TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Xxxx XXXX, Pages 000-000
S 0002-9947(XX)0000-0



                MONOIDAL MODEL CATEGORIES



                             MARK HOVEY


       Abstract.A monoidal model category is a model category with a closed
       monoidal structure which is compatible with the model structure. Given a
       monoidal model category, we consider the homotopy theory of modules over
       a given monoid and the homotopy theory of monoids.  We make minimal
       assumptions on our model categories; our results therefore are more gene*
 *ral,
       yet weaker, than the results of [SS97]. In particular, our results apply*
 * to the
       monoidal model category of topological symmetric spectra [HSS98].



                            Introduction

  A monoidal model category is a (closed) monoidal category that is also a model
category in a compatible way. Monoidal model categories abound in nature: ex-
amples include simplicial sets, compactly generated topological spaces, and cha*
 *in
complexes of modules over a commutative ring. The thirty-year long search for a
monoidal model category of spectra met success with the category of S-modules
of [EKMM97   ] and the symmetric spectra of [HSS98 ].
  Given any monoidal category, one has categories of monoids and of modules
over a given monoid. If we are working in a monoidal model category, we would
like these associated categories also to be model categories, so that we can ha*
 *ve
a homotopy theory of rings and modules.  The first results on this subject were
obtained in [SS97]. This paper is a followup to that paper. In [SS97], the auth*
 *ors
added the following three assumptions about a monoidal model category C:

 (a) Every object of C is small relative to the whole category;
 (b) C satisfies the monoid axiom; and
 (c) Given a monoid A and a cofibrant left A-module M, smashing over A with
     M takes weak equivalences of right A-modules to weak equivalences.

  The first two assumptions guarantee the existence of a model structure on the
category of monoids and on the category of modules over a given monoid.  The
third assumption guarantees that a weak equivalence of monoids induces a Quillen
equivalence of the corresponding module categories.
  All these assumptions are reasonable ones in any combinatorial situation, such
as simplicial sets, chain complexes, or simplicial symmetric spectra. However, *
 *for
any category of topological spaces the third assumption will fail, and the first
assumption is not known to be true and probably fails. Furthermore, in the cate*
 *gory
of topological symmetric spectra the second assumption is not known to hold.
  The goal of this paper, then, is to investigate what can be said when these
assumptions do not hold. After a preliminary section reminding the reader of so*
 *me
____________
   Received by the editors February 28, 1998.
   1991 Mathematics Subject Classification. 55P42, 55U10, 55U35.

                                             cO1997 American Mathematical Socie*
 *ty
                                  1



2                            MARK HOVEY


basic definitions and facts about model categories, we begin in the second sect*
 *ion
by showing that one always gets a model category of modules over a cofibrant
monoid. Furthermore, under a minor assumption on our model category C, we show
that a weak equivalence of cofibrant monoids induces a Quillen equivalence of t*
 *he
corresponding module categories. Also, a Quillen equivalence F of monoidal model
categories induces a Quillen equivalence between R-modules and F R-modules, for
R a monoid which is cofibrant in the domain of F . In the third section, we show
that, if the unit S in C is cofibrant, then, though we do not get a model categ*
 *ory
of monoids in general, we do at least get a homotopy category of monoids.  In
particular, given a general monoid A, we can find a cofibrant monoid QA and a
weak equivalence and homomorphism QA -!A. Then the model category QA-mod
is the homotopy invariant replacement for the category A-mod , which may not ev*
 *en
be a model category. We also show that the homotopy category of monoids is itse*
 *lf
homotopy invariant.  In particular, there is a homotopy category of monoids of
topological symmetric spectra, and this homotopy category is equivalent to the
homotopy category of monoids of simplicial symmetric spectra.
  The most obvious question left unaddressed in this paper concerns the category
of commutative monoids in a symmetric monoidal model category.  What do we
need to know to get a model structure on commutative monoids?  Can we get
a homotopy category of commutative monoids in any symmetric monoidal model
category? The author does not know the answer to these questions.
  The author would like to thank his coauthors Brooke Shipley and Jeff Smith.
This paper grew out of [HSS98 ], when the authors of that paper realized that
topological spaces are not as simple as they had originally thought.  The author
would also like to thank Gaunce Lewis and Peter May for helping him come to that
realization, which of course they have understood for years.

                              1. Basics

  We will have to assume some familiarity with model categories on the part of
the reader. A gentle introduction to the subject can be found in [DS95 ]. A more
thorough and highly recommended source is [Hir97, Part 2].  Other sources in-
clude [Hov97 ] and [DHK  ].
  In particular, in a model category C, we have a cofibrant replacement functor*
 * Q
and a fibrant replacement functor R. There is a natural trivial fibration QX -q*
 *!X,
and QX is cofibrant. Similarly, there is a natural trivial cofibration X -! RX *
 *and
RX is fibrant.
  Our basic object of study is a monoidal model category, which we now define. *
 *In
a monoidal category C, we will denote the monoidal product by ^ and the unit by
S. Note that in model category theory, functions seem to come in adjoint pairs.*
 * We
will therefore consider a closed monoidal category rather than a general monoid*
 *al
category.  This means that both functors X ^ - and - ^ X have right adjoints
natural in X.  For our purposes, the closed structure just guarantees for us th*
 *at
both functors X ^ - and - ^ X preserve colimits.

Definition 1.1.Suppose C is a closed monoidal category. Given maps f :A -!B
and g :X -! Y in C, define the pushout smash product f  g of f and g to be the
map (A ^ Y ) qA^X (B ^ X) -!B ^ Y .

Definition 1.2.Suppose C is a closed monoidal category which is also a model
category. Then C is a monoidal model category if the following conditions hold.



                      MONOIDAL MODEL CATEGORIES                     3


 (a) If f and g are cofibrations, so is f  g. If one of f or g is in addition a*
 * weak
     equivalence, so is f  g.
 (b) Both maps q ^ X :QS ^ X -! S ^ X ~=X and X ^ q :X ^ QS -! X are weak
     equivalences for all cofibrant X.

