Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
For a monoidal model category there is under mild conditions a natural model category structure on its category of monoids.
For a monoidal category with all colimits the category of monoids comes equipped (as discussed there) with a free functor/forgetful functor adjunction
Typically one uses on the transferred model structure along this adjunction, if it exists.
This is part of (SchwedeShipley, theorem 4.1).
If the symmetric monoidal model category
then the transferred model structure on monoids exists.
Suppose the transferred model structure exists on . By the discussion of free monoids at category of monoids we have that then pushouts of the form
exist in , for all in
Let the monoidal model category be
If is an acyclic cofibration in the model structure on , then the pushout as above is a weak equivalence in .
This is SchwedeShipley, lemma 6.2.
Under mild conditions on the model structure on monoids in is Quillen equivalent to that of A-infinity algebras in . See model structure on algebras over an operad for details.