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This is a terminology question. That will help me to write down my papers. In Marc Olschok's PhD available here (I cannot find it anymore on the Internet so it is in my webpage, the original URL is welcome), Cisinski's constructions are generalized to any locally presentable category equipped with a cartesian cylinder with respect to a cofibrant small weak factorization system (Theorem 2.2.5 page 25). I need a name for this structure because I have a lot of model categories constructed like that in my work. Here are the ideas:

  1. A cartesian model category: the terminology of cartesian model category is already used here.
  2. A Cisinski-Olschok model category: it is very long but it is my favorite choice.
  3. A cylindric model category: since there are cylinders in any model category, this terminology does not seem to be a good idea.

And I call the set of cofibrations $S$ in the statement of Theorem 2.2.5 the set of generating anodyne cofibrations. Any suggestion before I make a decision ?

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