For a closure operator in a category, an object X is called categorically compact provided for each object Y, the morphism \pi₂:X× Y--> Y preserves closed subobjects. This generalization of the Kuratowski-Mrówka theorem has many consequences in topology and algebra. Historically, this idea has its roots in the paper by E. Manes [Gen. Top. Appl. 4 (1974)] and has been developed extensively by D. Dikranjan and E. Giuli, principally for various topological settings. T. H. Fay [Comm. Alg. 16 (1988)] showed, for module categories, an intimate connection between categorical compactness and relative injectivity. Dikranjan and Giuli have thoroughly investigated the module situation [Comm. Alg. 19 (1991)].

For the class of locally nilpotent groups, if one declares a subgroup "closed" provided it is normal and the quotient is torsion-free, then a torsion-free (torsion-free plays the role of Hausdorff) locally nilpotent group is categorically compact if and only if each element has all n^th roots. Each torsion-free locally nilpotent group can be "densely" embedded in a categorically compact group (the Mal'cev completion) which behaves much like the completion of a metric space. Far reaching generalizations of this for other classes of torsion-free groups have been obtained by T. H. Fay and G. L. Walls [Appl. Cat. Struct. 3 (1995)]. It is interesting that topological notions have a great deal of power and influence in very algebraic settings.