  The second condition is a consequence of the first in case the unit S is cofi-
brant. This is usually the case, but S is not cofibrant in the category of S-mo*
 *dules
of [EKMM97   ]. Without the second condition, the homotopy category of a monoid*
 *al
model category would not be a monoidal category, because there would not be a
unit. With it, it is an exercise in derived functors, carried out in [Hov97 , C*
 *hapter
4], to verify that the homotopy category is indeed a monoidal category.
  Of course, we only need half the second condition in case C is symmetric mono*
 *idal,
as it usually is in our examples.
  We point out, following the insight of Stefan Schwede, that the second condit*
 *ion
in Definition 1.2 is equivalent to requiring that both maps X -! Hom `(QS; X) a*
 *nd
X -! Hom r(QS; X) are weak equivalences for all fibrant X, where Hom `and Hom r
are the two adjoints that define the closed structure on C. To see this, one ca*
 *n show
that both the Hom  conditions just defined and the ^ conditions of Definition 1*
 *.2
are equivalent to the unit axioms in the monoidal category Ho C.
  Examples of symmetric monoidal model categories include the categories of sim-
plicial sets, compactly generated topological spaces, S-modules [EKMM97   ], sy*
 *m-
metric spectra [HSS98 ], and topological symmetric spectra.
  The reader should note that, in a monodial model category, smashing with a
cofibrant object preserves cofibrations and trivial cofibrations, and hence als*
 *o weak
equivalence between cofibrant objects, by Ken Brown's lemma [DS95 , Lemma 9.9].
  We will now repeat some standard definitions.

Definition 1.3.A map f :A -! B in a category C is said to have the left lifting
property with respect to another map g :X -! Y if, for every commutative square

                             A  ----! X
                             ?         ?
                            f?y        ?yg


                             B  ----! Y

there is a lift h: B -! X making the diagram commute. We also say that g has
the right lifting property with respect to f in this situation.

  The following argument is often used in model category theory.

Proposition 1.4 (The Retract Argument).Let C be a category and let f = pi be
a factorization in C.

  1. If p has the right lifting property with respect to f then f is a retract *
 *of i.
  2. If i has the left lifting property with respect to f then f is a retract o*
 *f p.

Proof.We only prove the first part, as the second is similar. Since p has the r*
 *ight
lifting property with respect to f, we have a lift g :Y -! Z in the diagram


                             X  --i--! Z
                             ?         ?
                            f?y       p?y


                             Y  _______Y



4                            MARK HOVEY


This gives a diagram

                        X  _______X _______X
                        ?         ?        ?
                        f?y      i?y      f?y


                        Y  --g--! Z ---p-! Y

where the horizontal compositions are identity maps, showing that f is_a_retrac*
 *t of
i.                                                                   |__|

  In model category theory, one very often needs to construct factorizations. T*
 *he
standard way to do this is by the small object argument.

Definition 1.5.Suppose A is an object of a cocomplete category C. Suppose D
is a subcategory of C.  We say that A is small relative to D if there is a cofi*
 *nal
class S of ordinals such that, for all ff 2 S and for all colimit-preserving fu*
 *nctors
X :ff -!C such that each map Xfi-! Xfi+1is in D, the induced map

                 colimfi<ffC(A; Xfi) -!C(A; colimfi<ffXfi)

is an isomorphism.

  In this context, "cofinal" means that given any ordinal ff there is an ordinal
fi 2 S with ff  fi. For example, every set is small relative to the whole categ*
 *ory
of sets. For a finite set A, the class S is the collection of limit ordinals; f*
 *or a more
general set A one has to take a sparser collection of ordinals.

Definition 1.6.Suppose I is a collection of maps in a cocomplete category C.
Define I-injto be the class of all maps with the right lifting property with re*
 *spect
to I, and define I-cofto be the class of all maps with the left lifting property
with respect to I-inj. Define I-cellto be the class of all transfinite composit*
 *ions
of pushouts of I. That is, for any map f :A -! B in I-cellthere is an ordinal ff
and a colimit-preserving functor X :ff -! C such that X0 = A, the map X0 -!
colimfi<ffXfiis isomorphic to f, and each map Xfi-! Xfi+1is a pushout of a map
of I.

  Note that I-cell I-cof.

Theorem 1.7 (The Small Object Argument).Suppose I is a set of maps in a co-
complete category C, and suppose that the domains of I are small relative to I-*
 *cell.
Then there is a functorial factorization of every map in C into a map of I-cell
followed by a map of I-inj.

  We will not prove this theorem: see [Hir97, Section 12.4] or [Hov97 , Section*
 * 2.1].
Note that most authors include transfinite compositions of pushouts of coproduc*
 *ts
of I, but this is not necessary in view of [Hir97, Proposition 12.2.5]. Also, i*
 *f the
domains of I are small relative to I-cell, then every map of I-cofis a retract *
 *of a
map of I-cellby the retract argument, and furthermore the domains of I are small
relative to I-cof[Hir97, Theorem 12.4.21].
  We can now define a cofibrantly generated model category.

Definition 1.8.A model category C is cofibrantly generated if there is are sets*
 * I
and J of maps of C such that the following conditions hold.

  1. The domains of I are small relative to the cofibrations. The domains of J *
 *are
     small relative to the trivial cofibrations.



                      MONOIDAL MODEL CATEGORIES                     5


  2. The fibrations form the class J-inj. The trivial fibrations form the class*
 * I-inj.

  Every model category in common use is cofibrantly generated, such as the model
categories of topological spaces, simplicial sets, and chain complexes.  In a c*
 *ofi-
brantly generated model category, the cofibrations form the class I-cofand the
trivial cofibrations form the class J-cof.
  The big advantage of cofibrantly generated model categories is that they allo*
 *w us
to prove things by induction. Suppose we have some claim about cofibrant objects
A in a cofibrantly generated model category.  That claim is almost certain to be
preserved by retracts, so we can usually assume the map 0 -! A is in I-cell, so*
 * is
a transfinite composition of pushouts of maps of I.  We can then use transfinite
induction.
  We also remind the reader that functors between model categories come in ad-
joint pairs.  A functor F :C -! D between model categories with right adjoint
U is called a left Quillen functor (and U is called a right Quillen functor) if*
 * F
preserves cofibrations and trivial cofibrations. Equivalently, we can require t*
 *hat U
preserve fibrations and trivial fibrations. A Quillen pair induces a pair of ad*
 *joint
functors LF and RU on the homotopy categories. The functor LF is defined by
(LF )(X) = F (QX), where Q is the functorial cofibrant replacement functor (this
is a very good reason to assume these factorizations are part of the model stru*
 *cture,
as is done in [Hov97 ]). Simiilarly RU is defined by (RU)(X) = U(RX), where R
is the functorial fibrant replacement functor.
  A left Quillen functor F is called a Quillen equivalence if for all cofibrant*
 * A 2 C
and fibrant X 2 D, a map F A -!X is a weak equivalence if and only if its adjoi*
 *nt
A -!UX is a weak equivalence.
  The following lemma deserves to be better known than it is. Recall that a fun*
 *ctor
is said to reflect some property of morphisms if, given a morphism f, if F f ha*
 *s the
property so does f.

