\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*{discrete object}

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

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

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

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

\hypertarget{discrete_objects}{}\section*{{Discrete objects}}\label{discrete_objects}

\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{discrete_geometric_spaces}{Discrete geometric spaces}\dotfill \pageref*{discrete_geometric_spaces} \linebreak
\noindent\hyperlink{local_toposes}{Local toposes}\dotfill \pageref*{local_toposes} \linebreak
\noindent\hyperlink{Fibrations}{Topological categories, fibrations, and final lifts}\dotfill \pageref*{Fibrations} \linebreak
\noindent\hyperlink{in_simplicial_sets}{In simplicial sets}\dotfill \pageref*{in_simplicial_sets} \linebreak
\noindent\hyperlink{discrete_cellularcategorical_structures}{Discrete cellular/categorical structures}\dotfill \pageref*{discrete_cellularcategorical_structures} \linebreak
\noindent\hyperlink{ExamplesInInfinityToposes}{In $\infty$-toposes}\dotfill \pageref*{ExamplesInInfinityToposes} \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 \textbf{discrete space} is, in general, an object of a [[concrete category]] $Sp$ of spaces that is [[free functor|free]] on its own underlying set. More generally, the notion can be applied relative to any [[forgetful functor]].

\textbf{Note:} \emph{This page is about the ``[[cohesive]]'' or ``[[topology|topological]]'' notion of discreteness. In [[2-category theory]] the term ``discrete object'' is also often used for [[n-truncated object|0-truncated objects]]. For this usage, see \emph{[[discrete morphism]]} instead.}

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

A discrete space must, in particular, be a [[free object]] for the forgetful functor $U\colon Sp\to Set$, i.e. in the image of its [[adjoint functor|left adjoint]] $F: Set \to Sp$. However, this is not sufficient for it to be free on its \emph{own} underlying set; we must also require that the counit $F U X\to X$ be an [[isomorphism]].

Thus, we say that $U\colon Sp \to Set$ (or more generally, any functor) \textbf{has discrete spaces} or \textbf{discrete objects} if it has a [[fully faithful functor|fully faithful]] left adjoint. This ensures that the functor

\begin{displaymath}
Set \stackrel{F}\to Sp \stackrel{U}\to Set
\end{displaymath}
is ([[natural isomorphism|naturally isomorphic]] to) the [[identity functor]] on [[Set]]. This is true, for example, if $Sp$ is [[Top]], [[Diff]], [[Loc]], etc.

Assuming that $U$ is [[faithful functor|faithful]] (as it is when $Sp$ is a concrete category), we can characterise a discrete space $X$ as one such that every function from $X$ to $Y$ (for $Y$ any space) is a morphism of spaces. (More precisely, this means that every function from $U(X)$ to $U(Y)$ is the image under $U$ of a morphism from $X$ to $Y$.)

The dual notion is a [[codiscrete object]].

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

\hypertarget{discrete_geometric_spaces}{}\subsubsection*{{Discrete geometric spaces}}\label{discrete_geometric_spaces}

The best known example is a discrete [[topological space]], that is one, $X$, in which all subsets of $X$ are [[open subset|open]] in the topology. This is the [[discrete topology]] on $X$. If $X$ is discrete in this sense, then its [[diagonal map]] $\Delta: X \to X \times X$ is [[open map|open]]. The converse also holds: if the diagonal $\Delta(X)$ is open, then so is $i_x^{-1}(\Delta(X)) = \{x\}$ for any $x \in X$, where $i_x(y) \coloneqq (x, y)$.

This same space serves as a discrete object in many subcategories and supercategories of $Top$, from [[convergence space]]s (where the only proper filter that converges to a point is the free ultrafilter at that point) to (say) paracompact Hausdorff spaces or [[Diff|manifolds]] (because a discrete topological space has those properties).

It is also [[sober space|sober]] and thus serves as a discrete [[locale]], whose corresponding [[frame]] is the [[power set]] of $X$; see [[CABA]]. (Note that [[Loc]] is not concrete over [[Set]].). A locale is discrete if and only if $X \to X \times X$ is open and $X \to 1$ is also open. A locale that satisfies the latter condition is called [[overt locale|overt]]; note that every locale is $T_0$ while every topological space is overt. Moreover, in [[classical mathematics]], every locale is overt, but the notion is important when internalizing in [[toposes]].

A discrete [[uniform space]] $X$ has all [[reflexive relations]] as [[entourages]], or equivalently all [[covers]] as [[uniform cover]]s. It is the only uniformity (on a given set) whose underlying topology is discrete.

