6. Other kinds of fibrations [0012]

6.1. Cocartesian fibrations [0015]

Cocartesian fibrations are a dual notion to cartesian fibrations, in which the variance of indexing is reversed.

[0016] Definition 6.1·a (Cocartesian morphism).

Let $E$ be displayed over $B$, and let $f:x\to y \in B$; a morphism $\bar{f}:\bar{x}\to\Sub{f} \bar{y}$ in $E$ is called cocartesian over $f$ when for any $m:y\to u$ and $\bar{h}:\bar{x}\to\Sub{f;m} \bar{u}$ there exists a unique $\bar{m} : \bar{y}\to\Sub{m} \bar{u}$ with $\bar{f};\bar{m} = \bar{h}$:

We use a “pushout corner” to indicate $\bar{x}\to\bar{y}$ as a cocartesian morphism, a notation justified by Section 6.1.3 [0019].

[0017] Definition 6.1·b (Cocartesian fibration).

A displayed category $E$ over $B$ is a cocartesian fibration when for each $f : x \to y\in B$ and $\bar{x}\in E\Sub{x}$, there exists a displayed object $\bar{y}\in E\Sub{y}$ and a cocartesian morphism $\bar{f} : \bar{x}\to\Sub{f} \bar{y}$.

Remark. These are also known as opfibrations.

6.1.1. The total opposite of a displayed category [0018]

[001I] Construction 6.1.1·a.

Let $E$ be displayed over $B$; we define its total opposite $\TotOpCat{E}$ displayed over $\OpCat{B}$ as follows:

  1. An object of $\TotOpCat{E}\Sub{x}$ is given by an object of $E\Sub{x}$.

  2. Given $f : x \to y\in \OpCat{B}$, a displayed morphism $\bar{x}\to\Sub{f} \bar{y}$ in $\TotOpCat{E}$ is given by a displayed morphism $\bar{y}\to\Sub{f} \bar{x}$ in $E$.

Warning. Do not confuse this construction with Construction 2.6·a [001Z], which produces a displayed category over $B$ and not $\OpCat{B}$.

[001J] Exercise 6.1.1·b.

Let $E$ be displayed over $B$. Prove that the total category (Section 3.1 [000A]) $\TotCat{\TotOpCat{E}}$ is $\OpCat{\prn{\TotCat{E}}}$, and its projection functor is $\OpCat{\prn{p\Sub{E}}} : \OpCat{\TotCat{E}}\to\OpCat{B}$.

[001K] Exercise 6.1.1·c.

Let $E$ be displayed over $B$, and let $f:x\to y\in B$. Prove that a morphism $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ is cartesian over $f$ in $E$ if and only if $\bar{f}:\bar{y}\to\Sub{f}\bar{x}$ is cocartesian over $f$ in $\TotOpCat{E}$.

[001L] Exercise 6.1.1·d.

Prove that a displayed category $E$ is a cartesian fibration over $B$ if and only if $\TotOpCat{E}$ is a cocartesian fibration over $\OpCat{B}$.

6.1.2. Example: Revisiting the canonical self-indexing [002X]

Recall that the canonical self-indexing $\SelfIx{B}$ (Construction 2.1·a [001X]) of a category $B$ is a displayed category with $\SelfIx{B}\Sub{x} = \Sl{B}{x}$. As discussed in Exercise 2.1·b [001Y], $\SelfIx{B}$ is a cartesian fibration over $B$ if and only if $B$ has pullbacks. However, $\SelfIx{B}$ is unconditionally a cocartesian fibration.

[002Y] Exercise 6.1.2·a.

Prove that $\SelfIx{B}$ from [001X] is a cocartesian fibration for any category $B$.

6.1.3. Example: The dual self-indexing [0019]

Dually to the canonical self-indexing (Construction 2.1·a [001X]), every category $B$ can also be displayed over itself via its coslices $\CoSl{x}{B}$.

[002Z] Construction 6.1.3·a.

Let $B$ be a category. Define the displayed category $\overline{B}$ over $B$ as follows:

  1. For $x\in B$, define $\overline{B}\Sub{x}$ as the collection of pairs $(\bar{x}\in B,p\Sub{x}:x\to\bar{x})$.
  2. For $f : x\to y\in B$, define $\overline{B}\Sub{f}$ to be the collection of commuting squares in the following configuration:
[001M] Exercise 6.1.3·b.

Prove that $\overline{B}$ is a cocartesian fibration if and only if $B$ has pushouts.

[001N] Exercise 6.1.3·c.

Prove that the total category (Section 3.1 [000A]) of $\overline{B}$ is the arrow category $B^{\to}$, and the projection is the domain functor.

[0030] Exercise 6.1.3·d.

Prove that $\overline{B}$ is a cartesian fibration for any category $B$.

6.2. Right fibrations [0013]

[001O] Definition 6.2·a.

A cartesian fibration $E$ over $B$ is said to be a right fibration when all displayed morphisms in $E$ are cartesian.

Recall from Section 2.5 [0005] that for every $b\in B$, the collection of displayed objects $E\Sub{b}$ and vertical maps $E\Sub{1\Sub{b}}$ forms a category. When $E$ is a right fibration over $B$, this category is in fact a groupoid.

[001P] Theorem 6.2·b (Characterization of right fibrations).

A cartesian fibration $E$ over $B$ is a right fibration if and only if all its vertical maps are isomorphisms.

Proof. Suppose that $E$ is a right fibration over $B$, and fix $b\in B$, $\bar{b}\in E\Sub{b}$, and a vertical map $f:\bar{b}\to\Sub{1\Sub{b}} \bar{b}$. Using the hypothesis that $f$ is cartesian, it has a unique section $g:\bar{b}\to\Sub{1\Sub{b}} \bar{b}$ as follows: Likewise, because $g$ is cartesian, $f$ is the unique section of $g$; thus $f$ is an isomorphism in $E\Sub{b}$.

Conversely, suppose that $E$ is a cartesian fibration whose vertical maps are isomorphisms. Fix $f:x\to y \in B$ and an arbitrary displayed morphism $\bar{g}:\bar{x}\to\Sub{f}\bar{y}$. Then $\bar{g}$ is the precomposition of a cartesian lift $\bar{f}:\bar{x}\tick\to\Sub{f}\bar{y}$ with a vertical map: Because vertical maps are isomorphisms and $\bar{f}$ is cartesian, we can observe that $\bar{g}$ is cartesian as follows, writing $\bar{m} : \bar{u}\to\Sub{m} \bar{x}\tick$ for the unique factorization of $\bar{h}$ through $\bar{f}$ over $m$: