\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*{infinitesimal cohesive (infinity,1)-topos}

\begin{quote}%
This entry is about a variant of the concept of [[cohesive (∞,1)-topos]]. The definition here expresses an intuition not unrelated to that at [[infinitesimally cohesive (∞,1)-presheaf on E-∞ rings]] but the definitions are unrelated and apply in somewhat disjoint contexts.


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

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

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

\hypertarget{synthetic_differential_geometry}{}\paragraph*{{Synthetic differential geometry}}\label{synthetic_differential_geometry}

[[!include synthetic differential 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{super_groupoids}{Super $\infty$-groupoids}\dotfill \pageref*{super_groupoids} \linebreak
\noindent\hyperlink{FormalModuliProblems}{Formal moduli problems/ strong homotopy Lie algebras}\dotfill \pageref*{FormalModuliProblems} \linebreak
\noindent\hyperlink{GoodwillieTangentCohesion}{Goodwillie-tangent cohesion}\dotfill \pageref*{GoodwillieTangentCohesion} \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 [[cohesive (∞,1)-topos]] is \emph{infinitesimal cohesive} if all its objects behave like built from [[infinitesimally thickened points|infinitesimally thickened]] [[geometrically discrete ∞-groupoids]] in that they all have ``precisely one point in each cohesive piece''.

(There is an evident version of an infinitesimally cohesive 1-topos. In (\hyperlink{Lawvere07}{Lawvere 07, def. 1}) such is referred to as a ``[[quality type]]''. A hint of this seems to be also in (\hyperlink{Lawvere91}{Lawvere 91, p. 9})).

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

\begin{remark}
\label{InfinitesimalCohesion}\hypertarget{InfinitesimalCohesion}{}
A [[cohesive (∞,1)-topos]] $\mathbf{H}$ with its [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]] denoted $&#643; \dashv \flat \dashv \sharp$ is \emph{infinitesimal cohesive} if the canonical \href{cohesive+topos#CanonicalComparison}{points-to-pieces transform} is an [[equivalence]]

\begin{displaymath}
\flat \stackrel{\simeq}{\longrightarrow} &#643;
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The underlying [[adjoint triple]] $\Pi \dashv Disc \dashv \Gamma$ in the case of infinitesimal cohesion is an [[ambidextrous adjunction]]. Such a [[localization of a category|localization]] is called a ``quintessential localization'' in (\hyperlink{Johnstone96}{Johnstone 96}).

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

Given an [[(∞,1)-site]] with a [[zero object]], then the [[(∞,1)-presheaf (∞,1)-topos]] over it is infinitesimally cohesive. This class of examples contains the following ones.

\hypertarget{super_groupoids}{}\subsubsection*{{Super $\infty$-groupoids}}\label{super_groupoids}

[[super ∞-groupoids]] are infinitesimally cohesive over [[geometrically discrete ∞-groupoids]], while [[smooth super ∞-groupoids]] are cohesive over [[super ∞-groupoids]] and [[differential cohesion|differentially cohesive]] over [[smooth ∞-groupoids]]

\begin{displaymath}
\itexarray{
    cohesion &&
    SmoothSuper\infty Grpds 
    &\stackrel{\overset{\Pi^s}{\longrightarrow}}{\stackrel{\overset{Disc^s}{\leftrightarrow}}{\stackrel{\overset{\Gamma^s}{\longrightarrow}}{\underset{coDisc^s}{\leftarrow}}}}&
   Super \infty Grpds
    \\
    &&{}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    && 
    {}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    \\
    cohesion && 
    Smooth \infty Grpds
    &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftrightarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}&
    \infty Grpds
    \\
    \\
    && diff.\;cohesion && inf.\;cohesion
  }
\end{displaymath}
\hypertarget{FormalModuliProblems}{}\subsubsection*{{Formal moduli problems/ strong homotopy Lie algebras}}\label{FormalModuliProblems}

[[synthetic differential ∞-groupoids]] are cohesive over generalized [[formal moduli problems]]/[[L-∞ algebras]] (generalized meaning without the condition of vanishing on the point and of without the condition of being [[cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive sheaves in Lurie's sense]]) which in turn are infinitesimally cohesive over [[geometrically discrete ∞-groupoids]].

\begin{displaymath}
\itexarray{
    cohesion
    &&
    SynthDiff\infty Grpds 
    &\stackrel{\overset{\Pi^i}{\longrightarrow}}{\stackrel{\overset{Disc^i}{\leftrightarrow}}{\stackrel{\overset{\Gamma^i}{\longrightarrow}}{\underset{coDisc^i}{\leftarrow}}}}&
   L_\infty Alg^{op}_{gen}
    \\
    &&{}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    && 
    {}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    \\
    cohesion && Smooth \infty Grpds
    &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftrightarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}&
    \infty Grpds
    \\
     && diff.\;cohesion && inf.\;cohesion
  }
\end{displaymath}
See also at \emph{[[differential cohesion and idelic structure]]}.

\hypertarget{GoodwillieTangentCohesion}{}\subsubsection*{{Goodwillie-tangent cohesion}}\label{GoodwillieTangentCohesion}

A [[tangent (∞,1)-topos]] $T \mathbf{H}$ is infinitesimally cohesive over $\mathbf{H}$:

\begin{displaymath}
\itexarray{
    && Stab(\mathbf{H})
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
    &
    Stab(\infty Grpd) \simeq Spectra
    \\
    && \simeq && \simeq
    \\
    && T_\ast \mathbf{H} && T_\ast \infty Grpd
    \\
    && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}}
    \\
    \mathbf{H}
    &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}&
    T \mathbf{H}
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  &
  T \infty Grpd
  \\
  &&
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} 
  && 
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} 
  \\
  &&
    \mathbf{H}
    &
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  &
  \infty Grpd
  }
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[differential cohesion]]


\item [[differential cohesion and idelic structure]]



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

The notion of infinitesimal cohesion appears under the name ``quality type'' in def. 1 of

\begin{itemize}%
\item [[William Lawvere]], \emph{Axiomatic cohesion}, Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41--49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf})

\end{itemize}
An earlier hint of the same notion seems to be that on the bottom of p. 9 in

\begin{itemize}%
\item [[William Lawvere]], \emph{[[Some Thoughts on the Future of Category Theory]]} (1991)

\end{itemize}
The above examples of infinitesimal cohesion appear in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (2013)

\end{itemize}
Localization by an [[ambidextrous adjunction]] is also discussed in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Remarks on quintessential and persistent localizations}, Theory and Applications of Categories, Vol. 2, No. 8, 1996, pp. 90--99 (\href{http://www.tac.mta.ca/tac/volumes/1996/n8/2-08abs.html}{TAC})

\end{itemize}
[[!redirects infinitesimal cohesive (infinity,1)-toposes]] [[!redirects infinitesimal cohesive (infinity,1)-topoi]]

[[!redirects infinitesimal cohesion]] [[!redirects infinitesimally cohesive]]

[[!redirects infinitesimal cohesive (∞,1)-topos]] [[!redirects infinitesimal cohesive (∞,1)-toposes]] [[!redirects infinitesimal cohesive (∞,1)-topoi]]



\end{document}