Combinatorial and accessible weak model categories

In a previous work, we have introduced weakening of Quillen model categories called weak categories. They still allow all the usual constructions category theory, but are easier to construct and in some sense better behaved. this paper continue develop their general theory by introducing combinatorial accessible We give simple necessary sufficient conditions under which such can be extended into left and/or right semi-model category. As an application, recover Cisinski-Olschok generalize it also provide existence theorems for both Bousfield localization structures, combined with results above gives categories, generalizing Barwick. Surprisingly, show that any or always exists, without properness assumptions, is simultaneously category, necessarily being itself.