Lemma 1.9.  Suppose F :C -! D is a Quillen functor with right adjoint U. The
following are equivalent:

 (a) F is a Quillen equivalence.
 (b) F reflects weak equivalences between cofibrant objects and, for every fibr*
 *ant
     Y , the map F QUY -! Y is a weak equivalence.
 (c) U reflects weak equivalences between fibrant objects and, for every cofibr*
 *ant
     X, the map X -! URF X is a weak equivalence.
 (d) LF is an equivalence of categories.

Proof.Suppose first that F is a Quillen equivalence. Then, if X is cofibrant, t*
 *he
weak equivalence F X -! RF X gives rise to a weak equivalence X -! URF X.
Similarly, the weak equivalence QUX -!  UX gives rise to a weak equivalence
F QUX -!  X.  This shows that (a) implies half of (b) and (c).  Now suppose
f :X -! Y is a map between cofibrant objects such that F f is a weak equivalenc*
 *e.
Since both maps X -! URF X and Y  -! URF Y  are weak equivalences, f is a
weak equivalence if and only if URF f is a weak equivalence. Since F f is a weak
equivalence, R preserves weak equivalences, and U preserves weak equivalences
between fibrant objects, we find that f is a weak equivalence. Thus (a) implies*
 * (b),
and a similar argument shows that (a) implies (c).
  To see that (b) implies (d), note that the counit map (LF )(RU)X -! X is an
isomorphism by hypothesis. We must show that the unit map X -! (RU)(LF )X is



6                            MARK HOVEY


an isomorphism. But (LF )X -! (LF )(RU)(LF )X is inverse to the counit map of
(LF )X, so is an isomorphism. Since F reflects weak equivalences between cofibr*
 *ant
objects, this implies that QX -! QURF QX is a weak equivalence. Since Q reflects
all weak equivalencs, this implies that X -! URF QX = (RU)(LF )X is a weak
equivalence, as required. A similar proof shows that (c) implies (d).
  To see that (d) implies (a), note that (LF )X is isomorphic to F X in the hom*
 *o-
topy category when X is cofibrant, and similarly (RU)Y is isomorphic to UY in
the homotopy category when U is fibrant. So F X -! Y is a weak equivalence if
and only (LF )X -! Y is an isomorphism in the homotopy category. Since LF is an
equivalence of categories with adjoint RU, this is true if and only if X -! (RU*
 *)Y
is an isomorphism. But this holds if and only if X -! UY is a weak equivalence,_
as required.                                                          |__|

                             2. Modules

  In this section we investigate model categories of modules over a monoid A in*
 * a
cofibrantly generated monoidal model category C.

Theorem 2.1.  Suppose C is a cofibrantly generated monoidal model category with
generating cofibrations I and generating trivial cofibrations J. Let A be a mon*
 *oid
in C, and suppose the following conditions hold.

  1. The domains of I are small relative to (A ^ I)-cell.
  2. The domains of J are small relative to (A ^ J)-cell.
  3. Every map of (A ^ J)-cellis a weak equivalence.

Then there is a cofibrantly generated model structure on the category of left A-
modules, where a map is a weak equivalence or fibration if and only if it is a *
 *weak
equivalence or fibration in C.

  There is an obvious analogue of this theorem for right A-modules.

Proof.By adjointness, the fibrations of A-modules form the class (A ^ J)-inj, a*
 *nd
the trivial fibrations form the class (A ^ I)-inj. We therefore define the cofi*
 *brations
of A-modules to be the class (A^I)-cof, and take our generating trivial cofibra*
 *tions
to be A^J. Since each element of J is in I-cof, each element of A^J is in (A^I)*
 *-cof,
and so the maps of (A ^ J)-cofare cofibrations in A-mod .
  The category of A-modules is certainly bicomplete, with limits and colimits t*
 *aken
in C. The retract and two out of three axioms are immediate, as is the lifting *
 *axiom
for cofibrations and trivial fibrations. By assumption, the domains of I are sm*
 *all
relative to (A ^ I)-cell.  By adjointness, it follows that the domains of A ^ I*
 * are
small in A-mod relative to (A^I)-cell. Thus the small object argument gives us *
 *the
cofibration-trivial fibration half of the factorization axiom. Similarly, the d*
 *omains
of A ^ J are small in A-mod relative to (A ^ J)-cell. We can then factor any map
in A-mod into a map of (A ^ J)-cellfollowed by a fibration. We have already seen
that the maps of (A ^ J)-cellare cofibrations, and by assumption they are weak
equivalences. This gives the other half of the factorization axiom.
  For the remaining lifting axiom, suppose f is a cofibration and weak equivale*
 *nce.
Factor f = pi, where i 2 (A ^ J)-celland p is a fibration of A-modules. Since i
is a weak equivalence, and so is f, it follows that p is a weak equivalence. Th*
 *us f
has the left lifting property with respect to p. By the Retract Argument 1.4, f*
 * is
a retract of i, and so has the left lifting property with respect to all_fibrat*
 *ions_of
A-modules, as required.                                                |__|



                      MONOIDAL MODEL CATEGORIES                     7


Corollary 2.2.Suppose C is a cofibrantly generated monoidal model category, and
suppose A is a monoid which is cofibrant in C. Then there is a cofibrantly gene*
 *rated
model structure on the category of  (left, or right) A-modules, where a map is a
weak equivalence or fibration if and only if it is a weak equivalence or fibrat*
 *ion in
C. Furthermore, a cofibration of A-modules is a cofibration in C.

