NOTES ON ENRICHED MODEL CATEGORIES

BERTRAND GUILLOU

Let G be a compact Lie group and let U be a complete G-universe. We will be interested in describing an enrichment of a good category of G-spectra G𝒮 over a good category of spectra 𝒮. We first mention a general way of producing enrichments. The first statement appears as Proposition 6.3 of [EK].

Lemma 0.1. Let \xymatrix{𝒱_{1} \ar[r]^{𝕋} & 𝒱_{2} \ar[l]^{𝕌}} be an adjoint pair between symmetric monoidal cat-

egories. Suppose that ℳ is enriched and bitensored over 𝒱_{2}. If 𝕌 is lax symmetric monoidal then one can define an enrichment of ℳ over 𝒱_{1} by

ℳ^{𝒱_{1}} (M, N) := 𝕌ℳ^{𝒱_{2}} (M, N).

If moreover 𝕋 is strong symmetric monoidal then the definitions

V_{1} ⊗ M := 𝕋(V_{1}) ⊗ M,  and  M^{V_{1}} := M^{𝕋V_{1}}

make ℳ into a category enriched and bitensored over 𝒱_{1}.

Actually, the underlying category \underline{ℳ} ^{𝒱_{1}} defined by

\underline{ℳ} ^{𝒱_{1}} (M, N) := Hom_{𝒱_{1}} (U_{𝒱_{1}} , ℳ^{𝒱_{1}} (M, N))

need not agree with the original underlying category \underline{ℳ} ^{𝒱_{2}} if we only assume that R is lax symmetric monoidal. One merely has a comparison functor \underline{ℳ} ^{𝒱_{2}} → \underline{ℳ} ^{𝒱_{1}} . But with the further assumption that 𝕋(U_{𝒱_{1}} ) ≅ U_{𝒱_{2}} (for example, if 𝕋 is strong symmetric monoidal), one gets \underline{ℳ} ^{𝒱_{1}} = \underline{ℳ} ^{𝒱_{2}} .

Aside 0.2. A Yoneda argument allows one to deduce the following enriched adjunction isomorphism:

Lemma 0.3. Let \xymatrix{𝒱_{1} \ar[r]^{𝕋} & 𝒱_{2} \ar[l]^{𝕌}} be an adjoint pair between symmetric monoidal

categories such that 𝕋 is strong symmetric monoidal. Then one has binatural comparison isomorphisms

𝒱 ^{1}(V_{1}, 𝕌V_{2}) ≅ 𝕌𝒱_{2}(𝕋V_{1}, V_{2}).

0.1. Examples.

Example 0.4. Let G𝒰 be the category of G-spaces (spaces will always mean compactly generated spaces). This is closed symmmetric monoidal under cartesian product with diagonal G-action. The internal mapping spaces G𝒰 (X, Y) is defined to be the space of (nonequivariant) maps, on which G acts by conjugation. The functor of fixed points (−)^{G} : G𝒰 → 𝒰 has strong symmetric monoidal left adjoint ϵ^{∗}, which endows any space with the trivial G-action. Thus one gets an enrichment of G𝒰 over 𝒰 by G𝒰^{𝒰} (X, Y) = G𝒰 (X, Y)^{G}, and G𝒰 is also bitensored over 𝒰 .

Example 0.5. Consider the adjunction \xymatrix{sSet \ar[r]^{|−|} & 𝒰 \ar[l]^{S_{•}}} . Then the geometric re-

alization functor | − | is strong symmetric monoidal, and we conclude that any category enriched and bitensored over 𝒰 is canonically enriched and bitensored over sSet.

Example 0.6. The category Gℳ_{S_{G}}^{U^{G}} of ‘naive’ equivariant S-modules is closed symmetric monoidal. There is a fixed points functor (−)^{G} : Gℳ_{S_{G}}^{U^{G}} → ℳ_{S}^{U^{G}} which is defined levelwise, and it has a strong symmetric monoidal left adjoint ϵ^{∗}. Thus Gℳ_{S_{G}}^{U^{G}} is enriched and bitensored over ℳ_{S}^{U^{G}}.