Strictly speaking, there is no discrete [[metric space]] on any set with more than one element, because the forgetful functor has no left adjoint. However, there is a discrete \emph{extended} metric space, given by $d(x,y) = \infty$ whenever $x \ne y$. More usually, the term `discrete metric' is used when $d(x,y) = 1$ for $x \ne y$, which is discrete in the category of metric spaces of diameter at most $1$. (Comparing the [[adjoint functor theorem]], the problem with $Met$ is that it generally lacks infinitary [[product]]s; in contrast, $Ext Met$ and $Met_1$ are [[complete category|complete]].)

In [[Abstract Stone Duality]], a space is called \textbf{discrete} if the diagonal map $\delta: X \to X \times X$ is open, which corresponds to the existence of an [[equality]] relation on $X$; discrete spaces as described above correspond to discrete \emph{overt} spaces in ASD.

\hypertarget{local_toposes}{}\subsubsection*{{Local toposes}}\label{local_toposes}

Any [[local topos]] has discrete and [[codiscrete object|codiscrete]] objects. By definition, a local topos $\mathbf{H}$ comes with an [[adjoint triple]] of [[functors]]

\begin{displaymath}
\mathbf{H}
   \stackrel{\overset{Disc}{\hookleftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}}
  \mathbf{B}
\end{displaymath}
to a [[base topos]] $\mathbf{B}$ (for instance [[Set]]), for which both $Disc$ and $Codisc$ are fully faithful. Thus, a \emph{discrete object} is one in the [[essential image]] of the functor $Disc$. Note that $\Gamma$ is not generally faithful in this case.

Even more generally, $\mathbf{H}$ may be a [[local (∞,1)-topos]]. For more on the discrete objects in such a context see \emph{[[discrete ∞-groupoid]]} .

Equivalently, this [[adjoint triple]] induces an [[adjoint pair]] of [[modalities]]

\begin{displaymath}
(\flat \dashv \sharp)
  \coloneqq
  ( Disc \Gamma \dashv coDisc \Gamma)
  \,,
\end{displaymath}
the \emph{[[flat modality]]} and the \emph{[[sharp modality]]}. The discrete objects are precisely the [[modal types]] for the [[flat modality]]. The [[codiscrete objects]] are the modal types for the [[sharp modality]].

\hypertarget{Fibrations}{}\subsubsection*{{Topological categories, fibrations, and final lifts}}\label{Fibrations}

Every [[topological concrete category]] has discrete (and also codiscrete) spaces

More generally, if $U$ is an [[opfibration]] and $Sp$ has an initial object preserved by $U$, then $Sp$ has discrete objects: the discrete object on $X$ can be obtained as $i_!(0)$ where $0$ is the initial object of $Sp$ and $i\colon \emptyset \to X$ is the unique map from the initial object in $Set$ (or whatever underlying category). (Conversely, if $Sp$ has discrete objects and pushouts preserved by $U$, then $U$ is an opfibration.)

Discrete objects can also be characterized as [[final lifts]] for empty [[sinks]].

\hypertarget{in_simplicial_sets}{}\subsubsection*{{In simplicial sets}}\label{in_simplicial_sets}

The category [[sSet]] of [[simplicial sets]] is a [[local topos]] (in fact a [[cohesive topos]]).

\begin{itemize}%
\item A \emph{discrete object} in $sSet$ is precisely the [[nerve]] of a [[discrete groupoid]].


\item A \emph{[[codiscrete object]]} in $sSet$ is precisely the [[nerve]] of a [[codiscrete groupoid]].



\end{itemize}
\hypertarget{discrete_cellularcategorical_structures}{}\subsubsection*{{Discrete cellular/categorical structures}}\label{discrete_cellularcategorical_structures}

Often one calls a cellular structure, such as those appearing in [[higher category theory]], \emph{discrete} if it is in the [[essential image]] of the inclusion of [[Set]].

For instance, one may speak of a \emph{[[discrete category]]} as a category that is equivalent (or, in some cases, isomorphic) to one which has only [[identity]] morphisms. This concept has a generalization to a notion of [[discrete object in a 2-category]].

An alternative terminology for this use of ``discrete'' is \emph{[[truncated object|0-truncated]]}, or more precisely (0,0)-truncated. A discrete groupoid in this sense is a \emph{[[homotopy n-type|homotopy 0-type]]}, or simply a [[h-level|0-type]]. This terminology may be preferable to ``discrete'' in this context, notably when one is dealing with higher categorical structures that are in addition equipped with geometric structure. For instance, when dealing with a [[topological category]] there is otherwise ambiguity in what it means to say that it is ``discrete'': it could either mean that its underlying topological spaces (of objects and of morphisms) are discrete spaces, or it could mean that it has no nontrivial morphisms, but possibly a non-discrete topological space of objects.

\begin{remark}
\label{}\hypertarget{}{}
In some cases, the cellular notion of ``discreteness'' for higher categories can be seen as a special case of the \emph{spatial} notion of discreteness --- often the 1-category of shapes will have a functor to sets for which the cellularly discrete objects are the discrete objects in the sense considered on this page. For instance, this is the case for [[simplicial sets]], which form a [[local topos]] over [[Set]]. The discrete objects relative to this notion of cohesion are precisely the simplicial sets that are constant on a given ordinary set, hence those that are ``discrete'' in the cellular sense.

