\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*{direct category}

\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{standard_definition}{Standard definition}\dotfill \pageref*{standard_definition} \linebreak
\noindent\hyperlink{direct_versus_oneway_categories}{Direct versus one-way categories}\dotfill \pageref*{direct_versus_oneway_categories} \linebreak
\noindent\hyperlink{allowing_automorphisms}{Allowing automorphisms}\dotfill \pageref*{allowing_automorphisms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{model_structures}{Model structures}\dotfill \pageref*{model_structures} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{inside_the_simplex_category}{Inside the simplex category}\dotfill \pageref*{inside_the_simplex_category} \linebreak
\noindent\hyperlink{the_direct_category_of_corollas}{The direct category of corollas}\dotfill \pageref*{the_direct_category_of_corollas} \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 \emph{direct category} is a [[category]] in which the morphisms only go in ``one direction.''

A direct category can be thought of as a category of [[geometric shapes for higher structures]] which includes only the ``inclusions of faces,'' not any ``degeneracy'' maps. (The more general notion of [[Reedy category]] can also include degeneracies.) The objects of a direct category admit no nontrivial automorphisms, but the notion can also be generalized to allow for such automorphisms.

Alternately, a direct category can be thought of as a combinatorial representation of a \emph{single} geometric shape. In this case, each object is a ``face'' and each morphism is the inclusion of a lower-dimensional face in a higher-dimensional one. If $D$ is a direct category regarded as a category \emph{of} geometric shapes, then each $d\in D$ can be represented by the [[slice category]] $D/d$, a direct category regarded as a single geometric shape. (In general, however, $D/d$ may admit more automorphisms than $d$, so it may need to be equipped with a ``labeling'' or ``orientation'' to truly recapture the shape $d$.)

Finally, a direct category can also be thought of as a [[categorification]] of the notion of [[well-founded relation]] from [[posets]] to [[categories]].

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

\hypertarget{standard_definition}{}\subsubsection*{{Standard definition}}\label{standard_definition}

A category $D$ is a \textbf{direct category} if the following \emph{equivalent} conditions are satisfied.

\begin{itemize}%
\item $D$ contains no infinite descending chains of nonidentity morphisms $\cdots \to\cdot \to\cdot\to\cdot$ (including cycles of length $\gt 0$).
\item The relation $a\prec b$ on $ob(D)$ defined by ``there exists a nonidentity morphism from $a$ to $b$'' is [[well-founded relation|well-founded]].
\item There exists a function $d\colon ob(D)\to Ord$, where $Ord$ is the class of [[ordinals]], such that every nonidentity morphism of $D$ raises the degree.
\item There exists an identity-reflecting functor $d : D\to \mathbf{Ord}$, where $\mathbf{Ord}$ is the large poset of ordinals viewed as a category.
\item $D$ is a [[Reedy category]] (in particular an \emph{[[elegant Reedy category]]}) in which $D_-$ consists only of identity maps (or equivalently $D_+$ is all of $D$).

\end{itemize}
In particular, a [[poset]] is a direct category just when its strict order relation $\lt$ is well-founded. Thus, direct categories can be seen as a categorification of well-founded relations.

If $D^{op}$ is a direct category, then we say that $D$ is an \textbf{inverse category}.

\hypertarget{direct_versus_oneway_categories}{}\subsubsection*{{Direct versus one-way categories}}\label{direct_versus_oneway_categories}

Of course, a direct category can have no nonidentity [[endomorphisms]], since any such endomorphism would be a cycle of length 1. A category with this property is sometimes called a \textbf{one-way category}.

A direct category is also necessarily [[skeleton|skeletal]], since any nonidentity isomorphism and its inverse would form a cycle of length 2. The notion of generalized direct category, below, relaxes this requirement.

Conversely, a skeletal one-way category can have no cycles of nonidentity morphisms of any finite length, since by the one-way property all morphisms in such a cycle would be isomorphisms, hence endomorphisms by skeletality, and hence identities by the one-way property. Since any infinite chain in a [[finite category]] contains a cycle, we see that

\begin{itemize}%
\item A finite category is a direct category if and only if it is one-way and skeletal.

\end{itemize}
Since this condition is self-dual, we also see that

\begin{itemize}%
\item A finite category is direct if and only if it is inverse.

