An Involution on Rational S^{1}-Equivariant Spectra.

David Barnes

Abstract

We define a category with involution and the associated half-twisted category. We then apply the category of rational S^{1}-spectra and identify the half-twisted category as the category of rational cyclic O(2)-spectra.

1 An Involution on spaces and spectra

This section is based on the equation htht = 1 in 0(2), for t ∈ SO(2) and h ∈ O(2) \ SO(2) (so t is a rotation and h is a reflection). Conjugation by h considered as an automorphism of O(2) restricts to the inversion automorphism j of S0(2), j(t) = t^{−1}.

This gives a functor on topological spaces, j^{∗}, the pullback of an S0(2) space along j. For a general group homomorphism f: G → H, we can pull X, an H-space, back to a G-space by g · x = f(g) · x. Since this is a contravariant construction, we use an upper asterisk and call this G space f^{∗}X. The underlying set of f^{∗}X is, of course, the same and the action of f^{∗} on maps is effectively just a relabelling.

Now let us move to a marginally less trivial consequence, take X an O(2)-space then we can consider i^{∗}X, the restriction of X to an SO(2)-space along i: SO(2) → O(2). Any SO(2)-space Y has an inverse representation (which is the same space but with t ∈ SO(2) acting by its inverse) given by j^{∗}Y. We then have an SO(2) map h: i^{∗}X → j^{∗}i^{∗}X, t ∗ hv = t^{−1} · h · v = (ht) · v (∗ is the SO(2) action on j^{∗}i^{∗}X and · is the O(2) action on X and so gives the SO(2) action on i^{∗}X). The map h has an inverse, ĥ: j^{∗}i^{∗}X → i^{∗}X, ĥ(x) = h(x), from now on we will just write h for this map. In general there won’t be an invertible SO(2) map Y → j^{∗}Y.

We redraw some of the previous information for clarity, let X be an O(2)-space then we have the following commutative diagram, with ϕ giving the action of O(2) on X.

\xymatrix{SO(2) \ar[r]^{i} \ar[d]^{j} & O(2) \ar[r]^{ϕ} \ar[d]^{C_{h}} & Aut(X) \ar[d]^{C_{ϕ(h)}} \\ SO(2) \ar[r]^{i} & O(2) \ar[r]^{ϕ} & Aut(X)}

Where j is the inversion isomorphism, C_{h} is conjugation by h and and C_{ϕ(h)} is conjugation by ϕ(h). So we have the O(2)-spaces X = (X, ϕ), and C_{h}^{∗}X = (X, ϕ ∘ C_{h}) = (X, C_{ϕ(h)} ∘ ϕ). We also have SO(2)-spaces i^{∗}X = (X, ϕ ∘ i) and j^{∗}i^{∗}X = (X, ϕ ∘ i ∘ j) = (X, ϕ ∘ C_{h} ∘ i) = i^{∗}C_{h}^{∗}X, so j^{∗}i^{∗}X is an SO(2)-space in the image of the functor i^{∗}.

We now want to consider the analogue of this inversion in the category of rational S^{1}- equivariant spectra. The model we use is modules over a rational sphere spectrum S_{ℚ} ∈ S^{1}M (we will generally use a universe of form i^{∗}U for U a complete O(2) universe). So we have a functor j^{∗}: S^{1}M → S^{1}M induced from j, this also changes the universe, if we are indexed on i^{∗}U to begin, then we finish indexed on j^{∗}i^{∗}U (i^{∗}U with the opposite S^{1} action). Note that any reflection h ∈ O(2) gives an S^{1}- equivariant isomorphism h: i^{∗}U → j^{∗}i^{∗}U. We will investigate in a later section the relation between functors induced from maps of the underlying group and change of universe in a later section. We note the following:

j^{∗}S_{ℚ} = j^{∗}I_{U^{O(2)}}^{i^{∗}U}ε_{S^{1}}^{∗} H ℚ = I_{U^{O(2)}}^{j^{∗}i^{∗}U}j^{∗}ε_{S^{1}}^{∗} H ℚ = I_{U^{O(2)}}^{j^{∗}i^{∗}U}ε_{S^{1}}^{∗} H ℚ

and hence that j^{∗}S_{ℚ} ≅ I_{i^{∗}U}^{j^{∗}i^{∗}U}S_{ℚ}. This will not hold true in general for modules over S_{ℚ}. We can now consider the two j^{∗}S_{ℚ}-modules, I_{i^{∗}U}^{j^{∗}i^{∗}U}M and j^{∗}M for an S_{ℚ}-module M. Our involution is the functor I_{j^{∗}i^{∗}U}^{i∗U}j^{∗}.

