Monoidal model categories
by Mark Hovey
A monoidal model category is a model category with a compatible closed
monoidal structure. Such things abound in nature; simplicial sets and
chain complexes of abelian groups are examples. Given a monoidal model
category, one can consider monoids and modules over a given monoid. We
would like to be able to study the homotopy theory of these monoids and
modules. This question was first addressed by Stefan Schwede and Brooke
Shipley in "Algebras and modules in monoidal model categories", who
showed that under certain conditions, there are model categories of
monoids and of modules over a given monoid. This paper is a follow-up
to that one. We study what happens when the conditions of
Schwede-Shipley do not hold. This will happen in any topological
situation, and in particular, in topological symmetric spectra. We find
that, with no conditions on our monoidal model category except that it
be cofibrantly generated and that the unit be cofibrant, we still obtain
a homotopy category of monoids, and that this homotopy category is
homotopy invariant in an appropriate sense.