Comparing coequalizer and exact completions
Maria Cristina Pedicchio, Jiřı́ Rosický · Theory and applications of categories · 1999
. We characterize when the coequalizer and the exact completion of a category C with finite sums and weak finite limits coincide. Introduction Our aim is to compare two well known completions: the coequalizer completion C coeq of a small category C with finite sums (see [P]) and the exact completion of a small category C with weak finite limits (see [CV]). For a category C with finite sums and weak finite limits, C ex is always a full subcategory of C coeq . We characterize when the two completions are equivalent - it turns out that this corresponds to a finiteness condition expressed in terms of reflexive and symmetric graphs in C. 1. Two completions For a small category C with finite sums, the coequalizer completion of C is a category C coeq with finite colimits together with a finite sums preserving functor G C : C ! C coeq such that, for any finite sums preserving functor F : C !X into a finitely cocomplete category, there is a unique finite colimits preserving functor F : C...