Minimal bicompletion of a category
Dusko Pavlovic – 5 December 2001
The Dedekind-MacNeille completion of a poset is a complete (and cocomplete) lattice, generated and cogenerated by the original poset. The inclusion moreover preserves any meets and joins that the poset may have had already.
In his 1966 book on completions (Springer LNM vol 24) Lambek asked for a categorical generalization of the Dedekind-MacNeille completion. In 1974, Isbell has demonstrated that no extension of the group Z_4, where all objects are both limits and colimts of the original group, can be complete.
I describe a completion, based on the Chu construction, which satisfies a slightly weaker generation requirement, but still restricts to the Dedekind-MacNeille completion on posets.
(Btw, I tend to call it the Dedekind-MacChu completion, which could be an informative title for the talk, but may also be considered too silly if there are lots of serious people in the siminar.)