Spaces]], [[Jacob Lurie]] provides a coherent general picture of generalized notions of space in the context of [[higher geometry]] (often called ``derived geometry''). Here is an outline of the central aspects.

The \textbf{central ingredient} which we choose at the beginning to get a theory of [[higher geometry]] going is an [[(∞,1)-category]] $\mathcal{T}$ whose objects we think of as \textbf{model spaces} : the simplest objects exhibiting the geometric structures that we mean to consider.

\textbf{Examples for categories of model spaces}

\begin{itemize}%
\item with smooth structure

\begin{itemize}%
\item $\mathcal{T} =$ [[Diff]], the category of smooth [[manifold]]s;


\item $\mathcal{T} = \mathbb{L}$, the category of [[smooth locus|smooth loci]];



\end{itemize}

\item without smooth structure

\begin{itemize}%
\item $\mathcal{T} = (C Ring^{fin})^{op}$, the formal dual of [[CRing]]: the category of (finitely generated) algebraic [[affine scheme]]s;


\item $\mathcal{T} = (sC Ring^{fin})^{op}$, the formal dual of [[simplicial object]]s in [[CRing]];


\item $\mathcal{T} = (E_\infty Ring^{fin})^{op}$, the formal dual of [[E-∞ ring]]s: the category of (finitely generated) algebraic derived [[affine scheme]]s.



\end{itemize}


\end{itemize}
These [[(∞,1)-category|(∞,1)-categories]] $\mathcal{T}$ are naturally equipped with the structure of a [[site]] (and a bit more, which we won't make explicit for the present purpose). Following [[Jacob Lurie]] we call such a $\mathcal{T}$ a \textbf{(pre-)[[geometry (for structured (infinity,1)-toposes)|geometry]]} .

Every pregeometry $\mathcal{T}$ gives rise to a [[geometry (for structured (infinity,1)-toposes)|geometry]] $\mathcal{G}$ -- it's \emph{geometric envelope} . Roughly speaking, this contains not just the original test spaces but also their [[higher geometry|derived version]]. More on this below.

We want to be talking about generalized spaces \emph{modeled on} the objects of $\mathcal{G}$. There is a hierarchy of notions of what that may mean:

\textbf{Hierarchy of generalized spaces modeled on $\mathcal{G}$}

\begin{displaymath}
\itexarray{
    \mathcal{G} 
    &\stackrel{Spec^{\mathcal{G}}}{\hookrightarrow}&   
    Sch(\mathcal{G}) 
    &\hookrightarrow&
    \mathcal{L}Top(\mathcal{G})^{op} 
    &\hookrightarrow&      
    Sh_{(\infty,1)}(Pro(\mathcal{G}))
    \\
    \\
    model spaces
    &&
    spaces locally like model spaces
    &&
    concrete spaces coprobeable by model spaces
    &&
    spaces probeable by model spaces
    \\
    \\
    affine\;\mathcal{G}-schemes
    &&
    \mathcal{G}-schemes
    &&
    \mathcal{G}-structured\;(\infty,1)-toposes
    &&
    \infty-stacks\;on\;\mathcal{G}
    \\
    \stackrel{tame\;but\;restrictive}{\leftarrow} & 
    &&&&
    &
    \stackrel{versatile\;but\;possibly\;wild}{\to}
  }
\end{displaymath}
We explain what this means from right to left.

