Enriched categories and the Morita Context

David Barnes

Abstract

We show that the Quillen equivalence between S^{1}-spectra and E_{⊤}-modules is an involutary equivalence, and hence the half-twisted categories are Quillen equivalent. This is an important step in proving that the category of rational cyclic O(2)-spectra is Quillen equivalent to a twisted version of the standard category for S^{1} spectra A(S^{1}).

1 Basic Enriched Categories

We begin by providing some material from [Kel05], later we will use a different construction (closed modules in the sense of [Hov99, ch 4]) but it is the enrichment that is somehow the important concept. In particular, the reader might also like to consider [Dug06] or [Bor94].

Throughout ν will be a monoidal category, the unit of ν will be I and the product will be ⊗.

Definition 1.1 A ν-category A is a class of objects with a hom-object A(A, B) in ν for each pair of objects. For each triple of objects there is a composition law: M: A(B, C) ⊗ A(A, B) → A(A, C), a morphism in ν. For each object there is an identity element: j: I → A(A, A), a morphism in ν. These must satisfy the usual five-fold associativity diagram and a pair of triangles describing the identity elements.

Definition 1.2 A ν-functor T: A → B of ν-categories is a functor T: Ob A → Ob B and for each pair of objects a map in ν: T = T_{A,B}: A(A, B) → B(T A, T B). These maps must satisfy the relations T M = M(T ⊗ T) and T j = j.

Definition 1.3 For ν-functors T, S: A → B a ν-natural transformation α: T → S is an Ob A-indexed family of components α_{A}: I → B(T A, SA) satisfying the naturality condition (in ν) M(α_{B} ⊗ T)l^{−1} = M(S ⊗ α_{A})r^{−1} where l and r are the left and right unit isomorphisms of ν.

The composite of β and α has components M(β_{A} ⊗ α_{A}). The composite Qα has components Qα_{A}, and αP has components (αP)_{D} = α_{P D}.

Definition 1.4 A ν-adjunction of ν-categories is an adjunction (F, G) consisting of ν-functors together with an isomorphism in ν

B(F A, B) ≅ A(A, GB).

We can now use the construction of [Day70], with precise wording taken from [GS06, 3.5].

Definition 1.5 Assume that A is a symmetric monoidal ν-category. Then we can define a box product of ν-functors T, S: A → ν.

T □_{A}S(a) = ∫ ^{x,y} A(x ∧ y, a) ⊗ T (x) ⊗ S(y)

This is the left Kan extension of ⊗ along ∧ the monoidal product of A.

Lemma 1.6 Consider a ν-functor F: C → D between ν-categories and let x ∈ C, then provided the following coend exists there is an isomorphism in D:

∫ ^{c∈C} F (c) ⊗ C(c, x) ≅ F (x)

which is natural in F and x.

Proof We can rewrite the left-hand-side of the above equation into a coequaliser of coproducts:

∐_{c,c′} F (c) ⊗ C(c, c′) ⊗ C(c′, x)\overrightarrow{→} ∐_{c} F (c) ⊗ C(c, x)

with one arrow corresponding to composition and the other to the action

F (c) ⊗ C(c, c′) → F (c) ⊗ D(F (c), F (c′)) → F (c′).

The action map then gives a map in D ∐_{c} F (c)⊗C(c, x) → F (x), that this coequalises the two maps follows from the definition of a ν-functor. Given any test object T for this colimit we can use x-factor of the given map α: ∐_{c} F (c) ⊗ C(c, x) → T in the composite F (x) → F (x) ⊗ C(x, x) → T to give a map F (x) → T. That this map is unique and satisfies the required commutativity relation follows easily.

Corollary 1.7 Consider a ν-functor F: C^{op} → D between ν-categories and let x ∈ C, then provided the following coend exists there is an isomorphism in D:

∫ ^{c∈C} F (c) ⊗ C(x, c) ≅ F (x)

which is natural in F and x.

Lemma 1.8 Consider a ν-functor F: C → D between ν-categories and let x ∈ C, then provided the following end exists there is an isomorphism in D:

∫_{c∈C} \underline{Hom}(C(x, c), F (c)) ≅ F (x)

which is natural in F and x.

Proof This works in the same way as the previous lemma, we have the map F (x) → \underline{Hom}(C(x, c), F (c)) (which is the adjoint of F (x) ∧ C(x, c) → F (c)), this gives us the map from F (x) into the end.

Corollary 1.9 Consider a ν-functor F: C^{op} → D between ν-categories and let x ∈ C, then provided the following end exists there is an isomorphism in D:

∫_{c∈C} \underline{Hom}(C(c, x), F (c)) ≅ F (x)

which is natural in F and x.

The next lemma is [Bor94, Proposition 6.3.1], which we strongly recommend as a text for understanding and working with enriched categories.

Lemma 1.10 (Enriched Yoneda) Take ν to be a symmetric closed monoidal category. Let C be a ν-category, then for any c ∈ C and any ν-functor F: C → ν then ν –Nat(C(c, −), F) exists is naturally isomorphic to F (c).

Corollary 1.11 Take ν to be a symmetric closed monoidal category. Let C be a νcategory, then for any c ∈ C and any ν-functor F: C^{op} → ν then ν –Nat(C(−, c), F) exists is naturally isomorphic to F (c).

When we wish to consider a tensoring of categories we will use the description as found in [Hov99, Chapter 4]. To avoid confusion with the previous information and to match the notation of [Hov99] we let C be a symmetric monoidal category [Hov99, 4.1.4] with product ⊗ and unit 1_{C}.

Definition 1.12 A C-module is a category D with a bifunctor C ×D → D ([Hov99, 4.1.6]). The symmetry of C allows us to use left-modules and right-modules interchangeably. A C-module functor between C-modules D and E is a functor F: D → E with a natural isomorphism c ⊗ F d ≅ F (c ⊗ d). A (symmetric) Calgebra is a category D with a (symmetric) monoidal structure and a (symmetric) monoidal functor i_{D}: C → D. A (symmetric) C-algebra functor between Calgebras D and E is a (symmetric) monoidal functor F: D → E with a monoidal natural isomorphism F ∘ i_{D} → i_{E}.

