\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{space and quantity}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{Isbell}{Isbell duality: global functions and spectrum}\dotfill \pageref*{Isbell} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{cartesian_test_spaces_diffeological_spaces_and_smooth_algebras}{Cartesian test spaces: diffeological spaces and smooth algebras}\dotfill \pageref*{cartesian_test_spaces_diffeological_spaces_and_smooth_algebras} \linebreak
\noindent\hyperlink{higher_space_and_higher_quantity}{Higher space and higher quantity}\dotfill \pageref*{higher_space_and_higher_quantity} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Consider a [[category]] $C$ whose [[objects]] are thought of as \emph{[[spaces]]} of sorts (``test spaces''), and whose [[morphisms]] are regarded as [[homomorphisms]] between these spaces.

There is then a general notion of

\begin{itemize}%
\item \textbf{spaces} modeled on $C$ that are \emph{testable} or \emph{probe-able} by objects of $C$;


\item \textbf{quantities} with values in $C$.



\end{itemize}
Very generally, following \hyperlink{Lawvere86}{Lawvere 86}:

\begin{itemize}%
\item a generalized [[space]] modeled on the objects of $C$ is a [[presheaf]] on $C$, i.e. a [[functor]] of the form

$X \;\colon\; C^{op} \to$ [[Set]]:

we think of each such presheaf as being a rule that assigns to each test space $U \in C$ the set $X(U)$ of allowed maps from $U$ \emph{into} the would-be space $X$ (this is really the perspective of [[functorial geometry]], originally due to \href{functorial+geometry#Grothendieck65}{Grothendieck 65});


\item a generalized [[quantity]] modeled on $C$ is a [[copresheaf]] on $C$, i.e. a [[functor]] of the form

$A \;\colon\; C \to Set$:

we think of each such copresheaf $A$ as a rule that assigns to each test space $U \in C$ the set $A(U)$ of allowed maps \emph{from} the would-be space $A$ into $U$, hence as the collection of $U$-valued \emph{functions} on $A$. Since a function on a point is a ``quantity'', these are generalized quantities.



\end{itemize}
One may view the \emph{[[Yoneda lemma]]} and the resulting \emph{[[Yoneda embedding]]} as expressing consistency conditions on this perspective: The [[Yoneda lemma]] says that the prescribed rule for how to test a generalized space $X$ by a test space $U$ turns out to coincide with the actual maps from $U$ to $X$, when $U$ is itself regarded as a generalized space, and the [[Yoneda embedding]] says that, as a result, the nature of maps between test spaces does not depend on whether we regard these as test spaces or as generalized spaces.

Beyond this \emph{automatic} consistency condition, guaranteed by [[category theory]] itself, typically the admissible (co)presheaves that are regarded as generalized spaces and quantities are required to respect one more consistency condition:

\begin{itemize}%
\item If $C$ carries the structure of a [[site]], one asks a generalized space to be a presheaf $X = PSh(C) = [C^{op},Set]$ that respects the way objects in $C$ are [[covering|covered]] by other objects. These are the \emph{[[sheaves]]}. The [[category of sheaves]]

$Sh(C) \hookrightarrow PSh(C)$

is the [[topos]] of spaces modeled on objects in $C$. More details on how to think of sheaves as generalized spaces is at [[motivation for sheaves, cohomology and higher stacks]].


\item Given any generalized spaces, functions out of it are expected to respect [[product]]s of coefficient objects, in that a function with values in $U \times V$ is the same as a pair of functions, one with values in $U$, one with values in $V$. Hence one is typically interested in copresheaves that preserve at least [[product]]

$CoSh(C) \hookrightarrow CoPSh(C)$.



\end{itemize}
\hypertarget{details}{}\subsubsection*{{Details}}\label{details}

As indicated in \hyperlink{Lawvere86}{Lawvere 86, from p. 17 on}

\begin{itemize}%
\item (generalized) spaces;


\item (generalized) quantities (e.g. [[algebras of functions]]);


\item the [[duality]] between the two;



\end{itemize}
which underlies much of mathematics is at its heart controlled by the following elementary [[category theory|category theoretic]] reasoning:

Let $S$ be some category whose objects we want to think of as certain simple spaces on which we want to model more general kinds of spaces. For instance $S = \Delta$, the simplicial category, or $S =$ [[CartSp]].

