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\section*{(infinity,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{introduction}{Introduction}\dotfill \pageref*{introduction} \linebreak
\noindent\hyperlink{1CatIntro}{For 1-category theorists}\dotfill \pageref*{1CatIntro} \linebreak
\noindent\hyperlink{HomotopyIntro}{For homotopy theorists}\dotfill \pageref*{HomotopyIntro} \linebreak
\noindent\hyperlink{Definition}{Definitions}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{AxiomaticCharacterization}{Via generation by strict $n$-categories}\dotfill \pageref*{AxiomaticCharacterization} \linebreak
\noindent\hyperlink{strict_categories}{Strict $n$-categories}\dotfill \pageref*{strict_categories} \linebreak
\noindent\hyperlink{generation_by_strict_categories}{Generation by strict $n$-categories}\dotfill \pageref*{generation_by_strict_categories} \linebreak
\noindent\hyperlink{universal_presentation}{Universal presentation}\dotfill \pageref*{universal_presentation} \linebreak
\noindent\hyperlink{PresentationByThetaSpaces}{Presentation by $\Theta_n$-spaces and $n$-fold complete Segal spaces}\dotfill \pageref*{PresentationByThetaSpaces} \linebreak
\noindent\hyperlink{ViaEnrichment}{Via $\infty$-enrichment}\dotfill \pageref*{ViaEnrichment} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{presentation_by_segal_categories}{Presentation by Segal $n$-categories}\dotfill \pageref*{presentation_by_segal_categories} \linebreak
\noindent\hyperlink{ByEnrichedModelCategories}{Presentation by enriched model categories}\dotfill \pageref*{ByEnrichedModelCategories} \linebreak
\noindent\hyperlink{ViaInternalization}{Via $\infty$-internalization}\dotfill \pageref*{ViaInternalization} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{PresentationByCompleteSegal}{Presentation by $n$-fold complete Segal spaces}\dotfill \pageref*{PresentationByCompleteSegal} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Generators}{Generators}\dotfill \pageref*{Generators} \linebreak
\noindent\hyperlink{truncated_objects}{Truncated objects}\dotfill \pageref*{truncated_objects} \linebreak
\noindent\hyperlink{moduli}{Moduli}\dotfill \pageref*{moduli} \linebreak
\noindent\hyperlink{WebOfQuillenEquivalences}{Web of Quillen equivalent model category presentations}\dotfill \pageref*{WebOfQuillenEquivalences} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{extra_structure_and_properties}{Extra structure and properties}\dotfill \pageref*{extra_structure_and_properties} \linebreak
\noindent\hyperlink{monoidal_categories}{$\mathcal{O}$-Monoidal $(\infty,n)$-categories}\dotfill \pageref*{monoidal_categories} \linebreak
\noindent\hyperlink{categories_with_all_adjoints}{$(\infty,n)$-Categories with all adjoints}\dotfill \pageref*{categories_with_all_adjoints} \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}

In [[higher category theory]] an \emph{$(\infty,n)$-category} may be thought of as

\begin{itemize}%
\item an [[n-category]] up to [[coherence|coherent]] [[homotopy theory|homotopy]];


\item an [[(n,r)-category|(r,n)-category]] for $r = \infty$;


\item a [[weak omega-category]] for which all [[k-morphisms]] with $k \gt n$ are [[equivalences]].



\end{itemize}
Accordingly, the notion of $(\infty,n)$-categories is a joint generalization of \emph{[[categories]]}, \emph{[[2-categories]]}, \emph{[[3-categories]]}, \emph{[[4-categories]]}, etc. and of \emph{[[∞-groupoids]]} / \emph{[[homotopy types]]} and \emph{[[(∞,1)-categories]]}. From the point of [[homotopy theory]] they are a generalization to \emph{[[directed homotopy theory]]}, from the point of view of [[homotopy type theory]] they are a generalization to \emph{[[directed homotopy type theory]]}.

There are two main [[recursion|recursive]] definitions of $(\infty,n)$-categories:

\begin{enumerate}%
\item by iterated [[enriched (∞,1)-category|(∞,1)-enrichment]]

\begin{displaymath}
Cat_{(\infty,n)} \simeq (\cdots((Cat_{(\infty,0)} Cat) Cat) \cdots) Cat
\end{displaymath}

\item by iterated [[category object in an (∞,1)-category|(∞,1)-internalization]]



\end{enumerate}
\begin{displaymath}
Cat_{(\infty,n)} \simeq Cat(\cdots(Cat(Cat_{(\infty,0)}))\cdots)
    \,.
\end{displaymath}
There is also a fairly simple axiomatization of the [[(∞,1)-category]] $Cat_{(\infty,n)}$ itself, as something \emph{[[generators and relations|generated]]} by [[strict n-categories]].

Then there is also a plethora of [[model category]] structures that [[presentable (∞,1)-category|present]] the [[(∞,1)-category]] $Cat_{(\infty,n)}$ of all $(\infty,n)$-categories, which means that there are many (and many different) very explicit ways to describe them.

A central result of $(\infty,n)$-category theory is the proof of the [[cobordism hypothesis]], which revolves around the \emph{[[(∞,n)-category of cobordisms]]}. This turns out to be the [[free construction|free]] \emph{[[symmetric monoidal (∞,n)-category]]} \emph{[[(∞,n)-category with duals|with duals]]} and provides deep relations between [[algebraic topology]], [[higher algebra]] and [[extended topological quantum field theory]]. Other fundamental examples of $(\infty,n)$-categories, also in this context, are [[(∞,n)-categories of spans]] and of [[(∞,n)-vector spaces]].

While the subject is still young, visible at the horizon is its role in [[higher topos theory]]. Where [[(∞,1)-toposes]] regarded as [[(∞,1)-categories of (∞,1)-sheaves]]/[[∞-stacks]] are by now fairly well understood, it is clear that the [[(∞,2)-categories]] of [[(∞,2)-sheaves]] -- such as the [[codomain fibration]]/[[indexed category|self-indexing]] of an [[(∞,1)-topos]] -- will form an [[(∞,2)-topos]] in generalization of the non-homotopic notion of [[2-topos]]. And so on.

\hypertarget{introduction}{}\subsection*{{Introduction}}\label{introduction}

Here are some introductory words for readers unfamiliar with the general idea. Other readers should skip \hyperlink{Definition}{ahead}.

\begin{itemize}%
\item \emph{\hyperlink{1CatIntro}{Introduction for 1-category theorists}}


\item \emph{\hyperlink{HomotopyIntro}{Introduction for homotopy theorists}}



\end{itemize}
\hypertarget{1CatIntro}{}\subsubsection*{{For 1-category theorists}}\label{1CatIntro}

This section assumes that the reader is well familiar with [[category theory]] and maybe with [[strict omega-categories]] but in need of some introductory words on $(\infty,n)$-categories.