\hypertarget{spaces_probeable_by_model_spaces_stacks}{}\subsubsection*{{spaces probeable by model spaces: $\infty$-stacks}}\label{spaces_probeable_by_model_spaces_stacks}

An object $X$ \emph{probeable} by objects of $\mathcal{G}$ should come with an assignment

\begin{displaymath}
X : (U \in \mathcal{G}) \mapsto (X(U) \in \infty Grpd)
\end{displaymath}
of an [[∞-groupoid]] of possible ways to probe $X$ by $U$, for each possible $U$, natural in $U$. More precisely, this should define an object in the [[(∞,1)-category of (∞,1)-presheaves]] on $\mathcal{G}$

\begin{displaymath}
X \in PSh(\mathcal{G}) 
   = 
   Funct(\mathcal{G}^{op}, \infty Grpd)
\end{displaymath}
But for $X$ to be \emph{consistently} probeable it must be true that probes by $U$ can be reconstructed from overlapping probes of pieces of $U$, as seen by the [[coverage|topology]] of $\mathcal{G}$. More precisely, this should mean that the [[(∞,1)-presheaf]] $X$ is actually an object in an [[(∞,1)-category of (∞,1)-sheaves]] on $\mathcal{G}$

\begin{displaymath}
X \in 
  Sh(\mathcal{G}) \stackrel{}{\hookrightarrow}
  PSh(\mathcal{G})
  \,.
\end{displaymath}
Such objects are called [[∞-stack]]s on $\mathcal{G}$. The [[(∞,1)-category]] $Sh(\mathcal{G})$ is called an [[∞-stack]] [[(∞,1)-topos]].

A supposedly pedagogical discussion of the general philosophy of [[∞-stacks]] as probeable spaces is at [[motivation for sheaves, cohomology and higher stacks]].

The [[∞-stack]]s on $\mathcal{G}$ that are used in the following are those that satisfy [[descent]] on [[?ech cover]]s. But we will see [[(∞,1)-topos]]es of [[∞-stack]]s that may satisfy different descent conditions, in particular with respect to [[hypercover]]s. Every [[∞-stack]] [[(∞,1)-topos]] has a [[hypercompletion]] to one of this form.

For concretely working with [[hypercomplete (∞,1)-topos]]es it is often useful to use [[models for ∞-stack (∞,1)-toposes]] in terms of the [[model structure on simplicial presheaves]].

\begin{displaymath}
\itexarray{
    Sh^{hc}_{(\infty,1)}(C) 
   &\stackrel{\stackrel{\;\;\;\;\;lex\;\;\;\;\;\;}{\leftarrow}}  
  {\hookrightarrow}&
     PSh_{(\infty,1)}(C)
    && \text{general abstract def of (\infty,1)-topos}
    \\
    \uparrow^{\simeq} &&
    \uparrow^{\simeq}
    && \text{Lurie's theorem}
    \\
    ([C^{op}, SSet]_{loc})^\circ
    &\stackrel{\stackrel{Bousfield\;loc.}{\leftarrow}}{\to}&
    ([C^{op}, SSet]_{glob})^\circ
    &&
    \text{model category of simplicial presheaves}
  }
\end{displaymath}
\begin{uremark}
This discussion here is glossing over all set-theoretic size issues. See [[Structured Spaces|StSp, warning 2.4.5]].

\end{uremark}
\hypertarget{concrete_spaces_coprobeable_by_model_spaces_structured_toposes}{}\subsubsection*{{concrete spaces co-probeable by model spaces: structured $(\infty,1)$-toposes}}\label{concrete_spaces_coprobeable_by_model_spaces_structured_toposes}

Spaces probeable by $\mathcal{G}$ in the above sense can be very general. They need not even have a \emph{concrete underlying space} , even for general definitions of what \emph{that} might mean.

\textbf{(Counter-)Example} For $\mathcal{G} =$ [[Diff]], for every $n \in \mathbb{N}$ we have the [[∞-stack]] $\Omega_{cl}^n(-)$ (which happens to be an ordinary [[sheaf]]) that assigns to each manifold $U$ the set of closed [[differential form|n-form]]s on $U$. This is important as a generalized space: it is something like the rational version of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}, n)$. But at the same time this is a ``wild'' space that has exotic properties: for instance for $n=3$ this space has just a single point, just a single curve in it, just a single surface in it, but has many nontrivial probes by 3-dimensional manifolds.

