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\begin{document}

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\section*{n-category}

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

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{categories_of_categories}{Categories of $n$-categories}\dotfill \pageref*{categories_of_categories} \linebreak
\noindent\hyperlink{defn}{Definitions}\dotfill \pageref*{defn} \linebreak
\noindent\hyperlink{list_of_definitions}{List of definitions}\dotfill \pageref*{list_of_definitions} \linebreak
\noindent\hyperlink{comparisons}{Comparisons}\dotfill \pageref*{comparisons} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

For $n \in \mathbb{N}$, an \emph{$n$-category} is like

\begin{itemize}%
\item an $n$-truncated [[directed space]] in which $(k \leq n)$-dimensional paths have a direction, while all higher dimensional paths are reversible and parallel higher dimensional paths are homotopic;


\item a $n$-fold higher analog of what a [[category]] is to a [[set]]



\end{itemize}
$n$-Categories are the main subject of [[higher category theory]], and give the $n$-[[About|Lab]] its name. In their modern formulation in [[homotopy theory]] they are known as \emph{[[(∞,n)-categories]]} (see there for more details).

Semi-formally, $n$-categories can be described as follows. An $n$-category is an [[∞-category]] such that all $(n+1)$-morphisms are [[equivalence]]s, and all parallel pairs of $j$-morphisms are equivalent for $j > n$. (One says that the $\infty$-category is \emph{trivial} in degree greater than $n$.) This is the same thing as an $(n,n)$-category in the sense of $(n,r)$-[[(n,r)-category|categories]].

Up to equivalence, you may assume that all equivalent pairs of $j$-morphisms for $j > n$ are in fact equal, and many authorities include this as a requirement. On the other hand, you can also write down a definition of $n$-category from scratch (without passing through $\infty$-categories), and then this question never comes up. The point is that you don't talk about $j$-morphisms for $j > n$; you stop at $n$-morphisms.

On the $n$Lab, the term ``$n$-category'' usually means a \emph{weak} $n$-category, in which the compositions of cells obeys the usual associativity, unit, and exchange laws only up to coherent [[equivalence]]. This sort of $n$-category is somewhat tricky to define; there are a number of proposals, not yet shown to be equivalent. By contrast, [[strict n-categories]] are easy to define, but are not sufficient for most examples when $n\ge 3$ (see [[semistrict n-category]]).

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

\begin{itemize}%
\item A $0$-category is a [[set]].


\item A $1$-category is an ordinary [[category]].


\item A $2$-[[2-category|category]] is (depending on how strict was your initial notion of $\infty$-category) either a [[strict 2-category]] or a [[bicategory]].



\end{itemize}
One also speaks of $(-1)$-[[(-1)-category|categories]] and $(-2)$-[[(-2)-category|categories]], but these concepts are not as well behaved.

\hypertarget{categories_of_categories}{}\subsection*{{Categories of $n$-categories}}\label{categories_of_categories}

Just as the collection of all ([[small category|small]]) sets is the prototypical example of a category, so the collection of all small $n$-categories is the prototypical example of an $(n+1)$-category.

Actually, if you define things cleverly, then you can get an $(n+1)$-category of \emph{all} $n$-categories. If one assumes the [[Grothendieck universe|Axiom of Universes]], then there is a sequence of Grothendieck universes

\begin{displaymath}
U_0 \subset U_1 \subset U_2 \subset \cdots
\end{displaymath}
and we can say a set is \textbf{$U_i$-small} if it is an element of $U_i$. This allows us to make the following definitions:

\begin{itemize}%
\item $\Set$ is the category of all $U_0$-small sets;
\item $\Cat$ is the 2-category of all $U_1$-small categories;
\item $2\Cat$ is the 3-category of all $U_2$-small 2-categories;
\item etc.

\end{itemize}
This is a convenient way to settle size questions once and for all for finite $n$, but it doesn't really work for $\infty$-categories.

For more, see the discussion at \href{http://groups.google.com/group/sci.logic/browse_thread/thread/6891c85c2d9b2caf}{sci.logic}.

\hypertarget{defn}{}\subsection*{{Definitions}}\label{defn}

Here is a list of (some of) the proposed definitions of (weak) $n$-category, with references, and also a list of (some of) the comparisons that have been done.

\hypertarget{list_of_definitions}{}\subsubsection*{{List of definitions}}\label{list_of_definitions}

Many of these definitions are actually ``truncations'' of definitions of (weak) [[∞-categories]] (aka [[∞-categories]]). Some others are truncations of a definition of [[(∞,n)-categories]]. A nice overview of (many) of these can be found in Tom Leinster's paper ``A survey of definitions of $n$-category.''

