Proposition [efr-TQ3W]

\int _X \mathcal {A}(X) \to \mathcal {C} is a Grothendieck fibration, and the pseudofunctor it induces is equivalent to \mathcal {A}

Proof

Given f: X \to Y and \bar {Y} \in \mathcal {A}(Y), it is clear that the map {\mathcal {A}(f)(\bar {Y}) \choose X} \to {\bar {Y} \choose Y} given by f, 1_{\mathcal {A}(f)(\bar {Y})} is locally Cartesian---the required bijection is the definition of maps in the Grothendieck construction. But it's straightforward to see that these compose.

Context

Fibrations [efr-3AEA]

In category theory, there are many families of categories indexed by the objects of some other category. For example, for each commutative ring, we have the category \mathsf {Mod}(R). Given a ring homomorphism \phi : R \to S, there is an induced restriction of scalars functor \phi ^*: \mathsf {Mod}(S) \to \mathsf {Mod}(R) (given simply by composing the module structure by \phi ), and this is (contravariant) functorial, assembling into a functor \mathsf {Mod}(-): \mathsf {CRing}^\mathrm {op} \to \mathsf {Cat}.

In most cases, one can not expect strict functoriality as above. From an abstract point of view, it makes sense that one should really ask only for a natural isomorphism \phi ^*\psi ^* \simeq (\psi \phi )^*, up to some coherence conditions. This assembles into a so-called pseudofunctor into the 2-category \mathsf {Cat}.

From a concrete point of view, there are many natural families of categories which arise as pseudofunctors. For example, restriction of scalars always has a left adjoint (extension of scalars, given by M \mapsto M \otimes _R S, viewing S as an R-module via the map \phi )---since adjoints compose (that is, if F \vdash G and F' \vdash G', then FF' \vdash G'G) this must be functorial up to natural isomorphism, but this is the best we can promise.

To avoid the higher categorical algebra involved in working with pseudofunctors, Grothendieck introduced the notion of fibration in Reference [grothendieck-descent-fibrations].

Definition [efr-Z7EF]

Let p: \mathcal {D} \to \mathcal {C} be a functor. Given X \in \mathcal {C}, write \mathcal {D}_X for the (strict) pullback \{x\} \times _\mathcal {C} \mathcal {D}. Explicitly, this consists of the objects in \mathcal {D} with p(A) = X and the morphisms with p(f) = 1_X.

Let f: X \to Y \in \mathcal {C} be a morphism.

  1. A map \bar {f}: \bar {X} \to \bar {Y} with p(\bar {f}) = f is locally Cartesian if for each \bar {X}' with p(\bar {X}') = X, postcomposition with \bar {f} induces a bijection \{g: \bar {X}' \to \bar {X} \mid p(g) = 1_X\} \xrightarrow {\sim } \{g' : \bar {X'} \to \bar {Y} \mid p(g') = f\}
  2. A map is Cartesian if for every g: Z \to X and \bar {Z} with p(\bar {Z}) = Z, there is a bijection \{\bar {g} : \bar {Z} \to \bar {X} \mid p(\bar {g}) = g\} \to \{\bar {g}' : \bar {Z} \to \bar {Y} \mid p(\bar {g'} = fg)\}, note that every Cartesian map is locally Cartesian (take g = 1_X)
  3. p is a Grothendieck fibration (or just fibration) if, for every \bar {Y} \in \mathcal {D} such that p(\bar {Y}) = Y, there exists a Cartesian map \bar {f}: \bar {X} \to \bar {Y} (for some \bar {Y}) so that p(\bar {f}) = f

Lemma [efr-Z4L4]

p: \mathcal {D} \to \mathcal {C} is a Grothendieck fibration if and only if every f admits a locally Cartesian lift, and the class of locally Cartesian morphisms in \mathcal {D} is stable under composition.

Theorem [efr-QHXA]

Let \mathcal {D} \to \mathcal {C} be a Grothendieck fibration. For every f: X \to Y, \bar {Y} \in \mathcal {D}_Y, select a Cartesian lift f^*\bar {Y} \to \bar {Y} of f. Then there is a unique extension of f^* to a functor \mathcal {D}_Y \to \mathcal {D}_X so that the squares

commute. With this, the assignment X \mapsto \mathcal {D}_X, f \mapsto f^* assembles into a pseudofunctor \mathcal {C}^\mathrm {op} \to \mathsf {Cat}.

