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Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one situation to the other.

But there are very few examples of model categories where every object is both fibrant and cofibrant ("bifibrant").

The only example I know is when one starts with a strict 2-category with strict 2-limits and 2-colimits there are model structures on its underlying 1-category where every object is bifibrant and whose Dwyer-Kan localization is equivalent to the $2$-category itself (where one drop non-invertible 2-cells). So this typically applies to the canonical model category on Cat or on Groupoids. I'm not even sure it can be used to model things like the $2$-category of categories with finite limits.

I don't believe there are that many other examples. But I have never seen any obstruction for this. So:

Is there any example of a model category where every object is bifibrant whose localization is not a $2$-category?

Is every presentable $\infty$-category represented by a model category where every object is bifibrant? If (as I expect) this is not the case, can we give an explicit 'obstruction' or an example to show it isn't the case?

Edit : The first question has been completely answered, but I havn't accepted answer so far because I'm still hoping to get an answer (positive or negative) to the second question.

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  • $\begingroup$ Hi Simon. Just wanted to say, I haven't forgotten your email, and I do plan to reply. I've just been really busy. Sorry! $\endgroup$ – David White Mar 29 at 11:18
  • $\begingroup$ Trivial examples: any complete and cocomplete category with the isomorphisms as weak equivalences, and all morphisms are both fibrations and cofibrations. $\endgroup$ – Daniel Robert-Nicoud Mar 29 at 16:36
  • $\begingroup$ @DanielRobert-Nicoud : This one is a special case of the 2-categorical example. $\endgroup$ – Simon Henry Mar 29 at 17:15
  • $\begingroup$ I think your second question is much harder than your first one, so it might be better to ask them as separate questions. $\endgroup$ – Mike Shulman Mar 30 at 14:46
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Another example is given by Strom's model structure on topological spaces where

  • Fibrations: Hurewicz fibrations,
  • Weak equivalences : (strong) homotopy equivalences.
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An example of a different sort is the model structure on $R$-mod, whose homotopy category is the stable module category. A great reference is Theorem 2.2.12 in Hovey's book Model Categories. In this reference, $R$ is taken to be quasi-Frobenius. This model structure is generalized to work for any ring in the thesis of Daniel Bravo, and a resulting paper of Bravo-Gillespie-Hovey. But, you lose the property about all objects being bifibrant.

Another example is the projective (or injective) model structure on $Ch(R)$ where $R$ is a field, and where we take chain complexes to be bounded (e.g. always non-negative degree, or you could do cochain complexes in non-positive degree). A great reference is Quillen's Rational Homotopy Theory. See also Section 2.3 of Hovey's book, but this is for the situation of unbounded chain complexes. The point is that, for the projective model structure, the fibrations are surjections, and as Lemma 2.3.6 shows, bounded below complexes of projective modules are cofibrant, and if $R$ is a field (or semi-simple ring) then all modules are projective.

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Look at this paper for some examples: Strong cofibrations and fibrations in enriched categories by R. Schwänzl and R. M. Vogt.

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