Let MR be a right module over the associative ring R (with 1). Assume one has an expression for M as a colimit (direct limit) of a system
over the (directed) poset D. A natural way to get M as a colimit of the family ﹛F∞ —” Fβ|∞, fβ ∈ E﹜ for some subset £ of D is to take E cofinal in D. However, if one is concerned about the cardinality of the set E, cofinal subsets may be too large. Let us look at a specific example. Lazard  has shown that any flat MR is a direct limit of finitely generated free R-modules. The cardinality of his indexing set depends on the cardinality of M.