We now want to add still more structure: to consider categories enriched, tensored and cotensored over another category. We let C be a closed (symmetric) monoidal category [Hov99, 4.1.13] with product ⊗ and unit 1_{C}, the internal function object of C will be F_{C} (recall that the adjective closed means that we have an adjunction of two variables rather than just a bifunctor), see also [Dug06, Appendix A].

Definition 1.13 A closed C-module is a category D with an adjunction of two variables ([Hov99, 4.1.12]) C × D → D. The symmetry of C allows us to use leftmodules and right-modules interchangeably. Thus we have three bifunctors

\begin{matrix}⊗ : & C × D & ⟶ & D \\ \underline{Hom}_{D} : & C^{op} × D & ⟶ & D \\ \underline{Hom}_{C} : & D^{op} × D & ⟶ & C \end{matrix}

In [Hov99] the notation Hom_{r} is used for \underline{Hom}_{C} and Hom_{l} for \underline{Hom}_{D}. We also have the adjunctions

C(c, \underline{Hom}_{C}(d, d′)) ≅ D(c ⊗ d, d′) ≅ D(d, \underline{Hom}_{D}(c, d′))

There is the following result [Dug06, Lemma A.2] which extends the above isomorphisms of sets to isomorphisms of C.

Lemma 1.14 Let D be a closed C-module, then for any d, d′ in D and c in C, there are canonical isomorphisms in C

F_{C}(c, \underline{Hom}_{C}(d, d′)) ≅ \underline{Hom}_{C}(d ⊗ c, d′) ≅ \underline{Hom}_{C}(d, \underline{Hom}_{D}(c, d′))

which, after applying C(1_{C}, −), reduce to the isomorphisms of the definition above.

As (briefly) mentioned after [Hov99, 4.1.12] the correct notion of a ‘map’ of Cmodules is an adjoint pair of functors such that the left adjoint is a C-module functor. We note the following result ([Dug06, Lemma A.7] which supports this argument.

Lemma 1.15 Take an adjunction L : D ⇆ E : R between closed C-modules, then the following are equivalent:

(i). Natural isomorphisms in E, c ⊗ Ld ≅ L(c ⊗ d), which reduce to the canonical isomorphisms for c = 1_{C}.

(ii). Natural isomorphisms in D, \underline{Hom}_{D}(c, Re) ≅ R \underline{Hom}_{E}(c, e) which reduce to the canonical isomorphisms for c = 1_{C}.

(iii). Natural isomorphisms in C, \underline{Hom}_{C}(d, Re) ≅ \underline{Hom}_{C}(Ld, e) which after applying C(I_{C}, −) reduce to the adjunction between L and R.

Proof Take any c, d, e in C, D, E respectively. Then

E(c ⊗ Ld, e) ≅ E(Ld, \underline{Hom}_{E}(c, e)) ≅ D(d, R \underline{Hom}_{E}(c, e))
E(L(c ⊗ d), e) ≅  D(c ⊗ d, Re)  ≅ D(d, \underline{Hom}_{E}(c, Re))

so since an object in a category is determined (up to natural isomorphism) by the maps out of (or the maps into) that object, we can see that the first two conditions are equivalent. The next two collections of isomorphisms show (respectively) that the first and last conditions are equivalent and that the second and third are equivalent.

C(c, \underline{Hom}_{C}(d, Re)) ≅  D(c ⊗ d, Re)  ≅ E(L(c ⊗ d), e)
C(c, \underline{Hom}_{C}(Ld, e)) ≅  E(c ⊗ Ld, e)
C(c, \underline{Hom}_{C}(d, Re)) ≅  D(c ⊗ d, Re)  ≅ D(d, \underline{Hom}_{D}(c, Re))
C(c, \underline{Hom}_{C}(Ld, e)) ≅ E(Ld, \underline{Hom}_{E}(c, e)) ≅ D(d, R \underline{Hom}_{E}(c, e)))

The statement about reductions are similarly routine.

Definition 1.16 An adjunction of closed C-modules is an adjoint pair satisfying one (and hence all) of the above criteria and a pair of coherence diagrams, see [Hov99, Definition 4.1.7] or [Dug06, Appendix A]. A closed (symmetric) C-algebra is a category D with a closed (symmetric) monoidal structure and a (symmetric) monoidal adjunction i_{D} : C ⇆ D : j_{C} [Hov99, Definition 4.1.14]. An adjunction of closed (symmetric) C-algebras between C-algebras D and E is a (symmetric) monoidal adjunction L : D ⇆ E : R with a monoidal natural isomorphism L ∘ i_{D} → i_{E} (so L is a C-algebra functor).

Lemma 1.17 ([Dug06]) Take a strong monoidal adjunction L : C ⇆ D : R. Then a D-module category M can be given the structure of a C-module category by setting

X ⊗ c = X ⊗ Lc, F_{C}(c, Y) = F (Lc, Y) and \underline{Hom}_{C}(X, Y) = R \underline{Hom}_{D}(X, Y)

note that the direction of the adjunction is essential to this lemma.

Corollary 1.18 If there is a strong (symmetric) monoidal adjunction L : C\overrightarrow{⟵}D : R between closed (symmetric) monoidal categories then C and D are a closed (symmetric) C-algebras and (L, R) is an adjunction of closed (symmetric) C-algebras.

We now link these structures back to the definitions of [Kel05].

Proposition 1.19 If (L, R) is an adjunction of closed C-modules then both L and R are C-functors.