Note that all of these functors are strong monoidal, hence they preserve the module structures and the smash product of modules. Now we give a lemma which is central to this section, as it tells us that in certain cases we can replace the above complicated functor with the more intuitive (but generally non-monoidal) change of universe functors f_{∗} and f^{∗}. This result is [May98, Lemma 3.5].

Lemma 1.1 Let f: U → U′ be an isomorphism of G-universes. The there are natural isomorphisms

f_{∗}E ≅ I_{U}^{U′}E f^{∗}E′ ≅ I_{U′}^{U}E′

for E ∈ G𝒮 U[𝕃] and E′ ∈ G𝒮 U′[𝕃′].

To make things a little clearer we will use the notation f_{†} and f^{†} for these change of universe functors.

Theorem 1.2 For an O(2)-spectrum X indexed on an O(2)-universe U, we have a map (natural in O(2)-maps of X)

h: i^{∗}X → h^{†}j^{∗}i^{∗}X.

Proof We begin with the definition of the map on the levels of i^{∗}X, then show this is a map of spectra and a map of i ∗ 𝕃-spectra, S_{O(2)}-modules and S_{ℚ}-modules in turn.

Take an indexing space i^{∗}V of i^{∗}U, then we have (for any fixed reflection h ∈ O(2)) a map

h: (i^{∗}X)(i^{∗}V) = i^{∗}(X(V)) ⟶ j^{∗}i^{∗}(X(V)) = (h^{†}j^{∗}i^{∗}X)(i^{∗}V).

The last equality is simple: V is a sub-inner-O(2)-product space of U, hence hV = V as sets, but as usual the action of S^{1} is inverted, so hV = j^{∗}V as S^{1}-inner product spaces.

Now we show that this gives a map of spectra: take i^{∗}W ⊂ i^{V} and recall that the structure maps for h^{†}j^{∗}i^{∗}X are given by j^{∗}i^{∗}σ ∘ h ∧ 1

S^{i^{∗}W −i^{∗}V} ∧ (h^{†}j^{∗}i^{∗}X)(i^{∗}V) ⟶^{h∧1} S^{j^{∗}i^{∗}W −j^{∗}i^{∗}V} ∧ (h†j^{∗}i^{∗}X)(i^{∗}V)
= j^{∗}i^{∗}(S^{W −V} ∧ X(V))
⟶^{j^{∗}i^{∗}σ} j^{∗}i^{∗}(X(W))

The relevant equation to check is that j^{∗}i^{∗}σ ∘ h ∧ 1 ∘ 1 ∧ h = hi^{∗}σ (recall that as maps of the underlying sets, j^{∗}i^{∗}σ = i^{∗}σ = σ) and this is certainly true — X is an O(2) spectrum.

We have an action of ℐ (i^{∗}U, i^{∗}U) on h^{†}j^{∗}i^{∗}X using the composition map

c: ℐ (i^{∗}U, i^{∗}U) × ℐ (j^{∗}i^{∗}U, i^{∗}U) → ℐ (j^{∗}i^{∗}U, i^{∗}U).

Write ℐ (i^{∗}U, i^{∗}U) ⋉ h^{†} as

ℐ (i^{∗}U, i^{∗}U) ⋉ (ℐ (j^{∗}i^{∗}U, i^{∗}U) ⋉_{ℐ (j^{∗}i^{∗}U,j^{∗}i^{∗}U)} )

there is a natural isomorphism from this functor to

(ℐ (i^{∗}U, i^{∗}U) × ℐ (j^{∗}i^{∗}U, i^{∗}U)) ⋉_{ℐ (j^{∗}i^{∗}U,j^{∗}i^{∗}U)}

and here we can use the composition above to obtain a natural transformation to h^{†}. We will call this composite ξ′: ℐ (i^{∗}U, i^{∗}U) ⋉ h^{†} → h^{†}.

