Geodesic
A categorical concept of completion of objects
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 1, pp. 131-147.

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We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.
Classification : 18A32, 18A40, 18B30, 18G05, 54B30, 54E15
Keywords: firm reflection; (sub-)firm class; injective object; (co)-essential morphism 
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Brümmer, G. C. L.; Giuli, E. A categorical concept of completion of objects. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 1, pp. 131-147. http://geodesic.mathdoc.fr/item/CMUC_1992__33_1_a15/