\end{itemize}
An infinite category can be one-way and skeletal without being direct or inverse, such as the [[poset]] $\mathbb{Z}$ with its usual ordering. However, if it additionally has \textbf{finite fan-out}, meaning that for each object $x$ there are altogether only finitely many morphisms with [[source]] $x$ (and arbitrary target), then it must be an inverse category, for any infinite non-cyclic chain would induce infinitely many distinct morphisms out of any of its objects by composition. In [[FOLDS]], skeletal one-way categories with finite fan-out are called \textbf{simple categories} and used as signatures; thus

\begin{itemize}%
\item Any simple category (in the sense of FOLDS) is an inverse category.

\end{itemize}
However, a category can be inverse without having finite fan-out. Let $S$ be an infinite set and consider the poset $S \cup \{\infty\}$, with $S$ having the discrete ordering (no nonidentity arrows) and $\infty$ being less than every element of $S$. This does not have finite fan-out, since $\infty$ is the source of infinitely many distinct arrows, but it is inverse, since there are clearly no chains of nonidentity arrows of length $\gt 1$.

\hypertarget{allowing_automorphisms}{}\subsubsection*{{Allowing automorphisms}}\label{allowing_automorphisms}

Some categories of geometric shapes, such as the [[tree category]] $\Omega$ and the [[cycle category]] $\Lambda$, include automorphisms of their objects. By analogy with the notion of [[generalized Reedy category]], we can define $D$ to be a \textbf{generalized direct category} by replacing ``identity'' with ``isomorphism'' in the above definition. Thereby we obtain the following equivalent conditions for $D$ to be a generalized direct category.

\begin{itemize}%
\item $D$ contains no infinite descending chains of noninvertible morphisms $\cdots \to\cdot \to\cdot\to\cdot$.
\item The relation $a\prec b$ on $ob(D)$ defined by ``there exists a noninvertible morphism from $a$ to $b$'' is [[well-founded relation|well-founded]].
\item There exists a function $d\colon ob(D)\to Ord$, where $Ord$ is the class of [[ordinals]], such that every noninvertible morphism of $D$ raises the degree.
\item $D$ is a [[generalized Reedy category]] in which $D_-$ consists only of isomorphisms (or equivalently $D_+$ is all of $D$).

\end{itemize}
Of course, $\Omega$ and $\Lambda$ are not generalized direct categories themselves, since they have degree-lowering degeneracies as well, but their full subcategories of coface maps are generalized-direct. More generally:

\begin{itemize}%
\item If $R$ is any generalized Reedy category, then $R_+$ is a generalized direct category.

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

\hypertarget{model_structures}{}\subsubsection*{{Model structures}}\label{model_structures}

Every direct category (and every inverse category) is in particular a [[Reedy category]], in fact an \emph{[[elegant Reedy category]]}. Therefore whenever $M$ is a [[model category]] there is a [[Reedy model structure]] on $M^D$. In the case of direct and inverse categories, these model structures are even easier to describe, since either the latching or the matching objects are degenerate.

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

\hypertarget{inside_the_simplex_category}{}\subsubsection*{{Inside the simplex category}}\label{inside_the_simplex_category}

The [[wide subcategory]] $\Delta_+$ of the [[simplex category]] on the injective map (the co-face maps) is direct. Its [[presheaves]] are [[semi-simplicial objects]]/[[semi-simplicial sets]] as opposed to [[simplicial objects]]/[[simplicial sets]].

\hypertarget{the_direct_category_of_corollas}{}\subsubsection*{{The direct category of corollas}}\label{the_direct_category_of_corollas}

We now define a ``[[universal property|universal]]'' generalized direct category which contains ``all'' [[geometric shapes for higher structures]]. This is based on (\hyperlink{Borisov}{Borisov}).

A functor $f\colon D\to E$ between generalized direct categories is called a \textbf{dependency} if it is equivalent to the inclusion of a [[sieve]], or equivalently if it is a [[fully faithful functor]] and a [[discrete fibration]] in the generalized sense of Street. If $D$ and $E$ are [[skeleton|skeletal]] (as they must be if they are \emph{non-generalized} direct categories), then any dependency must be isomorphic to the inclusion of a sieve. One also usually works only with skeletal generalized direct categories, although the definition does not require it.