Now we must check that the following commutes:

\xymatrix{ℐ (i^{∗}U, i^{∗}U) ⋉ i^{∗}X \ar[r]^{i^{∗}ξ} \ar[d]^{1∧h} & i^{∗}X \ar[d]^{h} \\ ℐ (i^{∗}U, i^{∗}U) ⋉ h^{†}j^{∗}i^{∗}X \ar[r]^{ξ′} & h^{†}j^{∗}i^{∗}X}

This is easiest to do by considering an alternative (but equivalent) construction of the ℐ (i^{∗}U, i^{∗}U) action. We have an isomorphism of functors

(ℐ (i^{∗}U, i^{∗}U) × {h^{−1}})⋉ ≅ ℐ (i^{∗}U, i^{∗}U) ⋉ h^{†}

and we can apply conjugation by h (more properly by h^{−1}) to obtain a homeomorphism of SO(2)-spaces:

C_{h}: ℐ (i^{∗}U, i^{∗}U) × {h^{−1}} → {h^{−1}} × ℐ (j^{∗}i^{∗}U, j^{∗}i^{∗}U).

We combine these to give a structure map as follows:

ℐ (i^{∗}U, i^{∗}U) ⋉ h^{†}j^{∗}i^{∗}X → ({h^{−1}} × ℐ (j^{∗}i^{∗}U, j^{∗}i^{∗}U)) ⋉ j^{∗}i^{∗}X → h^{†}j^{∗}i^{∗}X

the first map is of course the composite of the above isomorphisms, the second is h^{†}j^{∗}i^{∗}ξ.

It is now clear that the above square commutes precisely when h is a map of 𝕃spectra, and that this occurs if and only if

hξ(θ) = h^{†}j^{∗}i^{∗}(ξ(hθh)) ∘ 1 ⋉ h = h^{†}j^{∗}i^{∗}(ξ(θ)) ∘ (C_{h} ⋉ h)

for all θ ∈ ℐ (i^{∗}U, i^{∗}U), which is precisely the condition that ξ is O(2)-equivariant, which it is in this case.

We offer an equation which may shed some light on the previous paragraphs: for each θ ∈ ℐ (i^{∗}U, i^{∗}U) we must have a map of non-equivariant spectra

ξ′(θ): h^{†}j^{∗}i^{∗}X → θ^{†}h^{†}j^{∗}i^{∗}X = (hθ)^{†}j^{∗}i^{∗}X = (hθhh)^{†}j^{∗}i^{∗}X = h^{†}(hθh)^{†}j^{∗}i^{∗}X

Thankfully we can now turn to easier considerations: is h a map of S_{O(2)}-modules? Since the functor h^{†}j^{∗} is strong monoidal, h^{†}j^{∗}i^{∗}X is a module over h^{†}j^{∗}i^{∗}S. We have a map of ring spectra h: i^{∗}S → h^{†}j^{∗}i^{∗}S and we can use this map to give h^{†}j^{∗}i^{∗}X an i^{∗}S-module structure. With this in place we have to check that the following diagram commutes.

\xymatrix{i^{∗}S ∧ i^{∗}X \ar[r]^{i^{∗}ν} \ar[d]^{1∧h} & i^{∗}X \ar[d]^{h} \\ i^{∗}S ∧ h^{†}j^{∗}i^{∗}X \ar[r]^{ν′} & h^{†}j^{∗}i^{∗}X}

Of course, since ν′ = i^{∗}ν(h ∧ 1) this diagram commutes precisely when ν is O(2)- equivariant.

Repeat this procedure with S_{ℚ} as the ring spectrum (recall that h on this ring spectrum is the identity) to see that h is a map of S_{ℚ}-modules if and only if the corresponding square commutes, which is precisely the requirement that the module action map is O(2)-equivariant.

One excuse for such a lengthy proof is that we have proved much more:

Proposition 1.3 An SO(2) spectrum Y can be given the structure of an O(2) spectrum if there exists a map of order two Y → h^{†}j^{∗}Y.

Definition 1.4 A map of order two is a map of spectra w: X → h^{†}j^{∗}X such that (h^{†}j^{∗}w) ∘ w = 1_{X} (and hence w ∘ (h^{†}j^{∗}w) = 1_{h^{†}j^{∗}X}).

