Algebras and modules
in monoidal model categories
Stefan Schwede and Brooke E. Shipley1
Abstract: We construct model category structures for monoids
and modules in symmetric monoidal model categories, with appli
cations to symmetric spectra and spaces.
1991 AMS Math. Subj. Class.: primary 55U35, secondary 18D10
1 Summary
This paper gives a general approach for obtaining model category structures for*
* algebras or
modules over some other model category. Technically, what we mean by an `algebr*
*a' is a monoid
in a symmetric monoidal category. Of course, the symmetric monoidal and model *
*category
structures have to be compatible, which leads to the definition of a monoidal m*
*odel category,
see Definition 2.1. To obtain a model category structure of algebras we have to*
* introduce one
further axiom, the monoid axiom (Definition 2.2). A filtration on certain pusho*
*uts of monoids
(see Lemma 5.2) is then used to reduce the problem to standard model category a*
*rguments
based on Quillen's "small object argument". Our main result is stated in Theore*
*m 3.1.
This approach was developed in particular to apply to the category of symmetric*
* spectra defined
in [HSS] and to spaces in [Sch2]. In both of these categories we thus obtain m*
*odel categories
for the associative monoids, the Rmodules for any monoid R, and the Ralgebras*
* for any
commutative monoid R. A significant shortcut is possible if the underlying mon*
*oidal model
category has the special property that all objects are fibrant, see Remark 3.5.*
* This is not true
for our main examples, symmetric spectra and spaces. It does hold though, in t*
*he monoidal
model categories of simplicial abelian groups, chain complexes, or Smodules (i*
*n the sense of
[EKMM]).
We assume that the reader is familiar with the language of homotopical algebra *
*(cf. [Q], [DS])
and with the basic ideas concerning monoidal and symmetric monoidal categories *
*(cf. [MacL,
VII], [Bor, 6]) and triples (also called monads, cf. [MacL, VI.1], [Bor, 4]).
Acknowledgments. We would first like to thank Charles Rezk for conversations wh*
*ich led us to
the filtration that appears in Lemma 5.2. We also benefited from several conver*
*sations about
this project with Bill Dwyer, Mark Hovey and Manos Lydakis. We would also like *
*to thank Bill
Dwyer, Phil Hirschhorn, and Dan Kan for sharing the draft of [DHK] with us. In *
*Appendix A
we recall the notion of a cofibrantly generated model category from their book.
_______________________________1
Research partially supported by an NSF Postdoctoral Fellowship
1
2 Monoidal model categories
A monoidal model category is essentially a model category with a compatible clo*
*sed symmetric
monoidal product. The compatibility is expressed by the pushout product axiom b*
*elow. In this
paper we always require a closed symmetric monoidal product although for exposi*
*tory ease we
refer to these categories as just `monoidal' model categories. One could also *
*consider model
categories enriched over a monoidal model category with certain compatibility r*
*equirements
analogous to the pushout product axiom or the simplicial axiom of [Q, II.2]. F*
*or example,
closed simplicial model categories [Q, II.2] are such compatibly enriched categ*
*ories over the
monoidal model category of simplicial sets.
We also introduce the monoid axiom which is the crucial ingredient for lifting *
*the model category
structure to monoids and modules. Examples of monoidal model categories satisf*
*ying the
monoid axiom are given in Section 4.
Definition 2.1A model category C is a monoidal model category if it is endowed *
*with a closed
symmetric monoidal structure and satisfies the following pushout product axiom.*
* We will
denote the symmetric monoidal product by ^, the unit by I and the internal Hom *
*object by
[; ].
Pushout product axiom. Let A ! B and K ! L be cofibrations in C. Then th*
*e map
A^ L [A^ K B ^K ! B ^L
is also a cofibration. If in addition one of the former maps is a weak equivale*
*nce, so is the latter
map.
If C is a category with a monoidal product ^ and I is a class of maps in C, we *
*denote by I ^C
the class of maps of the form
A^ Z  ! B ^Z
for A ! B a map in I and Z an object of C. We also denote by Icofregthe clas*
*s of
maps obtained from the maps of I by cobase change and composition (possibly tra*
*nsfinite, see
Appendix A.) These maps are referred to as the regular Icofibrations.
Definition 2.2A monoidal model category C satisfies the monoid axiom if every m*
*ap in
({acyc. cofibrations}^ C)cofreg
is a weak equivalence.
Note that if C has the special property that every object is cofibrant, then th*
*e monoid axiom is
a consequence of the pushout product axiom. However, this special situation rar*
*ely occurs in
practice.
In Appendix A we recall cofibrantly generated model categories. In these model*
* categories
fibrations can be detected by checking the right lifting property against a set*
* of maps, called
generating acyclic cofibrations, and similarly for acyclic fibrations. This is *
*in contrast to general
model categories where the lifting property has to be checked against the whole*
* class of acyclic
cofibrations. In cofibrantly generated model categories, the pushout product a*
*xiom and the
monoid axiom only have to be checked for the generating (acyclic) cofibrations:
2
Lemma 2.3 Let C be a cofibrantly generated model category endowed with a close*
*d symmetric
monoidal structure.
1.If the pushout product axiom holds for the generating cofibrations and the *
*generating
acyclic cofibrations, then it holds in general.
2.Let J be a set of generating acyclic cofibrations. If every map in (J ^C)c*
*ofregis a weak
equivalence, then the monoid axiom holds.
Proof: For the first statement consider a map i : A ! B in C. Denote by G(i) t*
*he class of
maps j :K ! L such that the pushout product
A^ L [A^ K B ^K  ! B ^L
is a cofibration. This pushout product has the left lifting property with resp*
*ect to a map
f :X ! Y if and only if j has the left lifting property with respect to the map
p : [B; X] ! [B; Y ] x[A;Y[]A; X]:
Hence, a map is in G(i) if and only if it has the left lifting property with re*
*spect to the map p
for all f :X ! Y which are acyclic fibrations in C.
G(i) is thus closed under cobase change, transfinite composition and retracts. *
*If i : A ! B
is a generating cofibration, G(i) contains all generating cofibrations by assum*
*ption; because
of the closure properties it thus contains all cofibrations, see Lemma A.1. Rev*
*ersing the roles
of i and an arbitrary cofibration j : K ! L we thus know that G(j) contains al*
*l generating
cofibrations. Again by the closure properties, G(j) contains all cofibrations, *
*which proves the
pushout product axiom for two cofibrations. The proof of the pushout product be*
*ing an acyclic
cofibration when one of the constituents is, follows in the same manner.
For the second statement note that by the small object argument, Lemma A.1, eve*
*ry acyclic
cofibration is a retract of a transfinite composition of cobase changes along t*
*he generating
acyclic cofibrations. Since transfinite compositions of transfinite composition*
*s are transfinite
compositions, every map in ({acyc. cofibrations}^ C)cofregis thus a retract of*
* a map in (J ^C)
cofreg. __*
*_
3 Model categories of algebras and modules
In this section we state the main theorem, Theorem 3.1, which constructs model *
*categories for
algebras and modules. The proof of this theorem is delayed to section 5. Exampl*
*es of model
categories for which this theorem applies are given in section 4. We end this s*
*ection with two
theorems which compare the homotopy categories of modules or algebras over weak*
*ly equivalent
monoids.
We consider a symmetric monoidal category with product ^ and unit I. A monoid i*
*s an object
R together with a "multiplication" map R ^R ! R and a "unit" I ! R which sati*
*sfy
certain associativity and unit conditions (see [MacL, VII.3]). R is a commutati*
*ve monoid if the
multiplication map is unchanged when composed with the twist, or the symmetry i*
*somorphism,
of R ^R. If R is a monoid, a left Rmodule ("object with left Raction" in [Mac*
*L, VII.4]) is an
3
object N together with an action map R ^N ! N satisfying associativity and uni*
*t conditions
(see again [MacL, VII.4]). Right Rmodules are defined similarly.
Assume that C has coequalizers. Then there is a smash product over R, denoted M*
* ^RN, of
a right Rmodule M and a left Rmodule N. It is defined as the coequalizer, in *
*C, of the two
maps M ^R ^N !!M ^N induced by the actions of R on M and N respectively*
*. If R is
a commutative monoid, then the category of left Rmodules is isomorphic to the *
*category of
right Rmodules, and we simply speak of Rmodules. In this case, the smash prod*
*uct of two
Rmodules is another Rmodule and smashing over R makes Rmod into a symmetric *
*monoidal
category with unit R. If C has equalizers, there is also an internal Hom object*
* of Rmodules,
[M; N]R. It is the equalizer of two maps [M; N] !![R ^M; N]. The first *
*map is induced by
the action of R on M, the second map is the composition of R ^ : [M; N] ! [R *
*^M; R ^N]
followed by the map induced by the action of R on N.
For a commutative monoid R, an Ralgebra is defined to be a monoid in the categ*
*ory of R
modules. It is a formal property of symmetric monoidal categories (cf. [EKMM, V*
*II 1.3]) that
specifying an Ralgebra structure on an object A is the same as giving A a mono*
*id structure
together with a monoid map f :R ! A which is central in the sense that the fol*
*lowing diagram
commutes. switch id^f
R ^A _____Aw^R _____Aw^A
 