In the classical theory for instance of [[ringed space]]s or [[diffeological space]]s a \emph{concrete underlying space} is taken to be a [[topological space]]. But this in turn is a bit \emph{too} restrictive for general purposes: a topological space is the same as a [[localic topos]]: a [[category of sheaves]] on a [[category of open subsets]] of a [[topological space]]. The obvious generalization of this to [[higher geometry]] is: an [[n-localic (∞,1)-topos]] $\mathcal{X}$.

This makes us want to say and make precise the statement that

A \textbf{concrete [[∞-stack]]} $X$ is one which has an \emph{underlying} [[(∞,1)-topos]] $\mathcal{X}$:

the collection of $U$-probes of $X$ is a [[subobject]] of the collection of [[(∞,1)-topos]]-morphisms from $U$ to $\mathcal{X}$:

\begin{displaymath}
X(U) \subset \mathcal{L}Top(\mathcal{G})^{op}(Sh_{\infty}(U),\mathcal{X})
\end{displaymath}
We think of $\mathcal{X}$ as the [[(∞,1)-topos]] of [[∞-stack]]s on a category of open subsets of a would-be space $X$, only that this would be space $X$ might not have an independent existence as a space apart from $\mathcal{X}$. The available entity closest to it is the [[terminal object]] ${*}_{\mathcal{X}} \in \mathcal{X}$.

To say that $\mathcal{X}$ is \emph{modeled on $\mathcal{G}$} means that among all the [[∞-stack]]s on the would-be space a [[structure sheaf]] of functions with values in objects of $\mathcal{G}$ is singled out: for each object $V \in \mathcal{G}$ there is a [[structure sheaf]] $\mathcal{O}(-,V) \in \mathcal{X}$, naturally in $V$.

This yields an [[(∞,1)-functor]]

\begin{displaymath}
\mathcal{O} : \mathcal{G} \to \mathcal{X}
  \,.
\end{displaymath}
We think of $X$ as being a concrete space \emph{co-probebale} by $\mathcal{G}$ (we can map from the concrete $X$ into objects of $\mathcal{G}$).

Such an $\mathcal{O}$ is a \emph{consistent} collection of coprobes if coprobes with values in $V$ may be reconstructed from co-probes with values in pieces of $V$.

More precisely:

\begin{udef}
\textbf{($\mathcal{G}$-structure, [[Structured Spaces|StSp, def. 1.2.8]])}

An [[(∞,1)-functor]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ is a \textbf{$\mathcal{G}$-valued structure sheaf} on the [[(∞,1)-topos]] if

\begin{itemize}%
\item it preserves finite [[limit]]s


\item and sends covering coproducts $(\coprod_i U_i) \to U$ to [[effective epimorphism]]s.



\end{itemize}
A pair $(\mathcal{X}, \mathcal{O})$ of an [[(∞,1)-topos]] $\mathcal{X}$ equipped with $\mathcal{G}$-valued [[structure sheaf]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ we call a [[structured (∞,1)-topos]].

\end{udef}
In summary:

A \textbf{concrete [[∞-stack]] $X$ modeled on $\mathcal{G}$} is

\begin{itemize}%
\item an [[(∞,1)-topos]] $\mathcal{X}$ (``of $\infty$-stacks on $X$'')


\item equipped with a $\mathcal{G}$-valued structure sheaf $\mathcal{O}$ in the form of a finite limits and cover preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$.



\end{itemize}
The fundamental \textbf{example} for [[structured (∞,1)-topos]]es are provided by the objects of $\mathcal{G}$ themselves, which are canonically equipped with a $\mathcal{G}$-structure as follows.

\begin{utheorem}
\textbf{([[Structured Spaces|StSp, thm. 2.1.1]])}

Let $f : \mathcal{G} \to \mathcal{G}'$ be a morphism of [[geometry (for structured (infinity,1)-toposes)|geometries]], then the obvious [[(∞,1)-functor]] $f^* : \mathcal{L}Top(\mathcal{G}) \to \mathcal{L}Top(\mathcal{G}')$ admits a [[left adjoint]]

\begin{displaymath}
f^*
  :
  \mathcal{L}Top(\mathcal{G}') 
   \stackrel{\leftarrow}{\to} 
  \mathcal{L}Top(\mathcal{G}) 
  :
  Spec_{\mathcal{G}}^{\mathcal{G}'}
\end{displaymath}
called the \textbf{relative spectrum functor}.