Lemma 1.5 The functor h^{†}j^{∗} preserves weak equivalences, cofibrations and fibrations. For an O(2)-spectrum X, with reflection map h: i^{∗}X → h^{†}j^{∗}i^{∗}X the composite (h^{†}j^{∗}h) ∘ h is the identity map of X.

Proof First we show how h^{†}j^{∗} preserves weak equivalences. Recall that we have an isomorphism S^{n} ≅ h^{†}j^{∗}S^{n} and consider the following commutative diagram.

\xymatrix{[S^{n}, h^{†}j^{∗}X]^{H} \ar[d]^{≅} \ar[r] & [S^{n}, h^{†}j^{∗}Y]^{H} \ar[d]^{≅} \\ [h^{†}j^{∗}S^{n}, X]^{H} \ar[d]^{≅} \ar[r] & [h^{†}j^{∗}S^{n}, Y]^{H} \ar[d]^{≅} \\ [S^{n}, X]^{H} \ar[r]^{≅} & [S^{n}, Y]^{H}}

The first pair of horizontal maps is the fact that h^{†}j^{∗} is self-adjoint, the second induced from the above isomorphism of S^{n}. It now suffices to prove that h^{†}j^{∗} preserves cofibrations (the rest will follow by adjointness). Inversion gives an isomorphism SO(2)/H_{+} → j^{∗}SO(2)/H_{+} and hence we have an isomorphism E → h^{†}j^{∗}E for a (generalised) S-module E. Hence we have an isomorphism of maps between the generating cofibration E → CE and its image h^{†}j^{∗}E → h^{†}j^{∗}CE, hence the later is also a cofibration and so h^{†}j^{∗} preserves cofibrations.

Now we consider the second statement, take an indexing space i^{∗}V and consider the composition

(h^{†}j^{∗}h) ∘ h: i^{∗}(X(V)) → j^{∗}i^{∗}(X(V)) → i^{∗}(X(V))

and we note that (on the level of sets) this is simply a double application of the automorphism h to (the O(2)-space) X.

2 Categories With Involution

Definition 2.1 A category with involution (𝒞 , τ) is a category 𝒞 with a functor τ: 𝒞 → 𝒞 such that τ^{2} = 1 (or so that τ^{2} is naturally isomorphic to 1).

It follows, of course, that τ is both a left and a right adjoint.

Definition 2.2 In a category with involution (𝒞 , τ), a map of order two is a map f: A → τA such that τf = f^{−1}.

Definition 2.3 Given a category with involution (𝒞 , τ), we define the associated half-twisted category τ ⋉ 𝒞 to be the category with objects w: A → τA (also denoted (A, w)), such that w ∘ τw = 1_{τA} (so w is a map of order two). A morphism of such objects is a commutative square:

\xymatrix{A \ar[d]^{w} \ar[r]^{f} & B \ar[d]^{w′} \\ τA \ar[r]^{τf} & τB}

and we denote such a morphism by the map of the first factor: f.

Remark 2.4 There is an adjunction

Fun(𝒞 × 𝒟, ℰ) ≅ Fun(𝒞 , fun(𝒟, ℰ)

for (small) categories 𝒞 , 𝒟, ℰ. Applying this in the case of the group categories G and H (these each have one object and arrows correspond to group elements), we see that the category of G × H-objects in 𝒞, Fun(G × H, 𝒞 is isomorphic to the category of G-objects in the category of H-objects of 𝒞 Fun(G, fun(H, 𝒞). We are constructing language to ask the question: when are G ⋉ Fun(H, 𝒞) and Fun(G ⋉ H, 𝒞) isomorphic? Of course since equivariant spectra aren’t a diagram category we will have to work a little harder to obtain the equivalent result in this case.

Proposition 2.5 Assuming that 𝒞 has coproducts, denoted ∨, there is an adjoint pair of functors

𝔻 : 𝒞 \overrightarrow{⟵}τ ⋉ 𝒞 : ℙ

The left adjoint is the free functor, and it acts on objects as 𝔻X = X ∨τX → τX ∨X with twist w the interchange of factors map. The right adjoint is projection onto the first factor ℙ(w: A → τA) = A. These functors act on morphisms in the obvious way.

