While is “completion” a very general term for a number of notions in mathematics, many of them are in the sense of completions under some sort of limits. In that case, sometimes one uses the term cocompletion for a dual case of completing under colimits. For example, free cocompletion and ind-cocompletion.

  • R. Betti, Cocompleteness over coverings, J. Austral. Math. Soc. Ser. A, 39 (1985), pp. 169–177

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