f ^id mult.
u u
A ^A __________________Awmult.
Now we can state our main theorem. It essentially says that monoids, modules an*
*d algebras in
a cofibrantly generated, monoidal model category C again form a model category *
*if the monoid
axiom holds. (See Appendix A for the definition of a cofibrantly generated mode*
*l category.) To
simplify the exposition, we assume that all objects in C are small (refer to Ap*
*pendix A) relative
to the whole category. This last assumption can be weakened as indicated in A.5*
*. The proofs
will be delayed until the last section.
In the categories of monoids, left Rmodules (when R is a fixed monoid), and R*
*algebras (when
R is a commutative monoid) a morphism is defined to be a fibration or weak equi*
*valence if it
is a fibration or weak equivalence in the underlying category C. A morphism is *
*a cofibration if
it has the left lifting property with respect to all acyclic fibrations.
Theorem 3.1 Let C be a cofibrantly generated, monoidal model category. Assume f*
*urther that
every object in C is small relative to the whole category and that C satisfies *
*the monoid axiom.
1.Let R be a monoid in C. Then the category of left Rmodules is a cofibrantl*
*y generated
model category.
2.Let R be a commutative monoid in C. Then the category of Rmodules is a cof*
*ibrantly
generated, monoidal model category satisfying the monoid axiom.
3.Let R be a commutative monoid in C. Then the category of Ralgebras is a co*
*fibrantly
generated model category. If the unit I of the smash product is cofibrant i*
*n C, then every
cofibration of Ralgebras whose source is cofibrant in C is also a cofibrat*
*ion of Rmodules.
In particular, any cofibrant Ralgebra is cofibrant as an Rmodule.
If in part (3) of the theorem we take R to be the unit of the smash product, we*
* see that in
particular the category of monoids in C forms a model category.
4
Remark 3.2 The full strength of the monoid axiom is not necessary to obtain a m*
*odel category
of Rmodules for a particular monoid R. In fact, to get hypothesis (1) of Lemma*
* A.3 for R
modules, one need only know that every map in
({acyc. cofibrations}^ R) cofreg
is a weak equivalence. This holds, independent of the monoid axiom, if R is cof*
*ibrant in the
underlying category C. For then the pushout product axiom implies that smashin*
*g with R
preserves acyclic cofibrations.
The following theorems concern comparisons of homotopy categories of modules an*
*d algebras.
The homotopy theory of Rmodules and Ralgebras should only depend on the weak *
*equivalence
type of the monoid R. To show this for Rmodules we must require that the funct*
*or ^ RN take
any weak equivalence of right Rmodules to a weak equivalence in C whenever N i*
*s a cofibrant
left Rmodule. In all of our examples this added property of the smash product *
*holds. For the
comparison of Ralgebras, we also require that the unit of the smash product is*
* cofibrant.
Theorem 3.3 Assume that for any cofibrant left Rmodule N, ^ RN takes weak equ*
*ivalences
of right Rmodules to weak equivalences in C. If R ~!S is a weak equivalence o*
*f monoids, then
the total derived functors of restriction and extension of scalars induce equiv*
*alences of homotopy
categories
Ho(Rmod) ~= Ho (Smod) :
Proof: This is an application of Quillen's adjoint functor theorem ([Q, I.4 Thm*
*. 3] or [DS,
Thm. 9.7]). The weak equivalences and fibrations are defined in the underlying*
* symmetric
monoidal category, hence the restriction functor preserves fibrations and acycl*
*ic fibrations. By
assumption, for N a cofibrant left Rmodule
N ~=R ^RN ! S ^RN
is a weak equivalence. Thus if Y is a fibrant left Smodule, an Rmodule map N *
*! Y is a
weak equivalence if and only if the adjoint Smodule map S ^RN ! Y is a weak e*
*quivalence.
[DS, Thm. 9.7] then gives the desired result. *
* ___
Theorem 3.4 Suppose that the unit I of the smash product is cofibrant in C and *
*that for any
cofibrant left Rmodule N, ^ RN takes weak equivalences of right Rmodules to *
*weak equiva
lences in C. Then for a weak equivalence of commutative monoids R ~! S, the to*
*tal derived
functors of restriction and extension of scalars induce equivalences of homotop*
*y categories
Ho(Ralg) ~= Ho (Salg) :
Proof: The proof is similar to the one of the previous theorem. Again the rig*
*ht adjoint
restriction functor does not change underlying objects, so it preserves fibrati*
*ons and acyclic
fibrations. Since cofibrant Ralgebras are also cofibrant as Rmodules (Thm. 3.*
*1 (3)), for any
cofibrant Ralgebra the adjunction morphism is again a weak equivalence. So [DS*
*, Thm. 9.7]
applies one more time. _*
*__
Remark 3.5 Some important examples of monoidal model categories have the proper*
*ty that
all objects are fibrant. This greatly simplifies the situation. If there is a*
*lso a simplicial or
topological model category structure and if a simplicial (resp. topological) tr*
*iple T acts, then
the category of T algebras is again a simplicial (topological) category, so it*
* has path objects.
Hence hypothesis (2) of Lemma A.3 applies. One example of this situation is the*
* category of
Smodules in [EKMM]. Lemma A.3 (2) should be compared to [EKMM, Thm. VII 4.7].
5
Remark 3.6 We point out again that in our main examples, symmetric spectra and *
*spaces,
not all objects are fibrant, which is why we need a more complicated approach. *
*In the fibrant
case, one gets model category structures for algebras over all reasonable (e.g.*
* continuous or
simplicial) triples, whereas our monoid axiom approach only applies to the free*
* Rmodule and
free Ralgebra triples. The category of commutative monoids often has a model c*
*ategory struc
ture in the fibrant case (e.g. commutative simplicial rings or commutative Sal*
*gebras [EKMM,
Cor. VII 4.8]). In contrast, for spaces and symmetric spectra, the category of*
* commutative
monoids can not form a model category with fibrations and weak equivalences def*
*ined in the
underlying category. For if such a model category structure existed, one could *
*choose a fibrant
replacement of the unit S0 inside the respective category of commutative monoid*
*s. Evaluating
this fibrant representative on 1+ 2 op, or at level 0 respectively, would give *
*a commutative
simplicial monoid weakly equivalent to QS0. This would imply that the space QS0*
* is weakly
equivalent to a product of EilenbergMacLane spaces, which is not the case. The*
* homotopy cat
egory of commutative monoids in symmetric spectra is still closely related to E*
*1 ring spectra,
though.
4 Examples
Simplicial sets.
The category of simplicial sets has a wellknown model category structure estab*
*lished by D.
Quillen [Q, II 3, Thm. 3]. The cofibrations are the degreewise injective maps, *
*the fibrations are
the Kan fibrations and the weak equivalences are the maps which become homotopy*
* equivalences
after geometric realization. This model category is cofibrantly generated. The *
*standard choice
for the generating (acyclic) cofibrations are the inclusions of the boundaries *
*(resp. horns) into
the standard simplices. Here every object is small with respect to the whole ca*
*tegory.
