A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C)

In the sixties, Grothendieck developed the theory of pro-objects over a category. The fundamental property of the category $Pro(C)$ is that there is an embedding $C \stackrel{c}{\rightarrow} Pro(C)$, $Pro(C)$ is closed under small cofiltered limits, and these are free in the sense that for any category $E$ closed under small cofiltered limits, pre-composition with $c$ determines an equivalence of categories $Cat(Pro(C),E)_+ \simeq Cat(C,E)$, (the $+$ indicates the full subcategory of the functors that preserve cofiltered limits). In this work we develop a 2-dimensional pro-object theory. Given a 2-category $\mathcal{C}$, we define the 2-category $2$-$\mathcal{P}ro(\mathcal{C})$ whose objects we call 2-pro-objects. We prove that $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ has all the expected basic properties adequately relativized to the 2-categorical setting, including the corresponding universal property. We give an adecuate definition of closed 2-model 2-category and demonstrations of its basic properties. We leave for a future work the construction of its homotopy 2-category. Finally, we prove that our 2-category $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ has a closed 2-model 2-category structure provided that $\mathcal{C}$ has one. Part of the motivation of this work was to develop a conceptual framework to handle the $\check{C}$ech nerve in homotopy theory, in particular in strong shape theory. The $\check{C}$ech nerve is indexed by the categories of covers and of hypercovers, with cover refinements as morphisms, which are not filtered categories, but determine 2-filtered 2-categories on which the $\check{C}$ech nerve is also defined, sends 2-cells into homotopies, and determines a 2-pro-object of simplicial sets. Usually, the $\check{C}$ech nerve has to be considered as a pro-object in the homotopy category, loosing the information encoded in the explicit homotopies.