Proof.For definiteness, we work with left A-modules.  Since A is cofibrant in C,
every map of A ^ I is a cofibration, so (A ^ I)-cell I-cof. Since the domains o*
 *f I
are small relative to I-cof, they are certainly small relative to (A^I)-cell. S*
 *imilarly,
(A ^ J)-cell J-cof, so the domains of J are small relative to (A ^ J)-cell,_and*
 *_the
maps of (A ^ J)-cellare weak equivalences.                                |__|

  As an example, we would like to consider the model category of topological
spaces. This is not, however, a monoidal model category, since the functor X x -
need not have a right adjoint unless X is locally compact Hausdorff. To get aro*
 *und
this, it is usual to consider some version of compactly generated spaces.  We u*
 *se
the definitions of [Lew78 , Appendix A], the properties of which are summarized
in [HSS98 , Section 6.1].  In particular, a subset U of a topological space X is
compactly open if, for every continuous f :K -! X, where K is compact Hausdorff,
the preimage f-1 (U) is open. The space X is called a k-space if every compactly
open space is open. A space is called compactly generated if it is both a k-spa*
 *ce
and weak Hausdorff; i.e., for every continuous f :K -! X where K is compact
Hausdorff, the image f(K) is closed. Then both the category K of k-spaces and
the category T of compactly generated spaces are cofibrantly generated symmetric
monoidal model categories.
  In order to understand these categories a little better, we need the following
lemma.

Lemma 2.3.  Suppose C is a cofibrantly generated monoidal model category, such
that the domains and codomains of the generating trivial cofibrations J are fib*
 *rant.

Suppose in addition that there is an object I 2 C and a factorization S q S -(i*
 *0;i1)---!
I -! S of the fold map, where (i0; i1) is a cofibration, such that the induced *
 *map
X ^ I -! X is a weak equivalence for all X. Then C satisfies the monoid axiom:
that is, every map of (C ^ J)-cellis a weak equivalence.

Proof.We only sketch the proof. We can define a map f :A -! B to be a strong
deformation retract if there is a retraction r :B -! A such that rf = 1A and a
homotopy H :B ^ I -! B such that Hi0 = rf and Hi1 = 1B . Here we are using
the specific object I in the hypothesis of the lemma. In particular, B ^ I will*
 * not
be a cylinder object for B in general. Nevertheless, one can check that any str*
 *ong
deformation retract us a weak equivalence, and furthermore, that the class of s*
 *trong
deformation retracts is closed under smashing with an arbitrary object, pushout*
 *s,
and transfinite compositions. Furthermore, following the argument of [Qui67, pg.
2.5], we can see that each map of J is a strong deformation retract. Thus every
map of (C ^ J)-cellis a strong deformation retract, and hence a weak equivalenc*
 *e,_
as required.                                                          |__|

  In particular, this applies to both the category K of k-spaces and the catego*
 *ry
T of compactly generated topological spaces, where in this case I is the usual *
 *unit
interval.  So both K and T satisfy the monoid axiom.  However, the smallness
conditions of Theorem 2.1 do not appear to hold in K. In any topological catego*
 *ry,



8                            MARK HOVEY


the best one can usually do for smallness is that every object is small relativ*
 *e to
the inclusions.  But inclusions are not preserved very well, so we do not know
that the maps of (A ^ I)-cellare inclusions. In T, however, closed inclusions a*
 *re
preserved by almost every construction one ever makes (see [HSS98 , Section 6.1*
 *],
based on [Lew78 , Appendix A]). In particular, the maps of (A ^ I)-celland of
(A ^ J)-cellare closed inclusions. Therefore, we do get model categories of mod*
 *ules
over an arbitrary monoid in T.
  In order for the model category of A-modules to be useful, it must be both
homotopy invariant in appropriate senses and have good properties. We begin with
the homotopy invariance.

Theorem 2.4.  Suppose C is a cofibrantly generated monoidal model category such
that the domains of the generating cofibrations can be taken to be cofibrant. S*
 *uppose
f :A -! A0 is a weak equivalence of monoids which are cofibrant in C. Then the
induction functor induced by f and its right adjoint, the restriction functor, *
 *define
a Quillen equivalence from the model category of left (resp. right) A-modules t*
 *o the
model category of left (resp. right) A0-modules.

Proof.Again, we work with left modules for definiteness. In this case the induc*
 *tion
functor takes M to A0^A M.  The restriction functor obviously preserves weak
equivalences and fibrations, so is a right Quillen functor. Furthermore, the re*
 *stric-
tion functor reflects weak equivalences as well.  It follows from Lemma 1.9 that
induction is a Quillen equivalence if and only if for all cofibrant A-modules M*
 *, the
map M -! A0^A M is a weak equivalence. Because the category of A-modules is
cofibrantly generated, we may as well assume that M is the colimit of a colimit-
preserving functor ff -! A-mod , where ff is an ordinal, M0 = 0 and each map
Mfi-! Mfi+1is a pushout of a map of A ^ I.
  We will prove by transfinite induction that the map ifi:Mfi-! A0^A Mfiis
a weak equivalence for all fi  ff, taking Mff= colimff<fiMfi= M. Getting the
induction started is easy, since M0 = 0.  For the successor ordinal case, suppo*
 *se
that ifiis a weak equivalence. We have a pushout diagram

                          A ^ K ----!  A ^ L
                            ??          ?
                            y           ?y

                           Mfi  ----!  Mfi+1

where K -! L is some map of I.  Both horizontal maps are cofibrations in C.
Furthemore, because K and L are by assumption cofibrant in C, each object in the
diagram is cofibrant in C. By applying the functor A0^A -, we get an analogous
pushout diagram

                       A0^ K   ----!    A0^ L
                         ??               ?
                         y                ?y

                     A0^A Mfi  ----! A0^A Mfi+1

where again the horizontal maps are cofibrations in C, and each object is cofib*
 *rant
in C.  There is a map from the first pushout square to the second, which by the
induction hypothesis is a weak equivalence on the lower left square.  Since K is
cofibrant, smashing with K preserves trivial cofibrations in C, and hence, by K*
 *en
Brown's lemma [DS95 , Lemma 9.9], preserves weak equivalences between cofibrant



                      MONOIDAL MODEL CATEGORIES                     9


objects of C. Thus the map of pushout squares is also a weak equivalence on the
upper left corner, and by similar reasoning, on the upper right corner. Dan Kan*
 *'s
cubes lemma (see [Hov97 , Section 5.2] or [DHK  ]) then shows that it is a weak
equivalence on the lower right corner as well. Thus ifi+1is a weak equivalence.
  Now consider the limit ordinal case. Here we assume iflis a weak equivalence
for all fl < fi. We then have a map of sequences

       M0     ----!    M1    ----!  : :-:---!    Mfl   ----!  : : :
        ?               ?                         ?
       i0?y           i1?y                      ifl?y


    A0^A M0   ----! A0^A M1  ----!  : :-:---! A0^A Mfl ----!  : : :

where the vertical maps are all weak equivalences, and the horizontal maps are
cofibrations of cofibrant objects. Then [Hir97, Proposition 18.4.1] implies tha*
 *t_ifi
is also a weak equivalence, as required.                                   |__|


  The author learned the following example from Neil Strickland.