Proof Sowing this for the right adjoint takes a rather large diagram chase, see [Dug06, Proposition A.9] for a proof of the statement. That the left adjoint is a C-functor follows from the statement for the left adjoint and [Bor94, Proposition 6.7.2]. The C-structures on F and G arise from the unit and counit:

\underline{Hom}_{C}(d′, d) → \underline{Hom}_{C}(d′, RLd) ≅ \underline{Hom}_{C}(Ld′, Ld)
\underline{Hom}_{C}(e, e′) → \underline{Hom}_{C}(LRe, e′) ≅ \underline{Hom}_{C}(Re, Re′)

where d ∈ D and e ∈ E so that there is a unit map in D d → RLd and a counit map LRe → e in E.

Remark 1.20 Let C be a (symmetric) monoidal category, then we can give C the structure of a closed C-module or a closed (symmetric) C-algebra. If D is a closed (symmetric) C-algebra, then it is certainly a closed C-module. So now we can ask: when is a closed C-module D a closed (symmetric) C-algebra? Obviously the primary requirement is that D should be a (symmetric) monoidal category, but we also need an associativity relation: c⊗(d∧d′) ≅ (c⊗d)∧d′ satisfying the appropriate coherence diagrams. Then we have a strong monoidal (symmetric) adjunction

(−) ⊗ 1_{D} : C\overrightarrow{⟵}D \underline{Hom}(1_{D}, −)

making D into a closed (symmetric) C-algebra. This is precisely analogous to the case of modules and algebras over a ring. When considering model structures the only additional condition is to require that the adjunction is a Quillen pair, which is automatic when 1_{D} is cofibrant.

2 Spectral Enrichments

Now we turn to [SS03b, 3.3.1 and A.1.1], which specialises to the case of enrichments over symmetric spectra. We will also add the model structure conditions by considering the Quillen bifunctors of [Hov99, Chapter 4]. The construction of symmetric spectra is in [HSS00], sometimes it will be helpful for us to use the positive model structure on symmetric spectra from [MMSS01], but the definitions below are the same in each case. It should be clear how to replace the category of symmetric spectra with any other closed symmetric monoidal model category (which satisfies the monoid axiom) in the following. Note that to prove Theorem 2.6 we use the result: smashing with a cofibrant object preserves weak equivalences in Sp^{Σ}.

A spectral model category is a model category M that is a Sp^{Σ}-model category in the sense of [Hov99, 4.2.18]. See also [SS03b, 3.5.1], so M is a closed Sp^{Σ}-module with some conditions on the model structure behaviour. A spectral left Quillen functor is a left Quillen functor that is also a Sp^{Σ}-module functor, it preserves (up to natural isomorphism) the tensor operation ([Hov99, 4.1.7], note that we are assuming compatibility with the associativity isomorphisms). A spectral right Quillen functor is a right Quillen functor R that preserves (up to natural isomorphism) the cotensor operation: \underline{Hom}_{C}(K, RX) ≅ R \underline{Hom}_{D}(K, X).

As (briefly) mentioned after [Hov99, 4.1.12] the correct notion of a ‘map of spectral model categories’ is a spectral adjunction of categories as we define below see [Hov99, 4.1.14]. When we have an adjunction of spectral model categories we can ask for the relation of [SS03b, 3.9.2]: a natural isomorphism of symmetric spectra

\underline{Hom}_{C}(A, RX) ≅ \underline{Hom}_{D}(LA, X).

So a spectral adjunction is an adjunction of closed Sp^{Σ}-modules. This a relation will make both L and R into Sp^{Σ}-functors in the sense of enriched categories above. We call such a pair a spectral Quillen pair if the functors (L, R) are also a Quillen pair. A spectral Quillen equivalence is a spectral Quillen pair that is a Quillen equivalence. We also consider closed (symmetric) monoidal spectral model categories, a closed monoidal model category D with a (symmetric) monoidal adjunction Sp^{Σ} \overrightarrow{⟵} D. An adjunction of closed (symmetric) monoidal spectral model categories is a (symmetric) monoidal Quillen pair L : D \overrightarrow{⟵} E : R such that L ∘ i_{D} is naturally isomorphic to i_{E} by a (symmetric) monoidal natural transformation.

We introduce some language for categories enriched over symmetric spectra which don’t necessarily have model structures or aren’t tensored or cotensored over symmetric spectra.

Definition 2.1 A spectral category is a category O enriched over the category of symmetric spectra Sp^{Σ}, i.e. O is an Sp^{Σ}-category. A spectral functor is an Sp^{Σ}- functor and a spectral adjunction is an Sp^{Σ}-adjunction. A (right) O-module is a contravariant spectral functor O → Sp^{Σ}. A morphism of O-modules is an Sp^{Σ}-natural transformation.

We spell out the above requirements for an O-module M. For every object of O we have a symmetric spectrum M(o), and we must have coherently unital and associative maps of symmetric spectra

M(o) ∧ O(o′, o) → M(o′)

for pairs of objects o, o′ in O. A morphism of such modules is a collection of maps of symmetric spectra M(o) → N(o) which are strictly compatible with the above action.

The category of O-modules will be written mod– O. We also have a notion of a free module: F_{o}, F_{o}(o′) = O(o′, o). If the category O is a monoidal category then the category of O-modules is a monoidal category via the box product.

Remark 2.2 The definitions above fit neatly into the framework of [Bor94, Section 6.2], If O is an Sp^{Σ}-category then so is O^{op} is an Sp^{Σ}-category [Bor94, Proposition 6.2.2]. Then we have the following definitions from [Bor94, Proposition 6.3.1]:

Category of left O-modules O –mod Sp^{Σ}[O, Sp^{Σ}]
Category of right O-modules mod– O Sp^{Σ}[O^{op}, Sp^{Σ}]

the categories of covariant and contravariant spectral functors from O to Sp^{Σ} with morphisms given by those natural transformations strictly compatible with the action and enriched morphism given by the symmetric spectrum of spectral natural transformations. The above categories are spectral categories.

The following result is [SS03b, Theorem A.1.1].

Theorem 2.3 Let O be a spectral category, then the category of O-modules with object-wise stable equivalences, object-wise stable fibrations and cofibrations as necessary gives a cofibrantly generated spectral model structure. The collection of free modules give a set of compact generators.

The following definition and proposition are in [SS03a, Theorem 6.1].