Proof See 3.3 for a more detailed construction of 𝔻X. First let us understand what information is contained in the map

\xymatrix{X ∨ τX \ar[d]^{w} \ar[r]^{(f,g)} & A \ar[d]^{w′} \\ τX ∨ X \ar[r]^{(τf,τg)} & τA}

Thus we have the requirement w′ ∘ (f, g) = (τf, τg) ∘ w, but (τf, τg) ∘ w = (τg, τf), hence τg = w′f, so g = τ(w′f). Note that w′g = w′τw′τf = τf since w′τw′ = 1. It follows that a map as above determines and is determined by a map f: X → A in 𝒞, hence we have our adjunction.

Lemma 2.6 Assuming that 𝒞 has products, denoted ∏, there is an adjoint pair of functors

ℙ : τ ⋉ 𝒞 \overrightarrow{⟵}𝒞 : 𝔻′

The right adjoint and acts on objects as 𝔻X = X ∏ τX → τX ∏ X with twist w the interchange of factors map. These functors act on morphisms in the obvious way.

Lemma 2.7 Assume that 𝒞 has equalisers, then there is an adjoint pair of functors

ε : 𝒞 \overrightarrow{⟵}1 ⋉ 𝒞 : F ix.

Assume that 𝒞 has coequalisers, then there is an adjoint pair of functors

Orb : 1 ⋉ 𝒞 \overrightarrow{⟵}𝒞 : ε.

Proof For an object A of 𝒞, or (A, u) ∈ 1 ⋉ 𝒞 we set

εA =  1: A → A
F ix(A, u) = A^{u} =  eq(A\overrightarrow{⟶}^{u}_{1}A)
Orb(A, u) = A/u = coeq(A\overrightarrow{⟶}^{u}_{1}A).

Since 1 ⋉ 𝒞 is simply the objects of 𝒞 with an action of the group of order two, we have the usual triple of functors ε which gives an object of 𝒞 a trivial action, F ix which is the categorical fixed point object and Orb which is the categorical orbit set.

Remark 2.8 So when we consider 1 ⋉ 𝒞 we recovered the usual collection of functors: the forgetful functor 1 ⋉ 𝒞 → 𝒞 with it left and right adjoints and the trivial action functor 𝒞 → 1 ⋉ 𝒞 with its left and right adjoints.

Definition 2.9 An involutary functor (F, α): (𝒞 , τ) → (𝒟, σ) consists of a functor F: 𝒞 → 𝒟 and a natural transformation α: F τ → σF such that σα ∘ ατ = 1.

Note that the identity functor, with the identity natural transformation 1: 1_{𝒞} ˙→1_{𝒞} is an involutary functor.

Lemma 2.10 An involutary functor (F, α): (𝒞 , τ) ⟶ (𝒟, σ) passes to a functor (τ, σ) ⋉ F: τ ⋉ 𝒞 ⟶ σ ⋉ σ which we call the half-twisted functor, we will often just call this functor F.

Proof We define (τ, σ) ⋉ F on an object w: A → τA to be the composite map α_{A} ∘ F w: F A → σF A. To see that this is an object of the category we draw the following commutative diagram

\xymatrix{ & F A \ar[d]^{F w} \ar[dr] \ar[ddl]_{=} \\  & F τA \ar[r]^{α_{A}} \ar[d]^{F τw} & σF A \ar[d]^{σF w} \ar[dr] \\ F A \ar[r]^{=} & F τ^{2}A \ar[r]^{α_{τA}} & σF τA \ar[r]^{σα_{A}} & F τ^{2}A \ar[r]^{=} & F A}

it is clear that the equation σα ∘ ατ = 1 allows this involutary functor to pass to a half-twisted functor. For a map f: (w: A → τA) → (w′: A′ → τA′) in τ ⋉ 𝒞, we make the definition: (τ, σ) ⋉ F f = F f (labelling maps by their first factor).

Definition 2.11 For a pair of involutary functors (F, α): (𝒞 , τ) ⟶ (𝒟, σ) and (G, β): (𝒟, σ) ⟶ (ℰ , ρ) we define the composite to be

(G ∘ F, βF ∘ Gα): (𝒞 , τ) ⟶ (ℰ , ρ).

For a map f: (w: A → τA) → (w′: A′ → τA′) in τ ⋉𝒞, we obtain the diagram below which will explain the definition.

