The author and Tomer Schlank studied a much weaker homotopical structure than a model category, which we called a "weak cofibration category". We showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The main purpose of this paper is to give sufficient intrinsic conditions on a weak cofibration category for this two out of three property to hold. We consider an application to the category of compact metrizable spaces, and thus obtain a model structure on its ind-category. This model structure is defined on a category that is closely related to a category of topological spaces and has many convenient formal properties. A more general application of these results, to the (opposite) category of separable $C^*$-algebras, appears in a paper by the author, Michael Joachim and Snigdhayan Mahanta.
Keywords: Ind-categories, model categories, cofibration categories, simplicially enriched categories, compact Hausdorff spaces
2010 MSC: 55U35, 55P05, 55U10, 54B30, 18G55, 18B30, 18C35, 18D20
Theory and Applications of Categories, Vol. 32, 2017, No. 17, pp 620-651.