Definition 2.4 For a spectral functor of spectral categories Ψ: O ⟶ R, we have a restriction of scalars functor

Ψ^{∗}: mod– R ⟶ mod– O, M ↦ M ∘ Ψ.

This has a left adjoint (−) ∧_{O} R the extension of scalars functor. It is defined as an enriched coend and is similar to many other constructions in this work,

N ∧_{O} R = ∫ ^{o} N(o) ∧ F_{Ψ(o)}

we give a construction below as a coequalizer of a pair of R-module homomorphisms.

⋁_{o,p∈O} N(p) ∧ O(o, p) ∧ F_{Ψ(o)}\overrightarrow{⟶} ⋁_{q∈O} N(q) ∧ F_{Ψ(q)}

Proposition 2.5 For a spectral functor of spectral categories Ψ: O ⟶ R there is a Quillen pair

(−) ∧_{O} R : mod– O\overrightarrow{⟵} mod– R : Ψ^{∗}.

We now add an extra condition on O to ensure that mod– O is in fact a closed monoidal model category with unit Id. Note that we will use the fact that in Sp^{Σ} (or Sp_{+}^{Σ}) smashing with a cofibrant object preserves weak equivalences.

Theorem 2.6 If the category O has a closed symmetric monoidal product then mod– O is a closed symmetric monoidal model category satisfying the monoid axiom.

Proof The monoidal product is the box product,

M□N(o) = ∫ ^{p,q} O(o, p ∧ q,) ⊗ M(p) ⊗ N(q)

from [GS06, Section A.2]. The unit is the module \underline{Hom}(−, 𝕊), which is easy to prove using the fact that one can interchange coends and 1.7. It is then a matter of formality that the internal function object exists, it is defined as

\underline{Hom}_{□}(M, N)(o) = Sp^{Σ} –Nat (M, N(o ∧ −)) = ∫_{p} \underline{Hom} (M(p), N(o ∧ p)).

Now we show that the pushout product and monoid axiom hold, to do so we must understand the generating cofibrations, using the proof [SS03a, Theorem 6.1] we identify mod– O with a category of algebras over a triple.

Let 𝕀_{O} be the Sp^{Σ}-category with the same objects as O and maps given by 𝕀_{O}(o, o) = 𝕊 and a point otherwise. There is a canonical map of Sp^{Σ}-categories u: 𝕀_{O} → O, which acts as the unit on 𝕀_{O}(o, o). The adjoint pair of restriction and extension of scalers (2.4), gives the required triple T ([Mac71, Chapter VI]) and the algebras over this triple are O modules. Now mod– 𝕀_{O} is simply an O-indexed product of copies of Sp^{Σ} hence we can give it the product model structure [Hov99, 1.1.6]. The generating (acyclic) cofibrations for mod– 𝕀_{O} are given by maps f: A → B concentrated in factor o where f is a generating (acyclic) cofibration for Sp^{Σ}. Hence a generating (acyclic) cofibration for mod– O has form A ∧ F_{o} → B ∧ F_{o} where f is a generating (acyclic) cofibration for Sp^{Σ}. In fact this argument shows that smashing with a free module takes (acyclic) cofibrations of Sp^{Σ} to (acyclic) cofibrations of mod– O.

The monoid axiom now follows from the monoid axiom for Sp^{Σ} (for the positive case see [MMSS01, Theorem 14.2]). Take a generating acyclic cofibration A∧F_{o} → B∧F_{o} and any module M now for p ∈ O we have ((A∧F_{o})□M)(p) = A∧(F_{o}□M)(p) so this must be a weak equivalence of symmetric spectra, hence (A∧F_{o})□M → (B∧F_{o})□M is a weak equivalence in mod– O. Since pushouts and transfinite compositions of O-modules are constructed object-wise, the rest of the monoid axiom follows.

As part of the pushout product axiom we show that for any module M

ĉ\underline{Hom}(−, Id)□M → M

is a weak equivalence. The important step is in identifying ĉ\underline{Hom}(−, Id) with ĉ𝕊 ∧ \underline{Hom}(−, Id), the latter is cofibrant as an O-module since ĉ𝕊 is cofibrant in Sp^{Σ}. The map

ĉ𝕊 ∧ \underline{Hom}(−, Id) → \underline{Hom}(−, Id)

is a weak equivalence since ĉ𝕊 ∧ X → X is a weak equivalence for any symmetric spectrum X (smashing with cofibrant objects in Sp^{Σ} preserves weak equivalences). So we have reduced the problem to proving that ĉ𝕊∧M → M is a weak equivalence, but this is obvious from the above statements.

The pushout product also requires the identity (A ∧ F_{o})□(B ∧ F_{p}) ≅ (A ∧ B) ∧ F_{o∧p}, the result now follows from the pushout product axiom for symmetric spectra and the fact that smashing with a free module preserves (acyclic) cofibrations as noted above.

Definition 2.7 Given a pair of spectral functors F, G: C → D between spectral categories, the symmetric spectrum of spectral natural transformations ([Bor94, Proposition 6.3.1]) is defined as

Sp^{Σ} –Nat(F, G) = ∫_{c} \underline{Hom}(F c, Gc).

As an equaliser of products we have

Sp^{Σ} –Nat(F, G) ⟶ ∏_{c∈C} \underline{Hom}(F c, Gc)\overrightarrow{⟶} ∏_{c,c′∈C} \underline{Hom}(C(c, c′) ⊗ F c, Gc′)

where \underline{Hom} internal function object for symmetric spectra.

Remark 2.8 The above is the general definition when working with right modules over O one must remember to account for the change in variance. So for right Omodules M and N the symmetric spectrum of morphisms of O-modules is given below.

Sp^{Σ} –Nat(M, N) ⟶ ∏_{c∈C} \underline{Hom}(Mc, Nc)\overrightarrow{⟶} ∏_{c′,c∈C} \underline{Hom}(\underline{Hom}(c, c′) ⊗ Mc, Nc′)

Proposition 2.9 The category of O-modules is tensored and cotensored over symmetric spectra.