The cartesian product of simplicial sets is symmetric monoidal with unit the di*
*screte onepoint
simplicial set. The pushout product axiom is wellknown in this case, (see [GZ,*
* IV Prop. 2.2],
[Q, II 3, Thm. 3]). Since every simplicial set is cofibrant, the monoid axiom f*
*ollows from the
pushout product axiom. A monoid in the category of simplicial sets under cartes*
*ian product is
just a simplicial monoid, i.e., a simplicial object of ordinary unital and asso*
*ciative monoids. So
the main theorem, Theorem 3.1 (3), recovers Quillen's model category structure *
*for simplicial
monoids [Q, II 4, Thm. 4, and Rem. 1, p. 4.2].
spaces and symmetric spectra
These two examples are new. In fact, the justification for writing this paper i*
*s to give a unified
treatment of why monoids and modules in these categories form model categories.*
* Here we
only give an overview; for the details the reader may consult [Se], [BF], [Ly] *
*and [Sch2] in
the case of spaces, and [HSS] in the case of symmetric spectra. The particular*
* interest in
these categories comes from the fact that they model stable homotopy theory. Th*
*e homotopy
category of symmetric spectra is equivalent to the usual stable homotopy catego*
*ry of algebraic
topology. In the case of spaces, one obtains the stable homotopy category of *
*connective
(i.e., (1)connected) spectra. Monoids in either of these categories are thus *
*possible ways of
defining `brave new rings', i.e., rings up to homotopy with higher coherence co*
*nditions. Another
approach to this idea consists of the Salgebras of [EKMM].
6
spaces. spaces were introduced by G. Segal [Se] who showed that they give ris*
*e to a homo
topy category equivalent to the usual homotopy category of connective spectra. *
*A. K. Bousfield
and E. M. Friedlander [BF] considered a bigger category of spaces in which the*
* ones introduced
by Segal appeared as the special spaces. Their category admits a simplicial mo*
*del category
structure with a notion of stable weak equivalence giving rise again to the hom*
*otopy theory of
connective spectra. Then M. Lydakis [Ly] showed that spaces admit internal fun*
*ction objects
and a symmetric monoidal smash product with nice homotopical properties. Small*
*ness and
cofibrant generation for spaces is verified in [Sch2], as well as the pushout *
*product and the
monoid axiom. The monoids in this setting are called Gammarings.
Symmetric spectra. The category of symmetric spectra, Sp , is described in [HSS*
*]. There it
is also shown that this category is a cofibrantly generated, monoidal model cat*
*egory, and that
the associated homotopy category is equivalent to the usual homotopy category o*
*f spectra. For
symmetric spectra over the category of simplicial sets every object is small wi*
*th respect to the
whole category. The monoid axiom and the fact that smashing with a cofibrant le*
*ft Rmodule
preserves weak equivalences between right Rmodules are verified in [HSS]. The *
*monoids in this
setting are called symmetric ring spectra.
Fibrant examples: simplicial abelian groups, chain complexes and Smodules
These are the examples of monoidal model categories in which every object is fi*
*brant. With
this special property it is easier to lift model category structures since the *
*(often hard to verify)
condition (1) of the lifting lemma A.3 is a formal consequence of fibrancy and *
*the existence
of path objects, see the proof of A.3. For example, the commutative monoids som*
*etimes form
model categories in these cases. The pushout product and monoid axioms also hol*
*d in these
examples, but since the fibrancy property deprives them of their importance, we*
* will not bother
to prove them.
Simplicial abelian groups. The model category structure for simplicial abelian*
* groups was
established by Quillen [Q, II.6]. The weak equivalences and fibrations are defi*
*ned on underlying
simplicial sets. The cofibrations are the retracts of the free maps (see [Q, II*
* p. 4.11, Rem. 4]).
This model category is cofibrantly generated and all objects are small. The (de*
*greewise) tensor
product provides a symmetric monoidal product for simplicial abelian groups. Th*
*e unit for this
product is the integers, considered as a constant simplicial abelian group. A m*
*onoid then is
nothing but a simplicial ring. These have path objects given by the simplicial *
*structure. This
means that for a simplicial ring R the simplicial set Hom([1]; R) of maps of th*
*e standard
1simplex into the underlying simplicial set of R is naturally a simplicial rin*
*g. The model
category structure for simplicial rings and simplicial modules was established *
*by Quillen in [Q,
II.4, Thm. 4] and [Q, II.6].
Chain complexes. The category of nonnegatively graded chain complexes over a c*
*ommutative
ring k forms a model category, see [Q, II p. 4.11, Remark 5], [DS, Section 7]. *
*The weak equiv
alences are the maps inducing homology isomorphisms, the fibrations are the map*
*s which are
surjective in positive degrees, and cofibrations are monomorphisms with degreew*
*ise projective
cokernels. This model category is cofibrantly generated and every object is sma*
*ll. The cate
gory of unbounded chain complexes over k, although less well known, also forms *
*a cofibrantly
generated model category with weak equivalences the homology isomorphism and fi*
*brations the
epimorphisms, see [HPS], remark after Thm. 9.3.1. The cofibrations here are sti*
*ll degreewise
split injections, but their description is a bit more complicated than for boun*
*ded chain com
plexes. The following remarks refer to this category of Zgraded chain complexe*
*s of kmodules.
7
The graded tensor product of chain complexes is symmetric monoidal and has adjo*
*int internal
homcomplexes. A monoid in this symmetric monoidal category is a differential g*
*raded algebra
(DGA). Every complex is fibrant and associative DGAs have path objects. To cons*
*truct them,
we need the following 2term complex denoted I. In degree 0, I consists of a fr*
*ee kmodule
on two generators [0] and [1]. In degree 1, I is a free kmodule on a single ge*
*nerator . The
differential is given by d = [1]  [0]. This complex becomes a coassociative a*
*nd counital
coalgebra when given the comultiplication
: I ! I k I
defined by ([0]) = [0] [0]; ([1]) = [1] [1]; () = [0] + [1]. The counit m*
*ap I ! k
sends both [0] and [1] to 1 2 k. The two inclusions k ! I given by the generat*
*ors in degree 0
and the counit are maps of coalgebras. Note that the comultiplication of I is n*
*ot cocommutative
(this is reminiscent of the failure of the AlexanderWhitney map to be commutat*
*ive).
For any coassociative, counital differential graded coalgebra C, and any DGA A,*
* the internal
Homchain complex HomCh(C; A)* becomes a DGA with multiplication
f . g = A O (f g) O C
where A is the multiplication of A and C is the comultiplication of C. In part*
*icular,
HomCh(I; A) is a DGA, and it comes with DGA maps from A and to A x A which make
it into a path object. In this way we recover the model category structure for *
*associative DGAs
over a commutative ring, first discovered by J. F. Jardine [J]. Our approach is*
* a bit more gen