Example 0.7. We are really after an enrichment of Gℳ_{S_{G}}^{U} over ℳ_{S}^{U^{G}}. This involves change of universe between U and its G-fixed universe U^{G}. In ([MM], §VI.2), two different change of universe adjunctions are discussed (i : U^{G} ↪ U is the inclusion):

\xymatrix{Gℳ_{S_{G}}^{U^{G}} \ar[r]^{i_{∗}} & Gℳ_{S_{G}}^{U} \ar[l]^{i^{∗}}} and \xymatrix{Gℳ_{S_{G}}^{U^{G}} \ar[r]^{I_{U^{G}}^{U}} & Gℳ_{S_{G}}^{U} \ar[l]^{I_{U}^{U^{G}}}} .

The first pair (i_{∗}, i^{∗}) is a Quillen pair but is unfortunately not monoidal. On the other hand, (I_{U^{G}}^{U}, I_{U}^{U^{G}}) is a pair of strong symmetric monoidal equivalences, but these functors are not homotopically well-behaved.

Nevertheless, we now have the adjoint pair

\xymatrix{ℳ_{S}^{U^{G}} \ar[r]^{ϵ^{∗}} & Gℳ_{S_{G}}^{U^{G}} \ar[l]^{(−)^{G}} \ar[r]^{I_{U^{G}}^{U}} & Gℳ_{S_{G}}^{U} \ar[l]^{I_{U}^{U^{G}}}}

in which the left adjoint is strong symmetric monoidal. Thus we obtain an enrichment and bitensoring of Gℳ_{S_{G}}^{U} over ℳ_{S}^{U^{G}}. On the other hand, due to the lack of homotopical control over the change of universe functors used here, it is not clear that this is the correct enrichment, from the homotopical standpoint.

We should point out that the above troubles with change of universe functors do not arise when working in the category of orthogonal spectra. That is, the change of universe adjunction (i_{∗}, i^{∗}) for orthogonal spectra is simultaneously sufficiently monoidal and homotopically well-behaved. We will return to this in §2.

The ultimate goal is as follows. For the moment, let 𝒮 denote a good category of spectra (so either ℳ_{S} or orthogonal spectra) and let G𝒮 be the resulting good category of (genuine) equivariant spectra. We wish first, as above, to describe G𝒮 as enriched and bitensored over 𝒮 and furthermore to make G𝒮 into a 𝒮 -model category. Let 𝒟 ⊂ G𝒮 be the full 𝒮 -subcategory spanned by the orbit spectra Σ^{∞}G/H_{+}. Then we wish to verify that one has the projective model structure on the category of 𝒮 -presheaves 𝒫(𝒟, 𝒮) and furthermore that the natural adjunc-

tion [figure: 𝒫(𝒟, 𝒮); 𝕋; G𝒮; 𝕌] is a Quillen equivalence of 𝒮 -model categories.

The above considerations suggest that, at least in regarding to describing G𝒮 as a 𝒮 -model category, the framework of orthogonal spectra is more suitable. We do this in §2. On the other hand, when attempting to verify that we obtain a Quillen equivalence between 𝒫(𝒟, 𝒮) and G𝒮, the fact that not all orthogonal spectra are fibrant will be a serious hindrance. S-modules avoid this nuisance, but the above considerations suggest that we must use the homotopically badly behave change of universe functors I_{U^{G}}^{U}, which will obstruct the construction of a 𝒮 -model structure on G𝒮.

1. Model structures

Suppose that we are in the situation of Lemma 0.1 and that now 𝒱_{1} and 𝒱_{2} are monoidal model categories and ℳ is a 𝒱_{2}-model category. It is natural to ask under what conditions on the adjuunction (T, U) the resulting 𝒱_{1}-category ℳ^{𝒱_{1}} becomes a 𝒱_{1}-model category. One might guess that it is enough for the adjuction (T, U) to be a Quillen pair, and this is almost correct.

Definition 1.1 ([Ho], 4.2.6). A symmetric monoidal model category 𝒱 is a model category with a closed symmetric monoidal structure satisfying the following two axioms:

(1) (pushout product) Given cofibrations A ⟶^{i} B and C ⟶^{j} D, the induced

map

B ⊗ C ∪_{A⊗C} A ⊗ D ⟶^{i □j} B ⊗ D

(referred to as the pushout product of i and j) is a cofibration which is a weak equivalence if either i or j is a weak equivalence.