Proof The tensor and cotensor are given object-wise, so for a symmetric spectrum A and a (right) O-module M

(A ⊗ M)(o) = A ∧ M(o) \underline{Hom}(A, M)(o) = \underline{Hom}(A, M(o)).

Now we show that these are compatible with the enrichment, so that:

Sp^{Σ} –Nat(M, \underline{Hom}(A, N)) ≅ Sp^{Σ} –Nat(A ⊗ M, N) ≅ \underline{Hom}(A, Sp^{Σ} –Nat(M, N)).

Writing this out in terms of ends we require a pair of relations

∫_{o} \underline{Hom}(A ∧ M(o), N(o)) ≅ ∫_{o} \underline{Hom}(M(o), \underline{Hom}(A, N(o)))
∫_{o} \underline{Hom}(A, \underline{Hom}(M(o), N(o))) ≅ \underline{Hom}(A, ∫_{o} \underline{Hom}(M(o), N(o)))

the first of which is certainly true and the second follows from [Bor94, Proposition 6.7.3]: the associativity of a Sp^{Σ} ensures that (A ∧ (−), \underline{Hom}(A, −)) is a Sp^{Σ}- adjunction.

In truth the above enrichment, tensoring and cotensoring over symmetric spectra actually arises from giving mod– O the structure of a Sp^{Σ}-algebra.

Theorem 2.10 There is a symmetric strong monoidal Quillen pair

(−) ⊗ F_{Id} : Sp^{Σ}\overrightarrow{⟵} mod– O : \underline{Hom}(F_{Id}, −)

thus mod– O is an Sp^{Σ}-algebra. This structure is compatible with the enrichment, tensoring and cotensoring above. The result remains true in the case of a Sp_{+}^{Σ} category or any other symmetric monoidal model category.

Proof We clearly have a symmetric strong monoidal pair (using 2.9), we must now check that (−) ⊗ F_{Id} preserves (acyclic) cofibrations, we have already done so in 2.6.

To recover the tensoring one simply needs to use the basic properties of coends (such as commutation with left adjoints and interchange of factors) and 1.7 to see:

A ⊗ M ≅ (A ⊗ F_{Id})□M.

For the cotensoring the same kind of argument suffices to show

\underline{Hom}(A, M) ≅ \underline{Hom}_{□}(A ⊗ F_{Id}, M)

but we need 1.9 since we are now in a ‘right-handed’ case. The enrichment can be identified as

\underline{Hom}(M, N) ≅ \underline{Hom}(F_{Id}, \underline{Hom}_{□}(M, N))

and this is proved in the same way as the rest. The last statement of the theorem is clear, at no point have we actually used anything about the model category other than what we have assumed, in particular we don’t need the unit to be cofibrant, nor do we need the free modules to be cofibrant.

Remark 2.11 It is instructive to look at this result in a little more generality. For each o ∈ O there is a Quillen pair

(−) ∧ F_{o} : Sp^{Σ}\overrightarrow{⟵} mod– O : ev_{o}

where ev_{o}(M) = M(o), the evaluation functor. To show that this is an adjoint pair we use the enriched Yoneda lemma 1.10,

mod– O(A ∧ F_{o}, M) ≅ Sp^{Σ}(A, \underline{Hom}(F_{o}, M)) ≅ Sp^{Σ}(A, M(o))

we can now use 2.6 to see that this is a Quillen pair. Alternatively, we can show this more directly: since fibrations and weak equivalences are defined object-wise, each ev_{o} is a right Quillen functor.

Definition 2.12 For a spectral category D and a full subcategory E(G) defined by an object set G there is a functor (sometimes called the tautological functor)

\underline{Hom}(G, −): D ⟶ mod– E(G)

\underline{Hom}(G, d)(g) = \underline{Hom}(g, d). This has a left adjoint − ∧_{E(G)} G, and for a module M this is given by an enriched coequaliser of the following diagram (or one could describe it as an enriched coend).

⋁_{g,h∈G} M(h) ∧ E(G)(g, h) ∧ g\overrightarrow{⟶} ⋁_{g∈G} M(g) ∧ g

Proposition 2.13 When D is a closed symmetric monoidal spectral category and the object subset G is closed under the monoidal product, the Quillen pair

(−) ∧_{E(G)} G : \overrightarrow{⟵}D : \underline{Hom}(G, −)

is an adjunction of closed symmetric monoidal spectral model categories.

Proof The left adjoint is symmetric monoidal by the following series of isomorphisms, where we most definitely use the assumption that D is a symmetric monoidal spectral category.

(M□N) ⊗_{E} G = ∫ ^{g∈G} ∫ ^{x,y} (E(g, x ∧ y) ∧ M(x) ∧ N(y)) ⊗ g
= ∫ ^{x,y} M(x) ∧ N(y) ⊗ (∫ ^{g∈G} E(g, x ∧ y))
≅ ∫ ^{x,y} M(x) ∧ N(y) ⊗ (x ∧ y)
≅ ∫ ^{x,y}(M(x) ⊗ x) ∧ (N(y) ⊗ y)
≅ ∫ ^{x}(M(x) ⊗ x) ∧ ∫ ^{y}(N(y) ⊗ y)
≅ (∫ ^{x} M(x) ⊗ x) ∧ (∫ ^{y} N(y) ⊗ y)
= M ⊗_{E} G ∧ N ⊗_{E} G

It is clear that the left adjoint preserves the Sp^{Σ}-algebra maps, that is:

(−) ⊗_{E} G ∘ (−) ∧ \underline{Hom}(−, Id) ≅ i_{D}

where i_{D}: Sp^{Σ} → D is the left adjoint of the Quillen pair giving D a closed symmetric monoidal spectral category structure.

We also need the following to be a weak equivalence in Sp^{Σ}.

