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

Abstract

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 $\mathsf{C}\stackrel{c}{⟶}\mathsf{Pro}(\mathsf{C})$, 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 $2-\mathcal{P}ro(\mathcal{C})$ whose objects we call 2-pro-objects. We prove that $2-\mathcal{P}ro(\mathcal{C})$ 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.

This is a version of the article "A theory of 2-Pro-objects, Cahiers de topologie et géométrie différentielle catégoriques, Vol LV, 2014", in which we have added more details in several proofs, and utilized the elevators calculus graphical notation

References in corpus (1)

• A construction of 2-filtered bicolimits of categories