(2) (unit) If QU ⟶^{r} U is a cofibrant replacement for the unit object, then the natural map QU ⊗ X ⟶^{r⊗1} U ⊗ X is a weak equivalence for every cofibrant X.

The appropriate notion of map of symmetric monoidal model categories is given by the following definition, due to Hovey (he only cosiders the strong monoidal variant).

Definition 1.2 ([Ho], 4.2.16). Given symmetric monoidal model categories 𝒱_{1} and

𝒱_{2}, a monoidal Quillen adjunction from 𝒱_{1} to 𝒱_{2} is a Quillen pair \xymatrix{𝒱_{1} \ar[r]^{T} & 𝒱_{2} \ar[l]^{U}}

such that T is lax monoidal and such that the map T QU_{1} ⟶^{T (q_{U_{1}} )} T U_{1} is a weak equivalence, where QU_{1} ⟶^{q_{U_{1}}} U_{1} is cofibrant replacement. Similarly, a strong monoidal Quillen adjunction is a monoidal Quillen adjunction such that T is strong symmetric monoidal.

As one might expect, it is not difficult to show the following.

Proposition 1.3 ([Ho], 4.3.3). If \xymatrix{𝒱_{1} \ar[r]^{T} & 𝒱_{2} \ar[l]^{U}} is a monoidal Quillen adjunc-

tion between symmetric monoidal model categories, then the derived adjunction

\xymatrix{Ho𝒱_{1} \ar[r]^{𝕃T} & Ho𝒱_{2} \ar[l]^{ℝU}} is lax symmetric monoidal. Similarly, if (T, U) is a strong

monoidal Quillen adjunction, then 𝕃T is strong symmetric monoidal.

The proof is straightforward, so we do not include it, but we point out that the requirement that T (q_{U_{1}}) is a weak equivalence in 𝒱_{2} is used in order to define the map

U_{2} → T (U_{1}) ⟵^{∼} T (QU_{1}) = 𝕃T (U_{1})

in Ho(𝒱_{2}).

Definition 1.4 ([LM], 3.3). If 𝒱 is a symmetric monoidal model category, a 𝒱 - model category ℳ is a 𝒱 -category ℳ with a model structure on \underline{ℳ} satisfying

(1) (Quillen’s SM7) Given a cofibration A ⟶^{i} B and a fibration X ⟶^{p} Y in \underline{ℳ} , the induced map

ℳ (i, p) : ℳ (B, X) → ℳ (A, X) ×_{ℳ (A,Y)} ℳ (B, Y)

in 𝒱 is a fibration which is a weak equivalence if either i or p is and

(2) (unit) For each cofibrant A and fibrant B in ℳ, the map

Hom_{Ho(\underline{ℳ} )}(A, B) → Hom_{Ho(𝒱 )} (U, ℳ (A, B))

induced from

Hom\underline{ℳ} (A, B) = Hom_{𝒱} (U, ℳ (A, B)) → Hom_{Ho(𝒱 )} (U, ℳ (A, B))

by the universal property of localisation is a bijection.

Remark 1.5 ([LM], 3.5). If ℳ is also tensored or cotensored over 𝒱 , then the unit axiom may be rephrased in an equivalent form along the lines of the unit axiom from the definition of a monoidal model category. For example, in the tensored situation, an equivalent formulation is that for every cofibrant X ∈ ℳ, the map X ⊗QU → X ⊗U is a weak equivalence, where QU → U is a cofibrant replacement for the unit object.

The following result answers the question raised at the beginning of this section. As noted in the remark following Lemma 0.1, it is desirable to require that T preserves the unit object up to isomorphism even if it does not preserve the monoidal product.

Proposition 1.6. Let T : 𝒱_{1} ⇄ 𝒱_{2} : U be a Quillen pair between symmetric monoidal model categories such that U is lax monoidal, the map T QU_{1} → T U_{1} is a weak equivalence, and T is strongly unital, in the sense that T (U_{1}) ≅ U_{2}. Suppose that ℳ is a 𝒱_{2}-model category. Then the 𝒱_{1} enrichment described in Lemma 0.1 makes ℳ into a 𝒱_{1}-model category.

In particular, the above result holds if (T, U) is a strong monoidal Quillen adjunction.

Corollary 1.7. Any topological model category has a canonical structure of a simplicial model category.