eral, since we can define similar path objects for associative DGAs over a fixe*
*d commutative
DGA, and for modules over a fixed DGA A. We thus also get model categories in t*
*hose cases.
However, since the basic differential graded coalgebra I is not cocommutative, *
*this does not
provide path objects for commutative DGAs.
Smodules. The model category of Smodules, MS, is described in [EKMM, VII 4.6*
*]. This
model category structure is cofibrantly generated (see [EKMM, VII 5.6 and 5.8])*
*. To ease
notation, let Fq = S ^L L1q(), the functor from topological spaces to MS that *
*is used to
define the model category structure on Smodules. In our terminology, the gener*
*ating (acyclic)
cofibrations are obtained by applying Fq to the generators for topological spac*
*es, Sn ! CSn
(CSn ! CSn ^ I+), where CX is the cone on X. The associative monoids are the S*
*algebras.
The difficult part for showing that model category structures can be lifted to *
*the categories
of modules and algebras in this case is verifying the smallness hypothesis. Thi*
*s is where the
"Cofibration Hypothesis" comes in, see [EKMM, VII 5.2]. The underlying category*
* of Smodules
is a topological model category, see [EKMM, VII 4.4] and the triples in questio*
*n are continuous.
Hence, Remark 3.5 applies to give path objects, recovering [EKMM, VII 4.7], in *
*particular the
model category structures for Ralgebras and Rmodules. Our module comparison *
*theorem
3.3 recovers [EKMM, III 4.2]. Our method of comparing algebra categories over *
*equivalent
commutative monoids does not apply here because the unit of the smash product i*
*s not cofibrant.
5 Proofs
Proof of Theorem 3.1 (1). The category of Rmodules is also the category of alg*
*ebras over
the triple TR where TR(M) = R^M. The triple structure for TR comes from the mul*
*tiplication
R ^ R ! R. This theorem is a direct application of Lemma A.3 since by the mono*
*id axiom,
the JTcofibrations are weak equivalences. *
* ___
8
Proof of Theorem 3.1 (2). The model category part is Theorem 3.1 (1). By Lemma *
*2.3, it
suffices to check the pushout product axiom and the monoid axiom for the genera*
*ting (acyclic)
cofibrations. Every generating (acyclic) cofibration is induced from C by smash*
*ing with R, i.e.
it is of the form
R ^A ! R ^B
for A ! B a(n) (acyclic) cofibration in C. In the pushout product of two such *
*maps, one R
smash factor cancels due to using ^R , so that the pushout product is again ind*
*uced from a
pushout product of (acyclic) cofibrations in C, where the pushout product axiom*
* holds. This
gives the pushout product axiom for ^R .
If J is a set of generating acyclic cofibrations in C, the set of generating ac*
*yclic cofibrations in
the category of Rmodules (called JT above) consists of maps of J smashed with *
*R. We thus
have the equality JT ^R(Rmod) = J ^C. Since the forgetful functor Rmod ! C p*
*reserves
colimits (it has a right adjoint [R; ]), (JT ^(Rmod))cofregis a subset of (J*
* ^C)cofreg. The
monoid axiom for C thus implies the monoid axiom for Rmod. *
* ___
Proof of Theorem 3.1 (3). This proof is much longer than the previous ones; it *
*occupies
the rest of the paper. The main ingredient here is a filtration of a certain p*
*ushout in the
monoid category. This filtration is also needed to prove the statement about co*
*fibrant monoids.
The crucial step only depends on the weak equivalences and cofibrations in the *
*model category
structure. Hence we formulate it in a more general context. The hope is that it*
* can also be
useful in a situation where one only has something weaker than a model category*
*, without a
notion of fibrations. The following definition captures exactly what is needed.
Definition 5.1An applicable category is a symmetric monoidal category C equippe*
*d with two
classes of morphisms called cofibrations and weak equivalences, satisfying the *
*following axioms.
oC has pushouts and filtered colimits. The monoidal product preserves colimi*
*ts in each of
its variables.
oAny isomorphism is a weak equivalence and a cofibration. Weak equivalences *
*are closed
under composition. Cofibrations and acyclic cofibrations are closed under t*
*ransfinite com
position and cobase change.
oThe pushout product and monoid axiom are satisfied.
Of course, any monoidal model category which satisfies the monoid axiom is appl*
*icable. We
are essentially forgetting all references to fibrations since they play no role*
* in the following
filtration argument. Note that the notion of regular cofibrations as defined in*
* Definition 2.2
and Appendix A still makes sense in an applicable category. In the following le*
*mma, let I (resp.
J) be the class of those maps between monoids in C which are obtained from cofi*
*brations (resp.
acyclic cofibrations) in C by application of the free monoid functor, see (*) b*
*elow.
Lemma 5.2 If C is an applicable category, any regular Jcofibration is a weak *
*equivalence in
the underlying category C. If the unit I of the smash product is cofibrant, th*
*en any regular
Icofibration whose source is cofibrant in C is a cofibration in the underlying*
* category C.
Proof of Theorem 3.1 (3), assuming lemma 5.2. By the already established part *
*(2)
of Theorem 3.1, the category of Rmodules is itself a cofibrantly generated, mo*
*noidal model
category satisfying the monoid axiom. Also if I is cofibrant in C, then R, the *
*unit for ^R , is
9
cofibrant in Rmod. So we can assume that the commutative monoid R is actually *
*equal to the
unit I of the smash product, thus simplifying terminology from "Ralgebras" to *
*"monoids".
To use Lemma A.3 here we need to recognize monoids in C as the algebras over th*
*e free monoid
triple T . For an object K of C, define T (K) to be
T (K) = I q K q (K ^K) q : :q:K ^nq : : : (*)
One can think of T (K) as the `tensor algebra'. Using that ^ distributes over t*
*he coproduct,
T (K) has a monoid structure given by concatenation. The functor T is left adj*
*oint to the
forgetful functor from monoids to C. Hence T is also a triple on the category *
*C and the T 
algebras are precisely the monoids.
Because the monoidal product is closed symmetric, ^ commutes with colimits. He*
*nce, the
underlying functor of T commutes with filtered colimits, as required for Lemma *
*A.3. The
condition on the regular cofibrations is taken care of by Lemma 5.2. Let f : M *
*! N be a
cofibration of monoids with M cofibrant in C. Every cofibration of monoids is a*
* retract of a
regular Icofibration with I as in Lemma 5.2. Hence f is a retract of a regular*
* Icofibration
with source cofibrant in C, hence is a cofibration in C. In particular, a cofib*
*rant monoid is a
monoid M such that the unit map I ! M is a cofibration of monoids. Since I is *
*cofibrant, this
implies that the unit map is an underlying cofibration. Hence, M is cofibrant i*
*n the underlying
category C. __*
*_
Proof of lemma 5.2 The main ingredient is a filtration of a certain kind of pus*
*hout in the
monoid category. Consider a map K ! L in C, a monoid X and a monoid map T (K) *
*! X.
We want to describe the pushout in the monoid category of the diagram
T (K)_____wT (L)