\xymatrix{GF A \ar[d]^{GF w} \ar[r]^{GF f} & GF A′ \ar[d]^{GF w′} \\ GF τA \ar[d]^{Gα_{A}} \ar[r]^{GF τf} & GF τA′ \ar[d]^{Gα_{A′}} \\ GσF A \ar[d]^{β_{F A}} \ar[r]^{GσF f} & GσF A′ \ar[d]^{β_{F A′}} \\ ρGF A \ar[r]^{ρGF f} & ρGF A′}

Definition 2.12 An involutary natural transformation η: (F, α) ˙ ⟶(G, β) between involutary functors (F, α), (G, β): (𝒞 , τ) ⟶ (𝒟, σ) is a natural transformation η: F ˙→G such that ση ∘ α = β ∘ ητ.

Lemma 2.13 An involutary natural transformation η: (F, α) ˙ ⟶(G, β) passes to a natural transformation between the half twisted functors

(τ, σ) ⋉ η: (τ, σ) ⋉ F ˙ ⟶(τ, σ) ⋉ G

called the half-twisted natural transformation, we will often abuse notation and just write η for this.

Proof All we need do is draw the diagram below, which will give the definition of (τ, σ) ⋉ η, explain the requirement ση ∘ α = β ∘ ητ and make it clear (through the naturality of each sqaure) that this will be a natural transformation between the half twisted functors.

\xymatrix{F A \ar[d]^{F w} \ar[r]^{η_{A}} & GA \ar[d]^{F w} \\ F τA \ar[d]^{α_{A}} \ar[r]^{η_{τA}} & GτA \ar[d]^{β_{A}} \\ σF A \ar[r]^{ση_{A}} & σGA}

Definition 2.14 An involutary adjunction ((F, α), (G, β), η, ϵ) between two categories with involution (𝒞 , τ) and (𝒟, σ) is an adjunction (F, G, η, ϵ) consisting of involutary functors (F, α): (𝒞 , τ) ⟶ (𝒟, σ) and (G, β): (𝒟, σ) ⟶ (𝒞 , τ) and involutary natural transformations η: 1_{𝒞} ˙ ⟶(GF, βF ∘ Gα) and ϵ: (F G, αG ∘ F β) ˙ ⟶1_{𝒟} .

Lemma 2.15 An involutary adjunction ((F, α), (G, β), η, ϵ) passes to an adjunction of the half-twisted categories. ((τ, σ)⋉F, (σ, τ)⋉G, (τ, σ)⋉η, (σ, τ)⋉ϵ). Furthermore, this gives the commutative adjoint square below.

\xymatrix{τ ⋉ 𝒞 \ar[r]^{(τ,σ)⋉F} \ar[d]^{ℙ} & σ ⋉ 𝒟 \ar[l]^{(σ,τ)⋉G} \ar[d]^{ℙ} \\ 𝒞 \ar[r]^{F} \ar[u]^{𝔻} & 𝒟 \ar[l]^{G} \ar[u]^{𝔻}}

Proof The various assumptions allow us to conclude that F and G pass to the halftwisted categories and that we have a unit and counit. We must check the following pair of equations

((σ, τ) ⋉ G (σ, τ) ⋉ ϵ) ∘ ((τ, σ) ⋉ η (σ, τ) ⋉ G) = 1_{G}
((τ, σ) ⋉ F (τ, σ) ⋉ η) ∘ ((σ, τ) ⋉ ϵ (τ, σ) ⋉ F ) = 1_{F}

but this is immediate from our definitions and the fact that we started with an adjunction. For the commutativity statement simply note that ℙ∘(σ, τ)⋉G = G∘ℙ.

Definition 2.16 A monoidal category with involution is a category with a closed monoidal product (⊗, F, ϕ), a functor τ such that (𝒞 , τ) is a category with involution and there exists

(i). an involutary natural transformation m: τc ⊗ τd → τ(c ⊗ d)

(ii). an isomorphism i: τ𝕀 → 𝕀 such that τi ∘ i = 1 (𝕀 the unit of ⊗).

That is, we require τ to be a strong monoidal functor (τ, m, i) with involutary natural transformation m and an order two isomorphism i.

