3
$\begingroup$

If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating cofibrations is a cofibration, that the Leibniz product of a generating cofibration with a generating acyclic cofibration is an acyclic cofibration, and that the unit axiom is satisfied.

The proof is a little involved, so I'd like to have a source to cite this from. Where can I find it in the literature?

I'm happy to assume that the unit is cofibrant.

$\endgroup$
5
$\begingroup$

This is Corollary 4.2.5 of Hovey's book. The proof is not that involved.

$\endgroup$
  • $\begingroup$ I see. It's cleaner to check lifting properties directly, rather than checking closure under cobase change, transfinite composition, and retract. Thanks! $\endgroup$ – Tim Campion Mar 8 at 15:13
  • 2
    $\begingroup$ Not only cleaner, but it doesn't require any smallness property on the classes of "generating" (acyclic) cofibrations or the objects involved. $\endgroup$ – Reid Barton Mar 8 at 15:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.