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, thus producing an M-complete category.
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].”
[4] 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.
[5] S. Awodey and H. Forssell, Algebraic models of intuitionistic theories of sets and classes, Theory and Applications of Categories 15(1): 147-163, 2005.
This is part of algebraic set theory.
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