Theorem 2.17 The half-twisted category of an monoidal category with involution (𝒞 , τ) with monoidal structure given by (⊗, F, ϕ, m, i) is a monoidal category. If the monoidal product on 𝒞 is symmetric and τ is symmetric, then so is the product on τ ⋉ 𝒞.

Proof We will use (⊗, F, ϕ) to denote the monoidal product on the half-twisted category as decorating these symbols further would be horrific.

We begin with defining the action of the functor ⊗ on a pair of objects u: A → τA and v: B → τB to be

m_{A,B} ∘ u ⊗ v: A ⊗ B → τA ⊗ τB → τ(A ⊗ B)

the action on maps is obvious from this definition (this is an object of the category since m is an involutary natural transformation). The unit is the object i: 𝕀 → τ𝕀, since i is a map of order two, this is also an object of the category.

Now one must check that i: 𝕀 → τ𝕀 defines a unit, that the product as above is associative, and that certain coherence diagrams are satisfied see either [Mac71, VII] of [Hov99, Chapter 4]. This is all routine and follows from our assumption that τ is strong monoidal, most coherence diagrams are easy to check, but one can use the ‘coherence theorem’ of [Mac71, VII] to note that all the diagrams that we are checking commute. Thus we have a monoidal product on τ ⋉ 𝒞, and it is clear that this product is symmetric provided ⊗ is symmetric on 𝒞 and τ is a symmetric monoidal functor.

Now we move to showing that this is a closed monoidal structure. The functors −⊗τB and τ(τ −⊗B) are naturally isomorphic, hence their right adjoints F (τB, −) and τF (B, τ−) (respectively) are isomorphic, and both behave as right adjoints to − ⊗ τB. Thus the result [Mac71, Corollary 1 IV.1] implies that there is a canonical natural isomorphism between these two right adjoints. A small amount of adjustment will give a canonical natural isomorphism f_{A,B}: F (τA, τB) → τF (A, B), since this is canonical (and we have the corresponding result for m) it follows that τf ∘ f(τ, τ) = 1, i.e. f is an involutary natural transformation.

Alternatively one can construct this natural transformation via the following diagram (note that since τ^{2} = 1, a general object of 𝒞 can be written as τA for suitable A).

\xymatrix{hom(τA, F (τB, τC)) \ar[r]_{ϕ_{τA,τB,τC}^{−1}} & hom(τA ⊗ τB, τC) \ar[r]_{τm_{A,B}} & hom(τ(A ⊗ B), τC) \ar[d]_{τ} \\ hom(τA, τF (B, C)) & hom(A, F (B, C)) \ar[l]^{τ} & hom(A ⊗ B, C) \ar[l]^{ϕ_{A,B,C}}}

Take A = τF (τB, τC) and follow the identity map round the diagram to obtain the natural transformation f: F (τ−, τ−) → τF (−, −).

Consider the following pair of triangles, it follows from the construction of the natural transformation f above that the left hand triangle commutes if and only if the right hand triangle commutes.

\xymatrix{τA ⊗ τB \ar[r]^{g} \ar[d]_{m_{A,B}} & τC & τA \ar[r]^{ϕg} \ar[dr]_{τϕk} & F (τB, τC) \ar[d]^{f_{B,C}} \\ τ(A ⊗ B) \ar[ur]_{τk} &  &  & τF (B, C)}

We draw another picture, showing three squares, in which if any one commutes, so does the other two.

\xymatrix{A ⊗ B \ar[dd]_{u⊗v} \ar[r]^{k} & C \ar[dd]^{w} & A \ar[r]^{ϕk} \ar[dd]_{u} & F (B, C) \ar[d]^{F (1,w)} & A \ar[r]^{ϕk} \ar[dd]_{u} & F (B, C) \ar[dd]^{F (τv,w)} \\  &  &  & F (B, τC) \\ τA ⊗ τB \ar[r]_{g} & τC & τA \ar[r]_{ϕg} & F (τB, τC) \ar[u]_{F (v,1)} & τA \ar[r]_{ϕg} & F (τB, τC)}

Finally we combine these into a pair of diagrams, with one commuting precisely when the other does, and hence we have our adjunction on the half-twisted category (g = w ∘ k ∘ (τu ⊗ τv)).

