\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*{internal (co-)limit}

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ExamplesInfinityGroupoidal}{$\infty$-Groupoidal homotopy limits and colimits of $\infty$-groupoids}\dotfill \pageref*{ExamplesInfinityGroupoidal} \linebreak
\noindent\hyperlink{BorelConstructionHomotopyQuotients}{Borel construction and homotopy-quotients, -invariants, -coinvariants}\dotfill \pageref*{BorelConstructionHomotopyQuotients} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Speaking in the [[internal language]] of a given [[category]] $\mathbf{H}$, one may try to formulate [[universal constructions]] such as of [[limits]]/[[colimits]] internally.

Given the system of [[slice categories]] $\mathbf{H}_{/X}$ of $\mathbf{H}$ (or more generally any [[indexed category]] $X \mapsto \mathbf{H}_X$), then one may regard $X$ as the shape of a discrete [[internal diagram]] in $\mathbf{H}$, and regard the objects in the slice $\mathbf{H}_{/X}$ as discrete actual [[internal diagrams]] of this shape.

If $\mathbf{H}$ happens to have a (small) [[object classifier]] $\mathbf{h}$, then this is particularly evocative, as then (small) such slice objects $\hat F$ over $X$ [[equivalence|equivalent]] to morphisms $F \colon X \to \mathbf{h}$, hence are directly analogous to external diagrams in the form of [[functors]] $\mathcal{X}\to \mathbf{H}$.

Now if $\mathbf{H}$ has good enough properties as a (self-)[[indexed category]] in that it is a [[hyperdoctrine]] with [[dependent sum]] $\underset{X \stackrel{f}{\to} Y}{\sum} \colon \mathbf{H}_{/X} \to \mathbf{H}_{/Y}$ and [[dependent product]] $\underset{X \stackrel{f}{\to} Y}{\prod} \colon \mathbf{H}_{/X} \to \mathbf{H}_{/Y}$ operations, then it makes sense to define the \emph{internal colimit} of a discrete diagram as above as

\begin{displaymath}
\underset{\longrightarrow}{\lim} F \coloneqq \underset{X}{\sum} \hat F
\end{displaymath}
and the \emph{internal limit} as

\begin{displaymath}
\underset{\longleftarrow}{\lim} F \coloneqq \underset{X}{\prod} \hat F
  \,.
\end{displaymath}
Of course with $X$ being, internally, a discrete diagram shape these are, so far, just internal [[coproducts]] and internal [[products]], respectively (which is of course the source of the terminology ``dependent sum'' and ``dependent product'').

It is more or less straightforward to extend the above from $\mathbf{H}$ to $Cat(\mathbf{H})$, the collection of [[internal categories]] in $\mathbf{H}$, then consider for a given internal category (hence [[internal diagram]] shape) $C \in Cat(\mathbf{H})$ the objects in $Cat(\mathbf{H})_{/C}$ as not-necessarily discrete diagrams, and proceed as before. For various specific choices of context $\mathbf{H}$, this is considered in (\hyperlink{Johnstone}{Johnstone, below prop. B2.3.20}, \hyperlink{Pisani09}{Pisani 09, p.18, p. 23}, \hyperlink{HoTTBook}{HoTTBook, section 6.12}).

To see internally the expected [[universal property]] of the above definition, consider the [[internal diagram|internal]] $X$-diagram constant on any $A \in \mathbf{H}$, namely $X^\ast A \in \mathbf{H}_{/X}$. If there is an [[object classifier]] $\mathbf{h}$ then this corresponds indeed to the [[constant map]] $X \to \ast \stackrel{\vdash A}{\to} \mathbf{h}$. Accordingly an internal [[cone]] with tip $A$ over the diagram $F$ as above is a morphism of $X$-diagrams (hence in $\mathbf{H}_{/X}$) of the fomr

\begin{displaymath}
X^\ast A \longrightarrow \hat F
  \,.
\end{displaymath}
Now by the defining [[adjunction]] $(X^\ast \dashv \underset{X}{\prod})$ of the [[dependent product]], such morphisms are equivalent to morphisms

\begin{displaymath}
A \longrightarrow \underset{X}{\prod} \hat F
\end{displaymath}
in $\mathbf{H}$, hence by the above definition to morphisms

\begin{displaymath}
A \longrightarrow \underset{\longleftarrow}{\lim} F
\end{displaymath}
from $A$ into the internal colimit.