u
X
The pushout P will be obtained as the colimit, in the underlying category C, of*
* a sequence
X = P0 ! P1 ! . ..! Pn ! . .:.
If one thinks of P as consisting of formal products of elements from X and from*
* L, with relations
coming from the elements of K and the multiplication in X, then Pn consists of *
*those products
where the total number of factors from L is less than or equal to n. For ordin*
*ary monoids,
this is in fact a valid description, and we will now translate this idea into t*
*he elementfree form
which applies to general symmetric monoidal categories.
As indicated above we set P0 = X and describe Pn inductively as a pushout in C.*
* We first
describe an ndimensional cube in C; by definition, such a cube is a functor
W : P({1; 2; : :;:n}) ! C
from the poset category of subsets of {1; 2; : :;:n} and inclusions to C. If S *
* {1; 2; : :;:n} is a
subset, the vertex of the cube at S is defined to be
W (S) = X ^C1 ^X ^C2 ^: :^:Cn ^X
10
with ae
Ci = KL ifii62fSi 2 S:
All maps in the cube W are induced from the map K ! L and the identity on the *
*X factors.
So at each vertex a total of n+ 1 smash factors of X alternate with n smash fac*
*tors of either
K or L. The initial vertex corresponding to the empty subset has all Ci's equal*
* to K and the
terminal vertex corresponding to the whole set has all Ci's equal to L. For exa*
*mple for n = 2,
the cube is a square and looks like
X ^K ^X ^K ^X _____wX ^K ^X ^L ^X
 
