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.
Categories: math.CT