Ordinary [[category theory]] provides various powerful tools for generating higher order structures, among them notably

\begin{enumerate}%
\item [[enriched category theory|enrichment]]


\item [[internalization]].



\end{enumerate}
Here we are interested in higher order \emph{[[categories]]}, so we consider [[Cat]] itself as a 1-categorical context for either of these procedures. Since [[Cat]] naturally a [[cartesian monoidal category]]

\begin{displaymath}
(\mathcal{V}, \otimes) \coloneqq (Cat, \times)
\end{displaymath}
we may form the [[category of V-enriched categories]] $\mathcal{V}Cat \coloneqq Cat Cat$. A $Cat$-category consists of

\begin{itemize}%
\item a collection of [[objects]];


\item for each pair of objects $A$, $B$ a \emph{category} of morphisms, hence to be thought of as collection of ordinary morphisms $A \to B$ together with \emph{morphisms between these morphisms}: [[2-morphisms]];


\item such that composition is a \emph{functor} on these hom-categories.



\end{itemize}
This is the structure of a \emph{[[strict 2-category]]}. We have that

\begin{displaymath}
Cat Cat \simeq Str 2 Cat
  \,.
\end{displaymath}
is the category of strict 2-categories.

By general results of [[enriched category theory]] (or by immediate inspection), this is still a [[cartesian monoidal category]] and so we may iterate this and consider now the enriching category

\begin{displaymath}
(\mathcal{V}, \otimes) \coloneqq (Cat Cat, \times)
\end{displaymath}
and construct again $\mathcal{V}Cat$, which now is

\begin{displaymath}
(Cat Cat) Cat \simeq Str 3 Cat
\end{displaymath}
the category of strict \emph{[[3-categories]]}. It continues this way, and so for every $n \in \mathbb{N}$ the $n$-fold iterated enrichment of $Cat$ is the category

\begin{displaymath}
Str n Cat \simeq  (\cdots ((Cat Cat)Cat) \cdots) Cat
\end{displaymath}
of \emph{[[strict n-categories]]}. The [[inductive limit]] of this construction finally is the category of [[strict omega-categories]].

While this easily generates [[higher category theory|higher categorical structures]], it does so, as the terminology indicates, only in a very restrictive way: while every [[2-category]] still happens to be [[equivalence of 2-categories|equivalent]] to a [[strict 2-category]], already the general [[3-category]] is no longer equivalent to a strict 3-category, and the discrepancy only increases with $n$.

But inspection in the case of [[2-categories]] already shows what the problem is: in a [[bicategory|weak 2-category]] structural relations such as [[associativity]] and [[unitality]] no longer hold as equations but only \emph{up to} an \emph{invertible} [[2-morphism]], whereas objects in $Str 2 Cat \simeq Cat Cat$, by definition of [[enriched category]], satisfy these relations \emph{strictly} -- therefore the name.

But this problem directly corresponds to an evident shortcoming of the very starting point of the above recursive construction: that construction regarded [[Cat]] as a 1-category in order to fit it into the standard formulation of [[enriched category theory]]; however [[Cat]] is naturally rather a [[2-category]] itself. The enrichment procedure should be allowed to make use of this extra structure. On the other hand, as we have just seen, the failure of $Cat Cat$ to model all of [[2Cat]] is only in the lack of \emph{invertible} 2-morphisms. Therefore what should really matter for the improved enrichment is just the [[(2,1)-category]] underlying [[Cat]], which is the 2-category consisting of all [[categories]], all [[functors]] between them, but only \emph{[[natural isomorphism]]} instead of all [[natural transformations]] between those.

This way one does arrive at a suitable refined notion of enrichment over the [[(2,1)-category]] $Cat$, and interpreted this way one does finds that $Cat Cat$ then indeed produces all of [[2Cat]].

However, this only fixed the first step of the above recursive definition. In the next step we want $(2 Cat)Cat$ to produce all [[3-categories]], but their associativity and unitalness now involves invertible [[coherence]] \emph{[[3-morphisms]]} which do not appear in enriched $(2,1)$-category theory. And so on, as the recursion proceeds.

This shows that the natural starting point for a construction of [[n-categories]] by recursive enrichment must be a conception of 1-[[category theory]] which knows already about \emph{invertible} [[k-morphisms]] for all $k$. The notion of category where all 1-categorical operations are relaxed up to \emph{invertible} higher morphisms is that of \emph{[[(∞,1)-category]]}. And this now turns out to be a good starting point for producing $n$-categories by recursive enrichment.

If we then just replace in the above the naive [[Cat]] with [[(∞,1)Cat]], then the simple formula

\begin{displaymath}
Cat_{(\infty,n)} \coloneq (\cdots ((Cat_{(\infty,1)} Cat_{(\infty,1)})Cat_{(\infty,1)}) \cdots) Cat_{(\infty,1)}
\end{displaymath}
does produce a good general notion of $n$-categories, these are the \emph{$(\infty,n)$-categories} discussed here.

There is also an alternative road to the same conclusion: another standard procedure for producing higher order structures from the 1-category [[Cat]] is to consider [[internal categories]] in $Cat$. For $E$ a category with [[finite limits]], write $Cat(E)$ for the category of $E$-[[internal categories]], and hence $Cat(Cat)$ for the category of $Cat$-internal categories.

This gives \emph{[[double categories]]}

\begin{displaymath}
DoubleCat \simeq Cat(Cat)
\end{displaymath}
and hence again not quite the [[2-categories]] that we are after. But it is of interest to note that now there are \emph{two} problems, not just the one above: while a $Cat$-internal category again has strict [[associativity]] and [[unitality]], instead of the desired version up to an invertible 2-morphism, in another direction it is more general than a [[strict 2-category]]: the latter only corresponds to those special double categories for which the ``vertical'' and the ``horizontal'' 1-morphisms come from the same 1-category and have sufficiently many degenerate 2-morphisms between them.

The first problem turns out to be solved as before: instead of working with the 1-category [[Cat]] we should already regard that as a [[(2,1)-category]] and then formulate \emph{internal (2,1)-categories} in straightforward generalization of the ordinary notion. For the second problem it turns out that one needs to slightly enhance that straightforward generalization and add a condition known (somewhat undescriptively) as \emph{[[complete Segal space|completeness]]}. But if this is understood then (as discussed in detail at [[internal category in an (∞,1)-category]]) the simple idea of iterated internalization does work out and we obtain $(\infty,n)$-categories by

\begin{displaymath}
Cat_{(\infty,n)} \simeq Cat(\cdots(Cat(Cat_{(\infty,0)}))\cdots)
  \,.
\end{displaymath}
\hypertarget{HomotopyIntro}{}\subsubsection*{{For homotopy theorists}}\label{HomotopyIntro}

This section assumes that the reader is well familiar with [[homotopy theory]] and maybe with [[(∞,1)-category theory]] but in need of some introductory words on $(\infty,n)$-categories.