ĉ\underline{Hom}(−, Id) ⊗_{E} G → \underline{Hom}(−, Id) ⊗_{E} G ≅ 𝕊

Here we use the characterisation ĉ\underline{Hom}(−, Id) = ĉ𝕊 ∧ \underline{Hom}(−, Id) and then the result is immediate from the above.

The following result ([SS03b, 3.9.3]) is important, though we will sometimes use slightly altered versions to deal with cases when not all the objects are cofibrant. In practice, we will work in a situation where both this result and Proposition 2.13 hold.

Theorem 2.14 Let D be a spectral model category and G a set of cofibrant and fibrant objects of D.

(i). The tautological functor \underline{Hom}(G, −): D ⟶ mod– E(G) is the right adjoint of a spectral Quillen pair.

(ii). If G is a set of compact generators for D then the above Quillen pair is a spectral Quillen equivalence.

3 Involutions on mod– E_{SO(2)}

We begin with the work of [GS06, Section 4], the authors are forced to work in the category of S_{SO(2)}^{ℚ}-modules, modules over the rational SO(2) equivariant sphere spectrum.

Theorem 3.1 The categories S_{SO(2)}^{ℚ} –mod, S_{O(2)}^{ℚ} –mod and S_{𝒞}^{ℚ} –mod are all spectral model categories, in fact they are also closed ĉ_{CR} H ℚ –mod-modules or equally they are algebras over Sp_{+}^{Σ} and ĉ_{CR} H ℚ –mod.

Proof Each of these categories is a closed symmetric monoidal category, hence they are algebras over themselves, we can now use the following series of adjoints to pull this structure back to ĉ_{CR} H ℚ –mod and Sp_{+}^{Σ} (symmetric spectra with the positive model structure).

\xymatrix{Sp_{+}^{Σ} \ar[r]^{F} & ĉ_{CR} H ℚ –mod \ar[r]^{Λ∘|−|} \ar[l]^{U} & Λ|ĉ_{CR} H ℚ| –mod \ar[r]^{ε^{♯}} \ar[l]^{S∘Φ} & S_{SO(2)}^{ℚ} –mod \ar[l]^{(−)^{SO(2)}}}

Where ε^{♯} is the strong monoidal inflation functor including the relevant change of universe. Thus for for two SO(2)-equivariant spectra X and Y the ĉ_{CR} H ℚ-module function object is given by ΦSF_{S_{ℚ}}(X, Y)^{SO(2)} and the symmetric spectrum function object is the underlying symmetric spectrum of this object.

Remark 3.2 Recall that Quillen invariance of modules holds in the positive model structure of symmetric spectra ([MMSS01, Theorem 14.5]). Hence the categories ĉ_{CR} H ℚ –mod and H ℚ –mod are Quillen equivalent. But induction, the left adjoint of the pair, goes from ĉ_{CR} H ℚ –mod to H ℚ –mod so we are unable to pull the enrichment over this adjunction.

Consider the homogenous spaces Σ^{∞}SO(2)/H_{+} where H runs over all subgroups of SO(2), these are a set of generators for M𝒮_{SO(2)}. Now we smash these with the rational SO(2)-equivariant sphere spectrum, S_{SO(2)}^{ℚ} to obtain a set of generators for

S_{SO(2)}^{ℚ} –mod.

We can now apply the idempotents and take cofibrant replacements (in the category of S_{SO(2)}^{ℚ}-modules) to obtain the basic cells σ_{H} = Qe_{H}S_{SO(2)}^{ℚ} ∧ Σ^{∞}SO(2)/H_{+}, since all spectra are fibrant in this model category, we have a set of cofibrant-fibrant objects which we call BC_{SO(2)}. We have the following result, [Gre99, 2.1.5], which implies that this collection is a generating set.

Lemma 3.3 The homogenous spaces can be obtained from the basic cells.

Σ^{∞}SO(2)/H_{+} ≃ ⋁_{K⊆H} σ_{K}

We have an alternative construction of the basic cells, which will be of use later. First, for each H ∈ 𝒞, construct the homotopy colimit e_{H}S_{SO(2)}^{ℚ} (in the category of

S_{SO(2)}^{ℚ}-modules), this gives the S_{SO(2)}^{ℚ}-module e_{H}S_{SO(2)}^{ℚ} ∧ Σ^{∞}SO(2)/H_{+}. Consider the map j: Σ^{∞}SO(2)/H_{+} → h^{†}j^{∗}Σ^{∞}SO(2)/H_{+}, which induces a map of order two

e_{H}S_{SO(2)}^{ℚ} ∧ Σ^{∞}SO(2)/H_{+} → h^{†}j^{∗}(e_{H}S_{SO(2)}^{ℚ} ∧ Σ^{∞}SO(2)/H_{+})

(since h^{†}j^{∗}S_{SO(2)}^{ℚ} = S_{SO(2)}^{ℚ} and h^{†}j^{∗}e_{H} = e_{H}). Now we apply the cofibrant replace-

ment functor of W ⋉ S_{ℚ}^{SO(2)} –mod to produce a map of cofibrant-fibrant objects

σ_{H} → h^{†}j^{∗}σ_{H} (in S_{ℚ}^{SO(2)} –mod). Since any two cofibrant replacements are weakly equivalent this construction is as good as any other and the preceding lemma remains valid in this case.

Now define B̄C_{SO(2)} to be the closure of BC_{SO(2)} under smash products with the unit included. By monoidality all non-unit objects are cofibrant and all objects are fibrant. Recall that we are using the smash product of S_{SO(2)}^{ℚ}-modules here. The full subcategory of MS_{SO(2)} with object set B̄C_{SO(2)} will be denoted E_{SO(2)}.

Theorem 3.4 The Quillen pair

(−) ∧_{E_{SO(2)}} G : MS_{SO(2)}\overrightarrow{⟵} mod– E_{SO(2)} : \underline{Hom}(G, −)

is a monoidal Quillen equivalence [GS06, Theorem 4.1].

