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\begin{document}

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\section*{strict 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{strict $n$-category} is a [[strict omega-category]] all whose [[k-morphisms]] for $k \gt n$ are identities.

The [[category]] $n Cat$ of strict $n$-categories and [[n-functors]] between them can also be defined [[induction|inductively]] by

\begin{itemize}%
\item starting by setting $0 Cat :=$ [[Set]];


\item noticing that [[Set]] is canonically a (symmetric, in fact cartesian) [[closed monoidal category]] such that one can consider [[enriched category|categories enriched]] over it;


\item noticing that for $V$ any [[complete category|complete]] and [[cocomplete category|cocomplete]] closed monoidal category, also $V Cat$ (the category of $V$-[[enriched categories]]) has these same properties;


\item finally setting, recursively,

\begin{displaymath}
(n+1)Cat := (n Cat) Cat
  \,.
\end{displaymath}


\end{itemize}
The category $Str\omega Cat$ of strict $\omega$-categories can then in turn be defined as a suitable [[colimit]] of the categories $n Cat$.

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

\begin{defn}
\label{GauntStrictNCategories}\hypertarget{GauntStrictNCategories}{}
Write $Str n Cat$ for the 1-[[category]] of [[strict n-categories]].

Write

\begin{displaymath}
Str n Cat_{gaunt} \hookrightarrow  Str n Cat
\end{displaymath}
for the [[full subcategory]] on the \emph{gaunt $n$-categories}, those $n$-categories whose only invertible [[k-morphisms]] are the identities.

\end{defn}
This subcategory was considered in (\hyperlink{Rezk}{Rezk}). The term ``gaunt'' is due to (\hyperlink{BarwickSchommerPries}{Barwick, Schommer-Pries}). See prop. \ref{GauntIs0Truncated} below for a characterization intrinsic to $(\infty,n)$-categories.

\begin{example}
\label{Globes}\hypertarget{Globes}{}
For $k \leq n$ the $k$-[[globe]] is gaunt, $G_k \in Str n Cat_{gaunt} \hookrightarrow \in Str n Cat$.

Write

\begin{displaymath}
\mathbb{G}_{\leq n} \hookrightarrow Str n Cat_{gaunt}
\end{displaymath}
for the [[full subcategory]] of the [[globe category]] on the $k$-globes for $k \leq n$.

Being a [[subobject]] of a gaunt $n$-category, also the [[boundary]] of a globe $\partial G_k \hookrightarrow G_k$ is gaunt, i.e. the $(k-1)$-[[skeleton]] of $G_k$.

\end{example}
\begin{defn}
\label{Suspension}\hypertarget{Suspension}{}
Write

\begin{displaymath}
\sigma_k : Str (k) Cat \to Str (k+1) Cat
\end{displaymath}
for the ``categorical suspension'' functor which sends a strict $k$-category to the object $\sigma(X) \in Str (k+1) Cat \simeq (Str k Cat)Cat$ which has precisely two objects $a$ and $b$, has $\sigma(C)(a,a) = \{id_a\}$, $\sigma(C)(b,b) = \{id_b\}$, $\sigma(C)(b,a) = \emptyset$ and

\begin{displaymath}
\sigma(C)(a,b) = C
  \,.
\end{displaymath}
\end{defn}
We usually suppress the subscript $k$ and write $\sigma^i = \sigma_{k+i} \circ \cdots \circ \sigma_{k+1} \circ \sigma_k$, etc.

\begin{example}
\label{}\hypertarget{}{}
The $k$-[[globe]] $G_k$ is the $k$-fold suspension of the 0-globe (the point)

\begin{displaymath}
G_k = \sigma^k(G_0)
  \,.
\end{displaymath}
The [[boundary]] $\partial G_k$ of the $k$-globe is the $k$-fold suspension of the empty category

\begin{displaymath}
\partial G_k = \sigma^k(\emptyset)
  \,.
\end{displaymath}
Accordingly, the boundary inclusion $\partial G_k \hookrightarrow G_k$ is the $k$-fold suspension of the initial morphism $\emptyset \to G_0$

\begin{displaymath}
(\partial G_k \hookrightarrow G_k) = \sigma^k(\emptyset \to G_0)
  \,.
\end{displaymath}
\end{example}
\begin{prop}
\label{}\hypertarget{}{}
The category $Str n Cat_{gaunt}$ is a [[locally presentable category]] and in fact a [[locally finitely presentable category]].

