nLab ideal completion

Related pages

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, 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.

  • Steve Awodey, Henrik Forssell and Michael A. Warren, Algebraic models of sets and classes in categories of ideals PDF

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