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In [BaSc2], the author and Tomer Schlank introduced a much weaker homotopicalstructure than a model category, which we called a "weak cofibration category".We further showed that a small weak cofibration category induces in a naturalway a model category structure on its indcategory, provided the ind-categorysatisfies a certain two out of three property. The main purpose of this paperis to give sufficient intrinsic conditions on a weak cofibration category forthis two out of three property to hold. We consider an application to thecategory of compact metrizable spaces, and thus obtain a model structure on itsind-category. This model structure is defined on a category that is closelyrelated to a category of topological spaces and has many convenient formalproperties. A more general application of these results, to the (opposite)category of separable $C^*$-algebras, appears in a paper by the author, MichaelJoachim and Snigdhayan Mahanta [BJM].

The two out of three property in ind-categories and a convenient model category of spaces