\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*{vertical categorification}

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

\hypertarget{categorification}{}\paragraph*{{Categorification}}\label{categorification}

[[!include categorification - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher Category Theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{as_a_section_of_decategorification}{As a section of decategorification}\dotfill \pageref*{as_a_section_of_decategorification} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{as_internalization_in_}{As internalization in $n Cat$}\dotfill \pageref*{as_internalization_in_} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{as_homotopy_coherent_resolution}{As homotopy coherent resolution}\dotfill \pageref*{as_homotopy_coherent_resolution} \linebreak
\noindent\hyperlink{examples_3}{Examples}\dotfill \pageref*{examples_3} \linebreak
\noindent\hyperlink{contrast_to_horizontal_categorification}{Contrast to horizontal categorification}\dotfill \pageref*{contrast_to_horizontal_categorification} \linebreak
\noindent\hyperlink{homotopification_versus_laxification}{Homotopification versus laxification}\dotfill \pageref*{homotopification_versus_laxification} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Roughly speaking, \emph{vertical categorification} is a procedure in which structures are generalized from the context of [[set theory]] to [[category theory]] or from category theory to [[higher category theory]].

What precisely that means may depend on circumstances and authors, to some extent. The following lists some common procedures that are known as categorification. They are in general different but may in cases lead to the same categorified notions, as discussed in the examples.

See also [[categorification in representation theory]].

\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

\hypertarget{as_a_section_of_decategorification}{}\subsubsection*{{As a section of decategorification}}\label{as_a_section_of_decategorification}

What has a more specific definition is the process of [[decategorification]]: this concretely quotients out [[k-morphism]]s in a [[n-category]] to produce a $k$-category for some $k \lt n$, in particular a (possibly large) [[set]] (the set of isomorphism classes or equivalence classes of objects) when $k = 0$.

One may understand \emph{vertical categorification} as any operation that is a [[section]] of this [[decategorification]] operation, and many examples in the literature are of this kind.

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

\begin{itemize}%
\item The maybe archetypical example of categorification as something that is taken by decategorification to the identity operation is the categorification of the [[set]] $\mathbb{N}$ of [[natural numbers]] to the [[category]] [[FinSet]] of [[finite sets]]:

The set $\mathbb{N}$ is a [[0-category]], while $FinSet$ is a [[1-category]]. The [[decategorification|isomorphism classes]] of $FinSet$ however are in canonical bijection with the elements of $\mathbb{N}$:

\begin{displaymath}
\mathbb{N} \simeq FinSet_\sim .
\end{displaymath}
So $\mathbb{N}$ is the [[decategorification]] of $FinSet$ and accordingly $FinSet$ is a (vertical) categorification of $\mathbb{N}$.

From the point of view of [[category theory]], there is some justification to the converse of this statement: the study of the natural numbers is nothing but the study of the isomorphism classes in $FinSet$. In a way, the very notion of \textbf{counting} is about this. This point is made nicely in \href{http://arxiv.org/abs/math.QA/9802029}{BaDo98}.

But there are other categorifications of the natural numbers, for instance the category $FinVect$ of finite dimensional [[vector spaces]] (over any given field). This is not equivalent to $FinSet$, but still, its set of isomorphism classes is also canonically in bijection with the natural numbers:

\begin{displaymath}
\mathbb{N} \simeq FinVect_\sim .
\end{displaymath}


\end{itemize}
\hypertarget{as_internalization_in_}{}\subsubsection*{{As internalization in $n Cat$}}\label{as_internalization_in_}

Common structures such as [[group]]s, [[ring]]s, etc. may be defined entirely diagrammatically as a collection of objects, morphisms and 2-morphisms in a [[category]] like [[Set]] (the 2-morphisms then are the identities required to hold between the given morphisms).

For instance a [[group]] is an object $G$ in [[Set]] equipped with a morphism $\mu : G \times G \to G$ and so on, and equipped with a 2-morphism from $\mu \circ Id_G \times \mu$ to $\mu \circ \mu \times Id_G$.

In such diagrammatic incarnation, these definitions may be [[internalization|internalized]] into other categories. For instance a group internal to [[Diff]] is a [[Lie group]].

But similarly one can also internalize in categories of higher categories. Let [[Cat]] be the category of (small) categories, regarded as an ordinary category. Then a group internal to [[Cat]] is a [[strict 2-group]]. This is thought of as a notion of a categorified group.