Example 2.5. Let us consider the category T* of compactly generated pointed
spaces. This is a monoidal model category under the smash product. Let A denote
the nonnegative natural numbers together with infinity, given the discrete topo*
 *logy.
With infinity as the base point, A is a monoid in T*, with unit 0. Let A0 be the
same set as A, but with the one-point compactification topology. Then A0is also*
 * a

topological monoid, with basepoint at infinity and unit 0, and the identity A f*
 *-!A0
is a homomorphism and weak equivalence of monoids. Note that A0is not cofibrant
as a topological space, so Theorem 2.4 does not apply. In fact, the induction f*
 *unctor
does not define a Quillen equivalence from A-modules to A0-modules. Indeed, take
M = A ^ S1.  Then the map M -! A0^A M is just the suspension of f.  The
suspension of A is an infinite wedge of circles, but the suspension of A0 is the
Hawaiian earring. In particular, f ^ S1 is not a weak equivalence, so induction*
 * can
not be a Quillen equivalence.

  In light of this example, it seems to the author that one should avoid consid-
ering modules over monoids which are not cofibrant in C, just we generally avoid
suspending non-cofibrant topological spaces. There are some categories, however,
where a weak equivalence of monoids always induces a Quillen equivalence of the
corresponding module categories. This is true in S-modules [EKMM97   , Theorem
3.8] and in simplicial symmetric spectra [HSS98 , Theorem 5.5.9]. See [SS97, Th*
 *eo-
rem 3.3].
  Theorem 2.4 shows that the model category of A-modules is homotopy invariant
under weak equivalences of cofibrant objects in C.  But we would also like the
model category of A-modules to be homotopy invariant under weak equivalences of
monoidal model categories C.
  For this to make sense, we need a notion of monoidal Quillen functor.

Definition 2.6.Suppose F :C -! D is a left Quillen functor between monoidal
model categories. Then F is a monoidal Quillen functor if F is monoidal and the
induced map F (QS) -!F S ~=S is a weak equivalence.

  This second condition is easy to overlook; it is essential in case the unit S*
 * of C
is not cofibrant in order to be sure LF is a monoidal functor.



10                           MARK HOVEY


Theorem 2.7.  Suppose F :C -!D is a monoidal Quillen equivalence of cofibrantly
generated monoidal model categories, with right adjoint U.  Suppose that A is a
monoid in C which is cofibrant. Then F induces a Quillen equivalence F :A-mod -!
F A-mod.

Proof.Since F is monoidal, F A is a monoid in D; since F preserves cofibrant
objects, F A is cofibrant in D. Given an A-module M, F M is an F A-module with
structure map F A ^ F M ~= F (A ^ M) -! F M.  Hence F does define a functor
F :A-mod -! F A-mod . Let j :X -! UF X denote the unit of the adjunction, and
let ": F UX -! X denote the counit. Given an F A-module N, UN is an A-module;
its structure map is given by the composite


              A ^ UN -j^1-!UF A ^ UN -! U(F A ^ N) -!UN

where the second map is adjoint to the composite
                            ~=                "^"
              F (UF A ^ UN) -! F UF A ^ F UN --! F A ^ N:

One can easily check that the resulting functor U :F A-mod -!  A-mod is right
adjoint to F .  The functor U clearly preserves fibrations and trivial fibratio*
 *ns,
so is a right Quillen functor. Furthermore, U reflects weak equivalences between
fibrant objects, since U does so as a functor from D to C. We are of course usi*
 *ng
Lemma 1.9, which tells us that we need only check that the map X -! ULF X is
a weak equivalence for cofibrant A-modules X.  Here L is a fibrant replacement
functor in F A-mod , not the fibrant replacement functor R in D.  Nevertheless,
there is a weak equivalence F X -! LF X, and since LF X is fibrant in D, there *
 *is
a map RF X -! LF X in D which is necessarily a weak equivalence. Thus the map
URF X -! ULF X is a weak equivalence in C. Since X is a cofibrant A-module,
and in particular cofibrant in C, the map X -! URF X is a weak equivalence. It_
follows that the map X -! ULF X is a weak equivalence as desired.           |__|


  We now discuss some of the properties of the model category A-mod .  If C is
a symmetric monoidal category, and A is a commutative monoid, then it is well
known that A-mod is also a symmetric monoidal category. We would like the model
structure on A-mod to be compatible with this symmetric monoidal structure.

Proposition 2.8.Suppose C is a cofibrantly generated symmetric monoidal model
category. In addition, suppose that either

  1. The unit S is cofibrant and A is a commutative monoid satisfying the condi-
     tions of Theorem 2.1; or
  2. A is a commutative monoid cofibrant in C.

Then the model category A-mod is a cofibrantly generated symmetric monoidal
model category.

Proof.It is well-known that A-mod is closed symmetric monoidal:  see [HSS98 ,
Section 2.2] for details. The symmetric monoidal structure is denoted ^A , and *
 *we
then have an analogous definition of A . Let I be the set of generating cofibra*
 *tions
of C and let J be the set of generating trivial cofibrations.  Then A ^ I is the
set of generating cofibrations of A-mod , and A ^ J is the set of generating tr*
 *ivial
cofibrations. We have

   (A ^ I) A (A ^ I) = (A ^ I)  I = A ^ (I  I)  A ^ (I-cof)  (A ^ I)-cof:



                      MONOIDAL MODEL CATEGORIES                    11


Thus f A g is a cofibration of A-modules if f and g are, using [SS97, Lemma
2.3].  See also [HSS98 , Corollary 5.3.5].  A similar argument shows that f A g
is a trivial cofibration if either f or g is.  Thus the pushout product part of*
 * the
definition of a monoidal model category holds. In the first case, we are done; *
 *since
S is cofibrant in C, A is cofibrant in A-mod .  In the second case, let QS be a
cofibrant replacement for the unit S in C, so that QS is cofibrant and we have a
weak equivalence QS -! S. Then A ^ QS is cofibrant in A-mod , and, since A is
cofibrant and C is monoidal, the A-module map A^QS -! A is a weak equivalence.
Thus A ^ QS is a cofibrant replacement for the unit A in A-mod . We must show
that, if M is a cofibrant A-module, then the map (A ^ QS) ^A M -! A ^A M = M
is still a weak equivalence. But (A ^ QS) ^A M ~=QS ^ M. Since M is cofibrant_
in C, the desired result holds.                                           |__|