This is clearly the internal version of the statement that is extrernally true [[Set]], that the [[limit]] of sets over a diagram is the set of all its [[cones]].

Dually, an internal [[cocone]] is a morphism

\begin{displaymath}
\hat F \longrightarrow X^\ast A
\end{displaymath}
in $\mathbf{H}_{/X}$, and by the defining [[adjunction]] $(\underset{X}{\sum} \dashv X^\ast)$ of the [[dependent sum]] this is equivalently a morphism

\begin{displaymath}
\underset{\longrightarrow}{\lim} F \longrightarrow A
\end{displaymath}
exhibiting internally the statement familiar externally from [[Set]] that maps out of colimits of a diagram are equivalently cocones under that diagram.

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

\hypertarget{ExamplesInfinityGroupoidal}{}\subsubsection*{{$\infty$-Groupoidal homotopy limits and colimits of $\infty$-groupoids}}\label{ExamplesInfinityGroupoidal}

The small [[object classifier]] of the [[(∞,1)-topos]] [[∞Grpd]] is $Core(\infty Grpd_{small})$ itself. Hence an $\infty$-groupoidal-shaped diagram of [[∞-groupoids]] is \emph{internally} in [[∞Grpd]] a discrete diagram $F \colon X\to Core(\infty Grpd_{small})$. The slice object classified by this is the pullback of the [[universal right fibration]], which is equivalently its [[(∞,1)-Grothendieck construction]] $\hat F = \int_X F$.

Accordingly the above gives the internal limits and colimits

\begin{displaymath}
\underset{\longleftarrow}{\lim} F = \underset{X}{\sum} \hat F
\end{displaymath}
and

\begin{displaymath}
\underset{\longrightarrow}{\lim} F = \underset{X}{\prod} \hat F
  \,.
\end{displaymath}
There are indeed also the correct external [[(∞,1)-colimits]] and [[(∞,1)-limits]], by the discussion \href{limit+in+a+quasi-category#WithValInooGrpd}{here}.

\hypertarget{BorelConstructionHomotopyQuotients}{}\subsubsection*{{Borel construction and homotopy-quotients, -invariants, -coinvariants}}\label{BorelConstructionHomotopyQuotients}

For $\mathbf{H}$ an [[(∞,1)-topos]] and $G$ an [[∞-group]] object, consider $X = \mathbf{B}G$ its [[delooping]]. Externally this is in general highly non-discrete, but internally this, being $\infty$-groupoidal, is a disrete diagram shape.

An [[internal diagram]] of this shape

\begin{displaymath}
\rho \;\colon\;  \mathbf{B}G \longrightarrow \mathbf{h}
\end{displaymath}
is equivalently an [[∞-action]] of $G$ on the object $V$ named by

\begin{displaymath}
\ast \to\mathbf{B}G \longrightarrow \mathbf{h}
  \,.
\end{displaymath}
Now the internal colimit

\begin{displaymath}
\underset{\longrightarrow}{\lim} \rho = \underset{\mathbf{B}G}{\sum} \hat \rho
  \simeq
   V/\!/G
\end{displaymath}
is the [[homotopy quotient]] of $V$ by $G$, equivalently the object of [[homotopy coinvariants]]. In $\mathbf{H} =$ [[∞Grpd]] this is given by the [[Borel construction]].

Similarly, the internal limit

\begin{displaymath}
\underset{\longleftarrow}{\lim} \rho = \underset{\mathbf{B}G}{\prod} \hat\rho
  \simeq
   Ext_G(\ast, V)
\end{displaymath}
is the [[homotopy invariants]] of the action.

More generally, the internal left and right [[Kan extension]] give the [[induced representation]] and [[coinduced representation]] constructions. See at \emph{[[∞-action]]} for more on this.

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

\begin{itemize}%
\item [[Peter Johnstone]], section B2 in \emph{[[Sketches of an Elephant]]}


\item [[Claudio Pisani]], \emph{Balanced Category Theory II} (\href{http://arxiv.org/abs/0904.1790}{arXiv:0904.1790})


\item [[Univalent Foundations Project]], section 6.12 of \emph{[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]}



\end{itemize}
[[!redirects internal (co-)limits]]

[[!redirects internal limit]] [[!redirects internal limits]]

[[!redirects internal colimit]] [[!redirects internal colimits]]



\end{document}