Stabilization of Model Categories

by Mark Hovey
Wesleyan University


Suppose C is a (nice enough) model category, and G: C --> C is a
left Quillen endofunctor of C.  Think of C as the category of pointed
topological spaces, and G as the suspension.  Then we construct a new
model category Sp(C,G), an embedding C --> Sp(C,G), and an extension of
G to a Quillen EQUIVALENCE of Sp(C,G).  Essentially, we have inverted
the functor G, up to homotopy.  When C is the category of pointed
topological spaces and G is the suspension, we recover the
Bousfield-Friedlander model category of spectra.  

The trouble with the Bousfield-Friedlander model category is that it is
not symmetric monoidal, and we have the same problem with Sp(C,G).  But
there is also the same fix.  Suppose C is a (nice enough) symmetric
monoidal model category, K is a cofibrant object of C, and D is a (nice
enough) C-model category.  Think of C as pointed simplicial sets, D as a
pointed simplicial model category, and K as the simplicial circle.  Then
we construct a model category Sp^Sigma(D,K), so that Sp^Sigma(C,K) is a
symmetric monoidal model category, Sp^Sigma(D,K) is a
Sp^Sigma(C,K)-model category, and smashing with K is a Quillen
equivalence on Sp^Sigma(D,K).  When C is pointed simplicial sets, and K
is S^1, we get the symmetric spectra of Hovey-Shipley-Smith. 

The method used is the Bousfield localization technology of Hirschhorn,
so the words "nice enough" mean "left proper cellular", though
occasionally we also need to assume to domains of the generating
cofibrations are cofibrant.