A theory of 2-pro-objects (with expanded proofs)

Grothendieck develops the theory of pro-objects over a category $\mathsf{C}$. The fundamental property of the category $\mathsf{Pro}(\mathsf{C})$ is that there is an embedding $\mathsf{C} \overset{c}{\longrightarrow} \mathsf{Pro}(\mathsf{C})$, the category $\mathsf{Pro}(\mathsf{C})$ is closed under small cofiltered limits, and these limits are free in the sense that for any category $\mathsf{E}$ closed under small cofiltered limits, pre-composition with $c$ determines an equivalence of categories $\mathcal{C}at(\mathsf{Pro}(\mathsf{C}),\,\mathsf{E})_+ \simeq \mathcal{C}at(\mathsf{C},\, \mathsf{E})$, (where the "$+$" indicates the full subcategory of the functors preserving cofiltered limits). In this paper we develop a 2-dimensional theory of pro-objects. Given a 2-category $\mathcal{C}$, we define the 2-category $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ 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 universal property corresponding to the one described above. We have at hand the results of $\mathcal{C}at$-enriched category theory, but our theory goes beyond the $\mathcal{C}at$-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.