3.2. Displayed categories from functors [000B]

In many cases, one starts with a functor $P:E\to B$; if it were meaningful to speak of equality of objects in an arbitrary category then there would be an obvious construction of a displayed category $P\Sub{\bullet}$ from $P$; we would simply set $P\Sub{x}$ to be the collection of objects $u\in E$ such that $Pu=x$. As it stands there is a more subtle version that will coincide up to categorical equivalence with the naïve one in all cases that the latter is meaningful.

  1. We define an object of $P\Sub{x}$ to be a pair $(u,\phi\Sub{u})$ where $u\in E$ and $\phi\Sub{u} : Pu\cong x$. It is good to visualize such a pair as a “crooked leg” like so:

  2. A morphism $(u,\phi\Sub{u})\to\Sub{f} (v,\phi\Sub{v})$ over $f : x \to y$ is given by a morphism $h : u\to v$ that lies over $f$ modulo the isomorphisms $\phi\Sub{u},\phi\Sub{v}$ in sense depicted below:

[001U] Exercise 3.2·a.

Suppose that $B$ is an internal category in $\mathbf{Set}$, i.e. it has a set of objects. Exhibit an equivalence of displayed categories between $P\Sub{\bullet}$ as described above, and the naïve definition which $E\Sub{x}$ is the collection of objects $u\in E$ such that $Pu = x$.

We have a functor $\TotCat{P\Sub{\bullet}}\to E$ taking a pair $(x,(u,\phi\Sub{u}))$ to $u$.

[001V] Exercise 3.2·b.

Explicitly construct the functorial action of $\TotCat{P\Sub{\bullet}}\to E$.

[001W] Exercise 3.2·c.

Verify that $\TotCat{P\Sub{\bullet}}\to E$ is a categorical equivalence.

3.2.1. Relationship to Street's fibrations [000C]

In classical category theory, cartesian fibrations are defined by Grothendieck (Revêtements Étales Et Groupe Fondamental (SGA 1), 1971) to be certain functors $E\to B$ such that any morphism $f:x\to Pv$ in $B$ lies strictly underneath a cartesian morphism in $E$. As we have discussed, this condition cannot be formulated unless equality is meaningful for the collection of objects of $B$.

There is an alternative definition of cartesian fibration (Street, 1980) that avoids equality of objects; here we require for each $f:x\to Pv$ a cartesian morphism $h:\InvImg{f}v \to v$ together with an isomorphism $\phi : P(\InvImg{f}v)\cong x$ such that $\phi^{-1};Ph = f$.

By unrolling definitions, it is not difficult to see that the displayed category $P\Sub{\bullet}$ is a cartesian fibration in our sense if and only if the functor $P:E\to B$ was a fibration in Street’s sense. Moreover, it can be seen that the Grothendieck construction yields a Grothendieck fibration $\TotCat{P\Sub{\bullet}}\to B$; hence we have introduced by accident a convenient destription of the strictification of Street fibrations into equivalent Grothendieck fibrations.