If we regard direct categories as a categorification of well-founded relations, then dependencies are a categorification of injective [[simulation]]s, i.e. the inclusions of initial segments.

Define a \textbf{corolla} to be a generalized direct category with a [[weak limit|weakly]] [[terminal object]]; we call this object the \textbf{vertex} of the corolla. If a generalized direct category is finite and skeletal, then it is a corolla if and only if it has a unique object which is not the source of any noninvertible morphism. Corollas are the direct categories which it is most natural to regard as \emph{single} geometric shapes; other direct categories are more like ``pasting diagrams'' of geometric shapes.

Let $Corolla$ be the category of [[small category|small]] skeletal corollas and dependencies.

\begin{theorem}
\label{UnivDirect}\hypertarget{UnivDirect}{}
$Corolla$ is a (large) generalized direct category.

\end{theorem}
\begin{proof}
Suppose that $\cdots \overset{d_2}{\to} C_2 \overset{d_1}{\to} C_1 \overset{d_0}{\to} C_0$ is a descending chain of noninvertible dependencies. Any dependency whose target is a corolla and whose image contains the vertex must be an equivalence, and hence an isomorphism if the corollas are skeletal; thus none of the dependencies $C_{n+1} \to C_{n}$ can have the vertex in their image. Let $c_n\in C_0$ be the image of the vertex $\star_n \in C_n$ under the composite $d_0\circ \dots \circ d_{n-1}$. Then we have a noninvertible morphism $c_{n+1} \to c_n$ for each $n$, arising from some map $d_n(\star_{n+1}) \to \star_n$ which exists since $\star_n$ is weakly terminal. This is a contradiction, since $C_0$ is a generalized direct category.

\end{proof}
Of course, the same is true for any subcategory of $Corolla$. In particular, it is very natural to consider only the category $FinCorolla$ of \emph{finite} corollas, which is moreover essentially small (though not finite).

We next observe that $Corolla$ is the ``universal'' generalized direct category in a certain sense. Let $D$ be any generalized direct category; then the slice category $D/d$ is a corolla for any $d\in D$. Moreover, if $f\colon d\to d'$ is a morphism in $D$ which is [[monomorphism|monic]], then the ``composition'' functor $\Sigma_f\colon D/d \to D/d'$ is a dependency. Thus, if every morphism in $D$ is monic, we have a functor $ext_D\colon D \to Corolla$ with $ext_D(d)= D/d$ and $ext_D(f)=\Sigma_f$.

Now if $f,g\colon d\to d'$ are parallel morphisms and $\Sigma_f=\Sigma_g$, then in particular $f = \Sigma_f(id_d) = \Sigma_g(id_d) = g$; thus $ext_D$ is [[faithful functor|faithful]]. Therefore, if $D$ is a direct category in which all morphisms are monic, and in which $D/d \cong D/d'$ implies $d=d'$ (which includes many examples), then $D$ is equivalent to a [[subcategory]] of $Corolla$.

This subcategory of $Corolla$ is usually \emph{not} full, however. In particular, for $d\in D$ the corolla $D/d$ will generally admit more automorphisms than $d$ has in $D$. For instance, if $D$ is a non-generalized direct category, then $d$ has no nontrivial automorphisms, whereas $D/d$ generally will. In particular, if the objects of $D$ have any sort of ``orientation'' or ``labeling,'' then this information is forgotten by the functor $ext_D$.

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

\begin{itemize}%
\item [[test category]]


\item [[filtered category]], [[sifted category]]



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

\begin{itemize}%
\item [[Dennis Borisov]], \emph{Comparing definitions of weak higher categories, I} (\href{http://arxiv.org/abs/0909.2534}{arXiv:0909.2534})

\end{itemize}
\begin{itemize}%
\item [[Clark Barwick]], \emph{On Reedy Model Cateogires} (\href{https://arxiv.org/abs/0708.2832}{arXiv:0708.2832})

\end{itemize}
[[!redirects direct category]] [[!redirects direct categories]] [[!redirects inverse category]] [[!redirects inverse categories]] [[!redirects one-way category]] [[!redirects one-way categories]] [[!redirects category with finite fan-out]] [[!redirects categories with finite fan-out]] [[!redirects categories with finite fan-outs]] [[!redirects simple category]] [[!redirects simple categories]]



\end{document}