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\begin{document}

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\section*{n-fibration}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{fibrations_versus_functors}{Fibrations versus functors}\dotfill \pageref*{fibrations_versus_functors} \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{idea}{}\subsection*{{Idea}}\label{idea}

An \textbf{$n$-fibration} is the version of a [[Grothendieck fibration]] appropriate for $n$-[[n-category|categories]].

The idea is that a functor $p:E\to B$ between $n$-categories is an $n$-fibration if the assignation $x\mapsto E_x = p^{-1}(x)$ of an object $x\in B$ to its fiber can be made into a (contravariant) functor from $B$ to the $(n+1)$-category $n Cat$.

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

(This definition is schematic, and needs to be adapted to be made precise for any particular definition of $n$-category.)

Let $p:E\to B$ be a functor between (weak) $n$-categories.

\begin{defn}
\label{}\hypertarget{}{}
A morphism $\phi:b\to a$ in $E$ is \textbf{cartesian} (relative to $p$) if for any $c\in E$, the following square:

\begin{displaymath}
\itexarray{E(c,b) & \overset{\phi\circ -}{\to} & E(c,a)\\
  ^p \downarrow && \downarrow ^p\\
  B(p c, p b) & \underset{p\phi\circ -}{\to} & B(p c, p a)}
\end{displaymath}
(which commutes, up to equivalence, by functoriality of $p$) is a (weak) [[pullback]] of $(n-1)$-categories.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
We say that $p:E\to B$ is an \textbf{$n$-fibration} (or just a \textbf{fibration}) if

\begin{enumerate}%
\item For any object $a\in E$ and morphism $f:x\to p a$ in $B$, there exists a cartesian $\phi:b\to a$ and an [[equivalence]] $p\phi \simeq f$ in the [[over category|slice]] $n$-category $B/p a$,
\item For any objects $a,b\in E$, the functor $p:E(b,a) \to B(p b, p a)$ is an $(n-1)$-fibration, and
\item For any $a,b,c\in E$, the square\begin{displaymath}
\itexarray{E(b,c)\times E(a,b) &\overset{\circ}{\to} & E(a,c)\\
^p\downarrow && \downarrow^p\\
B(p b, p c)\times B(p a, p b) & \underset{\circ}{\to} & B(p a, p c)}
\end{displaymath}
is a morphism of $(n-1)$-fibrations.



\end{enumerate}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
If $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ are $n$-fibrations, a commutative square

\begin{displaymath}
\itexarray{E_1 & \overset{h}{\to} & E_2\\
  p_1 \downarrow && \downarrow p_2\\
  B_1 & \underset{g}{\to} & B_2}
\end{displaymath}
is a \textbf{morphism of $n$-fibrations} if

\begin{enumerate}%
\item Whenever $\phi$ is cartesian for $p_1$, $h(\phi)$ is cartesian for $p_2$, and
\item For any $a,b\in E_1$, the square\begin{displaymath}
\itexarray{E_1(b,a) & \overset{h}{\to} & E_2(h b, h a)\\
p_1 \downarrow && \downarrow p_2\\
B_1(p_1 b, p_1 a)& \underset{g}{\to} & B_2(g p_1 b, g p_1 a)}
\end{displaymath}
is a morphism of $(n-1)$-fibrations.



\end{enumerate}
\end{defn}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item The definition is recursive in $n$, but if we unravel it, it makes perfect sense for $n=\omega$. That is, saying that $f$ is a fibration requires some things about cartesian 1-cells, and also that its action on hom-categories be a fibration--which in turn requires some things about cartesian 2-cells, and also that \emph{its} action on hom-categories be a fibration---which in turn which requires some things about cartesian 3-cells, and so on. After $\omega$ steps of unraveling, we are left with a list of conditions on cartesian $n$-cells for every $n$.

An equivalent, conciser way to say this is that we interpret the definition in the case $n=\omega$ as a [[coinductive definition]].


\item When $n=1$, this reduces to a [[Street fibration]], a weakened version of a [[Grothendieck fibration]]. We recover Grothendieck's original notion by requiring that for any $a\in E$ and $f:x\to p a$ in $B$, there exists a cartesian $\phi:b\to a$ such that $p\phi$ and $f$ are \emph{equal}. This condition violates the [[principle of equivalence]] as stated, but not if it is rephrased to apply to [[displayed categories]] instead.


\item When $n=2$, so that weak 2-categories are [[bicategories]], this notion of fibration can be found in \hyperlink{Buckley}{(Buckley)}. A strict version for [[strict 2-categories]] (though with one condition missing) was originally studied by \hyperlink{Hermida}{(Hermida)}.


\item In general, given any notion of (semi)strict $n$-category, we can expect to appropriately strictify the definition to make it correspond to stricter notions of pseudofunctor.



\end{itemize}
\hypertarget{fibrations_versus_functors}{}\subsection*{{Fibrations versus functors}}\label{fibrations_versus_functors}

If $p:E\to B$ is an $n$-fibration, we define a functor (or `$n$-pseudofunctor') from $B$ to $n Cat$ as follows. (Like the above definition, this is only a schematic sketch.)

\begin{itemize}%
\item Send $x\in B$ to the [[essential fiber]] $E_x$, whose objects are objects $a\in E$ equipped with a equivalence $p a \simeq x$.


\item For a morphism $f:y\to x$ in $B$, define $f^*:E_x\to E_y$ by choosing, for each $a\in E_x$, a cartesian $\phi:b\to a$ over $f$ and defining $f^*(a)=b$. The universal property of cartesian arrows makes $f^*$ a functor.