  Note that the Quillen equivalences of Theorems 2.4 and 2.7 are monoidal Quill*
 *en
equivalences in case the monoids involved are commutative.
  In case A is not commutative, there is still a closed action of C on A-mod ; *
 *M ^X
is an A-module if M is an A-module and X is arbitrary. This action also respects
the model structures: f  g is a cofibration of A-modules if f is a cofibration *
 *of
A-modules and g is a cofibration. Furthermore, f g is a weak equivalence if eit*
 *her
f or g is. However, we will have trouble with the unit unless we assume either A
or S is cofibrant, just as above.
  Since Hirschhorn's landmark treatment [Hir97], it has become clear that the r*
 *ight
collection of model categories to work with is the collection of left proper ce*
 *llular
model categories. Hirschhorn shows that one can perform Bousfield localization *
 *in
this setting.

Proposition 2.9.Suppose C is a left proper cellular monoidal model category.
Suppose A is a monoid which is cofibrant in C. Then the model category A-mod is
also left proper and cellular.

  Because the definition of cellular is technical, the proof of this propositio*
 *n would
take us too far afield. It can be proved by the methods of [Hov98 , Section 6].



                             3.Algebras

  In this section we study the category A-algof algebras over a commutative
monoid A in a cofibrantly generated symmetric monoidal model category C. Note
that in case A = S, an S-algebra is the same thing as a monoid in C. Furthermor*
 *e,
an A-algebra is just a monoid in the symmetric monoidal category A-mod . We use
this to reduce to the case of monoids.
  The obvious definitions to make for a model structure on A-algare the followi*
 *ng.
We define a homomorphism f :X -! Y  of A-algebras to be a weak equivalence
(fibration) if and only if f is a weak equivalence (fibration) in C.  Then defi*
 *ne f
to be a cofibration if and only if f has the left lifting property with respect*
 * to all
homomorphisms of A-algebras which are both weak equivalences and fibrations.
  Note that the forgetful functor A-alg-!`C has a left adjoint, the free algebra
functor T . Of course, T (X) = A ^   n0 X^n, where X^0 = S. We will usually
use this only when A = S.
  We begin by slightly generalizing the main result of [SS97].



12                           MARK HOVEY


Theorem 3.1.  Suppose C is a cofibrantly generated symmetric monoidal model cat-
egory, with generating cofibrations I and generating trivial cofibrations J. Su*
 *ppose
the following conditions hold.

  1. The domains of I are small relative to (C ^ I)-cell.
  2. The domains of J are small relative to (C ^ J)-cell.
  3. The maps of (C ^ J)-cellare weak equivalences; i.e., the monoid axiom hold*
 *s.

Let A be a commutative monoid in C. Then A-algis a cofibrantly generated model
category where a map of A-algebras is a weak equivalence or fibration if and on*
 *ly if
it is so in C.

Proof.We first point out that we can assume that A = S. Indeed, our conditions
guarantee that A-mod is a cofibrantly generated model category (see Theorem 2.1)
which is symmetric monoidal, and that the pushout product half of the definitio*
 *n of
a monoidal model category holds (see the proof of Proposition 2.8). Furthermore,
we have A-mod ^A (A^I) = A-mod ^I  C^I. Thus the domains of I are small in
C relative to (A-mod ^A (A^I))-cell, and so the domains of A^I are small in A-m*
 *od
relative to (A-mod ^A (A ^ I))-cell. Similarly, the domains of A ^ J are small *
 *in
A-mod relative to (A-mod ^A (A^J))-cell. And the maps of (A-mod ^A (A^J))-cell
are in particular maps of (C ^ J)-cell, so are weak equivalences. Thus the cate*
 *gory
A-mod satisfies the same conditions as does C (except for the second half of the
definition of a monoidal model category), and so we may as well assume that A =*
 * S.
  It is well-known that the category S-algis bicomplete, and the two out of thr*
 *ee
and retract axioms are immediate consequences of the definitions. The cofibrati*
 *on-
trivial fibration half of the lifting axiom is also immediate.  For the factori*
 *zation
axioms, we use the sets T (I) and T (J).  Adjointness guarantees that the cofi-
brations in S-algare the elements of T (I)-cof, and the trivial fibrations are *
 *the
elements of T (I)-inj.  Similarly, the fibrations are the elements of T (J)-inj*
 *.  To
understand the maps of T (I)-cell, we need to understand the pushout in S-algof*
 * a
map T (g): T (K) -! T (L) through a map T (K) -! X. This pushout is described

in [SS97, Lemma 5.2] as a countable composition of maps Pi-hi!Pi+1. Each map
hi is a pushout in C of X^(i+1)^ gi, where gi is very similar to the inclusion *
 *of
the fat wedge into the product. That is, the target of gi is L^(i), and the sou*
 *rce
is analogous to the subset of the smash product where at least one term is in K.
In any case, one can see from the fact that C is monoidal that gi is a cofibrat*
 *ion if
g 2 I, and is a trivial cofibration if g 2 J. It follows that the maps of T (I)*
 *-cellare
in (C^I-cof)-cell, which is contained in (C^I)-cof. Since the domains of I are *
 *small
relative to (C^I)-cell, they are also small relative to (C^I)-cofby [Hir97, The*
 *orem
12.4.21]. Thus, by adjointness, the domains of T (I) are small in S-algrelative*
 * to
T (I)-cell.  The small object argument then gives the desired factorization int*
 *o a
cofibration followed by a trivial fibration.
  A very similar argument shows that the domains of J are small relative to
T (J)-cell, and also that the maps of T (J)-cell, since they are in (C ^ J)-cof*
 *, are
weak equivalences. Thus the small object argument applied to T (J) gives the de-
sired factorization into a trivial cofibration followed by a fibration. The pro*
 *of that
trivial cofibrations have the left lifting property with respect to fibrations_*
 *then uses
the Retract Argument, as in the proof of Theorem 2.1.                      |__|

Example 3.2. For an example where Theorem 3.1 applies but the simpler version
of [SS97] does not, let C be the category of compactly generated spaces. We have



                      MONOIDAL MODEL CATEGORIES                    13


already seen that the monoid axiom holds here (see Lemma 2.3). One can also ver*
 *ify
that the elements of (C ^ I)-cellare closed inclusions, using the results of [L*
 *ew78 ,
Appendix A]. It follows that the domains of I are small relative to (C ^ I)-cel*
 *l, and
also that the domains of J are small relative to (C ^ J)-cell.  Thus we do get a
model category of monoids of compactly generated spaces. On the other hand, if
we let C be the category of k-spaces, the monoid axiom holds, but the necessary
smallness conditions do not, so far as we know. So we do not get a model catego*
 *ry
of monoids of k-spaces.