And under the decategorification functor $Cat \to Set$ every categorified group $\mathbf{G}$ in this sense maps to an ordinary group $G$ and one could speak of $\mathbf{G}$ being ``a categorification'' of $G$. But notice again how highly non-unique such categorification is. In particular, every ordinary group may trivially be regarded as a strict 2-group. As such every ordinary group is at the same time one of its own categorifications, in this sense.

Not every [[2-group]] is a [[strict 2-group]] that is a group object internal to the 1-category [[Cat]]. In general, a [[2-group]] is a \emph{weak} group object in [[Cat]]: where in [[Set]] all 2-morphisms in the diagrammatic description of the concept of group necessarily had to be identities, in [[Cat]] they could be taken to be non-trivial. But if they are, one will usually want 3-morphisms (which now are necessarily identities) to relate various combinations of these 2-morphisms.

One speaks of this process of categorification using weak internalization of diagrammatic descriptions as categorification of structures \emph{up to coherent higher equivalences} or \emph{up to coherent higher homotopies} . One way to make this systematic is discussed below.

\hypertarget{examples_2}{}\paragraph*{{Examples}}\label{examples_2}

In the sense of ``weakly internalizing in a higher categorical context'' we have the following examples of categorification:

\begin{itemize}%
\item the categorification of [[sheaf]] is [[stack]], 2-stack, 3-stack, \ldots{} [[∞-stack]].


\item accordingly, the categorification of ([[sheaf]]-)[[topos]] is ([[stack]]-)2-topos, \ldots{} [[∞-stack]] [[(∞,1)-topos]]


\item using these [[∞-stack]] [[(∞,1)-topos]] notions one can then speak of [[∞-space]]s [[Lie ∞-groupoid]]s and the like.


\item Since [[simplicial object]]s in any category usually [[model category|model]] [[higher category theory|higher categorial]] structures, many simplicial objects may be regarded as secretly being objects in [[higher category theory]]. For instance [[simplicial group]]s, being [[Kan complex]]es, may be thought of as [[∞-groupoid]]s equipped with a strict group structure, i.e. as \emph{strict $\infty$-groups}. By the [[Dold-Kan correspondence]] it follows then that many objects in [[homological algebra]] such as [[chain complex]]es are equivalent incarnations of these simplicial (abelian) groups. This way large parts of [[homological algebra]] may be regarded as being secretly about ``categorified linear algebra'' in some sense. For instance the [[dg-category]] of [[chain complex]]es in many contexts plays the role of a [[stable (∞,1)-category]] of ``$(\infty,1)$-vector spaces''. Notably [[L-∞-algebra]]s may be thought of as [[Lie algebra]]s weakly internalized in $(\infty,1)$-vector spaces understood in this sense.


\item When understanding higher linear algebra in this sense, there are important $\infty$-categorifications of notions like integral transforms, [[Fourier transform]]s etc. Examples of such categorified linear algebra are [[Fourier-Mukai transform]]s or more generally [[integral transform]]s. A general theory of this is described at [[geometric ∞-function theory]]. Also [[geometric Langlands]] duality fits into this context.


\item \href{Gram-Schmidt+process#CategorifiedGramSchmidtProcess}{categorified Gram-Schmidt process}


\item [[(∞,1)-category theory|(∞,1)-]]categorification of [[differential calculus]] is [[Goodwillie calculus]]

[[(∞,1)-category theory|(∞,1)-]]categorification of [[derivative]] is [[Goodwillie derivative]]

[[(∞,1)-category theory|(∞,1)-]]categorification of [[chain rule]] is [[Goodwillie chain rule]]

etc.



\end{itemize}
\hypertarget{as_homotopy_coherent_resolution}{}\subsubsection*{{As homotopy coherent resolution}}\label{as_homotopy_coherent_resolution}

The above process of categorification by coherently weak internalization into higher categorical contexts can be made systematic at least in some cases.

If the structures being defined are [[algebra over an operad|algebras over]] an [[operad]] $T$ one may think of regarding $T$ as an [[(∞,1)-operad]]s and then consider the structures of its algebras as algebras over an $(\infty,1)$-operad. These are infinity-categorified versions of the original structures.

