Related $n$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 $\omega$-groupoids, arXiv:0812.0298
Richard Garner, Ionads, J. Pure Appl. Algebra 216 (2012), no. 8-9, 1734–1747. (arXiv) (doi) (author-archived version of published copy)