3. The Grothendieck construction [0009]

3.1. The total category and its projection [000A]

Note that any displayed category $E$ over $B$ can be viewed as an undisplayed category $\TotCat{E}$ equipped with a projection functor $p\Sub{E}: \TotCat{E}\to B$; in this case $\TotCat{E}$ is called the total category of $E$.

  1. An object of $\TotCat{E}$ is given by a pair $(x,\bar{x})$ where $x\in B$ and $\bar{x}\in E\Sub{x}$.
  2. A morphism $(x,\bar{x})\to (y,\bar{y})$ in $\TotCat{E}$ is given by a pair $(f,\bar{f})$ where $f:x\to y$ and $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$.

The construction of the total category of a displayed category is called the Grothendieck construction.

[001T] Exercise 3.1·a.

Prove that the total category $\TotCat{\SelfIx{B}}$ of the canonical self-indexing (Section 2.1 [0003]) is the arrow category $B^{\to}$, and the projection is the codomain functor.

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.

3.3. Iteration and pushforward [000D]

It also makes sense to speak of categories displayed over other displayed categories; one way to formalize this notion is as follows. Let $E$ be displayed over $B$; we define a category displayed over $E$ to be simply a category displayed over the total category $\TotCat{E}$.

Now let $F$ be displayed over $E$ over $B$. Then we may regard $F$ as a displayed category $B\Sub{!}F$ over $B$ as follows:

  1. An object of $(B\Sub{!}F)\Sub{x}$ is a pair $(\bar{x},{\ddot{x}})$ with $\bar{x}\in E\Sub{x}$ and ${\ddot{x}}\in F\Sub{\bar{x}}$.
  2. A morphism $(\bar{x},{\ddot{x}})\to\Sub{f}(\bar{y},{\ddot{y}})$ is given by a pair $(\bar{f},{\ddot{f}})$ where $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ in $E$ and ${\ddot{f}}:{\ddot{x}}\to\Sub{\bar{f}} {\ddot{y}}$ in $F$.

By virtue of Section 3.2 [000B], we may define the pushforward of a displayed category along a functor. In particular, let $E$ be displayed over $B$ and let $U:B\to C$ be an ordinary functor; then we may obtain a displayed category $U\Sub{!}E$ over $C$ as follows:

  1. First we construct the displayed category $U\Sub{\bullet}$ corresponding to the functor $U:B \to C$.
  2. We recall that there is a canonical equivalence of categories $\TotCat{U\Sub{\bullet}}\to B$.
  3. Because $E$ is displayed over $B$, we may regard it as displayed over the equivalent total category $\TotCat{U\Sub{\bullet}}$ by change of base (Section 2.8 [0007]).
  4. Hence we may define the pushforward $U\Sub{!}E$ to be the displayed category $(U\Sub{\bullet})\Sub{!}E$ as defined above.