\hypertarget{examples_3}{}\paragraph*{{Examples}}\label{examples_3}

For instance this way

\begin{itemize}%
\item the notion of ordinary associative [[algebra]] is sent to that of $A_\infty$-[[A-∞-algebra|algebra]]


\item the notion of ordinary [[Lie algebra]] to $L_\infty$-[[L-∞-algebra|algebra]].



\end{itemize}
\hypertarget{contrast_to_horizontal_categorification}{}\subsection*{{Contrast to horizontal categorification}}\label{contrast_to_horizontal_categorification}

Some people also speak of [[horizontal categorification]] as categorification. This is to be distinguished from vertical categorification.

Some people just say `oidification' for horizontal categorification, in which case it is consistent to speak of vertical categorificaton as just \emph{categorification} .

\hypertarget{homotopification_versus_laxification}{}\subsection*{{Homotopification versus laxification}}\label{homotopification_versus_laxification}

(Vertical) categorification can often be usefully decomposed into two operations.

\begin{itemize}%
\item In \textbf{groupoidal categorification}, which may also be called \textbf{homotopification} or \textbf{groupoidification} (although the latter term also has a [[groupoidification|different meaning]], we allow objects to come with automorphisms, those automorphisms to come with automorphisms, and so on. In the limit, this involves replacing sets by [[∞-groupoids]] or [[homotopy types]] (hence the name ``homotopification'').


\item In \textbf{directed categorification}, which may also be called \textbf{directification} or \textbf{laxification}, we allow morphisms that were previously required to be invertible to instead be noninvertible (i.e. ``directed'').



\end{itemize}
This all looks good, with one complaint: I don't think anyone would consider laxification (passage from sets to posets, groupoids to categories, categories to $2$-posets, etc) to be `categorification'. That is, the term `directed categorification' works best for the \emph{combination} of groupoidal categorification followed by laxification. Do you agree, Mike? ---Toby

[[Mike Shulman]]: I actually think those are definitely a type of categorification. I can think of several places where people have explicitly categorified from categories to (1,2)-categories, either with some particular application in mind, or as a waypoint on the way to 2-categories. I can see that the term ``directed categorification'' might most easily be interpreted to refer to the combination, but I do think ``pure directification'' is also a form of categorification.

\emph{Toby}: I think that examples of `categorifying' form categories to $2$-posets would help me understand. (Clear examples of `categorifying' from $(\infty,n)$-categories to $(\infty,n+1)$-categories would be maximally motivating, if you have them too.)

Of course, $2$-posets can be thought of as special $2$-categories, and that may come even more easily than thinking of posets as special categories. But moving from categories to $2$-posets or from sets to posets seems like such a different thing. It probably doesn't help that I don't like the term [[horizontal categorification]] either; categories may be a common generalisation of groupoids, posets, and monoids, but out of all of those three originals, only moving from posets (or something even simpler such as sets) to categories seems like `categorification' to me; only that moves us higher in the hierarchy of $n$-categories.

[[Mike Shulman]]: I think moving from $(\infty,n)$-categories to $(\infty,n+1)$-categories is \emph{definitely} vertical categorification. It's exactly like moving from $n$-categories to $(n+1)$-categories in so many ways. In particular, the collection of all $(\infty,n)$-categories is an $(\infty,n+1)$-category. Or consider the notion of [[symmetric monoidal (∞,1)-category]]; I think this is clearly a categorification of the notion of symmetric monoidal ∞-groupoid, i.e. [[infinite loop space]]. Not only can a definition of symmetric monoidal (∞,1)-category obtained from a definition of symmetric monoidal ∞-groupoid in a straightforward way, but it also satisfies a version of the [[microcosm principle]]: you can define symmetric monoidal objects in any symmetric monoidal (∞,1)-category. The beauty of working in the ∞-context from the start is that you don't have to invent new coherence isomorphisms when you categorify, but I don't think that makes it any less categorification.

For the case of (2,1)-categories, here are a few papers that come to mind: * Power and Kinoshita, ``Lax naturality through enrichment'' * Carboni and Walters, ``Cartesian bicategories I'' * or even the discussion of the Burali-Forti paradox in a (2,3)-topos [[michaelshulman:category of all sets|here]].