2. Equivariant Orthogonal Spectra

Recall the category Gℐ 𝒮 of orthogonal G-spectra developped in [MM]. It is closed symmetric monoidal ([MM], Cor. II.3.2). There are two Quillen equivalent model structures on Gℐ 𝒮 whose weak equivalences are the π_{∗}-isomorphisms: the stable model structure and the positive stable model structure. The former satisfies the convenient property that the unit S_{G} is cofibrant, while the latter is the model structure that allows for a Quillen equivalence to equivariant S-modules as well as a model structure for commutative S_{G}-algebras.

Proposition 2.1. The category Gℐ 𝒮 of orthogonal G-spectra is a monoidal model category under both the stable and the positive stable model structures.

Proof. For the stable model structure, this follows immediately from ([MM], Theorem III.4.2) and the pushout-product axiom ([MM], Theorem III.7.5) since the unit S_{G} is already cofibrant. For the positive stable model structure, we cite ([MM], Theorem III.5.2) and again ([MM], Theorem III.7.5), and it remains to verify that if QS_{G} is a positive cofibrant replacement for S_{G} and X is any positive cofibrant orthognal G-spectrum, then QS_{G} ∧ X → S_{G} ∧ X ≅ X is a π_{∗}-isomorphism. It suffices to show this for a particular cofibrant replacement, for example F_{V} S^{V} with V^{G} ≠ 0. Also, by passage to colimits, it suffices to show this for X = F_{W} S^{n} with W^{G} ≠ 0. The desired result follows from ([MM], Lemma III.4.5). ■

Turning now to consideration of fixed point spectra, let i : U^{G} ↪ U be the inclusion of the G-fixed universe. We have already mentioned the adjoint pairs

[figure: ℐ^{U^{G}}𝒮; ϵ^{∗}; Gℐ^{U^{G}}𝒮; (−)^{G}; i_{∗}; Gℐ^{U}𝒮; i^{∗}]

Recall that i_{∗} and i^{∗} are both strong symmetric monoidal and are inverse equivalences ([MM], Theorem V.1.5). Moreover, this is a Quillen pair with respect to the stable and the positive stable model structures ([MM], Theorem V.1.6). In addition, ϵ^{∗} is strong symmetric monoidal and (ϵ^{∗}, (−)^{G}) are a Quillen pair with respect to both the stable and positive stable model structures ([MM], Proposition V.3.4). Thus, in order to apply Proposition 1.6, it remains to check that i_{∗}ϵ^{∗}(QS) → i_{∗}ϵ^{∗}(S) ≅ S_{G} is a weak equivalence in the (positive) stable model structure. Since S is already cofibrant in the stable model structure, we only need to check this for the positive stable model structure. As before, it suffices to check this for a particular cofibrant replacement, so let us choose QS = F_{n}S^{n} for some n > 0. But i_{∗}ϵ^{∗}(F_{V}^{U^{G}}S^{V}) = i_{∗}(F_{V}^{U^{G}}S^{V}) ≅ F_{V}^{U}S^{V} ([MM], V.1.4), so we are done. We have proved the following

Proposition 2.2. The adjunction (i_{∗}ϵ^{∗}, (i^{∗}(−))^{G}) together with Lemma 0.1 define a ℐ^{U^{G}}𝒮 -model structure on Gℐ^{U}𝒮.

For the remainder of the section, we will simply write 𝒮 for the category ℐ^{U^{G}}𝒮 of orthogonal spectra indexed on the universe U^{G} and G𝒮 for the category Gℐ^{U}𝒮 of equivariant orthogonal spectra indexed on the complete G-universe U. Let 𝒟 ⊆ G𝒮 be the full 𝒮 -subcategory spanned by the objects RF_{1}(Σ^{1}G/H_{+}), where R denotes fibrant replacement. We wish to show that 𝒮 -presheaves on 𝒟 admit a projective model structure and that the resulting 𝒮 -model category is Quillen equivalent to G𝒮. By ([GM], Theorem. 1.6), one has the projective model structure on 𝒫 = 𝒫(𝒟, 𝒮) because 𝒮 satisfies the monoid axiom ([MMSS], Prop. 12.5).

