We show that every small model category that satisfies certain size conditions can be completed to yield a combinatorial model category, and conversely, every combinatorial model category arises in this way. We will also see that these constructions preserve right properness and compatibility with simplicial enrichment. Along the way, we establish some technical results on the index of accessibility of various constructions on accessible categories, which may be of independent interest.
Keywords: cofibrant generation, closed model category, weak factorization system, locally presentable category, ind-object, filtered colimit
2010 MSC: 18G55, 55U35 (Primary) 18D35, 55P60 (Secondary)
Theory and Applications of Categories, Vol. 31, 2016, No. 2, pp 31-62.