  It is interesting to note that, in the situation of Theorem 3.1, the category*
 * A-alg
is a model category even though the category A-mod may not be a monoidal model
category unless either A or S is cofibrant.
  In any case, we are really interested in the case where the hypotheses of The*
 *o-
rem 3.1 do not hold. In this case, we obtain the following result.

Theorem 3.3.  Suppose C is a cofibrantly generated symmetric monoidal model
category. Suppose that A is either S or a commutative monoid which is cofibrant
in C. Then the category A-algis almost a model category, in the following preci*
 *se
sense.

  1. A-algis bicomplete and the two out of three and retract axioms hold.
  2. Cofibrations have the left lifting property with respect to trivial fibrat*
 *ions, and
     trivial cofibrations whose source is cofibrant in A-mod have the left lift*
 *ing
     property with respect to fibrations.
  3. Every map whose source is cofibrant in A-mod can be functorially factored
     into a cofibration followed by a trivial fibration, and also can be functo*
 *rially
     factored into a trivial cofibration followed by a fibration.

Furthermore, cofibrations whose source is cofibrant in A-mod are cofibrations in
A-mod, and fibrations and trivial fibrations are closed under pullbacks.

Proof.The category A-algis the category of monoids in A-mod , which itself is a
cofibrantly generated monoidal model category, by Proposition 2.8. Thus we can
assume that A = S. We have already seen that bicompleteness and the retract and
two out of three axioms hold. The lifting axiom for cofibrations and trivial fi*
 *brations
holds by definition. As before, adjointness implies that the trivial fibrations*
 * form
the class T (I)-inj, so the cofibrations form the class T (I)-cof. The fibratio*
 *ns form
the class T (J)-inj, so the elements of T (J)-cofhave the left lifting property*
 * with
respect to fibrations.  Recall from the proof of Theorem 3.1 that the pushout in

S-algof a map T (K) -T(g)--!T (L) through a map T (K) -! X of monoids is a

countable composition of maps Pi -fi!Pi+1, where fi is the pushout in C of a
map X^(i+1)^ gi. The map gi is, as we have said, a cofibration if g is so, and a
trivial cofibration if g is so. Thus, if X is cofibrant in C, a pushout in M(C)*
 * of a
map of T (I) through a map to X is a cofibration in C and a pushout of a map
of T (J) through a map to X is a trivial cofibration in C. Since the domains of*
 * I
are small relative to cofibrations in C, it follows by adjointness that the dom*
 *ains of
T (I) are small relative to transfinite compositions of pushouts of maps of T (*
 *I) as
long as the initial stage is cofibrant in C. The small object argument then all*
 *ows
us to functorially factor any map in S-algwhose source is cofibrant in C into a
cofibration followed by a trivial fibration. A similar argument, with T (J) rep*
 *lacing
T (I), allows us to functorially factor any map in S-algwhose source is cofibra*
 *nt in
C into a trivial cofibration followed by a fibration.



14                           MARK HOVEY


  Now, given any cofibration f in M(C) whose source is cofibrant, we can factor
f = pi, where i is a transfinite composition of pushouts of maps of T (I), and *
 *is
therefore a cofibration in C, and p is a trivial fibration. The retract argumen*
 *t shows
that f is a retract of i, and so is also a cofibration in C. Now suppose f is a*
 * trivial
cofibration, again with source cofibrant in C. Then we can factor f = qj, where
q is a fibration of monoids and j is a transfinite composition of pushouts of T*
 * (J).
Since j is also a weak equivalence, so is q. Thus f has the left lifting proper*
 *ty with
respect to q, and so f is a retract of j. Then f has the left lifting property_*
 *with
respect to fibrations, as desired.                                         |__|

  We should point out that it is not even clear that the cofibrations T (J) are*
 * weak
equivalences in M(C). This will be true if we can choose the domains of J to be
cofibrant.
  The information contained in Theorem 3.3 is enough to carry out most of the
usual constructions in model category theory, at least as long as we have some
element of A-algwhich is cofibrant in A-mod . For this, we must require that the
unit S is cofibrant in C. Then the initial object A of A-algis cofibrant in A-m*
 *od .
Hence there is a cofibrant replacement functor Q in A-algand a natural trivial
fibration QX -! X. There is a fibrant replacement functor R defined on A-algebr*
 *as
which are cofibrant in A-mod , in particular on cofibrant A-algebras, and a nat*
 *ural
trivial cofibration X -! RX. In particular, though the composite functor QR does
not make sense in general, the composite RQ does, and we find that the quotient
category of A-algobtained by inverting the weak equivalences is equivalent to t*
 *he
quotient category of the cofibrant and fibrant objects.
  One does have to be careful though. We can not recognize trivial cofibrations*
 * in
A-algas the class of maps with the left lifting property with respect to fibrat*
 *ions.
But a map in A-algwhose source is cofibrant in A-mod is a trivial cofibration i*
 *f and
only if it has the left lifting property with respect to all fibrations. Simila*
 *rly, one
does not know that a pushout of a trivial cofibration f in A-algthrough a map g
is a trivial cofibration unless both the source of f and the target of g are co*
 *fibrant
in A-mod . Cylinder and path objects need not exist unless the object in questi*
 *on
is cofibrant in A-mod .
  Nevertheless, we can follow the usual definitions in the standard constructio*
 *n of
the homotopy category of a model category. See for example [Hov97 , Chapter 1],
[DS95 ], or [Hir97, Chapters 8 and 9]. In order for the notion of left homotopy*
 * to
have any content, one must assume that the sources of one's maps are cofibrant *
 *in
A-mod . Similarly, for right homotopy, one must assume that the target is cofib*
 *rant
in A-mod . It is not clear that right homotopy is an equivalence relation if th*
 *e target
is cofibrant in A-mod and fibrant, as one would expect. One must first prove th*
 *at
the notions of left and right homotopy coincide if the source is cofibrant and *
 *the
target is cofibrant and fibrant. Then, since left homotopy is an equivalence re*
 *lation
in that situation, so is right homotopy.
  In the end, we obtain the following theorem.

