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By abstract construction of a combinatorial model category, I mean starting from a locally presentable category satisfying some assumptions, e.g. equipped with a cylinder or a cocylinder satisfying some special hypothesis, and from these data build a model category structure. The question now is:

What are the known abstract constructions of a combinatorial model category with all objects fibrant and such that not all maps are fibrations (to rule out the case of the discrete model structure) ?

The only example I am aware of is the third section of Marc Olschok's PhD, Model structures from balls.

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There are a ton of papers about what you are asking. Another is the thesis of Richard Williamson (arXiv:1304.0867v1). Also, Valery Isaev has a paper that produces a model structure with all objects fibrant, given some cylinder or path object information (https://arxiv.org/pdf/1312.4327.pdf). The thesis of Remy Tuyeres produces a model structure given some even more general category theoretic information. Maybe check the references of those three sources; I'll bet there are many others.

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  • $\begingroup$ I can't find Remy Tuyeres' PhD with Google, he's also not on the Mathematics Genealogy Project. Any link would be welcome. Richard Williamson 's paper does not seem to be published. How did you find these references ? You already knew them or you have a magic keyword in your pocket ? $\endgroup$
    – Philippe Gaucher
    Commented Nov 24, 2017 at 4:12
  • $\begingroup$ Remy's webpage is here: normalesup.org/~tuyeras, and he hosts his thesis. He's now a postdoc at MIT. I pretty much read everything I can about model categories as the papers come out, and happened to remember these names in this case. I know I've read other papers on this topic, but can't remember the author names. I figured they might be cited by the 3 I told you. $\endgroup$
    – David White
    Commented Nov 25, 2017 at 18:34
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Nikolaus has shown (see Cor 2.21) that every combinatorial model category where all trivial cofibrations are monic is Quillen equivalent to its category of algebraically-fibrant objects, in which every object is fibrant.

The category of algebraically-fibrant objects is a category where every object has specified lifts against generating trivial cofibrations, preserved by morphisms. For example, algebraically fibrant Kan complexes have specified lifts for horns.

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  • $\begingroup$ Interesting. The author only assumes that all trivial cofibrations are monic. $\endgroup$
    – Philippe Gaucher
    Commented Nov 21, 2017 at 10:42
  • $\begingroup$ Yes, it's an interesting hypothesis to me. I haven't looked closely enough to learn what role it plays in the construction. $\endgroup$
    – Tim Campion
    Commented Nov 21, 2017 at 13:56
  • $\begingroup$ I meant you forgot the word trivial, you should fix your post. This hypothesis is used in the construction of the left adjoint of the forgetful functor. $\endgroup$
    – Philippe Gaucher
    Commented Nov 21, 2017 at 15:10

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