The point about ``horizontal categorification'' is, of course, that the intuition of ``moving us higher in the hierarchy of $n$-categories'' is only applicable, basically by definition, to \emph{vertical} categorification. Horizontal categorification is \emph{supposed} to be something different. I'd rather not call that ``categorification'' either, but it is certainly a process of ``generalizing something to use categories.'' However I \emph{do} feel, as I've said, that ``directification'' does ``move us higher in the hierarchy of $n$-categories''---it's just that that ``hierarchy'' is not \emph{linear}! Remember the \href{http://ncatlab.org/nlab/show/%28n%2Cr%29-category#the_periodic_table_7}{periodic table} of (n,r)-categories.

\emph{Toby}: All right, yes, that helps. I'll sleep on it, but there's a good chance that you've convinced me.

If you like [[negative thinking]], then instead of saying that categorification `replaces [[sets]] by [[categories]]' (to quote Wikipedia), you can say that we replace [[truth values]] by sets, especially the truth values of [[equality|equations]]. That is, we acknowledge that there may be many different ways in which something may be true, and in particular many different ways in which two things may be the same. And then it is meaningful to ask whether two ways in which these things are the same are the same way (and if so, whether two ways that \emph{they} are the same are the same way, etc).

However, when we apply ``replace truth values by sets'' to the truth values of the equality relation of a set, we end up with a groupoid, since the equality of a set is symmetric. Thus, while two [[elements]] of a set simply may (or may not) be equal, two [[objects]] of a [[groupoid]] may be [[isomorphic]] in many different ways. And while two [[parallel morphisms|parallel isomorphisms]] in a groupoid may be equal, two parallel [[equivalences]] in a $2$-groupoid may be isomorphic in many different ways. Thus, this gives us groupoidal categorification, or homotopification.

To get from groupoids to categories, we need to also allow things which were previously invertible to be noninvertible, i.e. perform ``directification.'' We could also do this first starting from a set, obtaining a poset (in which the symmetric relation ``is equal to'' has been replaced by the non-symmetric one ``is less than or equal to''). Then when we homotopify a poset, we get a category: while one element $x$ of a [[poset]] may precede an element $y$, there may be many different [[morphisms]] from one object $x$ of a category to an object $y$.

This can also be understood naturally in the language of [[(n,r)-categories]]. Recall that an $(n,r)$-category can be defined as an [[∞-category]] in which all cells above dimension $r$ are invertible, and all cells above dimension $n$ are trivial. Thus, groupoidal categorification can be understood as increasing $n$ but keeping $r$ constant, while directification can be understood as increasing $r$ but keeping $n$ constant.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[categorification in representation theory]]


\item [[categorified Dold-Kan correspondence]]



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

The terminology goes back to

\begin{itemize}%
\item [[Louis Crane]], [[David Yetter]], \emph{Examples of categorification} (\href{http://arxiv.org/abs/q-alg/9607028}{arXiv:q-alg/9607028})

\end{itemize}
and was further amplified in

\begin{itemize}%
\item [[John Baez]], [[James Dolan|James Dolan]], \emph{Categorification}, in [[Ezra Getzler]], [[Mikhail Kapranov]] (eds.) \emph{Higher Category Theory}, Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998, pp. 1-36 (\href{http://arxiv.org/abs/math.QA/9802029}{arXiv:math.QA/9802029})

\end{itemize}
A bit of $n$-Caf\'e{} discussion on this subject can be found here:

\begin{itemize}%
\item [[John Baez]], \emph{\href{http://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html}{What is categorification?}}

\end{itemize}
A general notion of categorification for structures defined by [[cartesian monads]], which specializes to produce weak [[n-categories]] in the sense of Leinster, can be found here:

\begin{itemize}%
\item [[D. Bourn]], J. Penon, \emph{Cat\'e{}gorification de structures d\'e{}finies par monade cart\'e{}sienne}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques \textbf{46}, no. 1 (2005), p. 2-52, \href{http://www.numdam.org/numdam-bin/fitem?id=CTGDC_2005__46_1_2_0}{numdam}.

\end{itemize}
[[!redirects vertical categorification]] [[!redirects vertical categorifications]] [[!redirects vertical categorificaton]]

[[!redirects categorification]] [[!redirects categorifications]] [[!redirects categorify]] [[!redirects categorified]]

[[!redirects ∞-categorification]] [[!redirects ∞-categorifications]]



\end{document}