\end{prop}
(\hyperlink{BarwickSchommerPries}{B-PS, lemma 3.5})

\begin{remark}
\label{}\hypertarget{}{}
For $A,B$ two categories, a [[profunctor]] $A^{op} \times B \to Set$ is equivalently a functor $K \to G_1$ equipped with an identification $A \simeq K_0$ and $B \simeq K_1$.

\end{remark}
This motivates the following definition.

\begin{defn}
\label{}\hypertarget{}{}
A \emph{$k$-profunctor} / $k$-correspondence of strict $n$-categories is a morphism $K \to G_k$ in $Str n Cat$. The category of $k$-correspondences is the [[slice category]] $Str n Cat/ G_k$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The categories $Str n Cat_{gaunt}/G_k$ of $k$-correspondences between gaunt $n$-categories are [[cartesian closed category]].

\end{defn}
(\hyperlink{BarwickSchommerPries}{B-SP, cor. 5.4})

\begin{remark}
\label{}\hypertarget{}{}
By standard facts, in a [[locally presentable category]] $\mathcal{C}$ with [[finite limits]], a [[slice category|slice]] $\mathcal{C}/X$ is cartesian closed precisely if [[pullback]] along all morphisms $f : Y \to X$ with codomain $X$ preserves [[colimits]] (see at \emph{[[locally cartesian closed category]]} the section \emph{\href{locally%20cartesian%20closed%20category#EquivalentCharacterizations}{Cartesian closure in terms of base change and dependent product}}).

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Without the restriction that the codomain of $f$ in the above is a [[globe]], the pullback $f^*$ in $Str n Cat$ will in general fail to preserves colimits. For a simple example of this, consider the [[pushout]] diagram in [[Cat]] $\hookrightarrow Cat_{(\infty,1)}$ given by

\begin{displaymath}
\itexarray{
     \Delta[0] &\stackrel{\delta_1}{\to}& \Delta[1]
     \\
     {}^{\mathllap{\delta_0}}\downarrow && \downarrow^{\mathrlap{\delta_0}}
     \\
     \Delta[1] &\stackrel{\delta_2}{\to}& \Delta[2]
  }
  \,.
\end{displaymath}
Notice that this is indeed also a [[homotopy pushout]]/[[(∞,1)-pushout]] since, by remark \ref{GauntIs0Truncted}, all objects involved are 0-truncated.

Regard this canonically as a pushout diagram in the [[slice category]] $Cat_{/\Delta[2]}$ and consider then the pullback $\delta_1^* : Cat_{/\Delta[1]} \to Cat_{/\Delta[1]}$ along the remaining face $\delta_1 : \Delta[1] \to \Delta[2]$. This yields the diagram

\begin{displaymath}
\itexarray{
     \emptyset &\stackrel{}{\to}& \emptyset
     \\
     {}^{}\downarrow && \downarrow^{}
     \\
     \emptyset &\stackrel{}{\to}& \Delta[1]
  }
  \,,
\end{displaymath}
which evidently no longer is a pushout.

\end{example}
(See also the discussion \href{http://golem.ph.utexas.edu/category/2011/11/the_1category_of_ncategories.html#c040335}{here}).

\begin{defn}
\label{nCatGen}\hypertarget{nCatGen}{}
Write

\begin{displaymath}
Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}
\end{displaymath}
for the smallest [[full subcategory]] that

\begin{enumerate}%
\item contains the [[globe category]] $\mathbb{G}_{\leq n}$, example \ref{Globes};
\item is closed under [[retracts]] in $Str n Cat_{gaunt}$;
\item has all [[fiber products]] over [[globes]] (equivalently: such that all [[slice categories]] over globes have [[products]]).