A fundamental insight of [[homotopy theory]] is, of course, that the [[geometric shapes for higher structures|cellular shape]] of [[simplices]] naturally serves to model paths and higher [[homotopies]] in ``spaces'', which here really means: in [[homotopy types]]/[[∞-groupoids]]. In fact, the simplices see a bit more: since $\Delta[n]$ is naturally identified with the [[total order|linear category]] $\{0 \to 1 \to 2 \to \cdots \to n\}$ on $(n+1)$-objects, there is a \emph{direction} on the paths which form the [[simplicial skeleton|1-skeleton]] of a map $\Delta^n \to X$.

If $X$ is a [[topological space]]/[[simplicial set]]/[[homotopy type]], then this directedness in a way ``disappears up to equivalence'', in that for every such directed path there is also the reverse path, which is an inverse up to equivalence.

But it is straightforward to consider a slight generalization of this situation where we take $X$ to be such that \emph{not} all paths in it have inverses. Still thinking of $X$ as a homotopy type this may be thought of as modelling a \emph{[[directed homotopy theory|directed homotopy type]]}. For $X$ instead modeled as a simplicial set, this has been formalized by the concept of a \emph{[[quasi-category]]} or \emph{[[(∞,1)-category]]}. These are combinatorial models for \emph{[[directed homotopy theory|directed homotopy types]]} in direct generalization of how [[Kan complexes]] are combinatorial models for ordinary [[homotopy types]].

As the notation already suggests, the idea of $(\infty,n)$-category theory is that this generalization from [[∞-groupoids]] (``($\infty,0$)-categories'') to [[(∞,1)-categories]] is but the first step in a tower of higher generalizations, where in step $n$ one considers ``directed homotopy'' up to and including dimension $n$.

It is natural that such \emph{$(\infty,n)$-categories} should be probed by corresponding higher dimensional analogs of the objects in the [[simplex category]], the linear categories $\Delta[n] = \{0 \to 1 \to 2 \to \cdots \to n\}$ that support traditional homotopy theory. There are many such generalizations which one could consider. One which has proven to be useful are the objects in the $n$th [[Theta-category]] $\Theta_n$. Where the linear categories as above arise from gluing -- [[pasting]] -- of cellular intervals, the objects of $\Theta_n$ arise from [[pasting]] of $n$-dimensional cellular \emph{[[globes]]} (an interval being a 1-dimensional globe).

Accordingly, just as an [[∞-groupoid]]/[[homotopy type]] may be presented by a [[simplicial set]], hence a [[presheaf]] on the [[simplex category]] -- or more generally by a [[bisimplicial set|simplicial space]]-- satisfying some ([[Kan complex|Kan filler]]-)condition that encodes the existence of composites and inverses, so an [[(∞,n)-category]] may be presented by a presheaf of spaces on the $n$th [[Theta-category]], similarly subject to some conditions that ensure the existence of composites and inverses -- but only of inverses above dimension $n$.

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

There are various different ways of defining $(\infty,n)$-categories, which are all natural in their own right, and all equivalent to each other.

There is an axiomatic characterization of the [[(∞,1)-category]] of $(\infty,n)$-categories by [[generators and relations|generation]] from [[strict n-categories]]:

\begin{itemize}%
\item \hyperlink{AxiomaticCharacterization}{Definition via generation from strict n-categories}

\end{itemize}
Among the more direct definitions of $(\infty,n)$-categories one can roughly distinguish two flavors, those that build $(\infty,n)$-categories by \emph{[[enriched category|enrichment]]} over $(\infty,n-1)$-categories

\begin{itemize}%
\item \hyperlink{ViaEnrichment}{Definitions via enrichment}

\end{itemize}
and those that build them by [[internalization]] in the collection of $(\infty,n-1)$-categories

\begin{itemize}%
\item \hyperlink{ViaInternalization}{Definitions via internalization}.

\end{itemize}
\hypertarget{AxiomaticCharacterization}{}\subsubsection*{{Via generation by strict $n$-categories}}\label{AxiomaticCharacterization}

We discuss a characterization of the [[(∞,1)-category]] of $(\infty,n)$-categories as an $(\infty,1)$-category [[generators and relations|generated]] by [[strict n-categories]], due to (\hyperlink{BarwickSchommerPries}{Barwick, Schommer-Pries}).

The blueprint for the following construction is the traditional fact that a [[category]] is characterized by the fact that its [[nerve]] is a [[simplicial set]] which satisfies the [[Segal conditions]], which reflect the existence of composition in a category. Since the simplicial nerve is induced from the [[total order|linear categories]] $\Delta[n] = \{0 \to 1 \to 2 \to \cdots \to n\}$ this can be taken as saying that these linear categories \emph{generate} [[Cat]], subject to the condition that there exists composites.

The following discussion takes this point of view and generalizes it to a similar presentation of $(\infty,n)$-categories by very simple [[strict n-categories]].

\hypertarget{strict_categories}{}\paragraph*{{Strict $n$-categories}}\label{strict_categories}

The main definition is def. \ref{AxiomaticDefinition} below, which roughly says that the collection of $(\infty,n)$-categories is \emph{generated} from [[strict n-categories]] in a certain sense. Therefore we first need to fix some terminology and notions about strict $n$-categories and about the relevant notion of generation.

\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 category|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})

We are going to be interested in a full [[subcategory]] $Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}$, given below in def. \ref{nCatGen}, which knows about the higher [[profunctors]]/[[correspondence|correspondences]] between $n$-categories.

\begin{remark}
\label{}\hypertarget{}{}
For $A,B$ two categories, a [[profunctor]] $A^{op} \times B \to Set$ is equivalently a category [[slice category|over]] the 1-globe functor, hence a functor

\begin{displaymath}
\itexarray{
    K
    \\
    \downarrow
    \\
    G_1 & = \Delta[1]
  }
\end{displaymath}
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}).

The definition of $Cat_{(\infty,n)}$ below, def. \ref{AxiomaticDefinition}, will take this property to be one of the characteristic properties. Therefore consider

\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 below in \emph{\hyperlink{PresentationByThetaSpaces}{Presentation by Theta-spaces and by n-fold Segal spaces}}.