\item For a 2-cell $\alpha:f\to g:y\to x$ in $B$, define a transformation $\alpha^*:g^*\to f^*$ as follows. Given $a\in E_x$, we have a cartesian arrow $\phi:g^*a\to a$ over $g$. Now choose a cartesian 2-cell $\mu:\psi\to \phi$ over $\alpha$ in $E(g^*a,a)$. Since $p \psi = f$, $\psi$ factors essentially uniquely through the cartesian arrow $\chi:f^*a\to a$, giving a morphism $g^*a \to f^*a$; we define this to be the component of the transformation $\alpha^*$ at $a$.


\item and so on\ldots{}



\end{itemize}
Note that the functor we obtain is ``totally contravariant:'' it is contravariant on $k$-cells for all $1\le k\le n$.

Conversely, if we have a totally contravariant `$n$-pseudofunctor' from $B$ to $n Cat$, we define $p:E\to B$ by a generalization of the [[Grothendieck construction]] as follows:

\begin{itemize}%
\item The objects of $E$ over $x\in B$ are those of $F x \in n Cat$.


\item The morphisms of $E$ over $f:y\to x$ in $B$ from $b\in F y$ to $a\in F x$ are the morphisms from $b$ to $F_f(a)$ in $F y$.


\item The 2-cells of $E$ over $\alpha:f\to g:y\to x$ in $B$ from $\psi : b \to F_f(a)$ to $\phi : b \to F_g(a)$ are the 2-cells in $F y$ from $\psi$ to the composite $b \xrightarrow{\phi} F_g(a) \xrightarrow{F_\alpha(a)} F_f(a)$.


\item and so on\ldots{}



\end{itemize}
One expects that in this way, the $(n+1)$-category of fibered $n$-categories over $B$ is equivalent to the $(n+1)$-category of totally contravariant functors $B\to n Cat$. These constructions are known precisely only for $n=2$.

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

\begin{itemize}%
\item The prototypical example is that if $K$ is an $n$-category with finite limits (or at least pullbacks), then the $n$-category $Fib_K$ of \emph{internal} $(n-1)$-fibrations in $K$ should admit an $n$-fibration $cod : Fib_K \to K$. Of course this requires defining the notion of internal $(n-1)$-fibration in an $n$-category; this is usually done representably. For $n=2$ this gives the notion of [[fibration in a 2-category]], and the fact that $cod : Fib_K \to K$ is a 2-fibration is in \hyperlink{Hermida}{(Hermida)}. For $n=1$ it is just the standard fact that the [[codomain fibration]] is a fibration, i.e. every morphism in a 1-category is an ``internal 0-fibration''.

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

\begin{itemize}%
\item [[Grothendieck fibration]], [[Street fibration]]


\item [[fibration in a 2-category]]


\item A notion of fibration of [[(∞,1)-category|(∞,1)-categories]] exists in terms of [[Cartesian fibration]]s of [[simplicial set]]s. (See also [[left fibration]], and [[right fibration]] .)



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

\href{http://maggie.cs.queensu.ca/chermida}{Claudio Hermida} introduced 2-fibrations in:

\begin{itemize}%
\item Claudio Hermida, ``Some properties of Fib as a fibred 2-category,'' J. Pure Appl. Algebra 134 (1), 83--109, 1999; \href{http://maggie.cs.queensu.ca/chermida/papers/jpaa.ps.gz}{preprint version ps.gz}

\end{itemize}
In fact they also appeared earlier, in some form, in [[Gray-adjointness-for-2-categories|Gray's book]].

However, Hermida's definition was missing the stability of cartesian 2-cells under postcomposition, which is necessary for the ``Grothendieck construction'' turning a pseudofunctor into a fibration to have an inverse. This was rectified, and the definition generalized to bicategories, in

\begin{itemize}%
\item Igor Bakovic, ``Fibrations of bicategories'', \href{http://www.irb.hr/korisnici/ibakovic/groth2fib.pdf}{preprint}

\end{itemize}
\begin{itemize}%
\item Mitchell Buckley, ``Fibred 2-categories and bicategories'', \href{https://doi.org/10.1016/j.jpaa.2013.11.002}{doi}, \href{https://arxiv.org/abs/1212.6283}{arxiv}

\end{itemize}
A definition for strict $n$-categories due to Hermida is unpublished, but it is used and presented in another joint work with \href{http://www.math.mcgill.ca/bunge}{Marta Bunge}, presented at CATS07 at Calais:

\begin{itemize}%
\item Marta Bunge, ``Intrinsic $n$-stack completions over a topos,'' slides \href{https://www.researchgate.net/publication/230792798_Intrinsic_n-stack_completions_over_a_topos}{here}

\end{itemize}
$n$-pseudofunctors may be viewed (and defined) as [[anafunctor]]s. For $n$-groupoids such an approach to $n$-pseudofunctors has been studied in

\begin{itemize}%
\item \href{http://www-lmpa.univ-littoral.fr/~bourn}{D. Bourn}, Pseudo functors and non abelian weak equivalences, in ``Categorical algebra and its applications'', Springer LNM 1348 (1988), 55--70.

\end{itemize}
[[!redirects fibered n category]] [[!redirects fibered n-category]] [[!redirects strict fibered n-category]] [[!redirects fibered strict n-category]]

[[!redirects fibred n category]] [[!redirects fibred n-category]] [[!redirects strict fibred n-category]] [[!redirects fibred strict n-category]]

[[!redirects 2-fibration]] [[!redirects 2-fibrations]] [[!redirects fibered 2-category]] [[!redirects fibred 2-category]] [[!redirects fibered bicategory]] [[!redirects fibred bicategory]]

[[!redirects fibration of n-categories]] [[!redirects fibration of 2-categories]] [[!redirects fibration of bicategories]]



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