Someone should add some more references!

\begin{itemize}%
\item Classical explicit definitions of ``fully weak'' $n$-category exist for $n\le 4$. Weak 0-categories are [[sets]], weak 1-categories are simply [[categories]] (due to [[Sammy Eilenberg|Eilenberg]] and [[Saunders Mac Lane|Mac Lane]]), weak 2-categories are [[bicategories]] (due to [[Jean Benabou|Benabou]]), weak 3-categories are [[tricategories]] (due to [[Gordon]]--[[John Power|Power]]--[[Ross Street|Street]]), and weak 4-categories are [[tetracategories]] (due to [[Todd Trimble]]). Going on in this way is generally admitted to be infeasible beyond $n=4$.


\item [[Ross Street|Street's]] definition: an $n$-category is a [[simplicial set]] satisfying certain horn-filling conditions. See [[weak complicial set]] and [[simplicial model for weak ∞-categories]]. This is a truncation of a definition of $\omega$-category. It can be specialized to yield a notion of $(\infty,n)$-category. The resulting notion of $(\infty,1)$-category is a [[quasicategory]], and the resulting notion of $\infty$-groupoid is a [[Kan complex]].


\item [[John Baez|Baez]]--[[James Dolan|Dolan]] definition: an $n$-category is an [[opetopic set]] having enough $n$-universal fillers. Alternate definitions of opetopes (aka multitopes) have been given by [[Claudio Hermida|Hermida]]--[[Michael Makkai|Makkai]]--[[John Power|Power]] and [[Tom Leinster|Leinster]]; a comparison is due to [[Eugenia Cheng]], see \href{http://arxiv.org/abs/math.CT/0304277}{these} \href{http://arxiv.org/abs/math.CT/0304279}{three} \href{http://arxiv.org/abs/math.CT/0304284}{papers}. Makkai's version can do $\omega$.


\item [[Jacques Penon|Penon]]`s definition: (someone describe this please!) Penon's original definition turned out to be too strict (see \href{http://web.science.mq.edu.au/~mbatanin/Penon1.ps}{Batanin} and \href{http://arxiv.org/abs/0907.3961}{Cheng--Makkai}) because it used [[reflexive globular set|reflexive globular sets]], but a modification of it using [[globular sets]] is still a contender.


\item [[Michael Batanin|Batanin]]--[[Tom Leinster|Leinster]] definition: an $n$-category is an $n$-[[globular set]] with an action of a suitable [[globular operad]]. This is a truncation of a definition of $\omega$-category; see [[Batanin ∞-category]].


\item [[Todd Trimble|Trimble]]-style definition: An $n$-category is a category weakly enriched over $(n-1)$-categories, where the weakness is parametrized by an [[operad]]. This definition is inductive and thus cannot do $\omega$ in an obvious way, but it has been accomplished using terminal coalgebras; see [[Trimble n-category]]. Alternately, by starting with enrichment in [[spaces]] or [[simplicial sets]], one can obtain directly a notion of [[(∞,n)-category]]. The resulting notion of [[(∞,1)-category]] is an $A_\infty$-[[A∞-category|category]].


\item [[Tamsamani]]--[[Carlos Simpson|Simpson]] definition: An $n$-category is a simplicial object in $(n-1)$-categories satisfying object-discreteness and the [[Segal condition]]. This definition is inductive (it is a different way of formalizing ``iterated weak enrichment'') and thus cannot do $\omega$ in an obvious way. It does have a natural extension to $(\infty,n)$-categories, and the resulting notion of [[(∞,1)-category]] reduces to a [[Segal category]]. The iterated version of this is that of [[Segal n-category]]. This notion of ``weak enrichment'' in a [[cartesian model category]] is studied carefully in Simpson's book [[Homotopy Theory of Higher Categories]].


\item [[Ieke Moerdijk|Moerdijk]] and [[Ittay Weiss|Weiss]]`s definition uses yet another way of formalizing ``iterated weak enrichment,'' using [[dendroidal sets]] and [[quasi-operad]]s.


\item [[Andre Joyal|Joyal]]`s definition: An $n$-category is an $n$-[[cellular set]] satisfying horn-filling conditions. This definition can do $\omega$ by using $\omega$-cellular sets instead of $n$-cellular sets, and it can do $(\infty,n)$ by requiring different horn-filling conditions on $n$-cellular sets. The notion of [[(∞,1)-category]] one obtains in this way is a [[quasicategory]], and the resulting notion of $\infty$-groupoid is a [[Kan complex]]. For $n\gt 1$, however, the obvious ``horn-filling conditions'' are not quite right; [[Dimitri Ara]] has shown how to correct them (albeit not very explicitly), obtaining a definition he calls an [[n-quasicategory]], which form a model category Quillen equivalent to Rezk's definition (below).


