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I need to prove the existence of a model structure but I am still unable to formulate a definition of the class of weak equivalences. I have the following informations:

  1. The underlying category is locally finitely presentable
  2. I have the set of generating cofibrations and every object is cofibrant
  3. I have a conjectural set of generating trivial cofibrations
  4. I have a cylinder functor but, unfortunately, it is not a left adjoint (so I cannot use Olschok's theorems)
  5. This model structure is left determined and probably not simplicial
  6. I can prove that there exist objects which are not fibrant (so Isaev's theory cannot be used either).

Does it exist a method for this specific situation ? I tried to relate this category to other categories by adjunctions but the acyclicity condition is never satisfied. Of course, that would be much simpler if I could characterize conjecturally the class of weak equivalences.

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    $\begingroup$ I'm going to say something trivial that you know (it's just to determine what options are available), but you have all acyclic cofibrations, and you can determine acyclic fibrations from the generating cofibrations, right? Then weak equivalences should just be the morphisms that can be written as compositions of the two. So standard theorems about cofibrantly generated model categories could apply. But this may not be a very useful description of weak equivalences of course. $\endgroup$
    – Najib Idrissi
    Commented Aug 29, 2020 at 8:06
  • $\begingroup$ @NajibIdrissi I agree with you but the problem is precisely that I cannot figure out from that what is the definition of a weak equivalence. That is the problem. I will find out eventually, I asked the question just in case someone would know a generic method. $\endgroup$
    – Philippe Gaucher
    Commented Aug 29, 2020 at 8:29
  • $\begingroup$ Do you know what the fibrant objects ought to be? $\endgroup$
    – Alexander Campbell
    Commented Aug 29, 2020 at 9:16
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    $\begingroup$ In some model structures (e.g. Joyal) a full description of the weak equivalences is tricky. It might be valuable to concentrate on weak equivalences between fibrant objects -- from what you say, these should be the homotopy equivalences with respect to your cylinder. Perhaps this can be reformulated into something more easily checkable. And for instance, if you choose a fibrant replacement functor $F$, then you can describe the weak equivalences as those maps sent to w.e.'s (between fibrant objects) by $F$. Otherwise I'm not sure what to say with this level of information. $\endgroup$
    – Tim Campion
    Commented Aug 29, 2020 at 17:05
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    $\begingroup$ May I suggest to have a look to my last paper arxiv.org/abs/2005.02360 ? It gives a lots of criterion to get left,right and "weak" model categories from assumption of this kind. there is an emphasize on Cisinski-Olschok type construction, but many results go beyond this. Note that because you are in the case where every object is cofibrant, a left semi-model structure would be the same as a Quillen model structure. $\endgroup$
    – Simon Henry
    Commented Aug 30, 2020 at 2:26

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