\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*{points-to-pieces transform}

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

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

\hypertarget{discrete_and_concrete_objects}{}\paragraph*{{Discrete and concrete objects}}\label{discrete_and_concrete_objects}

[[!include discrete and concrete objects - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_and_basic_properties}{Definition and basic properties}\dotfill \pageref*{definition_and_basic_properties} \linebreak
\noindent\hyperlink{relation_to_points_to_copieces}{Relation to points to co-pieces}\dotfill \pageref*{relation_to_points_to_copieces} \linebreak
\noindent\hyperlink{RelationToAufhebung}{Relation to Aufhebung of the initial opposition}\dotfill \pageref*{RelationToAufhebung} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{BundleEquivalenceAndConcordance}{Bundle equivalence and concordance}\dotfill \pageref*{BundleEquivalenceAndConcordance} \linebreak
\noindent\hyperlink{InGlobalEquivariantHomotopyTheory}{In global equivariant homotopy theory}\dotfill \pageref*{InGlobalEquivariantHomotopyTheory} \linebreak
\noindent\hyperlink{in_tangent_cohesion_the_differential_cohomology_diagram}{In tangent cohesion: the differential cohomology diagram}\dotfill \pageref*{in_tangent_cohesion_the_differential_cohomology_diagram} \linebreak
\noindent\hyperlink{comparison_map_between_algebraic_and_topological_ktheory}{Comparison map between algebraic and topological K-theory}\dotfill \pageref*{comparison_map_between_algebraic_and_topological_ktheory} \linebreak
\noindent\hyperlink{in_infinitesimal_cohesion}{In infinitesimal cohesion}\dotfill \pageref*{in_infinitesimal_cohesion} \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}

In a [[cohesive (∞,1)-topos]] $\mathbf{H}$, the canonical [[natural transformation]]

\begin{displaymath}
\flat \to \Pi
\end{displaymath}
from the [[flat modality]] to the [[shape modality]] may be thought of as sending ``points to the pieces in which they sit''.

\hypertarget{definition_and_basic_properties}{}\subsection*{{Definition and basic properties}}\label{definition_and_basic_properties}

Notice the existence of the following canonical [[natural transformations]] induced from the structure of a cohesive topos (a special case of the construction at \emph{[[unity of opposites]]}).

\begin{defn}
\label{TransformationFromPointsToPieces}\hypertarget{TransformationFromPointsToPieces}{}
Given a [[cohesive topos]] $\mathcal{E}$ with ($&#643; \dashv \flat$) its ([[shape modality]] $\dashv$ [[flat modality]])-[[adjunction]], then the [[natural transformation]]

\begin{displaymath}
\flat X \longrightarrow X \longrightarrow &#643; X
\end{displaymath}
(given by the [[composition]] of the $\flat$-[[counit of a comonad|counit]] followed by the $&#643;$-[[unit of a monad|unit]]) may be called the transformation \textbf{from points to their pieces} or the \textbf{points-to-pieces-transformation}, for short.

If this is an [[epimorphism]] for all $X$, we say that \emph{pieces have points} or that the \emph{[[Nullstellensatz]]} is verified.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The $(f^\ast \dashv f_\ast)$-[[adjunct]] of the transformation from pieces to points, def. \ref{TransformationFromPointsToPieces},

\begin{displaymath}
\flat X \longrightarrow X \longrightarrow &#643; X
\end{displaymath}
is (by the rule of forming right [[adjuncts]] by first applying the [[right adjoint]] functor and then precomposing with the [[unit of an adjunction|unit]] and by the fact that the [[adjunct]] of a [[unit of an adjunction|unit]] is the [[identity]]) the map

