Displayed categories from functors [frct-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 \(P_{\bullet }\) from \(P\); we would simply set \(P_{x}\) 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 \(P_{x}\) to be a pair \((u,\phi _{u})\) where \(u\in E\) and \(\phi _{u} : Pu\cong x\). It is good to visualize such a pair as a “crooked leg” like so:
    2. A morphism \((u,\phi _{u})\xrightarrow [f]{} (v,\phi _{v})\) over \(f : x \to y\) is given by a morphism \(h : u\to v\) that lies over \(f\) modulo the isomorphisms \(\phi _{u},\phi _{v}\) in sense depicted below:

    Exercise 1.  [frct-001U]

      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 \(P_{\bullet }\) as described above, and the naïve definition which \(E_{x}\) is the collection of objects \(u\in E\) such that \(Pu = x\).

      We have a functor \(\widetilde {P_{\bullet }}\to E\) taking a pair \((x,(u,\phi _{u}))\) to \(u\).

      Exercise 2.  [frct-001V]

        Explicitly construct the functorial action of \(\widetilde {P_{\bullet }}\to E\).

        Exercise 3.  [frct-001W]

          Verify that \(\widetilde {P_{\bullet }}\to E\) is a categorical equivalence.

          4. Relationship to Street's fibrations [frct-000C]

            In classical category theory, cartesian fibrations are defined by Grothendieck 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 due to Street that avoids equality of objects; here we require for each \(f:x\to Pv\) a cartesian morphism \(h:f^{*}v \to v\) together with an isomorphism \(\phi : P(f^{*}v)\cong x\) such that \(\phi ^{-1};Ph = f\).

            By unrolling definitions, it is not difficult to see that the displayed category \(P_{\bullet }\) 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 \(\widetilde {P_{\bullet }}\to B\); hence we have introduced by accident a convenient destription of the strictification of Street fibrations into equivalent Grothendieck fibrations.

            References

              Reference. Fibrations in bicategories [street-1980]

              Reference. Revêtements étales et groupe fondamental (SGA 1) [sga-1]

              Le texte présente les fondements d’une théorie du groupe fondamental en Géométrie Algébrique, dans le point de vue “kroneckerien” permettant de traiter sur le même pied le cas d’une variété algébrique au sens habituel, et celui d’un anneau des entiers d’un corps de nombres, par exemple.

              The text presents the foundations of a theory of the fundamental group in Algebraic Geometry from the Kronecker point of view, allowing one to treat on an equal footing the case of an algebraic variety in the usual sense, and that of the ring of integers in a number field, for instance.

              Context

                The Grothendieck construction [frct-0009]

                  The total category and its projection [frct-000A]

                  • 2022

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

                  1. An object of \(\widetilde {E}\) is given by a pair \((x,\bar {x})\) where \(x\in B\) and \(\bar {x}\in E_{x}\).
                  2. A morphism \({(x,\bar {x})}\xrightarrow {{}}{(y,\bar {y})}\) in \(\widetilde {E}\) is given by a pair \((f,\bar {f})\) where \({x}\xrightarrow {{f}}{y}\) and \({\bar {x}}\xrightarrow [f]{\bar {f}}{\bar {y}}:\bar {x}\).

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

                  Exercise.  [frct-001T]

                    Prove that the total category \(\widetilde {\underline {B}}\) of the fundamental self-indexing is the arrow category \(B^{\to }\), and the projection is the codomain functor.

                    Displayed categories from functors [frct-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 \(P_{\bullet }\) from \(P\); we would simply set \(P_{x}\) 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 \(P_{x}\) to be a pair \((u,\phi _{u})\) where \(u\in E\) and \(\phi _{u} : Pu\cong x\). It is good to visualize such a pair as a “crooked leg” like so:
                      2. A morphism \((u,\phi _{u})\xrightarrow [f]{} (v,\phi _{v})\) over \(f : x \to y\) is given by a morphism \(h : u\to v\) that lies over \(f\) modulo the isomorphisms \(\phi _{u},\phi _{v}\) in sense depicted below:

                      Exercise.  [frct-001U]

                        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 \(P_{\bullet }\) as described above, and the naïve definition which \(E_{x}\) is the collection of objects \(u\in E\) such that \(Pu = x\).

                        We have a functor \(\widetilde {P_{\bullet }}\to E\) taking a pair \((x,(u,\phi _{u}))\) to \(u\).

                        Exercise.  [frct-001V]

                          Explicitly construct the functorial action of \(\widetilde {P_{\bullet }}\to E\).

                          Exercise.  [frct-001W]

                            Verify that \(\widetilde {P_{\bullet }}\to E\) is a categorical equivalence.

                            Relationship to Street's fibrations [frct-000C]

                              In classical category theory, cartesian fibrations are defined by Grothendieck 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 due to Street that avoids equality of objects; here we require for each \(f:x\to Pv\) a cartesian morphism \(h:f^{*}v \to v\) together with an isomorphism \(\phi : P(f^{*}v)\cong x\) such that \(\phi ^{-1};Ph = f\).

                              By unrolling definitions, it is not difficult to see that the displayed category \(P_{\bullet }\) 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 \(\widetilde {P_{\bullet }}\to B\); hence we have introduced by accident a convenient destription of the strictification of Street fibrations into equivalent Grothendieck fibrations.

                              Iteration and pushforward [frct-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 \(\widetilde {E}\).

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

                                1. An object of \((B_{!}F)_{x}\) is a pair \((\bar {x},{\ddot {x}})\) with \(\bar {x}\in E_{x}\) and \({\ddot {x}}\in F_{\bar {x}}\).
                                2. A morphism \((\bar {x},{\ddot {x}})\xrightarrow [f]{}(\bar {y},{\ddot {y}})\) is given by a pair \((\bar {f},{\ddot {f}})\) where \(\bar {f}:\bar {x}\xrightarrow [f]{}\bar {y}\) in \(E\) and \({\ddot {f}}:{\ddot {x}}\xrightarrow [\bar {f}]{} {\ddot {y}}\) in \(F\).

                                Using the displayed category induced by a functor, 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 \(U_{!}E\) over \(C\) as follows:

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

                                Backlinks

                                  Iteration and pushforward [frct-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 \(\widetilde {E}\).

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

                                    1. An object of \((B_{!}F)_{x}\) is a pair \((\bar {x},{\ddot {x}})\) with \(\bar {x}\in E_{x}\) and \({\ddot {x}}\in F_{\bar {x}}\).
                                    2. A morphism \((\bar {x},{\ddot {x}})\xrightarrow [f]{}(\bar {y},{\ddot {y}})\) is given by a pair \((\bar {f},{\ddot {f}})\) where \(\bar {f}:\bar {x}\xrightarrow [f]{}\bar {y}\) in \(E\) and \({\ddot {f}}:{\ddot {x}}\xrightarrow [\bar {f}]{} {\ddot {y}}\) in \(F\).

                                    Using the displayed category induced by a functor, 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 \(U_{!}E\) over \(C\) as follows:

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