\end{enumerate}
\end{defn}
(\hyperlink{BarwickSchommerPries}{B-SP, def. 5.6})

\begin{example}
\label{}\hypertarget{}{}
The following categories are naturally [[full subcategories]] of $Str n Cat_{gen}$

\begin{itemize}%
\item the $n$-fold [[simplex category]] $\Delta^{\times n}$;


\item the $n$th [[Theta-category]].



\end{itemize}
\end{example}
This is discussed in more detail in [[(infinity,n)-category]] in \emph{\href{(infinity,n}{Presentation by Theta-spaces and by n-fold Segal spaces}-category\#PresentationByThetaSpaces)}.

\begin{defn}
\label{FundamentalPushouts}\hypertarget{FundamentalPushouts}{}
The following [[pushouts]] in $Str n Cat$ we call the \textbf{fundamental pushouts}

\begin{enumerate}%
\item Gluing two $k$-[[globes]] along their [[boundary]] gives the boundary of the $(k+1)$-globle

\begin{displaymath}
G_k \coprod_{\partial C_{k-1}} G_k \simeq \partial G_{k+1}
\end{displaymath}

\item Gluing two $k$-globes along an $i$-face gives a [[pasting]] composition of the two globles

\begin{displaymath}
G_k \coprod_{G_i} G_k
\end{displaymath}

\item The [[fiber product]] of globes along non-degenerate morphisms $G_{i+j} \to G_i$ and $G_{i+k} \to G_i$ is built from gluing of globes by

\begin{displaymath}
G_{i+j} \times_{G_i} G_{i+k}
  \simeq
  (G_{i+j} \coprod_{G_i} G_{i+k}) \coprod_{\sigma^{i+1}(G_{j-1} \times G_{k-1})} (G_{i+k} \coprod_{G_i} G_{i+j})
\end{displaymath}

\item The [[interval groupoid]] $(a \stackrel{\simeq}{\to} b)$ is obtained by forcing in $\Delta[3]$ the morphisms $(0\to 2)$ and $(1 \to 3)$ to be identities and it is equivalent, as an $n$-category, to the 0-globe

$\Delta[3] \coprod_{\{0,2\} \coprod \{1,3\}} (\Delta[0] \coprod \Delta[0])
\stackrel{\sim}{\to} G_0$

and the analog is true for all suspensions of this relation

\begin{displaymath}
\sigma^k(\Delta[3]) \coprod_{\sigma^k\{0,2\} \coprod \sigma^k\{1,3\}} (G_k\coprod G_k)
  \stackrel{\sim}{\to} G_k
  \,.
\end{displaymath}


\end{enumerate}
We say a functor $i$ on $Str n Cat$ \emph{preserves} the fundamental pushouts if it preserves the first three classes of pushouts, and if for the last one the morphism $i(\sigma^k(\Delta[3])) \coprod_{i(\sigma^k\{0,2\}) \coprod i(\sigma^k\{1,3\})} (i(G_k \coprod G_k)) \to i(G_k)$ is an equivalence.

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

A strict 1-category is just a [[category]].

[[strict 2-category|Strict 2-categories]] are important, because the [[coherence theorem for bicategories]] states that every (``weak'') [[2-category]] is [[equivalence of categories|equivalent]] to a strict one, and also because many 2-categories, such as [[Cat]], are naturally strict. However, for $n\ge 3$, these two properties fail, so that strict $n$-categories become less useful (though not useless). Instead, one needs to use (at least) [[semistrict categories]].

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

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


\item [[strict category]]



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

With an eye towards the generalization to [[(∞,n)-categories]], strict $n$-categories are discussed in

\begin{itemize}%
\item [[Charles Rezk]], \emph{A cartesian presentation of weak n-categories} (\href{http://arxiv.org/abs/0901.3602}{arXiv:0901.3602})

\end{itemize}
and in section 2 of

\begin{itemize}%
\item [[Clark Barwick]], [[Chris Schommer-Pries]], \emph{On the Unicity of the Homotopy Theory of Higher Categories} (\href{http://arxiv.org/abs/1112.0040}{pdf}, \href{http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/}{unusual slides})

\end{itemize}
[[!redirects strict n-categories]] [[!redirects strict infinity-category]] [[!redirects strict infinity-categories]] [[!redirects strict ∞-category]] [[!redirects strict ∞-categories]]



\end{document}