 
u u
X ^L ^X ^K ^X _____wX ^L ^X ^L ^X:
Denote by Qn the colimit of the punctured cube, i.e., the cube with the termina*
*l vertex removed.
Define Pn via the pushout in C
Qn _____(Xw^L)^ n^ X
 
 
 
u u
Pn1 _________Pn:w
This is not a complete definition until we say what the left vertical map is. *
*We define the
map from Qn to Pn1 by describing how it maps a vertex W (S) for S a proper sub*
*set of
{1; 2; : : :;:n}. Each of the smash factors of W (S) which is equal to K is fir*
*st mapped into X.
Then adjacent smash factors of X are multiplied. This gives a map
W (S) ! X ^L ^X ^: :^:L ^X ;
where the right hand side has S + 1 smash factors of X and S  smash factors*
* of L. So the
right hand side maps further to PS, hence to Pn1 since S is a proper subset.
We have to check that these maps on the vertices of the punctured cube W are co*
*mpatible
so that they assemble to a map from the colimit, Qn. So let S be again a prope*
*r subset of
{1; 2; : :;:n} and take i 62 S. We have to verify commutativity of the diagram
W (S) _________(Xw^L)^ S^X _________PSw
 
 
 
u u
W (S [{i})_____(Xw^L)^ (S+1)^X_______PS+1:w
By definition, W (S) and W (S [{i}) differ at exactly one smash factor in the 2*
*ith position
which is equal to K for the former and equal to L for the latter. The upper lef*
*t map factors as
W (S)_________w(X ^L)^ a^ X ^K ^(X ^L)^ b^ X _____(Xw^L)^ S^X
11
where a (resp. b) is the number of elements in S which are smaller (resp. large*
*r) than i; in
particular a + b = S . The right map in this factorization pushes K into X an*
*d multiplies
the three adjacent smash factors of X. Hence the diagram in question is the com*
*posite of two
commutative squares
W (S)_____w(X ^L)^ a^ X ^K ^(X ^L)^ b^ X ______PSw
  
  
  
u u u
W (S [{i})_________(Xw^L)^ (S+1)^X___________wPS+1:
The right square commutes by the definition of PS+1.
We have now completed the inductive definition of Pn. We set P = colimPn, the c*
*olimit being
taken in C. P comes equipped with Cmorphisms X = P0 ! P and
L ~=I ^L ^I ! X ^L ^X ! P1 ! P
which make the diagram
K _____wL
 
 
 
u u
X _____wP
commute. There are several things to check:
(i) P is naturally a monoid so that
(ii) X ! P is a map of monoids and
(iii)P has the universal property of the pushout in the category of monoids.
Define the unit of P as the composite of X ! P with the unit of X. The multipl*
*ication of
P is defined from compatible maps Pn ^Pm ! Pn+m by passage to the colimit. The*
*se maps
are defined by induction on n + m as follows. Note that Pn ^Pm is the pushout i*
*n C in the
following diagram.
Qn ^((X ^L)m ^ X) [(Qn ^Qm)((X ^L)n ^X) ^Qm _____w((X ^L)n ^X) ^((X ^L)m ^ X)
 
 
 
 
 
 
 
u u
(Pn1 ^Pm ) [(Pn1^ Pm1)(Pn ^Pm1_)__________________Pnw^ Pm
The lower left corner already has a map to Pn+m by induction, the upper right c*
*orner is mapped
there by multiplying the two adjacent factors of X followed by the map (X ^L)n+*
*m ^ X !
Pn+m from the definition of Pn+m . We omit the tedious verification that this i*
*n fact gives a well
defined multiplication map and that the associativity and unital diagrams commu*
*te. Hence, P
is a monoid. Multiplication in P was arranged so that X ! P is a monoid map.
12
For (iii) , suppose we are given another monoid M, a monoidal map X ! M, and a*
* Cmap
L ! M such that the outer square in
K _____wL4
 