\end{utheorem}
For $\mathcal{G}$ any [[geometry (for structured (infinity,1)-toposes)|geometry]], write $\mathcal{G}_{disc}$ for the [[geometry (for structured (infinity,1)-toposes)|geometry]] obtained from this by forgetting its [[coverage|Grothendieck topology]] and instead using the discrete topology where only equivalences cover.

Notice that we may identify $\mathcal{G}_{disc}$-structures on the archetypical [[(∞,1)-topos]] [[∞Grpd]], being finite [[limit]]-preserving functors $\mathcal{G}_{disc}^{op} \to \infty Grpd$ with [[ind-object]]s in $\mathcal{G}^{op}$, hence with the opposite of [[pro-object]]s in $\mathcal{G}$. This gives a canonical inclusion

\begin{displaymath}
Pro(\mathcal{G}) 
    \hookrightarrow 
  \mathcal{L}Top(\mathcal{G})^{op}
  \,.
\end{displaymath}
\begin{udef}
\textbf{([[Structured Spaces|StSp, def. 2.1.2]])}

The composite [[(∞,1)-functor]]

\begin{displaymath}
Spec^{\mathcal{G}}
  : 
  Pro(\mathcal{G})^{op}
  \hookrightarrow 
  \mathcal{L}Top(\mathcal{G}_{disc})
  \stackrel{Spec_{\mathcal{G}}^{\mathcal{G}_{disc}}}{\to}
  \mathcal{L}Top(\mathcal{G})
\end{displaymath}
we call the \textbf{absolute spectrum functor}

\end{udef}
This [[category theory|general abstract]] description is reassuring, but we want a more concrete definition of what such $Spec^{\mathcal{G}} U$ is like:

\begin{udef}
\textbf{([[Structured Spaces|StSp, def. 2.2.9]])}

For every $U \in \mathcal{G}$ there is naturally induced a [[coverage|topology]] on the [[over category]] $Pro(\mathcal{G})/U$. Define the [[(∞,1)-topos]]

\begin{displaymath}
Spec U := Sh_{(\infty,1)}(Pro(\mathcal{G})/U)
  \,,
\end{displaymath}
naturally to be thought of as the [[(∞,1)-topos]] of [[∞-stack]]s \emph{on $U$} .

This is canonically equipped with a [[(∞,1)-functor]]

\begin{displaymath}
\mathcal{O}_{Spec X}
  : 
  \mathcal{G} \to Spec X
  \,.
\end{displaymath}
\end{udef}
And this is indeed the concrete underlying space produced by the absolute spectrum functor:

\begin{utheorem}
\textbf{[[Structured Spaces|StSp, prop. 2.2.11, thm. 2.2.12]])}

For every $U \in \mathcal{G}$ the pair $(Spec U, \mathcal{O}_{Spec U})$ is indeed a [[structured (∞,1)-topos]] and is indeed equivalent to the $Spec^{\mathcal{G}} U$ defined more abstractly above.

\end{utheorem}
\textbf{Example} For $\mathcal{G} = (C Ring^{fin})^{op}$ with the standard [[coverage|topology]] we have that 0-localic $\mathcal{G}$-structured spaces are \emph{[[locally ringed space]]s} , while $\mathcal{G}_{disc}$-structured 0-localic spaces are just arbitrary [[ringed space]]s. Applying the above machinery to this situaton gives a spectrum functor that takes a [[ring]] $R$ first to the [[ringed space]] $({*,R})$ and this then to the [[locally ringed space]] $(Spec R, R)$.

