2014/06/22 by M. Emilia Descotte, Eduardo J. Dubuc, Descotte, M. Emilia +1
Mathematics · #18D05 : 18A30 #Category Theory (math.CT) #FOS: Mathematics #math.CT #msc:18A30 #msc:18D05
paper · pdf · doi:10.48550/arxiv.1406.5762
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
arxiv created 2014/06/22 · arxiv updated 2014/06/24
Grothendieck develops the theory of pro-objects over a category C. The fundamental property of the category Pro(C) is that there is an embedding C \oversetc\longrightarrow Pro(C), the category Pro(C) is closed under small cofiltered limits, and these limits 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)+ ≃ Cat(C, 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 C, we define the 2-category 2\hbox-Pro(C) whose objects we call 2-pro-objects. We prove that 2\hbox-Pro(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 Cat-enriched category theory, but our theory goes beyond the Cat-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.