\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{generation_by_strict_categories}{}\paragraph*{{Generation by strict $n$-categories}}\label{generation_by_strict_categories}

Def. \ref{AxiomaticDefinition} considers an $(\infty,1)$-category \emph{generated} from $Str n Cat_{gen}$ in the following sense

\begin{defn}
\label{StrongGeneration}\hypertarget{StrongGeneration}{}
For $\mathcal{D}$ an [[(∞,1)-category]] with all small [[(∞,1)-colimits]], say that an [[(∞,1)-functor]]

\begin{displaymath}
f : \mathcal{C} \to \mathcal{D}
\end{displaymath}
\emph{strongly generates} $\mathcal{D}$ if its $(\infty,1)$-[[Yoneda extension]] on the [[(∞,1)-category of (∞,1)-presheaves]]

\begin{displaymath}
f : \mathcal{C} \stackrel{y}{\hookrightarrow} PSh_\infty(\mathcal{C}) \stackrel{Lan_y}{\to} \mathcal{D}
\end{displaymath}
is the reflector of a [[reflective sub-(∞,1)-category]]

\begin{displaymath}
\mathcal{D} \stackrel{\overset{Lan_y}{\leftarrow}}{\hookrightarrow}
  PSh_\infty(\mathcal{C})
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
By definition, a strongly generated $(\infty,1)$-category is in particular a [[presentable (∞,1)-category]].

\end{remark}
\begin{defn}
\label{AxiomaticDefinition}\hypertarget{AxiomaticDefinition}{}
An \textbf{$(\infty,1)$-category of $(\infty,n)$-categories} $Cat_{(\infty,n)}$ is an [[(∞,1)-category]] equipped with a [[full and faithful functor]]

\begin{displaymath}
i :  Str n Cat_{gen} \hookrightarrow \tau_{\leq 0}Cat_{(\infty,n)}
\end{displaymath}
from the generating strict $n$-categories, def. \ref{nCatGen} into its category of [[n-truncated object in an (infinity,1)-category|0-truncated]] objects, such that

\begin{enumerate}%
\item $Str n Cat_{gen} \to \tau_{\leq 0} Cat_{(\infty,n)} \hookrightarrow Cat_{(\infty,n)}$ \hyperlink{StrongGeneration}{strongly generates} $\mathcal{C}$;


\item $i$ preserves the \hyperlink{FundamentalPushouts}{fundamental pushout} relations;


\item the [[base change]] [[adjoint triple]] in $Cat_{(\infty,n)}$ exists along morphisms with codomain a [[globe]];



\end{enumerate}
and such that $\mathcal{C}$ is [[universal property|universal]] with respect to these properties in that for any other $j : Str n Cat_{gen} \hookrightarrow \mathcal{C}$ satisfying these three conditions it factors through $i$

\begin{displaymath}
j : Str n Cat \stackrel{i}{\to} Cat_{(\infty,n)} \stackrel{L}{\to} \mathcal{C}
\end{displaymath}
by an [[(∞,1)-functor]] $L$ which is the reflector of a [[reflective sub-(∞,1)-category|reflective inclusion]] $\mathcal{C} \hookrightarrow Cat_{(\infty,n)}$.

\end{defn}
(\hyperlink{BarwickSchommerPries}{B-SP, def. 6.8})

\begin{remark}
\label{}\hypertarget{}{}
By the first axiom, the localization demanded in the universal property is essentially unique. In particular, therefore, $Cat_{(\infty,n)}$ is defined uniquely, up to [[equivalence of (∞,1)-categories]]. For more on this see prop. \ref{AutomorphismInfinityGroup} below.

\end{remark}
\begin{remark}
\label{GauntIs0Truncted}\hypertarget{GauntIs0Truncted}{}
The gaunt $n$-categories, def. \ref{GauntStrictNCategories} are indeed among the [[n-truncated object in an (∞,1)-category|0-truncated]] objects: since we are looking at just the [[(∞,1)-category]] of $(\infty,n)$-categories, instead of more generally the $(\infty,n+1)$-category the non-invertible [[transfors]] between $n$-categories are disregarded and so if an object $X \in Cat_{(\infty,n)}$ has no non-trivial invertible cells, then for every other objeyt $Y$, the hom-$\infty$-groupoid $Cat_{(\infty,n)}(Y,X)$ is 0-truncated, hence is a set.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The first axiom in particular says that $Cat_{(\infty,n)}$ is a [[presentable (∞,1)-category]], and hence so are all its [[over-(∞,1)-category|slices]]. In view of this the [[adjoint (∞,1)-functor theorem]] says that the third condition is equivalent to [[(∞,1)-pullbacks]]

\begin{displaymath}
f^* : Cat_{(\infty,n)}/_{i(G_k)} \to Cat_{(\infty,n)}/X
\end{displaymath}
along morphisms of the form $X \to i(G_k)$ preserving [[(∞,1)-colimits]].

\end{remark}
\hypertarget{universal_presentation}{}\paragraph*{{Universal presentation}}\label{universal_presentation}

By def. \ref{AxiomaticDefinition} $Cat_{(\infty,n)}$ is [[equivalence of (∞,1)-categories|equivalent]] to a [[localization of an (∞,1)-category|localization]] of the [[(∞,1)-category of (∞,1)-presheaves]] on $Str n Cat_{gen}$. In fact, various subcategories of $Str n Cat_{gen}$ are already sufficient, notable the [[Theta-category]] $\Theta_n \hookrightarrow Str n Cat$ (discussed below in \ref{PresentationByThetaSpaces}). Here we discuss these [[presentable (∞,1)-category|presentations]].

\begin{defn}
\label{UniversalLocalizingClass}\hypertarget{UniversalLocalizingClass}{}
Let $S_{0} \subset Mor(PSh_\infty(Str n Cat_{gen}))$ be the class of morphism generated under fiber product $X \times_{G_k} (-)$ with objects $X \in Str n Cat_{gen}$ over globes by

\begin{enumerate}%
\item the morphisms that witness the \hyperlink{FundamentalPushouts}{fundamental pushout relations}


\item the initial morphism $\emptyset \to i(\emptyset)$ into presheaf represented by the empty category (which coincides with the initial presheaf on all objects except on the empty category, where it is the singleton).