In order to show that the 𝒮 -adjoint pair 𝕋 : 𝒫 ⇄ G𝒮 : 𝕌 is a Quillen equivalence, it suffices by ([GM], Prop. 3.6) to show that (1) 𝒟 is detecting in the sense of ([GM], Defn. 3.4) and (2) for every cofibrant X in 𝒫, the composite

X ⟶^{η} 𝕌𝕋X ⟶^{𝕌R} 𝕌R𝕋X

is a weak equivalence in 𝒫.

Lemma 2.3. For an orthogonal Ω-G-spectrum X, we have a weak equivalence (i^{∗}j^{∗}X)^{H} → G𝒮 (F_{1}(Σ^{1}G/H_{+}), X) in 𝒮, where j : H ↪ G. Moreover, the same is true after taking a fibrant replacement of F_{1}(Σ^{1}G/H_{+}).

Proof. Again, ([MM], Lemma III.4.5) allows us to deduce that the map of orthogonal G-spectra

F (G/H_{+}, X) → F (F_{1}(Σ^{1}G/H_{+}), X)

is a weak equivalence if X is an Ω-G-spectrum. Since the domain and codomain are Ω-G-spectra, we deduce that the resulting map

i^{∗}F (G/H_{+}, X)^{G} → i^{∗}F (F_{1}(Σ^{1}G/H_{+}), X)^{G}

is also a weak equivalence. Finally, we claim that i^{∗}F (G/H_{+}, X)^{G} ≅ (i^{∗}j^{∗}X)^{H}. This can be deduced from the identification F (G/H_{+}, X) ≅ F_{H}(G_{+}, j^{∗}X) ([MM], Prop. V.2.4), together with inspection of the following diagram

[figure: ℐ^{U^{G}}𝒮; ϵ^{∗}; ϵ^{∗}; Gℐ^{U^{G}}𝒮; (−)^{G}; j^{∗}; i_{∗}; Gℐ^{U}𝒮; i^{∗}; j^{∗}; Hℐ^{j^{∗}(U^{G})}𝒮; (−)^{H}; F_{H}(G_{+},−); i_{∗}; Hℐ^{U^{H}}𝒮; i^{∗}; i_{∗}; Hℐ^{j^{∗}U}𝒮 .; i^{∗}; F_{H}(G_{+},−)]

The diagram of right adjoints commutes since the diagram of left adjoints does so.

The last statement follows since X is fibrant and G𝒮 (−, X) takes acyclic cofibrations to acyclic fibrations. ■

To check that 𝒟 is detecting, we must suppose given a map f : X → Y of orthogonal Ω-G-spectra such that G𝒮 (RF_{1}(Σ^{1}G/H_{+}), X) → G𝒮 (RF_{1}(Σ^{1}G/H_{+}), Y) is a weak equivalence for all H. By the previous lemma, we have G𝒮 (RF_{1}(Σ^{1}G/H_{+}), X) ≃ X^{H} and similarly for Y. Moreover, ([MM], Prop. V.3.2) gives an isomorphism π_{∗}^{H}(X) ≅ π_{∗}(X^{H}) and similarly for Y. Thus if each induced map f^{H} : X^{H} → Y^{H} is a weak equivalence, we deduce that f is a weak equivalence. This shows that 𝒟 is detecting.

To check that G𝒮 is Quillen equivalent to 𝒫, it remains to show that for any cofibrant X ∈ 𝒫, the composite

X ⟶^{η} 𝕌𝕋X ⟶^{𝕌R} 𝕌R𝕋X

is a weak equivalence. At this point it is useful to consider equivariant S-modules. Recall the following result of [MM]:

Theorem 2.4 ([MM], Theorem IV.1.1). There is a strong symmetric monoidal functor ℕ : G𝒮 → Gℳ and a lax symmetric monoidal functor ℕ^{#} : Gℳ → G𝒮 such that (ℕ, ℕ^{#}) is a Quillen equivalence between G𝒮 (with the positive stable model structure) and Gℳ.

Remark 2.5. Note that all of the adjunctions in the string

\xymatrix{𝒮 \ar[r]^{ϵ^{∗}} & G𝒮^{U^{G}} \ar[l]^{(−)^{G}} \ar[r]^{i_{∗}} & G𝒮 \ar[l]^{i^{∗}} \ar[r]^{ℕ} & Gℳ \ar[l]^{ℕ^{#}}}

are well-behaved, and the above allows us to describe an 𝒮 -enrichment of Gℳ, making Gℳ into a 𝒮 -model category.

