An $n$-fibration is the version of a Grothendieck fibration appropriate for $n$-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 $n Cat$.
(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.
A morphism $\phi:b\to a$ in $E$ is cartesian (relative to $p$) if for any $c\in E$, the following square:
(which commutes, up to equivalence, by functoriality of $p$) is a (weak) pullback of $(n-1)$-categories.
We say that $p:E\to B$ is an $n$-fibration (or just a fibration) if
is a morphism of $(n-1)$-fibrations.
If $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ are $n$-fibrations, a commutative square
is a morphism of $n$-fibrations if
is a morphism of $(n-1)$-fibrations.
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 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.
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 equal. This condition violates the principle of equivalence as stated, but not if it is rephrased to apply to displayed categories instead.
When $n=2$, so that weak 2-categories are bicategories, this notion of fibration can be found in (Buckley). A strict version for strict 2-categories (though with one condition missing) was originally studied by (Hermida).
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.
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.)
Send $x\in B$ to the essential fiber $E_x$, whose objects are objects $a\in E$ equipped with a equivalence $p a \simeq x$.
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.
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 $a$.
and so on…
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:
The objects of $E$ over $x\in B$ are those of $F x \in n Cat$.
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 $F y$.
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 $b \xrightarrow{\phi} F_g(a) \xrightarrow{F_\alpha(a)} F_f(a)$.
and so on…
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 $n=2$.
A notion of fibration of (∞,1)-categories exists in terms of Cartesian fibrations of simplicial sets. (See also left fibration, and right fibration .)
Claudio Hermida introduced 2-fibrations in:
In fact they also appeared earlier, in some form, in 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
A definition for strict $n$-categories due to Hermida is unpublished, but it is used and presented in another joint work with Marta Bunge, presented at CATS07 at Calais:
$n$-pseudofunctors may be viewed (and defined) as anafunctors. For $n$-groupoids such an approach to $n$-pseudofunctors has been studied in