\end{enumerate}
Write $S$ for the strongly saturated class of morphisms (see [[reflective sub-(∞,1)-category]]) generated by $S_0$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The [[localization of an (∞,1)-category|localization]] of the [[(∞,1)-category of (∞,1)-presheaves]] over $Str n Cat_{gen}$, def. \ref{nCatGen} at the class of morphism $S$ from def. \ref{UniversalLocalizingClass} is a [[presentable (∞,1)-category|presentation]] of $Cat_{(\infty,n)}$, def. \ref{AxiomaticDefinition}:

\begin{displaymath}
Cat_{(\infty,n)} \simeq PSh_\infty(Str n Cat_{gen})[S^{-1}]
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, theorem 7.6}).

\begin{proof}
The three axioms of def. \ref{AxiomaticDefinition} are satisfied effectively by construction of $S$ (\ldots{}). Conversely, every localization satisfying the second and third axiom must invert the morphisms in $S$, hence must be a sub-localization.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This construction shows that the \hyperlink{FundamentalPushouts}{fundamental pushout relations} encode the \emph{composition} of [[k-morphisms]] in an $(\infty,n)$-category.

Let $X \in PSh_\infty(Str n Cat)$ be some object.

Firts, by the [[(∞,1)-Yoneda lemma]] the value of this [[(∞,1)-presheaf]] on a strict $n$-category $C$ is the $\infty$-groupoid of $(\infty,n)$-functors $C \to X$, [[natural equivalences]] between them, and so on.

And if $X$ is an $S$-[[local object]] then it has in particular the property that all the morphisms

\begin{displaymath}
Cat_{(\infty,n)}(
    i(G_k) \coprod_{i(G_j)} i(G_k)
    \to 
    i(G_jk \coprod_{ G_j } G_k)
    , 
    X
  )
\end{displaymath}
are equivalences of $\infty$-groupoids. So by the [[(∞,1)-Yoneda lemma]] this is equivalent to

\begin{displaymath}
X(G_k) \times_{X(G_i)} X(G_k)
  \to
  Cat_{(\infty,n)}(i(G_jk \coprod_{ G_j } G_k), X)
\end{displaymath}
being an equivalence. On the left this is the collection of all those pairs of $k$-globes in $X$ that touch at an $i$-boundary. On the right this is the collection of all [[k-morphisms]] in $X$ equipped with a choice of decomposing them into two $k$-morphisms touching at an $i$-boundary. So the statement that this morphism is an equivalence says that \emph{composition} of $k$-morphisms along $i$-boundaries exists in $X$.

\end{remark}
Various other presentations of $Cat_{(\infty,n)}$ are obtained by localizations over subcategories of\newline $i : Str n Cat_{restr} \hookrightarrow Str n Cat_{gen}$ at a set of morphisms $T \subset Mor(PSh_\infty(R))$. Write

\begin{displaymath}
PSh_\infty(Str n Cat_{restr})
   \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}}
  PSh_\infty(Str n Cat_{gen})
\end{displaymath}
for the induced [[essential geometric morphism]].

\begin{prop}
\label{SufficientConditionsForPresentation}\hypertarget{SufficientConditionsForPresentation}{}
The following conditions are sufficient in order that

\begin{displaymath}
Cat_{(\infty,n)} \simeq 
  PSh_\infty(Str n Cat_{gen})[S^{-1}]
  \stackrel{i^*}{\to} 
  PSh_\infty(Str n Cat_{restr})[T^{-1}]
\end{displaymath}
is an [[equivalence of (∞,1)-categories]]:

\begin{enumerate}%
\item $i^*(S_0) \subset T$


\item $i_!(T_0) \subset S$


\item [[generalized the|the]] [[unit of an adjunction|counit]] $id \to i^* i_!$ has components in $T$;


\item the $k$-[[globe]] $G_k$ is in the essential image of $i$, for each $0 \leq k \leq n$.



\end{enumerate}
\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, theorem 9.2})

\hypertarget{PresentationByThetaSpaces}{}\paragraph*{{Presentation by $\Theta_n$-spaces and $n$-fold complete Segal spaces}}\label{PresentationByThetaSpaces}

We discuss now presentations of $Cat_{(\infty,n)}$ over subcategories of $Str n Cat_{gen}$, according to prop. \ref{SufficientConditionsForPresentation}.

\begin{prop}
\label{}\hypertarget{}{}
The $n$th [[Theta category]] is a [[full subcategory]]

\begin{displaymath}
\Theta_n \hookrightarrow Str n Cat_{gen}
\end{displaymath}
and the localization of $PSh_\infty(\Theta_n)$ that defines the $(\infty,1)$-category $\Theta_n Space$ of $(\infty,n)$-[[Theta-spaces]] satisfies the conditions of prop. \ref{SufficientConditionsForPresentation}.

Hence $(\infty,n)$-[[Theta-spaces]] are a model for $(\infty,n)$-categories, in the sense of def. \ref{AxiomaticDefinition}:

\begin{displaymath}
\Theta_n Space \simeq Cat_{(\infty,n)}
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, theorem 11.15})

There is a further restriction from the objects of $\Theta_n$ to \emph{$n$-fold simplices} regarded as \emph{\href{Theta+category#EmbeddingOfGrids}{grid object}}, under the canonical embedding

\begin{displaymath}
\delta_n : \Delta^{\times n} \to \Delta^{\wr n} \simeq \Theta_n
\end{displaymath}
induced by the identification of the $n$th[[Theta-category]] (see there) with the $n$-fold [[categorical wreath product]] of the [[simplex category]] with itself.

\begin{prop}
\label{CompleteSegalOvernFoldSimplicialSetsIsPresentation}\hypertarget{CompleteSegalOvernFoldSimplicialSetsIsPresentation}{}
The inclusion

\begin{displaymath}
\Delta^{\times n} \stackrel{\delta_n}{\to} 
  \Theta_n \hookrightarrow Str n Cat_{gen}
\end{displaymath}
and the localization of $PSh_\infty(\Delta^{\times n})$ that defines the $(\infty,1)$-category $CSS(\Delta^{\times n})$ of \emph{[[n-fold complete Segal spaces]]} satisfies the conditions of prop. \ref{SufficientConditionsForPresentation}.

Hence [[n-fold complete Segal spaces]] are a model for $(\infty,n)$-categories, in the sense of def. \ref{AxiomaticDefinition}:

\begin{displaymath}
CSS(\Delta^{\times n}) \simeq Cat_{(\infty,n)}
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, theorem 12.6})

\begin{remark}
\label{}\hypertarget{}{}
Below in \hyperlink{PresentationByCompleteSegal}{Via ∞-Internalization -- Presentation by complete Segal spaces} is discussed that $n$-fold complete Segal spaces also naturally model an alternative definition of $(\infty,n)$-categories by \hyperlink{ViaInternalization}{iterated ∞-internalization}. Then prop. \ref{CompleteSegalOvernFoldSimplicialSetsIsPresentation} serves to show that this is equivalent to def. \ref{AxiomaticDefinition} above.

