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