\end{remark}
\hypertarget{ExamplesInInfinityToposes}{}\subsubsection*{{In $\infty$-toposes}}\label{ExamplesInInfinityToposes}

The definition of discrete objects has the evident generalization from [[category theory]] to [[(∞,1)-category theory]]/[[homotopy theory]]. One noteworthy aspect of discrete objects in the context of homotopy theory is that there they are intimately related to the notion of \emph{[[cohomology]]}.

For $\mathcal{X}$ an [[(∞,1)-sheaf (∞,1)-topos]] with [[global section]] [[geometric morphism]]

\begin{displaymath}
(\Delta \dashv \Gamma)
  \colon
  \mathcal{X}
  \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}
 \infty Grpd
\end{displaymath}
then for $X \in \mathcal{X}$ any object and $A \in$ [[∞Grpd]] any object, one says that

\begin{displaymath}
H(X,A)
  \coloneqq
  \pi_0 \mathcal{X}(X,\Delta(A))
  \in
  Set
\end{displaymath}
is the [[cohomology]] of $X$ [[cohomology with constant coefficients|with (locally) constant coefficients]] in $A$. (Here on the right $\mathcal{X}(-,-)$ denotes the [[(∞,1)-categorical hom-space]].)

Now if $\mathcal{X}$ \emph{has discrete objects} in the sense that $\Delta \colon \infty Grpd \to \mathcal{X}$ is a [[full and faithful (∞,1)-functor]], then it follows immediately from the definitions that the cohomology of discrete objects with constant coefficients in $\mathcal{X}$ equals the cohomology in [[∞Grpd]], which is standard ``[[nonabelian cohomology]]'':

\begin{displaymath}
(\mathcal{X} \, has\, discrete\, objects)
  \Leftrightarrow 
  ( \forall_{S,A \in \infty Grpd} \mathcal{X}(\Delta S, \Delta A) \simeq \infty Grpd(S,A) )
  \,.
\end{displaymath}
Conversely: \emph{the failure of the cohomology with constant coefficients of objects in the image of $\Delta$ to coincide with standard cohomology is a measure for $\Delta$ not respecting discrete objects.}

For example the [[natural numbers object]] $\mathbf{N} \simeq \Delta \mathbb{N}$ in the [[(∞,1)-sheaf (∞,1)-topos]] over some [[topological spaces]] fails to be a discrete object. Accordingly in this case the natural numbers object can have nontrivial higher cohomology with constant coefficients, see for instance (\hyperlink{Blass83}{Blass 83}, \hyperlink{Shulman13}{Shulman 13}).

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

\begin{itemize}%
\item \textbf{discrete space}


\item [[discrete group]]


\item [[discrete groupoid]]


\item [[discrete ∞-groupoid]]


\item [[codiscrete space]], [[Freyd cover]]


\item [[adjoint cylinder]]


\item [[category of being]]



\end{itemize}
[[!include cohesion - table]]

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

\begin{itemize}%
\item \emph{Discreteness, concreteness, fibrations, and scones}: \href{http://golem.ph.utexas.edu/category/2011/11/discreteness_concreteness_fibr.html}{blog post}


\item [[Andreas Blass]], \emph{Cohomology detects failures of the axiom of choice}, Trans. Amer. Math. Soc. 279 (1983), 257-269 (\href{http://www.ams.org/journals/tran/1983-279-01/S0002-9947-1983-0704615-7/}{web})



\end{itemize}
\begin{itemize}%
\item [[Mike Shulman]], \emph{Cohomology} on the [[homotopy type theory]] blog \href{http://homotopytypetheory.org/2013/07/24/cohomology/}{here}

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

[[!redirects discrete uniformity]] [[!redirects discrete uniformities]] [[!redirects discrete uniform structure]] [[!redirects discrete uniform structures]] [[!redirects discrete uniform space]] [[!redirects discrete uniform spaces]]

[[!redirects discrete metric]] [[!redirects discrete metrics]] [[!redirects discrete metric space]] [[!redirects discrete metric spaces]] [[!redirects discrete extended metric]] [[!redirects discrete extended metrics]] [[!redirects discrete extended metric space]] [[!redirects discrete extended metric spaces]]

[[!redirects discrete locale]] [[!redirects discrete locales]]

[[!redirects discrete object]] [[!redirects discrete objects]]

[[!redirects discrete]] [[!redirects discreteness]]

[[!redirects discrete type]] [[!redirects discrete types]]



\end{document}