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Consider a right Quillen adjoint which is not a categorical left adjoint which takes (trivial resp.) cofibrations between cofibrant objects to (trivial resp.) cofibrations between cofibrant objects.

Question: does this situation have a name ? Is it studied abstractly somewhere ?

I would like to have other examples like mine to help me to separate what is a general fact from what is particular to the objects I am studying.

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  • $\begingroup$ I don't know any name for this situation, and it seems pretty rare. Do you have any examples, other than when all morphisms are cofibrations? $\endgroup$ – David White May 27 '19 at 15:52
  • $\begingroup$ @DavidWhite Yes I have such examples. I am not surprised that it is rare. I wanted to be sure that indeed it is. $\endgroup$ – Philippe Gaucher May 27 '19 at 16:49
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    $\begingroup$ I don't know if this is relevant, but there is a relatively natural example of this situation: the forgetful functor from algebraically fibrant object of a model category C to C, as in Nikolaus paper (arxiv.org/abs/1003.1342), see Theorem 3.2 , point 5 in the paper. (this was mentioned to me by John Bourke very recently). I don't think this has been studied very much though. $\endgroup$ – Simon Henry May 27 '19 at 20:04
  • $\begingroup$ @SimonHenry Excellent ! Thanks. $\endgroup$ – Philippe Gaucher May 27 '19 at 20:59

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