\begin{displaymath}
(f_\ast X \longrightarrow f_! X)
  \coloneqq
  \left(
  f_\ast X \longrightarrow f_\ast f^\ast f_! X \stackrel{\simeq}{\longrightarrow} f_!X
  \right)
  \,.
\end{displaymath}
Observe that going backwards by applying $f^\ast$ to this and postcomposing with the $(f^\ast \dashv f_\ast)$-[[counit of an adjunction|counit]] is equivalent to just applying $f^\ast$, since by [[idempotent monad|idempotency]] of $\flat$ the counit is an [[isomorphism]] on the [[discrete object]] $f^\ast f_! X$. Therefore the points-to-pieces transformation and its adjunct are related by

\begin{displaymath}
\left(
    \flat X \longrightarrow X \longrightarrow &#643; X
  \right)
  = 
  f^\ast \left(
    f_\ast X \longrightarrow f_! X
  \right).
\end{displaymath}
Observe then finally that since $f^\ast$ is a [[full and faithful functor|full and faithful]] [[left adjoint|left]] and [[right adjoint]], the points-to-pieces transform is an [[epimorphism]]/[[isomorphism]]/[[monomorphism]] precisely if its [[adjunct]] $f_\ast X \longrightarrow f_! X$ is, respectively.

If this adjunct

\begin{displaymath}
\flat X 
    \overset{\epsilon^{\flat}}{\longrightarrow} 
  X 
    \overset{\eta^\sharp}{\longrightarrow}
  \sharp X
\end{displaymath}
is a monomorphism, we say that \emph{[[discrete objects]] are [[concrete objects|concrete]]}.

\end{remark}
\hypertarget{relation_to_points_to_copieces}{}\subsection*{{Relation to points to co-pieces}}\label{relation_to_points_to_copieces}

\begin{prop}
\label{PiecesHavePointsIfDiscreteObjectsAreConcrete}\hypertarget{PiecesHavePointsIfDiscreteObjectsAreConcrete}{}
\textbf{([[pieces have points]] iff [[discrete objects are concrete]])}

For a [[cohesive topos]] $\mathbf{H}$, the the following two conditions are equivalent:

\begin{enumerate}%
\item \emph{pieces have points}, i.e. $\flat X \to X \to &#643; X$ is an [[epimorphism]] for all $X \in \mathbf{H}$;


\item \emph{[[discrete objects]] are [[concrete objects|concrete]]}, i.e. $\flat X \overset{ \eta^{\sharp}_{\flat X} }{\longrightarrow} \sharp \flat X$ is a [[monomorphism]].



\end{enumerate}
\end{prop}
See at \emph{[[cohesive topos]]} \href{cohesive+topos#PiecesHavePointsEquivalentToDiscreteObjectsAreConcrete}{this prop.}.

\hypertarget{RelationToAufhebung}{}\subsection*{{Relation to Aufhebung of the initial opposition}}\label{RelationToAufhebung}

For a cohesive 1-topos, if the pieces-to-points transform is an [[epimorphism]] then there is [[Aufhebung]] of the initial opposition $(\emptyset \dashv \ast)$ in that $\sharp \emptyset \simeq \emptyset$ (\hyperlink{LawvereMenni15}{Lawvere-Menni 15, lemma 4.1}, see also \hyperlink{Shulman15}{Shulman 15, section 3}). Conversely, if the [[base topos]] is a [[Boolean topos]], then this Aufhebung implies that the pieces-to-points transform is an [[epimorphism]] (\hyperlink{LawvereMenni15}{Lawvere-Menni 15, lemma 4.2}).

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

\hypertarget{BundleEquivalenceAndConcordance}{}\subsubsection*{{Bundle equivalence and concordance}}\label{BundleEquivalenceAndConcordance}

Given an [[∞-group]] $G$ in a [[cohesive (∞,1)-topos]] $\mathbf{H}$, with [[delooping]] $\mathbf{B}G$, then for any other object $X$ the [[∞-groupoid]] $\mathbf{H}(X,\mathbf{B}G)$ is that of $G$-[[principal ∞-bundles]] with [[equivalences]] between them. Alternatively one may form the [[internal hom]] $[X,\mathbf{B}G]$. Applying the [[shape modality]] to this yields the $\infty$-groupoid $\mathbf{H}^\infty(X,\mathbf{B}G) \coloneqq  &#643; [X,\mathbf{B}G]$ of $G$-principal $\infty$-bundles and [[concordances]] between them. Alternatively, the [[flat modality]] applied to the internal hom is again just the external hom $\flat [X,\mathbf{B}G] \simeq \mathbf{H}(X,\mathbf{B}G)$.

