2.3. An alternative definition of fibration [0029]

Warning. Some authors including Grothendieck (Revêtements Étales Et Groupe Fondamental (SGA 1), 1971) give an equivalent definition of cartesian fibration that factors through a nonequivalent definition of cartesian morphisms. Such authors refer to our notion of cartesian morphism as hypercartesian (Streicher, 2018).

[002A] Definition 2.3·a (Hypocartesian morphisms).

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 hypocartesian over $f$ when for any $\bar{u}\in E\Sub{x}$ and $\bar{h}:\bar{u}\to\Sub{f} \bar{y}$ there exists a unique $i : \bar{u}\to\Sub{\Idn{x}} \bar{x}$ with $i;\bar{f} = \bar{h}$ as follows:

Cartesian morphisms are clearly hypocartesian (setting $u=x$ and $m=\Idn{x}$), but the converse does not hold. The problem is that in an arbitrary displayed category, hypocartesian morphisms may not be closed under composition.

[002C] Lemma 2.3·b (Hypocartesian = cartesian in a cartesian fibration).

Let $E$ be a cartesian fibration in the sense of Definition 2·c [0002], and let $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$ be displayed over $f:x\to y$. The displayed morphism $\bar{f}$ is cartesian if and only if it is hypocartesian.

Proof. Any cartesian map is clearly hypocartesian. To see that a hypocartesian map $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ in a cartesian fibration is cartesian, we consider the cartesian lift of $f:x\to y$ under $\bar{y}$:

As the cartesian lift $\bar{x}\tick\to \bar{y}$ is also hypocartesian, it follows that there is a unique vertical isomorphism identifying $\bar{x}$ with $\bar{x}\tick$ factoring $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$ through $\bar{f}\tick : \bar{x}\tick\to\Sub{f}\bar{y}$. Being cartesian over $f$ is clearly stable under isomorphism, hence we conclude that $\bar{f}$ is cartesian from the fact that $\bar{f}\tick$ is cartesian.

Grothendieck (Revêtements Étales Et Groupe Fondamental (SGA 1), 1971) defines a fibration in terms of (what we refer to as) hypocartesian morphisms rather than (what we refer to as) cartesian morphisms, and therefore imposes the additional constraint that the hypocartesian morphisms be closed under composition. In Lemma 2.3·c [002B] below, we verify that these two definitions of cartesian fibration coincide.

[002B] Lemma 2.3·c (Equivalence with Grothendieck's fibrations).

Let $E$ be displayed over $B$. Then $E$ is a cartesian fibration in the sense of Definition 2·c [0002] if and only if the following two conditions hold:

  1. Hypocartesian lifts. For each $f:x\to y\in B$ and $\bar{y}\in E\Sub{y}$ there exists a displayed object $\bar{x}\in E\Sub{x}$ and hypocartesian morphism $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$.
  2. Closure under composition. If $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ and $\bar{g}:\bar{y}\to\Sub{g}\bar{z}$ are hypocartesian, then $\bar{f};\bar{g}$ is hypocartesian.

Proof. Suppose first that $E$ is a cartesian fibration in our sense. Then $E$ has hypocartesian lifts because it has cartesian lifts. For closure under composition, fix hypocartesian $\bar{f},\bar{g}$; by Lemma 2.3·b [002C] we know that $\bar{f},\bar{g}$ are also cartesian and hence by Lemma 2.2·a [001H] so is the composite $\bar{f};\bar{g}$; therefore it follows that $\bar{f};\bar{g}$ is also hypocartesian.

Conversely, suppose that $E$ is a cartesian fibration in the sense of Grothendieck, and let $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ be the hypocartesian lift of $f:x\to y$ at $\bar{y}\in E\Sub{y}$; we shall see that $\bar{f}$ is also a cartesian lift of $f$ at $\bar{y}$ by constructing a unique factorization as follows: Let $\bar{m}:\bar{u}\tick\to\Sub{m}\bar{x}$ be the hypocartesian lift of $m$ at $\bar{x}$, where $\bar{u}\tick\in E\Sub{u}$. By hypothesis, the composite $\bar{m};\bar{f} : \bar{u}\tick\to\Sub{m;f}\bar{y}$ is hypocartesian, so $\bar{h}$ factors uniquely through $\bar{m};\bar{f}$ over $\Idn{u}$: The composite $i;\bar{m} : \bar{u}\to\Sub{m}\bar{x}$ is the required (cartesian) factorization of $\bar{h}$ through $\bar{f}$ over $m$. To see that $i;\bar{m}$ is the unique such map, we observe that all morphisms $\bar{u}\to\Sub{m}\bar{x}$ factor uniquely through $\bar{m}$ over $\Idn{u}$ as a consequence of $\bar{m}$ being hypocartesian.

[002D] Remark 2.3·d (Two ways to generalize pullbacks).

Hypocartesian [002A] and cartesian [0001] morphisms can be thought of as two distinct ways to generalize the concept of a pullback, depending on what one considers the essential properties of pullbacks. Hypocartesian morphisms more directly generalize the “little picture” universal property of pullbacks as limiting cones, whereas cartesian morphisms generalize the “big picture” dynamics of the pullback pasting lemma. As we have seen in Lemma 2.3·b [002C] these two notions coincide in any cartesian fibration; the instance of this result for the canonical self-indexing (Construction 2.1·a [001X]) verifies that pullbacks can be equivalently presented in terms of cartesian morphisms, as we have pointed out in Exercise 2.1·b [001Y].