\item [[Clark Barwick|Barwick]]`s definition (popularized by [[Jacob Lurie|Lurie]] in solving the Baez--Dolan [[cobordism hypothesis]]): an $(\infty,n)$-category is an $n$-fold [[simplicial object|simplicial]] [[topological space]] satisfying completeness and the [[Segal condition]]. See [[n-fold complete Segal space]]. An $n$-category is again defined as an $(\infty,n)$-category in which all $k$-cells are essentially unique for $k\gt n$. It is not clear whether this definition can do $\omega$. An $(\infty,1)$-category with this definition is also the same as a [[complete Segal space]].


\item [[Charles Rezk|Rezk]]`s definition: An $(\infty,n)$-category is a \emph{simplicial} $n$-cellular set satisfying [[fibrancy]], completeness, and the [[Segal condition]]. An $n$-category can then be defined as an $(\infty,n)$-category in which all $k$-cells are essentially unique for $k\gt n$. This definition can potentially do $\omega$, although it seems not to have been written down yet. An $(\infty,1)$-category with this definition is the same as a [[complete Segal space]]. See [[Theta space]].


\item [[Georges Maltsiniotis|G. Maltsiniotis]] has apparently extracted a definition of $\infty$-groupoid from [[Pursuing Stacks]] and generalized it to a definition of $\infty$-category; see \href{http://people.math.jussieu.fr/~maltsin/ps/infgrart.pdf}{this} and \href{http://people.math.jussieu.fr/~maltsin/ps/infctart.pdf}{this}.



\end{itemize}
\hypertarget{comparisons}{}\subsubsection*{{Comparisons}}\label{comparisons}

\begin{itemize}%
\item All definitions produce the correct well-known notion of [[1-category]], up to minor inessential details.


\item Since all the common definitions of [[(∞,1)-category]] are known to be equivalent (give references!), the definitions of Street, Trimble, Tamsamani--Simpson, Joyal, Barwick, and Rezk can be said to agree for $(\infty,1)$-categories.


\item Julie Bergner has \href{http://arxiv.org/abs/0710.2254}{shown} that all the notions of $\infty$-groupoid obtained from the common notions of $(\infty,1)$-category are equivalent, so the definitions of Street, Trimble, Tamsamani--Simpson, Joyal, Barwick, and Rezk can also be said to agree for $(\infty,0)$-categories.


\item It is known that the notions of $(n,0)$-category obtained from categories, bicategories, and tricategories model all [[homotopy n-types]] for $n\le 3$. Thus, in these cases, the classical definitions can be said to agree with those listed in the previous example.


\item In Tom Leinster's paper, proofs are sketched showing that the notion of [[2-category]] obtained in each case looks somewhat like the notion of [[bicategory]].


\item Nick Gurski has shown in ``Nerves of bicategories as stratified simplicial sets'' that Street's definition is correct for $n=2$ (that is, it agrees with bicategories).


\item Eugenia Cheng has \href{http://arxiv.org/abs/math/0304285}{shown} that the opetopic definition is correct for $n=2$ (that is, it agrees with bicategories).


\item Eugenia Cheng has more recently also \href{http://arxiv.org/abs/0809.2070}{shown} that from any sequence of operads used for iterated enrichment in a Trimble-style definition, one can construct a Batanin--Leinster-style globular operad whose algebras are the $n$-categories obtained in the Trimblean inductive manner. Not all globular operads can be obtained in this way, however, since those that arise have strict interchange.



\end{itemize}
Please add any other comparisons you are aware of!

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

\begin{itemize}%
\item [[0-category]], [[(0,1)-category]], [[Set]]


\item [[category]]

\begin{itemize}%
\item [[category object in an (∞,1)-category]]

\end{itemize}

\item [[2-category]], [[bicategory]]


\item [[3-category]], [[tricategory]]


\item [[4-category]], [[tetracategory]]


\item \textbf{$n$-category}

\begin{itemize}%
\item [[n-category object in an (∞,1)-category]]

\end{itemize}

\item [[(n,1)-category]]


\item [[(∞,1)-category]]


\item [[(∞,2)-category]]


\item [[(∞,n)-category]]


\item [[(n,r)-category]]



\end{itemize}
[[!redirects $n$-category]] [[!redirects n-categories]] [[!redirects weak n-category]] [[!redirects weak n-categories]]



\end{document}