The ideal completion of a category generalises the notion of ideal completion of a partially ordered set. It is formed by taking the category of ideals? of . It is the completion of under filtered colimits of monomorphisms.
See page 24 of An Outline of Algebraic Set Theory.
“The actual definition requires either some care in specifying choices of monomorphisms, as is done in , or a sheaf-theoretic approach as in .”
 S. Awodey, C. Butz, A. Simpson and T. Streicher, Relating first-order set theories, toposes and categories of classes. In preparation, 2007. Preliminary version available here.
 S. Awodey and H. Forssell, Algebraic models of intuitionistic theories of sets and classes, Theory and Applications of Categories 15(1): 147-163, 2005.