nLab
ideal completion

The ideal completion of a category CC generalises the notion of ideal completion of a partially ordered set. It is formed by taking the category of ideals of CC. It is the completion of CC 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 [4], or a sheaf-theoretic approach as in [5].”

This is part of algebraic set theory.