\end{remark}
\hypertarget{ViaEnrichment}{}\subsubsection*{{Via $\infty$-enrichment}}\label{ViaEnrichment}

\hypertarget{general}{}\paragraph*{{General}}\label{general}

There should be a general notion of \emph{[[enriched (∞,1)-category]]} (see there) over a [[monoidal (∞,1)-category]] $\mathcal{V}$. Write $\mathcal{V}Cat$ for the [[(∞,1)-category]] of $\mathcal{V}$-enriched $(\infty,1)$-categories.

\begin{defn}
\label{EnrichementDefinition}\hypertarget{EnrichementDefinition}{}
For $n \in \mathbb{N}$ write

\begin{displaymath}
Cat_{(\infty,n)} \coloneqq
  (((\infty Grpd Cat) Cat) \cdots) Cat
  \,.
\end{displaymath}
\end{defn}
\hypertarget{presentation_by_segal_categories}{}\paragraph*{{Presentation by Segal $n$-categories}}\label{presentation_by_segal_categories}

The notion of \emph{[[Segal n-categories]]} is a realization of the idea of \emph{weak enrichment} in a suitable [[model category]]. For nice enough model categories this can be further strictfied to just the notion of [[enriched model category]], discussed \emph{\hyperlink{ByEnrichedModelCategories}{below}}

(\ldots{})

\hypertarget{ByEnrichedModelCategories}{}\paragraph*{{Presentation by enriched model categories}}\label{ByEnrichedModelCategories}

(\ldots{})

\hypertarget{ViaInternalization}{}\subsubsection*{{Via $\infty$-internalization}}\label{ViaInternalization}

\hypertarget{general_2}{}\paragraph*{{General}}\label{general_2}

There is a general notion of \emph{[[internal category in an (∞,1)-category]]} $\mathcal{C}$ provided that

\begin{enumerate}%
\item $\mathcal{C}$ has [[finite limit|finite]] [[(∞,1)-limits]] -- in order to formulate the [[Segal condition]];


\item $\mathcal{C}$ is equipped with a ``choice of internal [[∞-groupoids]]'' -- in order to formulate the \href{complete%20Segal%20space#CompleteSegalSpaces}{completeness condition}.



\end{enumerate}
We can use this to define $Cat_{(\infty,n)}$ by iterative internalization.

\begin{defn}
\label{}\hypertarget{}{}
Write $Grpd(Cat_{(\infty,0)})$ for the catgeory of [[groupoid objects in an (∞,1)-category]] in $Cat_{(\infty,0)} \simeq$ [[∞Grpd]].

Assume we have already defined $Cat_{(\infty,n)}$, either by one of the methods above, or by the induction in the following. Then the canonical inclusion

\begin{displaymath}
Grpd(Cat_{(\infty,0)}) \hookrightarrow PreCat_{Grpd(Cat_{(\infty,0)})} (Cat_{(\infty,n)})
\end{displaymath}
into the $(\infty,1)$-category of [[simplicial objects]] $X_\bullet$ in $Cat_{(\infty,n)}$ that

\begin{enumerate}%
\item satisfy the [[Segal conditions]]


\item such that $X_0 \in Cat_{(\infty,0)}$



\end{enumerate}
has a [[right adjoint|right]] [[adjoint (∞,1)-functor]] $Core$.

\end{defn}
(\hyperlink{Lurie}{Lurie, prop. 1.1.14}).

\begin{defn}
\label{IteratedInternalization}\hypertarget{IteratedInternalization}{}
An $(\infty,n+1)$-category is an object $X \in PreCat_{Grpd(Cat_{(\infty,0)})}$ such that $Core(X) \in \infty Grpd \hookrightarrow Grpd(Cat_{(\infty,0)})$.

For $n \in \mathcal{N}$ the $(\infty,1)$-category of $(\infty,n)$-categories is

\begin{displaymath}
Cat_{(\infty,n)}
  \coloneqq  
  Cat^n(\infty Grpd)
  \coloneqq
  Cat(\cdots Cat(Cat_{(\infty,0)}) \cdots)
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Lurie}{Lurie, prop. 1.1.14}).

\begin{prop}
\label{}\hypertarget{}{}
The $(\infty,1)$-category $Cat_{(\infty,n)}$ given by def. \ref{IteratedInternalization} is [[equivalence of (∞,1)-categories|equivalent]] to that given by def. \ref{AxiomaticDefinition}.

\end{prop}
This is prop. \ref{CompleteSegalOvernFoldSimplicialSetsIsPresentation} in view of the presentation discussed \hyperlink{PresentationByCompleteSegal}{below}.

\hypertarget{PresentationByCompleteSegal}{}\paragraph*{{Presentation by $n$-fold complete Segal spaces}}\label{PresentationByCompleteSegal}

By the discussion \href{category+object+in+an+%28infinity,1%29-category#ModelCategoryPresentations}{here} at \emph{[[category object in an (∞,1)-category]]} we have

\begin{prop}
\label{}\hypertarget{}{}
Write $cSegal_0 \coloneqq sSet_{Quillen}$ for the standard [[model structure on simplicial sets]]. Then recursively for $n \in \mathbb{N}$, $n \geq 1$, there is a model structure on

\begin{displaymath}
cSegal_n \coloneqq [\Delta^{op}, cSegal_{n-1}]
\end{displaymath}
which [[presentable (infinity,1)-category|presents]] $Cat^n(\infty Grpd)$.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
$cSegal_n$ is equivalent to the $CSS(\Delta^{\times n})$ from prop. \ref{CompleteSegalOvernFoldSimplicialSetsIsPresentation}.

\end{prop}
(\ldots{})

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

\hypertarget{Generators}{}\subsubsection*{{Generators}}\label{Generators}

\begin{prop}
\label{}\hypertarget{}{}
The [[(∞,1)-category]] $Cat_{(\infty,n)}$ is generated under [[(∞,1)-colimits]] from the $k$-[[globes]] $G_k$ for $k \leq n$: every object is the [[(∞,1)-colimit]] over a diagram of globes.

\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, cor. 8.4})

\begin{prop}
\label{}\hypertarget{}{}
[[equivalence in an (∞,1)-category|Equivalences]] in the [[(∞,1)-category]] $Cat_{(\infty,n)}$ are detected on [[globes]]: a morphism $f : X \to Y$ in $Cat_{(\infty,n)}$ is an equivalence precisely if for all globes $G_{k \leq n}$ the induced morphism on [[derived hom-space|(∞,1)-categorical hom-spaces]]

\begin{displaymath}
Cat_{(\infty,n)}(G_k, f) : Cat_{(\infty,n)}(Y, f) \to Cat_{(\infty,n)}(X, f)
\end{displaymath}
is an equivalence of [[∞-groupoids]].