  4
  4
u u 4
X fl___wP " 4
flfl " 4
flflfl"446]
flffl
M
commutes. We have to show that there is a unique monoidal map P  ! M making t*
*he
entire square commute. These conditions in fact force the behavior of the comp*
*osite map
W (S) ! Pn ! P ! M. Since P is obtained by various colimit constructions fro*
*m these
basic building blocks, uniqueness follows. We again omit the tedious verificati*
*on that the maps
W (S) ! M are compatible and assemble to a monoidal map P ! M.
Now that we have established that P is the pushout of the original diagram of m*
*onoids, we
continue with the homotopical analysis of the constructed filtration, i.e. we w*
*ill verify that the
regular Jcofibrations are weak equivalences. Assume now that K ! L is an acyc*
*lic cofibration
in C. The cube W used in the inductive definition of Pn has n + 1 smash factors*
* of X at every
vertex which map by the identity everywhere. Using the symmetry isomorphism for*
* ^, these
can all be shuffled to one side and we get that the map Qn ! (X ^L)^ n^ X is i*
*somorphic to
Qn ^ X ^(n+1)! L^ n^ X ^(n+1):
Here Qn is the colimit of a punctured cube analogous to W , but with all the sm*
*ash factors
of X in the vertices deleted. By iterated application of the pushout product ax*
*iom, the map
Qn ~!L^ nis an acyclic cofibration. So by the monoid axiom, the map Pn1 ~!Pn*
* is a weak
equivalence. The map X = P0 ~!P is an instance of a transfinite composite (ind*
*exed by the
first infinite ordinal) of the kind of maps considered in the monoid axiom, so *
*it is also a weak
equivalence.
With the use of the filtration we just established that any pushout, in the cat*
*egory of monoids,
of a map in J is a countable composite of maps of the kind considered in the mo*
*noid axiom.
A transfinite composite of transfinite composites is again a transfinite compos*
*ite. Because
the forgetful functor from monoids to C preserves filtered colimits, this shows*
* that regular
Jcofibrations are weak equivalences.
It remains to prove the statement about regular Icofibrations under the assump*
*tion that the
unit I is cofibrant. We note that if in the above pushout diagram K ! L is a c*
*ofibration and
the monoid X is cofibrant in the underlying category, then
Qn ^X ^(n+1)! L^ n^X ^(n+1)
is a cofibration in the underlying category (by several applications of the pus*
*hout product
axiom). Thus also the maps Pn1 ! Pn and finally X = P0 ! P are cofibrations *
*in the
underlying category. Since the forgetful functor commutes with filtered colimi*
*ts, transfinite
composites of such pushouts in the monoid category are still cofibrations in th*
*e underlying
category C. __*
*_
13
A Cofibrantly generated model categories
We need to transfer model category structures to categories of algebras over tr*
*iples. In [Q, p.
II 3.4], Quillen formulates his small object argument, which is now the standar*
*d device for such
purposes. After Quillen, several authors have axiomatized and generalized the *
*small object
argument (see e.g. [Bl, Def. 4.4], [Cr, Def. 3.2] or [Sch1, Def. 1.3.1]). In ou*
*r context we will need
a transfinite version of the small object argument. An axiomatization suitable *
*for our purposes
is the `cofibrantly generated model category' of [DHK], which we now recall.
If a model category is cofibrantly generated, its model category structure is c*
*ompletely de
termined by a set of cofibrations and a set of acyclic cofibrations. The transf*
*inite version of
Quillen's small object argument allows functorial factorization of maps as cofi*
*brations followed
by acyclic fibrations and as acyclic cofibrations followed by fibrations. Most *
*of the model cate
gories in the literature are cofibrantly generated, e.g. topological spaces and*
* simplicial sets, as
are all the examples that appear in this paper.
Ordinals and cardinals. An ordinal fl is an ordered isomorphism class of well o*
*rdered sets; it
can be identified with the well ordered set of all preceding ordinals. For an o*
*rdinal fl, the same
symbol will denote the associated poset category. The latter has an initial obj*
*ect ;, the empty
ordinal. An ordinal is a cardinal if its cardinality is larger than that of an*
*y preceding ordinal.
A cardinal is called regular if for every set of sets {Xj}j2J indexed by a set*
* J of cardinality
less than Ssuch that the cardinality of each Xj is less than that of , then the*
* cardinality of
the union J Xj is also less than that of . The successor cardinal (the smalles*
*t cardinal of
larger cardinality) of every cardinal is regular.
Transfinite composition. Let C be a cocomplete category and fl a well ordered s*
*et which we
identify with its poset category. A functor V : fl ! C is called a flsequence*
* if for every limit
ordinal fi < fl the natural map colimV fi! V (fi) is an isomorphism. The map*
* V (;) !
colimflV is called the transfinite composition of the maps of V . A subcatego*
*ry C1 C is
said to be closed under transfinite composition if for every ordinal fl and eve*
*ry flsequence
V : fl ! C with the map V (ff) ! V (ff + 1) in C1 for every ordinal ff < fl, *
*the induced map
V (;) ! colimflV is also in C1. Examples of such subcategories are the cofibr*
*ations or the
acyclic cofibrations in a closed model category.
Relatively small objects. Consider a cocomplete category C and a subcategory C1*
* C closed
under transfinite composition. If is a regular cardinal, an object C 2 C is c*
*alled small
relative to C1 if for every regular cardinal and every functor V : ! C1 wh*
*ich is a
sequence in C, the map
colimHom C(C; V ) ! Hom C(C; colimV )
is an isomorphism. An object C 2 C is called small relative to C1 if there exi*
*sts a regular
cardinal such that C is small relative to C1.
Iinjectives, Icofibrations and regular Icofibrations. Given a cocomplete ca*
*tegory C and a
class I of maps, we denote
oby Iinj the class of maps which have the right lifting property with respe*
*ct to the maps
in I. Maps in Iinj are referred to as Iinjectives.
oby Icof the class of maps which have the left lifting property with respec*
*t to the maps
in Iinj. Maps in Icof are referred to as Icofibrations.
14
oby Icofreg Icof the class of the (possibly transfinite) compositions of p*
*ushouts of maps
in I. Maps in Icofregare referred to as regular Icofibrations.
Quillen's small object argument [Q, p. II 3.4] has the following transfinite an*
*alogue. Note that
here I has to be a set, not just a class of maps.
Lemma A.1 [DHK] Let C be a cocomplete category and I a set of maps in C whose *
*domains
are small relative to Icofreg. Then
othere is a functorial factorization of any map f in C as f = qi with q 2 I*
*inj and i 2 I
cofregand thus
oevery Icofibration is a retract of a regular Icofibration.
Definition A.2[DHK] A model category C is called cofibrantly generated if it is*
* complete and
cocomplete and there exists a set of cofibrations I and a set of acyclic cofibr*
*ations J such that
othe fibrations are precisely the Jinjectives;
othe acyclic fibrations are precisely the Iinjectives;
othe domain of each map in I (resp. in J) is small relative to Icofreg(resp*
*. Jcofreg).
Moreover, here the (acyclic) cofibrations are the I (J)cofibrations.
For a specific choice of I and J as in the definition of a cofibrantly generate*
*d model category,
the maps in I (resp. J) will be referred to as generating cofibrations (resp. g*
*enerating acyclic
cofibrations). In cofibrantly generated model categories, a map may be functori*
*ally factored
as an acyclic cofibration followed by a fibration and as a cofibration followed*
* by an acyclic
fibration.
Let C be a cofibrantly generated model category and T a triple on C. We want to*
* form a model
category on the category of algebras over the triple T , denoted T alg. Call a*
* map of T algebras
a weak equivalence (resp. fibration) if the underlying map in C is a weak equiv*
*alence (resp.
fibration). Call a map of T algebras a cofibration if it has the left lifting *
*property with respect
to all acyclic fibrations. The forgetful functor T alg! C has a left adjoint,*
* the free functor F T.
The following lemma gives two different situations in which one can lift a mode*
*l category on C
to one on T alg. We make no great claim to originality for this lemma. Other l*
*ifting theorems
for model category structures can be found in [Bl, Thm. 4.14], [CG, Thm. 2.5], *
*[Cr, Thm. 3.3],
[DHK, II 8.2], [EKMM, VII Thm. 4.7, 4.9].
Let X be a T algebra. We define a path object for X to be a T algebra XI tog*
*ether with
T algebra maps
X _______XIw~_____XwxwX
factoring the diagonal map, such that the first map is a weak equivalence and t*
*he second map
is a fibration in the underlying category C.
Lemma A.3 Assume that the underlying functor of T commutes with filtered direc*
*t limits. Let
I (J) be a set of generating cofibrations (resp. acyclic cofibrations) for the *
*cofibrantly generated
model category C. Let IT (resp. JT) be the image of these sets under the free T*
* algebra functor.
Assume that the domains of IT (JT) are small relative to ITcofreg(JTcofreg). *
*Suppose
15
1.every regular JTcofibration is a weak equivalence, or
2.every object of C is fibrant and every T algebra has a path object.
Then the category of T algebras is a cofibrantly generated model category with*
* IT (JT) the
generating set of (acyclic) cofibrations.
Proof: We refer the reader to [DS, 3.3] for the numbering of the model category*
* axioms. All
those kinds of limits that exist in C also exist in T alg, and limits are crea*
*ted in the underlying
category C [Bor, Prop. 4.3.1]. Colimits are more subtle, but since the underlyi*
*ng functor of T
commutes with filtered colimits, they exist by [Bor, Prop. 4.3.6]. Model catego*
*ry axioms MC2
(saturation) and MC3 (closure properties under retracts) are clear. One half of*
* MC4 (lifting
properties) holds by definition of cofibrations of T algebras.
The proof of the remaining axioms uses the transfinite small object argument, w*
*hich exists here
because of Lemma A.1, and the hypothesis about the smallness of the domains.
We begin with the factorization axiom, MC5. Every map in IT and JT is a cofibr*
*ation of
T algebras by adjointness. Hence any ITcofibration or JTcofibration is a co*
*fibration of T 
algebras. By adjointness and the fact that I is a generating set of cofibration*
*s for C, a map
is ITinjective precisely when the map is an acyclic fibration of underlying ob*
*jects, i.e., an
acyclic fibration of T algebras. Hence the small object argument applied to th*
*e set IT gives a
(functorial) factorization of any map in T alg as a cofibration followed by an*
* acyclic fibration.
The other half of the factorization axiom, MC5, needs hypothesis (1) or (2). A*
*pplying the
small object argument to the set of maps JT gives a functorial factorization of*
* a map in T 
alg as a regular JTcofibration followed by a JTinjective. Since J is a genera*
*ting set for the
acyclic cofibrations in C, the JTinjectives are precisely the fibrations among*
* the T algebra
maps, once more by adjointness. In case (1) we assume that every regular JTcof*
*ibration is a
weak equivalence on underlying objects in C. We noted above that every JTcofib*
*ration is a
cofibration in T alg. So we see that the factorization above is an acyclic cof*
*ibration followed by
a fibration.
In case (2) we can adapt the argument of [Q, II p.4.9] as follows. Let i : X *
*! Y be any
JTcofibration. We claim that it is a weak equivalence in the underlying catego*
*ry. Since X is
fibrant and fibrations are JTinjectives, we obtain a retraction r to i by lift*
*ing in the square
X _____wXid
 
