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I'm looking for a Quillen model category which model the $2$-category of 'small category with finite limits (and functors between them preserving finite limits)':

  • Left proper,
  • right proper,
  • Enriched in $\mathbf{Cat}$, the folk model structure on the category of categories.

To give an example, if I only require left proper or right proper but not both at the same time, it is farily standard:

  1. One can work with "chosen limits": One considers the $1$-category of small categories with chosen (up to equality) terminal object and fiber product. It is a strict $2$-category (so enriched in Cat), and it has a model structure lifted along the forgetful functor to $\mathbf{Cat}$. (i.e. fibration, trivial fibration and equivalences are the map whose image by the forgetfull functor are as such). Every object is fibrant, so this model structure is right proper. But I'm pretty sure that it is not left proper.

  2. One can work with a category of "limit sketches": one considers the $1$-category of small categories equiped with a class of marked cocone closed under isomorphisms. We equip it with its "trivial model structure" (weak equivalences are the equivalence that detect the marked cocone, fibrations are the isofibrations) and we take a left Bousfield localization in which the fibrant objects are the category with all small limits and in which the marked cocone are exactly the colimit cocones. In this one, every object is cofibrant, so it is left proper. But I can show it is not right proper.

I've made many attempt to construct such a model structure which is both left and right proper, and it always seem to fail.

So I'm wondering if there is a homotopy theoretic obstruction to have all these three properties at the same time...

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    $\begingroup$ I'm assuming you know this paper by Steve Lack? It's the best technology I know for this kind of question: arxiv.org/abs/math/0607646 $\endgroup$
    – David White
    Commented Jul 13, 2020 at 15:16
  • $\begingroup$ Yes, I should have linked it as a reference for example (1). But I can't really find an answer in this model. I tend to think that there is an actual homotopy theoretic obstruction to all three properties together, but I can't find it. $\endgroup$
    – Simon Henry
    Commented Jul 13, 2020 at 15:22

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