\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, cor. 8.5})

\hypertarget{truncated_objects}{}\subsubsection*{{Truncated objects}}\label{truncated_objects}

\begin{prop}
\label{GauntIs0Truncated}\hypertarget{GauntIs0Truncated}{}
The [[truncated object in an (∞,1)-category|truncated objects]] in the [[(∞,1)-category]] $Cat_{(\infty,n)}$ are precisely the \hyperlink{GauntStrictNCategories}{gaunt} [[strict n-categories]]

\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, cor. 8.6})

\begin{remark}
\label{}\hypertarget{}{}
That 0-truncated objects in the $Cat_{(\infty,n)}$ regarded as an $(\infty,1)$-category are gaunt is effectively the definition of 0-truncation in the absence of non-invertibles [[transfors]]. That these gaunt $(\infty,n)$-categories are then necessarily \emph{[[strict n-category|strict]]} reflects the fact that all the weakening, namely all the [[associators]] and [[unitors]] as well as all there [[coherence|coherences]] need to be invertible [[k-morphisms]], and hence must be trivial if there are no non-trivial such.

\end{remark}
\hypertarget{moduli}{}\subsubsection*{{Moduli}}\label{moduli}

\begin{prop}
\label{AutomorphismInfinityGroup}\hypertarget{AutomorphismInfinityGroup}{}
Let $Models_{(\infty,n)} \hookrightarrow \hat Cat_{(\infty,1)}$ be the [[core]] (maximal [[∞-groupoid]] inside) the full [[sub-(∞,1)-category]] of [[(∞,1)Cat]] on those that satisfy the definition \ref{AxiomaticDefinition}.

This is [[equivalence of (∞,1)-categories|equivalent]] to

\begin{displaymath}
Models_{(\infty,n)} \simeq B (\mathbb{Z}_2)^n,
\end{displaymath}
the [[delooping]] [[groupoid]] of the [[group]] $(\mathbb{Z}_2)^n$, the $n$-fold [[product]] of the [[group of order 2]] with itself.

The nontrivial element $\sigma \in \mathbb{Z}_2$ in the $k$th slot acts by passing to the $k$-opposite $(\infty,n)$-category.

\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, theorem 8.13})

\begin{remark}
\label{}\hypertarget{}{}
This means that

\begin{enumerate}%
\item the $(\infty,1)$-category $Cat_{(\infty,n)}$ from def. \ref{AxiomaticDefinition} is uniquely defined, up to [[equivalence of (∞,1)-categories]];


\item the [[automorphism ∞-group]] of $Cat_{(\infty,n)}$ in $\hat Cat_{(\infty,1)}$ is $(\mathbb{Z}_2)^n$, hence the only auto-equivalences are given by forming the $n$ analogs of forming an [[opposite (∞,1)-category]].



\end{enumerate}
\end{remark}
\begin{proof}
The idea is this:

One first observes that $Str n Cat_{gaunt}$ from def. \ref{GauntStrictNCategories} has $(\mathbb{Z}_2)^{\times n}$ worth of automorphisms, given by reversing the directions of the [[k-morphisms]].

For this,

\begin{enumerate}%
\item observe that the identity is the only [[natural transformation]] [[endomorphism]] on $Id : Str n Cat_{gaunt} \to Str n Cat_{gaunt}$: this can be checked on [[globes]] for which one observes that if a functor $G_n \to G_n$ is the identity on $\partial G_n$, then it is so also on the unique $n$-cell. (\hyperlink{BarwickSchomerPries}{B-SP, lemma 4.1})


\item observe that every autoequivalence of $Str n Cat_{gaunt}$ restricts to one on the [[globe category]] $\mathbb{G}_n$ (\hyperlink{BarwickSchomerPries}{B-SP, lemma 4.4}).


\item observe that the only autoequivalences of $\mathbb{G}_n$ are those that reverse the direction of the $k$-morphisms for $1 \leq k \neq n$, which with the above implies the same for all of $Str n Cat_{gaunt}$ (\hyperlink{BarwickSchomerPries}{B-SP, lemma 4.5}).



\end{enumerate}
Now us that, by the \hyperlink{Generators}{above discussion}, $Str n Cat_{gaunt}$ generates all of $Cat_{(\infty,n)}$ under [[(∞,1)-colimits]].

\end{proof}
\hypertarget{WebOfQuillenEquivalences}{}\subsubsection*{{Web of Quillen equivalent model category presentations}}\label{WebOfQuillenEquivalences}

We list [[model category]] structures that [[presentable (infinity,1)-category|present]] $Cat_{(\infty,n)}$ and [[Quillen equivalences]] between them.

In the following $A$ is an [[model category]] presenting $Cat_{(\infty,n-1)}$ that is an ``absolute distributor'' in the sense discussed at \emph{[[category object in an (∞,1)-category]]}. (That includes most of the model structures in the table, so that one can recurse over these constructions.)

\newline | [[model structure for Segal categories|projective structure for]] $A$-[[Segal categories]] | $\stackrel{inclusion}{\leftarrow}$ | $A$-[[enriched categories]] | | \hyperlink{Lurie}{Lurie, theorem 2.2.16} | | [[model structure for Segal categories|injective structure for]] $A$-[[Segal categories]] | $\stackrel{UnPre}{\to}$ | [[model structure for complete Segal spaces|complete Segal space objects]] in $A$ | | \hyperlink{Lurie}{Lurie, prop 2.3.1} | | [[Theta-space|Theta-(n-1)-space]]-[[Segal categories]] | | [[Theta-space|Theta-(n-1)-space]]-[[enriched categories]] | | (\hyperlink{BergnerRezk}{Bergner-Rezk, prop. 7.2}) | |$\vdots$| |$\vdots$| | |

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

\hypertarget{special_cases}{}\subsubsection*{{Special cases}}\label{special_cases}

\begin{itemize}%
\item [[∞-groupoid]] = $(\infty,0)$-category
\item [[(∞,1)-category]]
\item [[(∞,2)-category]]
\item etc \ldots{}

\end{itemize}
In addition,

\begin{itemize}%
\item [[(n,r)-category|(m,n)-categories]] can be obtained as particular $(\infty,n)$-categories whose $k$-cells are trivial for $k\gt m$.
\item In particular, [[n-categories]] = $(n,n)$-categories can be so obtained.