Proposition 3.5 The adjoint pair of restriction and co-induction given by CI = F_{SO(2)}(O(2)_{+}, −), R = i^{∗} are S^{ℚ} –mod^{U^{O(2)}}-functors and hence are spectral functors. Moreover this is an adjunction of closed symmetric monoidal spectral functors.

Proof The following diagram of adjoints commutes up to natural isomorphism.

\xymatrix{S_{SO(2)}^{ℚ} –mod^{i^{∗}U} \ar[rr]_{CI} \ar@{-->}[dr] &  & S_{𝒞}^{ℚ} –mod^{U} \ar[ll]_{R} \ar[dl] \\  & S^{ℚ} –mod^{U^{O(2)}} \ar@{-->}[ul]_{ε^{♯}} \ar[ur]^{ε^{♯}}}

Lemma 3.6 The functor τ is a spectral functor, moreover (τ, τ) is an adjunction of closed symmetric monoidal spectral functors.

Proof In fact we have a symmetric monoidal adjunction of closed S^{ℚ} –mod^{U^{O(2)}}- algebras, which is no more than easily checked statement : τε_{SO(2)}^{#} ≅ ε_{SO(2)}^{#}.

Corollary 3.7 The functor τ is a self-inverse map of ringoid spectra τ: E_{SO(2)} → τE_{SO(2)}.

Proof We define the set τE_{SO(2)} to be the full subcategory of S_{SO(2)}^{ℚ} –mod with object set τG_{SO(2)}. The result is then obvious and we do not introduce any new notation for the inverse functor τ: τE_{SO(2)} → E_{SO(2)}.

Lemma 3.8 There is an invertible map of ringoid spectra W: E_{SO(2)} → τE_{SO(2)}.

Proof On objects, W acts as τ, so W σ = τσ. On the homomorphism spectra W acts as

\underline{Hom}(τw, w′): \underline{Hom}(σ, σ′) → \underline{Hom}(τσ, τσ′)

where w: σ → τσ and w′: σ′ → τσ′ are the maps which arise from j: SO(2) → SO(2) (recall that σ is some smash product of the basic cells and we have defined w_{H} for each basic cell σ_{H}). It should be obvious that this defines a ringoid map. We denote inverse of this map as W^{−1} and this acts τ on objects and acts as on homomorphism spectra as \underline{Hom}(w, τw′).

Definition 3.9 We define an involution on mod– E_{SO(2)} by ρ = (τW)^{∗}. That this functor is self-inverse follows immediately from the relation

τ ∘ \underline{Hom}(τw, w′) = \underline{Hom}(w, τw′) ∘ τ.

Hence the pair (mod– E_{SO(2)}, ρ) give a monoidal model category with involution that satisfies the monoid axiom.

Remark 3.10 Using the coequaliser version of the box product it is easy to see that mod– E_{SO(2)} is a monoidal category with involution, one simply has to draw a rather large diagram and use the fact that τ and W are well behaved maps. It should also be clear that ρ gives an adjunction of closed symmetric monoidal spectral functors.

Lemma 3.11 The following diagram commutes for all g ∈ G_{SO(2)} and is natural in X.

\xymatrix{ & g ∧ \underline{Hom}(τg, X) \ar[dr]^{1∧τ} \\ g ∧ \underline{Hom}(g, X) \ar[dr]^{1∧τ} \ar[ur]^{1∧(τw_{g})^{∗}} &  & g ∧ \underline{Hom}(g, τX) \ar[ddd]^{eval} \\  & g ∧ \underline{Hom}(τg, τX) \ar[ur]^{(1)}^{1∧(w_{g})^{∗}} \\  & τg ∧ \underline{Hom}(τg, τX) \ar[dr]^{eval} \ar[u]^{τw_{g}∧1} \\ τg ∧ \underline{Hom}(g, X) \ar[dr]^{(2)}^{γ} \ar[ur]^{1∧τ} \ar[uuu]^{τw_{g}∧1} &  & τX \\  & τ(g ∧ \underline{Hom}(g, X)) \ar[ur]^{(4)}^{(3)}^{τeval}}

Proof Where γ is the natural transformation between τX ⊗ A → τ(X ⊗ A) and for a symmetric spectrum A and a rational SO(2)-spectrum X. Squares (1) and (2) obviously commute, square (4) commutes τ is a spectral functor (see also [Bor94, Proposition 6.4.5]). The fact that Square (3) commutes requires a little more consideration but essentially follows from axiom that evaluation and composition are compatible.

Lemma 3.12 The following square commutes up to a natural isomorphism, β. The pair (\underline{Hom}(G_{SO(2)}, −), β) define an involutary functor.

\xymatrix{S_{SO(2)}^{ℚ} –mod \ar[r]^{\underline{Hom}(G_{SO(2)},−)} \ar[d]^{τ} & mod– E_{SO(2)} \ar[d]^{ρ} \\ S_{SO(2)}^{ℚ} –mod \ar[r]^{\underline{Hom}(G_{SO(2)},−)} & mod– E_{SO(2)}}

Proof We consider an SO(2) spectrum X, this gives the module ρ \underline{Hom}(−, X) via the top, and via the bottom it maps to \underline{Hom}(−, τX). The natural isomorphism ρ \underline{Hom}(−, τX) → \underline{Hom}(−, X) is given by \underline{Hom}(w, 1) ∘ τ and from the diagram below (which is natural in X) it should be obvious that this natural transformation satisfies the necessary condition to make \underline{Hom}(G_{SO(2)}, −) into an involutary functor.

\xymatrix{\underline{Hom}(A, τX) ∧ \underline{Hom}(B, A) \ar[r]^{1∧τ∘W} \ar[d]^{τ∧1} & \underline{Hom}(A, τX) ∧ \underline{Hom}(B, A) \ar[r] & \underline{Hom}(B, τX) \ar[d]^{τ} \\ \underline{Hom}(τA, X) ∧ \underline{Hom}(B, A) \ar[d]^{w_{A}^{∗}∧1} &  & \underline{Hom}(τB, X) \ar[d]^{w_{B}^{∗}} \\ \underline{Hom}(A, X) ∧ \underline{Hom}(B, A) \ar[rr] &  & \underline{Hom}(B, X)}

It is easy to see that the above diagram commutes, one can divide it into a triple of diagrams, 2 of which clearly commute and the third commutes by the previous lemma.

