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The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e. weak equivalences and fibrations are created by this functor. It is known that this model structure is cofibrantly generated (by the left adjoint $c \circ Sd^2$ applied to the usual generators in $sSet$) and is Quillen equivalent to $sSet$ along the adjoint pair $(c \circ Sd^2, Ex^2 \circ N)$. Furthermore, $Cat$ has the structure of a category enriched, tensored, and cotensored over $sSet$ (one such structure is described in Rezk's note).

Is there such a structure on $Cat$ that satisfies Quillen's SM7 axiom to make the Thomason model structure into a simplicial model category?

Some sufficient conditions to check can be found in Lurie's Higher Algebra, Prop 4.1.8.3, but don't seem likely to be satisfied in this case.

Similarly, $Cat$ has two monoidal structures: the Cartesian one and the "funny" one.

Do either of the monoidal structures satisfy the pushout product axiom to make the Thomason model structure into a monoidal model category?

On the one hand, its homotopy category certainly is monoidal. On the other hand, the cofibrations don't seem to be super well understood (though, it is known that every cofibrant object is a poset), and I don't know much about the monoidal properties of the functors $Ex^2$ and $N$.

UPDATE: Tim's answer below has an attractive argument, but the comments lay out the ways it's wrong. First, Tim's argument fails between the two displayed diagrams. Secondly, even if the Cartesian structure did satisfy the pushout product axiom, the counterexample shows that one simplicially enriched structure on Cat (the one you get from applying the nerve functor to the Cartesian internal hom) fails the SM7 axiom. However, there does not seem to be a counterexample yet to the question of whether or not the Cartesian structure satisfies the pushout product axiom, and we still don't know if there exists a simplicial structure that satisfies the SM7 axiom. As far as I can tell, both questions in the current thread are still open.

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    $\begingroup$ $\operatorname{Ex^2}$ preserves products, as does the nerve. Both are right adjoints. Does this help? $\endgroup$ Nov 25, 2018 at 23:19
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    $\begingroup$ The natural way to do this would be to have the given Quillen adjunction be an equivalence of $\mathrm{SSet}$-enriched model categories, which would amount to it preserving the simplicial enrichment. The usual way this would happen would be if left adjoint preserved finite products, which it does not-for instance, look at a square. So I'm skeptical whether this can be done. $\endgroup$ Nov 26, 2018 at 4:36
  • $\begingroup$ What is the "funny" monoidal structure? $\endgroup$
    – Cihan
    Nov 29, 2018 at 15:08
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    $\begingroup$ @Cihan ncatlab.org/nlab/show/funny+tensor+product $\endgroup$
    – Tim Campion
    Dec 5, 2018 at 20:46
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    $\begingroup$ I just want to record here a lesson I learned from Kevin Carlson in my now-deleted answer: $sd\dashv Ex : sSet \to sSet$ is an adjunction, but it's not a simplicial adjunction! This sinks the attempted proof I had. $\endgroup$
    – Tim Campion
    Jan 12, 2019 at 2:49

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The answer to this question is NO. In particular, the discussion after Corollary 3.7 of George Raptis's paper Homotopy theory of posets provides a concrete example of a cofibration (namely, $f: [0] \to [1]$) such that the pushout product of $f$ with itself with respect to the Cartesian monoidal structure, is not a cofibration. I think this example also shows that the Thomason model structure is not a simplicial model category (fails the SM7 axiom). Furthermore, the same example shows the funny monoidal structure fails the pushout product axiom, because taking the funny pushout product of this $f$ with the identity on $[1]$ (which is a cofibrant object) yields a map from $[1]$ to a square, which is not a weak equivalence.

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