i  oeo 
 oer 
uoe uu
Y _____w*:
Y possesses a path object and i has the LLP with respect to fibrations. So a li*
*fting exists in
the square
X _____wYi_____wY I
 B BBC 
 B B 
 B 
uB B uu
Y ____________Ywx(Y:id;iOr)
This shows that in the homotopy category of C, iOr is equal to the identity map*
* of Y . Since maps
in C are weak equivalences if and only if they become isomorphisms in the homot*
*opy category
16
of C, this proves that i is a weak equivalence, and it finishes the proof of mo*
*del category axiom
MC5 under hypothesis (2).
It remains to prove the other half of MC4, i.e., that any acyclic cofibration *
*A v_____Bw~has
the LLP with respect to fibrations. In other words, we need to show that the ac*
*yclic cofibrations
are contained in the JTcofibrations. The small object argument provides a fact*
*orization
A v_____Ww~____wBw
with A ! W a JTcofibration and W ! B a fibration. In addition, W ! B is a w*
*eak
equivalence since A ! B is. Since A ! B is a cofibration, a lifting in
A _____Ww
v 
 oeo~
 oe 
uoe uu
B _____wBid
exists. Thus A ! B is a retract of a JTcofibration, hence a JTcofibration. *
* ___
Remark A.4 Hypothesis (2) can be weakened to the existence of a fibrant replac*
*ement functor
in the category of T algebras which interacts well with respect to the path ob*
*ject, see [Sch2,
Lemma A.3]. Quillen's argument in [Q, II p.4.9] in fact uses Kan's Ex1 functo*
*r as such a
fibrant replacement functor.
Remark A.5 To simplify the exposition, we assume that every object of C is sma*
*ll relative
to the whole category C when we apply lemma A.3 in the rest of this paper. This*
* holds for
spaces and symmetric spectra based on simplicial sets. If the underlying funct*
*or of the triple
T on C commutes with filtered direct limits, then so does the forgetful functor*
* from T algebras
to C. Hence by adjointness, every free T algebra is small relative to the who*
*le category of
T algebras, so the smallness conditions of lemma A.3 hold. Of course, if one i*
*s interested in a
category where not all objects are small with respect to all of C one must veri*
*fy those smallness
conditions directly.
17
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[Se] G. Segal: Categories and cohomology theories, Topology 13 (1974), 293*
*312
Fakult"at f"ur Mathematik Department of Mathematics
Universit"at Bielefeld University of Chicago
33615 Bielefeld, Germany Chicago, IL 60637, USA
schwede@mathematik.unibielefeld.de bshipley@math.uchicago.edu
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