Remark [efr-WKG8]

If \mathcal {D} \to \mathcal {C} is such that each morphism admits merely a locally Cartesian lift (but these do not compose,) it is called a prefibration. Note that this is unrelated to our notion of Markov prefibration (reading ahead a bit, the "Cartesian" maps in a Markov prefibration do compose, but they enjoy the unique lifting property only for a subset of morphisms). This clash of terminology is perhaps unfortunate, but other potential prefixes seemed inferior (quasi-, pseudo-, semi-).

Example [efr-D7W0]

Let \mathcal {C} be any category, and let \mathcal {C}^\to denote the arrow category. Then the codomain functor \mathcal {C}^\to \to \mathcal {C} is a fibration if and only if \mathcal {C} admits all pullbacks, and in this case the functors f^*: \mathcal {C}_Y \to \mathcal {C}_X, given f: X \to Y, are given by pullback along f.

The functors f^* are sometimes referred to as pullback, a convention we generally adopt. They are also sometimes called base-change functors.

When f: X \to Y and A \in \mathcal {D}_Y, we may write A_X for the object f^*A if there is no chance of confusion. (Compare that the choice of f is also suppressed in the notation A \times _Y X for a pullback)

We will not go into a comprehensive description of the theory of fibrations, but simply give a few basic results. We will give some examples in the next section. For a textbook treatment, see eg. Reference [jacobs-categorical-logic] (chapters 1, 9), or Reference [borceux-handbook] (chapter 8). Note that we will not give a formal definition of the term "pseudofunctor" here. See eg Reference [jacobs-categorical-logic], def. 1.4.4.

Definition [efr-KUW7]

Let \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat} be a pseudofunctor. Then there is a category \int _{X \in \mathcal {C}}\mathcal {A}(X) defined as follows:

  1. The objects are pairs {\bar {X}\in \mathcal {A}(X) \choose X \in \mathcal {C}}
  2. The morphisms {\bar {X} \choose X} \to {\bar {Y} \choose Y} are pairs f: X \to Y, f^\#: \bar {X} \to \mathcal {A}(f)(\bar {Y}) \in \mathcal {A}(X)
  3. Composition is given by the "chain rule" (f,f^\#) \circ (g,g^\#) = (fg, \mathcal {A}(g)(f^\#)g^\#)
There is an obvious forgetful functor \int _X \mathcal {A}(X) \to \mathcal {C}.

The category \int _X \mathcal {A}(X) is known as the Grothendieck construction of \mathcal {A}

Proposition [efr-TQ3W]

\int _X \mathcal {A}(X) \to \mathcal {C} is a Grothendieck fibration, and the pseudofunctor it induces is equivalent to \mathcal {A}

Proof

Given f: X \to Y and \bar {Y} \in \mathcal {A}(Y), it is clear that the map {\mathcal {A}(f)(\bar {Y}) \choose X} \to {\bar {Y} \choose Y} given by f, 1_{\mathcal {A}(f)(\bar {Y})} is locally Cartesian---the required bijection is the definition of maps in the Grothendieck construction. But it's straightforward to see that these compose.

Given a pseudofunctor \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat}, it is obvious that the assignment \mathcal {A}(-)^\mathrm {op} is pseudofunctorial as well (the required natural isomorphisms are just the formal opposites of the ones for \mathcal {A}). Applying this through the equivalence of fibrations and pseudofunctors leads to the fiberwise opposite of a fibration. Explicitly:

Corollary [efr-8LGE]

Let p: \mathcal {D} \to \mathcal {C} be a fibration. Then there exists a category \mathcal {D}^\mathrm {fop}, called the fiberwise opposite of \mathcal {D}, whose objects are the same as \mathcal {D}, and where a morphism X \to Y is a tuple (f: p(X) \to p(Y), f^\#: f^*Y \to X \in \mathcal {D}_{p(X)}).