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.


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

  • $\begingroup$ 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

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