\hypertarget{spaces_locally_like_model_spaces_generalized_schemes}{}\subsubsection*{{Spaces locally like model spaces: generalized schemes}}\label{spaces_locally_like_model_spaces_generalized_schemes}

We have seen that $\mathcal{G}$-[[structured (∞,1)-topos]]es are those general spaces modeled on $\mathcal{G}$ that are well-behaved in that at least they do have an ``underlying topological structure'' in the form of an underlying [[(∞,1)-topos]]. But such concrete spaces may still be very different from the model objects in $\mathcal{G}$.

In parts this is desireable: many objects that one would naturally build out of the objects in $\mathcal{G}$, such as mapping spaces $[\Sigma,X]$, are much more general than objects in $\mathcal{G}$ but do live happily in $\mathcal{L}Top(\mathcal{G})^{op}$.

But in many situations one would like to regard $\mathcal{G}$-[[structured (∞,1)-topos]]es that are not globally but \emph{locally} equivalent to objects in $\mathcal{G}$. This is supposed to be captured by the following definition.

\begin{udef}
\textbf{[[Structured Spaces|StSp, def. 2.3.9]]}

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

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


\item such that

\begin{itemize}%
\item this covers $\mathcal{X}$ in that the canonical morphism

\begin{displaymath}
(\coprod_i V_i) \to {*}_{\mathcal{X}}
\end{displaymath}
to the [[terminal object]] in $\mathcal{X}$ is an [[effective epimorphism]]


\item the [[structured (∞,1)-topos]]es \newline $(\mathcal{X}/V_i, \mathcal{O}|_{V_i})$ induced by the $V_i$ are model spaces in that there exists $\{U_i \in \mathcal{G}\}$ and equivalences

\begin{displaymath}
(\mathcal{X}/V_i, \mathcal{O}|_{V_i})
  \simeq
  Spec^{\mathcal{G}} U_i
\end{displaymath}


\end{itemize}


\end{itemize}
\end{udef}
\textbf{Examples}

\begin{quote}%
\textbf{warning} the following statements really pertain to pregeometries, not geometries. For the moment, this here is glossing over the difference between the two. See [[geometry (for structured (∞,1)-toposes)]] for the details.


\end{quote}
\begin{itemize}%
\item ordinary smooth [[manifold]]s are [[n-localic (infinity,1)-topos|0-localic]] [[Diff]]-[[generalized scheme]]s (Structured Spaces|StSp, ex. 4.5.2])


\item ordinary [[schemes]] are those $(CRing^{fin})^{op}$-[[generalized scheme]]s whose underlying [[(∞,1)-topos]] is [[n-localic (infinity,1)-topos|0-localic]] and whose [[structure sheaf]] is [[n-truncated object of an (infinity,1)-category|0-truncated]] (Structured Spaces|StSp, prop. 4.2.9])


\item [[Deligne-Mumford stack]]s are 1-localic $(CRing^{fin})_{et}^{op}$-[[generalized scheme]]s (Structured Spaces|StSp, prop. 4.2.9])


\item This last statement is then the basis for calling a general $(CRing^{fin})_{et}^{op}$-[[generalized scheme]] a \textbf{derived Deligne-Mumford stack}


\item Finally, to make contact with the application to the derived moduli stack of derived elliptic curves, it seems that in [[Spectral Schemes]] a derived Deligne-Mumford stack (with derived in the sense of having replaced ordinary commutative rings by [[E-∞ ring]]s) is gonna be a 1-localic $(E_\infty Ring^{fin})^{op}$-[[generalized scheme]].



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

\begin{itemize}%
\item [[space form]]


\item [[gros topos]],


\item [[geometric homotopy type theory]], [[cohesive homotopy type theory]]


\item [[spacetime]] ([[physics]])



\end{itemize}
[[!redirects spaces]]



\end{document}