Lemma 3.13 The left adjoint (−) ∧_{E_{SO(2)}} G_{SO(2)} is also an involutary functor.

Proof Pick M, an object of mod– E_{SO(2)}, then M ∧_{E_{SO(2)}} G_{SO(2)} is given by the coequaliser in 2.12. The natural transformation

α: (σ−) ∧_{E_{SO(2)}} G_{SO(2)} ⟶ τ((−) ∧_{E_{SO(2)}} G_{SO(2)})

is defined by the map of coequalisers given in the diagram below.

\xymatrix{⋁_{g,h∈G} M(h) ∧ E(G)(g, h) ∧ g \ar[r]^{μ∘τ∘W_{g,h}} \ar[r]_{eval} \ar[d]_{1∧τ∘W_{g,h}∧w_{g}} & ⋁_{g∈G} M(g) ∧ g \ar[d]^{1∧w_{g}} \\ ⋁_{g,h∈G} M(h) ∧ E(G)(g, h) ∧ τg \ar[r]^{μ} \ar[r]_{τ(eval)∘γ} & ⋁_{g∈G} M(g) ∧ τg}

To prove that the above does define a map of coequalisers it suffices to show that the two diagrams below commute.

\xymatrix{M(h) ∧ E(G)(g, h) \ar[r]^{μ∘τ∘W_{g,h}} \ar[d]_{1∧τ∘W_{g,h}} & M(g) \ar[d]^{1} & E(G)(g, h) ∧ g \ar[r]^{eval} \ar[d]_{τ∘W_{g,h}∧w_{g}} & h \ar[d]_{w_{h}} \\ M(h) ∧ E(G)(g, h) \ar[r]^{μ} & M(g) & E(G)(g, h) ∧ τg \ar[r]^{τ(eval)∘γ} & τh}

The left hand diagram automatically commutes, it is easy to check that the right hand diagram commutes by considering the definition of the map W and using Lemma 3.11. Note that this natural transformation clearly satisfies the necessary condition for it to be an involutary natural transformation.

Proposition 3.14 The adjunction is involutary.

Proof Begin with the counit, one can draw a large and unwieldy diagram of coequalisers which commutes due to Lemma 3.11. The unit is dealt with by drawing an equally large diagram, which reduces the result to checking that the diagram below commutes.

\xymatrix{M(k) \ar[r]^{coeval} \ar[d]_{coeval} & \underline{Hom}(k, k ∧ M(k)) \ar[d]^{(1∧w_{k})_{∗}} \\ \underline{Hom}(k, k ∧ M(k)) \ar[r]^{γ^{−1}∘(w_{k})^{∗}∘τ} & \underline{Hom}(k, τk ∧ M(k))}

This is the coevaluation version of Lemma 3.11 this commutes since the pair of squares below commute.

\xymatrix{M(k) \ar[r]^{coeval_{k}} \ar[d]_{coeval_{τk}} & \underline{Hom}(k, k ∧ M(k)) \ar[d]^{(1∧w_{k})_{∗}} \\ \underline{Hom}(τk, τk ∧ M(k)) \ar[r]^{(w_{k})^{∗}} & \underline{Hom}(k, τk ∧ M(k)) \\ M(k) \ar[r]^{coeval_{k}} \ar[d]_{coeval_{τk}} & \underline{Hom}(k, k ∧ M(k)) \ar[d]^{τ} \\ \underline{Hom}(τk, τk ∧ M(k)) \ar[r]^{γ_{∗}} & \underline{Hom}(τk, τ(k ∧ M(k)))}

Corollary 3.15 There is a symmetric monoidal Quillen equivalence between the half-twisted categories

τ ⋉ (S_{𝒞}^{ℚ} –mod) ≃ ρ ⋉ (mod– E_{SO(2)}).

Furthermore this is an adjunction of closed symmetric 1 ⋉ Sp_{+}^{Σ}-algebras.

Proof We must show that the diagram below which gives the algebra structure on the half-twisted categories commutes up to natural isomorphism.

\xymatrix{ρ ⋉ (mod– E_{SO(2)}) \ar[rr] \ar[dr] &  & τ ⋉ S_{SO(2)}^{ℚ} –mod \ar[ll] \ar@{-->}[dl]^{ℙ} \\  & 1 ⋉ Sp_{+}^{Σ} \ar[ul] \ar@{-->}[ur]}

Since the corresponding statement is true for the underlying categories all we need check is that the involutions are compatible, in the sense that for x ∈ Sp_{+}^{Σ} and E ∈ S_{SO(2)}^{ℚ} –mod we will have the following commutative diagrams.

\xymatrix{∫^{g}(\underline{Hom}(g, S_{ℚ}) ∧ x) ⊗ g \ar[r] \ar[d] & S_{ℚ} ⊗ x \ar[dd] & \underline{Hom}(S_{ℚ}, τE) \ar[r] \ar[d] & ℙτE \ar[d] \\ ∫^{g} ρ(\underline{Hom}(g, S_{ℚ}) ∧ x) ⊗ g \ar[d] &  & ρ \underline{Hom}(S_{ℚ}, E) \ar[r] & ℙE \\ τ ∫^{g}(\underline{Hom}(g, S_{ℚ}) ∧ x) ⊗ g \ar[r] & τ(S_{ℚ} ⊗ x)}

The left hand diagram commutes by the same arguments we used above to show that (−) ∧_{E_{SO(2)}} G_{SO(2)} is an involutary functor. The right hand diagram also commutes since on both sides we are using the relations between taking fixed points and applying τ.

Remark 3.16 There remains one detail to consider, to show that the natural transformations α and β are monoidal, so that the adjoint pair above really is a monoidal pair on the half-twisted categories. This is a similar check as in many of the cases and we omit it for that reason.

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