Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can consider the adjunction:

$$ F: C \leftrightarrows \mathcal{O}\text{-}alg(C): U $$

between $C$ and the category of $\mathcal{O}$-algebras.

Now if $C$ is a Quillen model category, we can ask whether we can transfer this model structure along this adjunction to get a model structure on $\mathcal{O}$-algebras where a map is a weak equivalence or fibration if it is so after forgetting back to $C$.

There are various known conditions on $\mathcal{O}$ and $C$ for when this is possible. Most of them are fairly mild conditions (like $C$ is cofibrantly generated, $\mathcal{O}$ satisfies some cofibrancy condition). However every treatment I have been able to find starts with a strong assumption that $C$ is a monoidal model category. This means there is a strong compatibility between the monoidal structure on $C$ and the model category sptructure. Specifically the pushout-product axiom is satisfied.

Now this is all well and good. After all, it seems reasonable to expect that in order to get a good model structure on $\mathcal{O}$-algebras you would have to start with a model structure which plays well with the monoidal structure on $C$. However I find myself in the situation where I would still like some sort of transfer result like this, but where $C$ is decidedly not a monoidal model category in this strong sense.

In my specific case, $C$ is a Bousfield localization of the model category of simplicial presheaves on a nice Reedy category (with, say the injective model structure). So $C$ is as nice as it could be: it is combinatorial (so cofibrantly generated), the cofibrations are the monomorphisms, in this case it is a simplicial model category, and all the domains of the gen. cofib/gen. acyclic cofibrations are cofibrant, etc.

The operad I want to use is (a simplicial version of) the $E_\infty$-operad, so satisfies very nice cofibrancy conditions.

The monoidal structure on $C$ is the Cartesian product. The terminal object (unit of this monoidal structure) is fibrant and cofibrant. However $C$ is not a Cartesian model category and it definitely fails to satisfy the pushout-product axiom.

So my question is:

Are there any general or specific conditions that would allow one to construct a reasonable model structure on the category $\mathcal{O}\text{-}alg(C)$, even when $C$ fails the pushout-product axiom? Specifically are there conditions which might apply to the case I vaguely described above?

  • $\begingroup$ A nice interval is often a replacement for the push-out product axiom, as in Berger-Moerdijk or even Schwede-Shipley. If you're working with presheaves on simplicial sets, you probably have it. $\endgroup$ – Fernando Muro Feb 5 '15 at 11:55
  • $\begingroup$ So in Schwede-Shipley, they do have a transfer result using path-objects (lem 2.3), but they require every object to be fibrant, which is really much too strong and not valid in my applications. In Berger-Moerdijk, if I understand correctly, they start off with a promising alternative collection of assumptions (Thm 3.2), but by the time they get to discussing model cats for algebras over an operad (sect. 4), they also make the blanket assumption that the model category is monoidal. $\endgroup$ – Chris Schommer-Pries Feb 5 '15 at 14:23
  • $\begingroup$ where by "monoidal" I mean that it satisfies the push-out product axiom. $\endgroup$ – Chris Schommer-Pries Feb 5 '15 at 14:37
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    $\begingroup$ Would Prop 3.1.5 from here work? dspace.mit.edu/bitstream/handle/1721.1/41793/… $\endgroup$ – Tyler Lawson Feb 5 '15 at 14:50
  • $\begingroup$ @TylerLawson: Oooh, that looks like it might do it. I have to think for a bit about the simplical structure on $\mathcal{O}$-algebras to see if those conditions are met, but this looks good. Thanks! $\endgroup$ – Chris Schommer-Pries Feb 5 '15 at 15:16

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