Related Lab entries: ionad, small object argument, identity type weak factorization system?
R. Garner, Cofibrantly generated natural weak factorisation systems, arXiv:math.CT/0702290.
R. Garner, Understanding the small object argument, Applied Cat. Structures, arXiv:math.CT/0712.0724.
Benno van den Berg, Richard Garner, Types are weak -groupoids, arXiv:0812.0298
R. Garner, Ionads: a generalisation of the notion of topological space, (web, arXiv)