In conclusion, in this situation the points-to-pieces transform is the canonical map

\begin{displaymath}
\mathbf{H}(X,\mathbf{B}G) \longrightarrow \mathbf{H}^\infty(X,\mathbf{B}G)
\end{displaymath}
from $G$-principal $\infty$-bundles with bundle equivalences between them, to $G$-principal $\infty$-bundles with concordances between them.

\hypertarget{InGlobalEquivariantHomotopyTheory}{}\subsubsection*{{In global equivariant homotopy theory}}\label{InGlobalEquivariantHomotopyTheory}

In [[global equivariant homotopy theory]] an incarnation of the points to pieces transform is the comparison map from [[homotopy quotients]] to ordinary [[quotients]]

\begin{displaymath}
X//G \longrightarrow X/G
\end{displaymath}
which in terms of the [[Borel construction]] is induced by the map $E G \to \ast$

\begin{displaymath}
E G \times_G X \longrightarrow \ast \times_G X = X/G
  \,.
\end{displaymath}
(see at [[global equivariant homotopy theory]] \href{global+equivariant+homotopy+theory#OrdinaryQuotientAndHomotopyQuotientViaCohesion}{this prop.})

\hypertarget{in_tangent_cohesion_the_differential_cohomology_diagram}{}\subsubsection*{{In tangent cohesion: the differential cohomology diagram}}\label{in_tangent_cohesion_the_differential_cohomology_diagram}

In a [[tangent cohesive (∞,1)-topos]] on [[stable homotopy types]] the points-to-pieces transform is one stage in a natural hexagonal [[long exact sequence]], the \emph{[[differential cohomology diagram]]}. See there for more.

\hypertarget{comparison_map_between_algebraic_and_topological_ktheory}{}\subsubsection*{{Comparison map between algebraic and topological K-theory}}\label{comparison_map_between_algebraic_and_topological_ktheory}

Applied to [[stable homotopy types]] in $Stab(\mathbf{H}) \hookrightarrow T\mathbf{H}$ the [[tangent cohesive (∞,1)-topos]] which arise from a [[symmetric monoidal (∞,1)-category]] $V \in CMon_\infty(Cat_\infty(\mathbf{H}))$ [[internal (∞,1)-category|internal]] to $\mathbf{H}$ under internal [[algebraic K-theory of a symmetric monoidal (∞,1)-category]], the points-to-pieces transform interprets as the [[comparison map between algebraic and topological K-theory]]. See there for more

\hypertarget{in_infinitesimal_cohesion}{}\subsubsection*{{In infinitesimal cohesion}}\label{in_infinitesimal_cohesion}

In [[infinitesimal cohesion]] the points-to-pieces transform is an [[equivalence]].

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

[[!include cohesion - table]]

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

\begin{itemize}%
\item [[William Lawvere]], [[Matías Menni]], \emph{Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness}, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. (\href{http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html}{TAC})


\item [[Mike Shulman]], \emph{Brouwer's fixed-point theorem in real-cohesive homotopy type theory}, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (\href{https://arxiv.org/abs/1509.07584}{arXiv:1509.07584}, \href{https://doi.org/10.1017/S0960129517000147}{doi:10.1017/S0960129517000147}) [[!redirects points-to-pieces transforms]]



\end{itemize}
[[!redirects points-to-pieces transformation]] [[!redirects points-to-pieces transformations]]

[[!redirects points-to-pieces-transformation]] [[!redirects points-to-pieces-transformations]]

[[!redirects points-to-pieces-transform]] [[!redirects points-to-pieces-transforms]]

[[!redirects pieces have points]] [[!redirects pieces-have-points]]

[[!redirects discrete objects are concrete]]



\end{document}