An ordinary [[manifold]], for instance, is a space required to be \emph{locally isomorphic} to an object in $S = CartSp$. But more generally, a space $X$ modeled on $S$ need only be \emph{probeable} by objects of $S$, giving a rule which to each test object $U \in S$ assigns the collection of admissible maps from $U$ to $X$, such that this assignment is well-behaved with respect to morphisms in $S$. Such an assignment is nothing but a [[presheaf]] on $S$, i.e. a contravariant functor

\begin{displaymath}
X : S^{op} \to Set
  \,.
\end{displaymath}
Therefore general spaces modeled on $S$ are nothing but presheaves on $S$:

\begin{displaymath}
Spaces_S := PSh(S)
  \,.
\end{displaymath}
Of course this is an extremely general notion of spaces modeled on $S$.

We have the [[Yoneda lemma|Yoneda embedding]] $S \hookrightarrow Spaces_S$ and using this we can say that the collection of \emph{functions} on a generalized space $X$ with values in $U \in S$ is

\begin{displaymath}
C(X,U) := Hom_{Spaces_S}(X,U)
  \,.
\end{displaymath}
This assignment is manifestly covariant in $U$, and hence more generally we can consider the functions on $X$, $C(X)$ to be a copresheaf on $S$, namely a covariant functor

\begin{displaymath}
C(X) := Hom(X,-) : S \to Sets
  \,.
\end{displaymath}
One can think of $C(X)$ as being a generalized quantity which may be \emph{co-probed} by objects of $S$.

In this vein, one can say, generally, that co-presheaves on $S$ are generalized quantities modeled on $S$, and we write

\begin{displaymath}
Quantities_S := CoPSh(S)
  \,.
\end{displaymath}
Given any such generalized quantity $A \in Quantities_S$, we can ask which generalized space it behaves like the algebra of functions on. This generalized space should be called $Spec(A)$ and can be defined as a presheaf by the assignment

\begin{displaymath}
Spec(A) 
  :
  U \mapsto Hom_{Quantities_S}(A, C(U))
  \,.
\end{displaymath}
In total this yields an adjoint pair of functors between generalized spaces and generalized quantities:

\begin{displaymath}
Spaces_S
\stackrel{\stackrel{C(-)}{\to}}{\stackrel{Spec(-)}{\leftarrow}}
  Quantities_S
  \,.
\end{displaymath}
(That this is an adjunction can be understood as a special case of [[abstract Stone duality]] induced by a [[dualizing object]].)

Lawvere refers to this [[adjoint pair]] as \textbf{[[Isbell conjugation]]}.

In conclusion, the grand [[duality]] between spaces and quantities is a consequence of the [[formal duality]] which reverses the arrows in the category $S$ of test spaces.

This story generalizes straightforwardly from [[presheaf|presheaves]] with values in [[Set]] to presheaves with values in other categories. Of relevance are in particular presheaves with values in the category [[Top]] of [[topological space]]s and presheaves with values in the category of [[spectrum|spectra]]. See the examples below.

\hypertarget{Isbell}{}\subsection*{{Isbell duality: global functions and spectrum}}\label{Isbell}

we describe the [[duality]] between space and quantity induced by forming

\begin{itemize}%
\item functions on spaces;


\item spectra of function algebras.



\end{itemize}
Let $V$ be a [[symmetric monoidal category]] and $C$ a $V$-[[enriched category]]. Write $[C^{op},V]$ for the [[enriched functor category]] and $j : C \to [C^{op},V]$ for the [[Yoneda embedding]].

There is canonically a $V$-[[adjunction]]

\begin{displaymath}
(\mathcal{O} \dashv Spec)  \;\;: \;\; [C, V]^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  [C^{op},V]
\end{displaymath}
the \textbf{[[Isbell adjunction]]}. Here

\begin{itemize}%
\item $\mathcal{O} := [C^{op},V](j(-), -)$ sends a presheaf $X$ to the copresheaf $U \mapsto [C^{op},V](X,j(U))$;


\item $Spec := [C,V]^{op}(j(-),-)$ sends a copresheaf $A$ to the presheaf $U \mapsto [C,V](A, j^{op}(U))$.



