\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*{solid functor}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{left_adjoint}{Left adjoint}\dotfill \pageref*{left_adjoint} \linebreak
\noindent\hyperlink{lifting_of_colimits}{Lifting of colimits}\dotfill \pageref*{lifting_of_colimits} \linebreak
\noindent\hyperlink{lifting_of_model_structures}{Lifting of model structures}\dotfill \pageref*{lifting_of_model_structures} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{solid functor} (also called a \emph{semi-topological functor}) is a [[forgetful functor]] $U\colon A\to X$ for which the structure of an $A$-object can be universally lifted along [[sinks]]. One can also say that $U$ has not just a [[left adjoint]] but all possible ``relative'' left adjoints.

When $U$ is solid, [[colimit]]s in $A$ can be constructed in a natural way out of colimits in $X$, and $A$ inherits strong cocompleteness properties from $X$.

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

Let $U\colon A\to X$ be a [[faithful functor]]. A \textbf{$U$-structured sink} is a [[sink]] in $X$ of the form $(U a_i \overset{f_i}{\to} x)$. Note that the indexing family $i\in I$ need not be a [[set]], it can be a [[proper class]]. A \textbf{[[semi-final lift]]} of such a $U$-structured sink consists of a morphism $x\overset{g}{\to} U b$ in $X$ such that

\begin{enumerate}%
\item Every composite $g \circ f_i\colon U a_i \to U b$ is in the image of $U$, i.e. is of the form $U(\tilde{g})$ for some $\tilde{g}\colon a_i\to b$ (necessarily unique, since $U$ is faithful), and


\item $g$ is [[initial object|initial]] with this property, i.e. for any other morphism $x \overset{g'}{\to} U b'$ such that each $g' \circ f_i$ is in the image of $U$, there exists a unique $h\colon b\to b'$ in $A$ such that $g' = U(h) \circ g$.



\end{enumerate}
Finally, $U$ is called \textbf{solid} if every $U$-structured sink has a semi-final lift.

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

\begin{itemize}%
\item Any [[topological functor]] is solid. Indeed, a functor $U$ is topological just when it has final lifts for all $U$-structured sinks, where a \emph{final lift} is a semi-final lift for which $g$ is an isomorphism.


\item Any [[monadic functor]] into $Set$ is solid.


\item A [[fully faithful functor]] is solid if and only if it has a left adjoint.


\item If $U\colon A\to X$ is faithful and has a left adjoint, and moreover $A$ is [[cocomplete category|cocomplete]] and [[well-powered category|well-copowered]], then $U$ is solid.


\item For $C$ a [[cofibrantly generated model category]] with monic generating cofibrations, the forgetful functor from [[model structure on algebraic fibrant objects|algebraic fibrant objects]] to $C$ is solid. See there for details.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{left_adjoint}{}\subsubsection*{{Left adjoint}}\label{left_adjoint}

For any $x\in X$, the empty family of morphisms into $x$ is a $U$-structured sink, and a semi-final lift for this family is a  $x\to U b$. Therefore, if $U$ is solid, then it has a [[left adjoint]].

\hypertarget{lifting_of_colimits}{}\subsubsection*{{Lifting of colimits}}\label{lifting_of_colimits}

Suppose that $U\colon A\to X$ is solid and let $F\colon D\to A$ be a [[diagram]] such that $U F$ has a [[colimit]] in $X$, consisting of a [[cocone]] $U F d_i \to c$. Let $c \to U e$ be a semi-final lift of this $U$-structured sink, for which we have induced morphisms $F d_i \to e$ in $A$. Since $U$ is faithful, these morphisms are a cocone under $F$, and the semi-finality makes it into a colimit in $A$.

Therefore, if $A$ is solid over $X$, then it admits all colimits which $X$ does. Moreover, if we understand colimits in $X$, and we understand the semi-final lifts, then we understand colimits in $A$.

In particular, if $X$ is [[cocomplete category|cocomplete]], then so is $A$. In fact, more is true: if $X$ is [[total category|total]], then so is $A$.

