nLab n-fibration

Idea

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 $n Cat$.

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.

Definition

A morphism $\phi:b\to a$ in $E$ is cartesian (relative to $p$) if for any $c\in E$, the following square:

$$\array{E(c,b) & \overset{\phi\circ -}{\to} & E(c,a)\\ ^p \downarrow && \downarrow ^p\\ B(p c, p b) & \underset{p\phi\circ -}{\to} & B(p c, p a)}$$

(which commutes, up to equivalence, by functoriality of $p$) is a (weak) pullback of $(n-1)$-categories.

Definition

We say that $p:E\to B$ is an $n$-fibration (or just a fibration) if

1. 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 slice $n$-category $B/p a$,
2. 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
3. For any $a,b,c\in E$, the square
   $$\array{E(b,c)\times E(a,b) &\overset{\circ}{\to} & E(a,c)\\ ^p\downarrow && \downarrow^p\\ B(p b, p c)\times B(p a, p b) & \underset{\circ}{\to} & B(p a, p c)}$$

   is a morphism of $(n-1)$-fibrations.

Definition

If $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ are $n$-fibrations, a commutative square

$$\array{E_1 & \overset{h}{\to} & E_2\\ p_1 \downarrow && \downarrow p_2\\ B_1 & \underset{g}{\to} & B_2}$$

is a morphism of $n$-fibrations if

1. Whenever $\phi$ is cartesian for $p_1$, $h(\phi)$ is cartesian for $p_2$, and
2. For any $a,b\in E_1$, the square
   $$\array{E_1(b,a) & \overset{h}{\to} & E_2(h b, h a)\\ p_1 \downarrow && \downarrow p_2\\ B_1(p_1 b, p_1 a)& \underset{g}{\to} & B_2(g p_1 b, g p_1 a)}$$

   is a morphism of $(n-1)$-fibrations.

Remarks

• 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.

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.)

• 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 $n=2$.

Examples

• 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 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 (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”.

Related concepts

• Grothendieck fibration, Street fibration

• fibration in a 2-category

• A notion of fibration of (∞,1)-categories exists in terms of Cartesian fibrations of simplicial sets. (See also fibration of quasi-categories, left fibration, and right fibration.)

References

The notion of 2-fibrations was introduced in:

• Claudio Hermida, Some properties of $Fib$ as a fibred 2-category, Journal of Pure and Applied Algebra 134 (1999) 83-109 [doi:10.1016/S0022-4049(97)00129-1, ps.gz]

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

• Igor Baković, Fibrations of bicategories (2010) [pdf, pdf]

• Mitchell Buckley, Fibred 2-categories and bicategories, Journal of Pure and Applied Algebra 218 6 (2014) 1034-1074 [arXiv:1212.6283, doi:10.1016/j.jpaa.2013.11.002]

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:

• Marta Bunge, “Intrinsic $n$-stack completions over a topos,” slides here

$n$-pseudofunctors may be viewed (and defined) as anafunctors. For $n$-groupoids such an approach to $n$-pseudofunctors has been studied in

• D. Bourn, Pseudo functors and non abelian weak equivalences, in “Categorical algebra and its applications”, Springer LNM 1348 (1988), 55–70.