\end{itemize}
If we assume that $C$ is [[copower|tensored]] over $V$, then that this is an adjunction may be seen in [[end]]/[[coend]]-calculus to express the [[hom-object]]s in the [[enriched functor category]] as follows. We compute

\begin{displaymath}
\begin{aligned}
    [C,V]^{op}(\mathcal{O}(X),A)
    & =
    \int_{u \in C} V(A(u), [C^{op},V](X,j(u)))
    \\
    & \simeq
    \int_{u \in C} V(A(u), [C^{op},V](\int^{v \in C} j(v) \cdot X(v),j(u)))   
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), [C^{op},V](j(v),j(u)))    
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), V(v,u))    
  \end{aligned}
  \,,
\end{displaymath}
where we used the [[Yoneda lemma]] $[C^{op},V](j(v),j(u)) \simeq V(v,u)$ and the [[co-Yoneda lemma]] $X \simeq \int^{v \in V} j(v) \cdot X(v)$ and the fact that the $V$-enriched hom sends colimits and coends in the first argument to limits and ends.

Analogously we find

\begin{displaymath}
\begin{aligned}
    [C^{op},V](X,Spec A)
    & =
    \int_{v \in C} V(X(v),[C,V](A, j^{op}(v)))
    \\
    & \simeq
    \int_{v \in C} V(X(v), [C,V](\int^{u \in C} j^{op}(u) \cdot X(v),j^{op}(u)))   
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), [C,V](j^{op}(v),j^{op}(u)))    
    \\
    & \simeq
    \int_{u, v \in C} V(A(u) \cdot X(v), V(v,u))    
  \end{aligned}
  \,,
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{cartesian_test_spaces_diffeological_spaces_and_smooth_algebras}{}\subsubsection*{{Cartesian test spaces: diffeological spaces and smooth algebras}}\label{cartesian_test_spaces_diffeological_spaces_and_smooth_algebras}

Consider the category of test spaces $C =$ [[CartSp]].

Then

\begin{itemize}%
\item spaces modeled on $C$ are [[generalized smooth space]]s such as [[diffeological space]]s;


\item quantities modeled on $C$ are [[smooth algebra]]s ($C^\infty$-rings).



\end{itemize}
The adjunction $(\mathcal{O} \dashv Spec)$ sends a smooth space to its smooth algebra of functions and a smooth algebra of functions to its ``spectrum''.

\hypertarget{higher_space_and_higher_quantity}{}\subsection*{{Higher space and higher quantity}}\label{higher_space_and_higher_quantity}

There are various specializations of interest on this

\begin{itemize}%
\item [[higher category theory|higher categorical]] version

\begin{itemize}%
\item [[∞-space]] modeled on $C$ is a [[simplicial presheaf]] on $C$, i.e. a functor $X : C^{op} \to$ [[SSet]].


\item [[∞-quantity]] modeled on $C$ is a cosimplicial copresheaf on $C$, i.e. a functor $X : C \to CoSSet$ .



\end{itemize}


\end{itemize}
With the advent of [[Higher Topos Theory]] abstract concepts of \emph{space and quantity} have been realized fully in the context of [[(∞,1)-topos]]es in terms of [[structured (∞,1)-topos]]es and [[generalized scheme]]s. For a summary see the tables at \href{http://ncatlab.org/nlab/show/A+Survey+of+Elliptic+Cohomology+-+the+derived+moduli+stack+of+derived+elliptic+curves#NotionsOfSpace}{notions of space}.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

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


\item [[generalized smooth space]]


\item [[geometry of physics -- categories and toposes]]



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

The general perspective is due to

\begin{itemize}%
\item [[William Lawvere]], \emph{Taking categories seriously}, Revista Colombiana de Matematicas, XX (1986) 147-178, reprinted in: Reprints in Theory and Applications of Categories, No. 8 (2005) pp. 1-24 (\href{http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html}{TAC})


\item [[William Lawvere]], \emph{Categories of space and quantity}, in: J. Echeverria et al (eds.), \emph{The Space of mathematics}, de Gruyter, Berlin, New York (1992) (\href{https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1992-categories-of-space-and-quantity.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item [[William Lawvere]], \emph{John Isbell's Adequate Subcategories}, TopCom \textbf{11} no.1 2006. (\href{http://at.yorku.ca/t/o/p/d/65.htm}{link})

\end{itemize}
[[!redirects quantity]] [[!redirects quantities]]

[[!redirects space and quantity]] [[!redirects duality between space and quantity]]

[[!redirects generalized space]] [[!redirects generalized spaces]]



\end{document}