\hypertarget{lifting_of_model_structures}{}\subsubsection*{{Lifting of model structures}}\label{lifting_of_model_structures}

The standard \emph{[[transferred model structure|transfer theorem]]} for [[model category|model structures]] states that if $U\colon A\to X$ is a functor such that

\begin{enumerate}%
\item $U$ has a left adjoint $F$,
\item $U$ is [[accessible functor|accessible]], i.e. preserves $\kappa$-[[filtered colimits]] for sufficiently large $\kappa$,
\item $X$ has a [[cofibrantly generated model category|cofibrantly generated model structure]], and
\item [[transfinite composition|Transfinite composites]] of [[pushouts]] of images under $F$ of generating acyclic cofibrations in $X$ become weak equivalences after applying $U$ (the \emph{acyclicity condition}),

\end{enumerate}
then $A$ has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by $U$. Using an argument of (\hyperlink{Nikolaus}{Nikolaus}) we can show:

\begin{utheorem}
Let $U\colon A\to X$ be an accessible solid functor, and assume that $X$ has a cofibrantly generated model structure and the following acyclicity condition:

\begin{itemize}%
\item If $F\colon D\to A$ is a [[filtered diagram]] and $U(F(d_i)) \to x$ is a cocone under $U\circ F$, each of whose legs is an acyclic cofibration in $X$, then the semifinal lift $x\to U(b)$ of this $U$-structured sink is also an acyclic cofibration.

\end{itemize}
Then the transfer theorem applies, so that $A$ has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by $U$.

\end{utheorem}
\begin{proof}
We have remarked above that $U$ has a left adjoint, and we assumed it to be accessible, so it remains to show that the given acyclicity condition implies the standard one.

We first show that [[pushout]]s in $A$ of images under $F$ of generating acyclic cofibrations become acyclic cofibrations (not just weak equivalences) upon applying $U$. Let $i\colon x\to y$ be a generating acyclic cofibration, and

\begin{displaymath}
\itexarray{F(x) & \overset{f}{\to} & a\\
  ^{F (i)}\downarrow \\
  F(y)}
\end{displaymath}
a diagram in $A$ of which we would like to take the pushout. Consider the pushout of the corresponding diagram in $X$:

\begin{displaymath}
\itexarray{x & \overset{\bar{f}}{\to} & U(a)\\
  ^{i}\downarrow && \downarrow^g\\
  y& \underset{h}{\to} & U(a) \sqcup_{x} y.}
\end{displaymath}
Since $X$ is a model category, $g$ is an acyclic cofibration. Therefore, if $U(a) \sqcup_x y \overset{k}{\to} U(b)$ is a semifinal lift of the singleton sink $\{g\}$, by assumption, $k$ is also an acyclic cofibration and thus so is the composite $U(a)\to U(b)$. But it is straightforward to verify that in fact, the map $a\to b$ of which this is the image (which exists by assumption) gives a pushout diagram in $A$:

\begin{displaymath}
\itexarray{F(x) & \overset{f}{\to} & a\\
  ^{F(i)}\downarrow && \downarrow\\
  F(y) & \underset{\bar{h}}{\to} & b.}
\end{displaymath}
If $U$ is not just accessible but [[finitary functor|finitary]], then it preserves all transfinite composites, so any transfinite composite of such pushouts in $A$ maps to a transfinite composite in $X$, and we know that transfinite composites of acyclic cofibrations in $X$ are acyclic cofibrations, so the desired acyclicity condition follows. In general, we can argue as follows: given a transfinite sequence $a_0\to a_1\to\dots$ in $A$ of such pushouts, its colimit (= composition) can be constructed as above by forgetting down to $X$, taking the colimit there, and then taking the semifinal lift. But since acyclic cofibrations in $X$ are closed under transfinite composites, the legs of the colimiting cocone in $X$ are acyclic cofibrations. Hence by the assumed acyclicity condition, so is the semifinal lift, and hence (by composition) so are the images in $X$ of the legs of the colimiting cone in $A$. This completes the proof.

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

\begin{itemize}%
\item [[Walter Tholen]], \emph{Semitopological functors. I}


\item and others\ldots{}



\end{itemize}
The example of algebraic fibrant objects and the argument entering the above \hyperlink{LiftingTheorem}{lifting theorem} appears in

\begin{itemize}%
\item [[Thomas Nikolaus]], \emph{Algebraic models for higher categories} (\href{http://arxiv.org/abs/1003.1342}{arXiv})

\end{itemize}
See also [[model structure on algebraic fibrant objects]].

[[!redirects solid functors]] [[!redirects semi-topological functor]] [[!redirects semitopological functor]] [[!redirects semi-topological functors]] [[!redirects semitopological functors]]



\end{document}