\end{itemize}
\hypertarget{specific_examples}{}\subsubsection*{{Specific examples}}\label{specific_examples}

\begin{itemize}%
\item One motivating example for $(\infty,n)$-categories is the [[(∞,n)-category of cobordisms]] which plays a central role in the formalization of the [[cobordism hypothesis]].


\item Another class of examples are [[(∞,n)-categories of spans]].



\end{itemize}
\hypertarget{extra_structure_and_properties}{}\subsection*{{Extra structure and properties}}\label{extra_structure_and_properties}

We discuss extra [[structure]] that an [[(∞,n)-category]] can carry and extra [[properties]] that it may enjoy.

\hypertarget{monoidal_categories}{}\subsubsection*{{$\mathcal{O}$-Monoidal $(\infty,n)$-categories}}\label{monoidal_categories}

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


\item [[symmetric monoidal (∞,n)-category]]



\end{itemize}
\hypertarget{categories_with_all_adjoints}{}\subsubsection*{{$(\infty,n)$-Categories with all adjoints}}\label{categories_with_all_adjoints}

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

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

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


\item [[category]]


\item [[2-category]]


\item [[3-category]]


\item [[n-category]]


\item [[(∞,0)-category]]


\item [[(∞,1)-category]]

\begin{itemize}%
\item [[table - models for (∞,1)-categories]]

\end{itemize}

\item [[(∞,2)-category]]


\item \textbf{(∞,n)-category}

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


\item [[n-category object in an (∞,1)-category]]


\item [[n-fold complete Segal space]]


\item [[Theta-space]], [[n-quasicategory]], [[model structure on cellular sets]]


\item [[symmetric monoidal (∞,n)-category]]



\end{itemize}

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


\item [[(∞,n)-sheaf]], [[(∞,n)-topos]]



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

Definition in terms of [[n-fold complete Segal spaces]] and [[Segal n-categories]] are due to the (unpublished) thesis

\begin{itemize}%
\item [[Clark Barwick]], \emph{$(\infty,n)$-$Cat$ as a closed model category} PhD (2005)

\end{itemize}
The definition in terms of [[Theta spaces]] is due to

\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}
An iterartive definition in terms of [[n-fold complete Segal spaces]] is given in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{$(\infty,2)$-Categories and the Goodwillie Calculus I} (\href{http://arxiv.org/abs/0905.0462}{arXiv:0905.0462})

\end{itemize}
A summary of definitions and some known comparison results can be found in

\begin{itemize}%
\item [[Julie Bergner]], \emph{Models for $(\infty,n)$-Categories and the Cobordism Hypothesis} , in [[Urs Schreiber]], [[Hisham Sati]] (eds.) \emph{[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]}, Proceedings of Symposia in Pure Mathematics, volume 83 AMS (2011) (\href{http://arxiv.org/abs/1011.0110}{arXiv:1011.0110})

\end{itemize}
A textbook account focusing on [[n-fold complete Segal spaces]] and related models is in

\begin{itemize}%
\item [[Simona Paoli]], \emph{Simplicial Methods for Higher Categories -- Segal-type Models of Weak $n$-Categories}, Springer 2019 (\href{https://doi.org/10.1007/978-3-030-05674-2}{doi:10.1007/978-3-030-05674-2}, \href{https://link.springer.com/content/pdf/bfm%3A978-3-030-05674-2%2F1.pdf}{toc pdf})

\end{itemize}
One axiomatic characterization is in

\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}{arXiv:1112.0040}, \href{http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/}{slides})

\end{itemize}
Comparison of models ($\Theta_{n+1}$-spaces and [[enriched (infinity,1)-categories]] in $\Theta_n$-spaces) is in

\begin{itemize}%
\item [[Julie Bergner]], [[Charles Rezk]], \emph{Comparison of models for $(\infty,n)$-categories} (\href{http://arxiv.org/abs/1204.2013}{arXiv:1204.2013})


\item [[Julie Bergner]], [[Charles Rezk]], \emph{Comparison of models for $(\infty,n)$-categories II} (\href{http://arxiv.org/abs/1406.4182}{arXiv:1406.4182})


\item [[Rune Haugseng]], \emph{On the equivalence between $\Theta_n$-spaces and iterated Segal spaces}, \href{https://arxiv.org/abs/1604.08480}{arXiv}


\item [[Julie Bergner]], \emph{A survey of models for $(\infty,n)$-categories} (\href{https://arxiv.org/abs/1810.10052}{arXiv:1810.10052})



\end{itemize}
A model for $(\infty,n)$-categories in terms of [[(∞,1)-sheaves]] on variant of a [[site]] of $n$-[[dimension|dimensional]] [[manifolds]] with [[embeddings]] between them is discussed in

\begin{itemize}%
\item [[David Ayala]], [[Nick Rozenblyum]], \emph{Weak $n$-categories are sheaves on iterated submersions of $\leq n$-manifolds} (in preparation)

\end{itemize}
previewed in

\begin{itemize}%
\item [[David Ayala]], \emph{Higher categories are sheaves on manifolds}, talk at \emph{\href{http://www.nd.edu/~cmnd/conferences/topology/}{FRG Conference on Topology and Field Theories}}, U. Notre Dame (2012) (\href{http://www.youtube.com/watch?v=8nm2ByS5NnY}{video})

\textbf{Abstract} [[topological chiral homology|Chiral]]/[[factorization homology]] gives a procedure for constructing a [[topological field theory]] from the data of an [[En-algebra]]. I'll explain a mulit-object version of this construction which produces a topological field theory from the data of an $n$-category with adjoints. This construction is a consequence of a more primitive result which asserts an equivalence between [[(infinity,n)-category|n-categories]] with adjoints and ``transversality sheaves'' on [[framed manifold|framed]] $n$-[[manifolds]] - of which there is an abundance of examples.



\end{itemize}
This lends itself to a model of \emph{[[(∞,n)-category with adjoints]]}. See there for more.

The first globular and algebraic models of $(\infty,n)$-categories is in

\begin{itemize}%
\item [[Camell Kachour]], Algebraic Definition of weak $(\infty,n)$-Categories, Published in Theory and Applications of Categories (2015), Volume 30, No. 22, pages 775-807: \href{http://tac.mta.ca/tac/volumes/30/22/30-22abs.html}{journal web site}

\end{itemize}
A model structure using [[complicial sets]] is in

\begin{itemize}%
\item [[Viktoriya Ozornova]], Martina Rovelli, Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces, \href{https://arxiv.org/abs/1809.10621}{arxiv}

\end{itemize}
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[[!redirects (∞,r)-category]] [[!redirects (∞,r)-categories]] [[!redirects (∞,k)-category]] [[!redirects (∞,k)-categories]]



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