Every object of Gℳ is fibrant, so it follows that Y → ℕ^{#}ℕY is a fibrant replacement for any positive cofibrant orthogonal G-spectrum Y. We deduce as in ([GM], Remark 3.8) that in order to obtain our Quillen equivalence between G𝒮 and 𝒫, it suffices to show that for any cofibrant X ∈ 𝒫, the unit

X → 𝕌ℕ^{#}ℕ𝕋X

of the composite adjunction (ℕ ∘ 𝕋, 𝕌 ∘ ℕ^{#}) is a weak equivalence.

Suppose that X = F_{G/K}F_{n}S^{m} with n > 0. Then we wish to show that for every orbit G/H, the map

(F_{G/K}F_{n}S^{m})(G/H) → (𝕌ℕ^{#}ℕ𝕋F_{G/K}F_{n}S^{m})(G/H)

is a weak equivalence of orthogonal spectra. The left-hand side is

G𝒮 (RF_{1}Σ^{1}G/H_{+}, RF_{1}Σ^{1}G/K_{+}) ∧ F_{n}S^{m}.

The lemma gives an equivalence G𝒮 (RF_{1}Σ^{1}G/H_{+}, RF_{1}Σ^{1}G/K_{+}) ≃ (RF_{1}Σ^{1}G/K_{+})^{H}, but we are not able to conclude that we have an equivalence after smashing with F_{n}S^{m}.

Let us analyze the right-hand side. First recall that 𝕋F_{d}X ≅ d ⊗ X and that the functor ℕ is strong symmetric monoidal. Thus we have

ℕ𝕋F_{G/K}F_{n}S^{m} ≅ ℕ(RF_{1}Σ^{1}G/K_{+}) ∧ ℕ(i_{∗}ϵ^{∗}F_{n}S^{m})

Assume now that the adjunction (ℕ, ℕ^{#}) satisfies a projection formula for cofibrant objects. That is, we ask that for cofibrant W ∈ G𝒮 and cofibrant Y ∈ Gℳ, the canonical map ℕ^{#}Y ∧ W → ℕ^{#}(Y ∧ ℕW) is an isomorphism. Then we have

ℕ^{#}ℕ𝕋F_{G/K}F_{n}S^{m} ≅ ℕ^{#}ℕ(RF_{1}Σ^{1}G/K_{+}) ∧ F_{n}S^{m}.

It follows that

G𝒮 (RF_{1}Σ^{1}G/H_{+}, ℕ^{#}ℕ𝕋F_{G/K}F_{n}S^{m}) ≃ (ℕ^{#}ℕ𝕋F_{G/K}F_{n}S^{m})^{H}
≅ (ℕ^{#}ℕ(RF_{1}Σ^{1}G/K_{+}) ∧ F_{n}S^{m})^{H}
≅ (ℕ^{#}ℕ(RF_{1}Σ^{1}G/K_{+}))^{H} ∧ F_{n}S^{m}
≃ (RF_{1}Σ^{1}G/K_{+})^{H} ∧ F_{n}S^{m}

It remains to identify this up to weak equivalence with G𝒮 (RF_{1}Σ^{1}G/H_{+}, RF_{1}Σ^{1}G/K_{+})∧ F_{n}S^{m}.

References

[EK] S. Eilenberg and G. M. Kelly, Closed Categories, in Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer-Verlag, Berlin-New York, 1966, 421-562.

[GM] B. J. Guillou and J. P. May, Enriched Model Categories and Diagram Categories, in preparation.

[Ho] M. Hovey, Model Categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999.

[LM] L. G. Lewis, Jr. and M. A. Mandell, Modules in Monoidal Model Categories, J. Pure App. Algebra 210 (2007), no. 2, 395-421.

[MM] M. A. Mandell and J. P. May, Equivariant Orthogonal Spectra and S-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755.

[MMSS] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model Categories of Diagram Spectra, Proc. London Math. Soc. 82 (2001), 441-512.

[SS] S. Schwede and B. Shipley, Algebras and Modules in Monoidal Model Categories, Proc. London Math. Soc. 80 (2000), 491-511.