A theory of 2-Pro-objects

DESCOTTE, M. EMILIA; DUBUC, EDUARDO J.

CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE CATEGORIQUES

Andree Ehresman

Lugar: Amiens; Año: 2014 p. 3 - 35

0008-0004

In [1], 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: C -->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-Pro(C) whose objects we call 2-pro-objects. We prove that 2-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.