Theorem 3.4.  Suppose C is a cofibrantly generated symmetric monoidal model
category, where the unit S is cofibrant. Suppose A is a commutative monoid which
is cofibrant in C.  Let Ho A-algdenote the category obtained from A-algby for-
mally inverting the weak equivalences. Then there is an equivalence of categori*
 *es
(A-alg)cf= ~-! Ho A-alg, where (A-alg)cf is the full subcategory of cofibrant a*
 *nd
fibrant A-algebras, and ~ denotes the homotopy equivalence relation. In particu*
 *lar,



                      MONOIDAL MODEL CATEGORIES                    15


Ho A-algexists. A map in A-algis an isomorphism in Ho A-algif and only if it is
a weak equivalence.

  We now show that the homotopy category of monoids is homotopy invariant.
The following theorem is analogous to Theorem 2.4.

Theorem 3.5.  Suppose C is a cofibrantly generated symmetric monoidal model cat-
egory such that the unit S is cofibrant and the domains of the generating cofib*
 *rations
can be taken to be cofibrant.  Suppose f :A -! A0 is a weak equivalence of com-
mutative monoids which are cofibrant in C. Then the derived functors of inducti*
 *on
and restriction define an adjoint equivalence of categories Ho A-alg-! HoA0-alg.

Proof.One can easily check that induction which takes M to A0^A M, defines
a functor from A-algebras to A0-algebras, which is left adjoint to the restrict*
 *ion
functor. The restriction functor obviously preserves fibrations and trivial fib*
 *rations,
and reflects weak equivalences between fibrant objects.  It follows that induct*
 *ion
preserves cofibrations, and those trivial cofibrations whose source is cofibran*
 *t in
A-mod . One can then easily check that the functor which takes M to A0^A QM is
the total left derived functor of induction, and that its right adjoint is the *
 *functor
which takes M to RQM. The situation is very similar to the usual Quillen functor
formalism, it is just that we need to apply Q before applying the fibrant repla*
 *cement
functor R since R is not globally defined.
  The same argument as in Lemma 1.9 applies, so we are reduced to showing that
the map M -! A0^A M is a weak equivalence for all cofibrant A-algebras M. But
since every cofibrant A-algebra is also a cofibrant A-module, Theorem 2.4_compl*
 *etes
the proof.                                                            |__|

  Then the following theorem is analogous to Theorem 2.7.

Theorem 3.6.  Suppose F :C -!D is a monoidal Quillen equivalence of cofibrantly
generated symmetric monoidal model categories. Suppose as well that S is cofibr*
 *ant
in C, and A is a commutative monoid which is cofibrant in C. Then F induces an
equivalence of categories LF : HoA-alg-! HoF A-alg.

Proof.Let U denote the right adjoint of F . It is clear that F defines a functo*
 *r from
A-algebras to F A-algebras. Indeed, F defines a monoidal functor from A-modules
to F A-modules. To see that U defines a functor going the other way, note that *
 *we
have a natural map UX ^A UY -! U(X ^FA Y ) adjoint to the composite

              F (UX ^A UY ) ~=F UX ^FA F UY -"^"-!X ^FA Y

Thus, if X is an A-algebra, so is UX; the unit of UX is adjoint to the unit of
X using the isomorphism. One can easily check that F :A-alg-! F A-algis left
adjoint to U :F A-alg-! A-alg.
  The functor U preserves fibrations and trivial fibrations of monoids. It foll*
 *ows
that F preserves cofibrations and those trivial cofibrations with source cofibr*
 *ant
in A-mod .  We can then define (LF )X = F (QX) as usual.  We have to define
(RU)X = U(RQX) however, since RX does not make sense in general. The usual
argument shows that LF : HoA-alg-! HoF A-algis left adjoint to RU.
  The functor U reflects weak equivalences between fibrant objects.  A similar
argument as in Lemma 1.9 implies that we need only check that the map X -!
URF X is a weak equivalence for cofibrant A-algebras X.  Note that F X is still
cofibrant, so we do not need to consider URQF X, though of course we could do so



16                           MARK HOVEY


without difficulty. Here R is a fibrant replacement functor in F A-alg. If we l*
 *et L
denote a fibrant replacement functor in F A-mod , then we know from Theorem 2.7
that X -! ULF X is a weak equivalence, since X is also cofibrant as an A-module.
A simple lifting argument shows that there is a weak equivalence LF X -! RF X in
F A-mod . Since U preserves weak equivalences between fibrant objects, this giv*
 *es
a weak equivalence ULF X -! URF X. Hence X -! URF X is a weak equivalence_
as well.                                                              |__|

Example 3.7. For example, we can take C to be the symmetric monoidal model
category of simplicial symmetric spectra [HSS98 ].  Here the monoid axiom holds
and everything is small relative to the whole category, so we get a model categ*
 *ory
of S-algebras. We take D to be the symmetric monoidal model category of topo-
logical symmetric spectra. We do not know whether the monoid axiom holds here.
Nonetheless, the geometric realization is a monoidal Quillen equivalence C -! D,
so defines an equivalence of categories Ho S-alg-! HoS-algbetween the homotopy
categories of S-algebras.

  Note that the homotopy category of an arbitrary model category has a closed
action by the homotopy category of simplicial sets [Hov97 , Chapter 5]. This re*
 *sult
will still hold for Ho R-alg; one can use the same approach as in [Hov97 , Chap*
 *ter 5],
replacing R by RQ everywhere. In particular, there are mapping spaces of monoid*
 *s.
The equivalence of Theorem 3.6 will preserve these mapping spaces.

                             References

[DHK]    W. G. Dwyer, P. S. Hirschhorn, and D. M. Kan, Model categories and gen*
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  Department of Mathematics, Wesleyan University, Middletown, CT
  E-mail address: hovey@member.ams.org