\xymatrix{A ⊗ B \ar[d]_{u⊗v} \ar[r]^{k} & C \ar[dd]^{w} & A \ar[r]^{ϕk} \ar[dd]_{u} & F (B, C) \ar[d]^{F (τv,w)} \\ τA ⊗ τB \ar[d]_{m_{A,B}} &  &  & F (τB, τC) \ar[d]^{f_{B,C}} \\ τ(A ⊗ B) \ar[r]_{τk} & τC & τA \ar[r]_{τϕk} & τF (B, C)}

Remark 2.18 The relevant parts of the above theorem hold even when we have a non-closed monoidal structure, i.e. when there is no internal function object, but this is not the case of interest to us.

Definition 2.19 An involutary monoidal adjunction is an involutary adjunction (L, α) : (𝒞 , σ)\overrightarrow{⟵}(𝒟, τ) : (R, β) such that L is op-monoidal and R is monoidal (in the weak sense), with monoidal natural transformations α and β.

So we have the usual result that this will pass to the half-twisted category we have required natural transformations L(c ⊗ c′) → Lc ⊗ Lc′, and Rd ⊗ Rd′ → R(d ⊗ d′) which are compatible with the natural transformations α and β in the sense of the following diagram.

\xymatrix{Lσ(c ⊗ c′) \ar[d] \ar[r] & Lσc ⊗ Lσc′ \ar[d] & Rτd ⊗ Rτd′ \ar[d] \ar[r] & Rτ(d ⊗ d′) \ar[d] \\ τL(c ⊗ c′) \ar[r] & τLc ⊗ τLc & σRd ⊗ σRd′ \ar[r] & σR(d ⊗ d′)}

Remark 2.20 Take a monoidal category with involution (C, σ) such that C is a closed ν-algebra and σ is a ν-algebra functor. This is not enough for σ ⋉ C to be a 1 ⋉ ν-algebra, we need i_{C} : ν\overrightarrow{⟵}C : j_{C} to be an involutary adjunction with respect to 1 on ν. We ask the to reader note that 1 ⋉ ν is not necessarily equivalent to ν since there may be non-identity maps of order two in ν. But we can use the functors of Remark 2.8 and the above to move between ν and σ ⋉ C in the usual equivariant ways. In turn we can ask for an adjunction of involutary closed ν-algebras. We define such an adjunction to be an involutary monoidal adjunction which is also an adjunction of closed ν-algebras, with the associated natural transformations now required to be involutary.

3 Model categories with involution

Definition 3.1 A model category with involution (M, τ) is a cofibrantly generated model category M with left Quillen functor τ: M → M such that τ^{2} = 1 (or so that τ^{2} is naturally isomorphic to 1). That is, (M, τ) is a category with involution.

It follows, of course, that τ is also a right Quillen functor and that τ preserves all weak equivalences (since a weak equivalence is a composite of a trivial cofibration followed by a trivial fibration).

Definition 3.2 An involutary Quillen functor is an involutary functor that is a Quillen functor.

Lemma 3.3 The category τ ⋉ M has all small limits and colimits.

Proof Take a diagram D (i.e. a small category) and a functor F: D → τ ⋉ M. We can form Colim_{D} ℙF (since M is bicomplete) and since τ is a left adjoint we have a canonical isomorphism τ Colim_{D} ℙF ≅ Colim_{D} τℙF. We also have a map Colim_{D} w: Colim_{D} ℙF → Colim_{D} τℙF we combine these maps to give an object of the half-twisted category in the diagram below.

\xymatrix{ & Colim_{D} ℙF \ar[d]^{Colim_{D} w} \ar[dr] \ar[ddl]_{=} \\  & Colim_{D} τℙF \ar[r] \ar[d]^{Colim_{D} τw} & τ Colim_{D} ℙF \ar[d]^{τ Colim_{D} w} \ar[dr] \\ Colim_{D} ℙF \ar[r]^{=} & Colim_{D} τ^{2}ℙF \ar[r] & τ Colim_{D} τℙF \ar[r] & τ^{2} Colim_{D} ℙF \ar[d]^{=} \\  &  &  & Colim_{D} ℙF}

The composite Colim_{D} ℙF → τ^{2} Colim_{D} ℙF is the identity since there is a unique isomorphism between any two colimits of a diagram and these two colimits are the same object of M. The case for a limit is identical since τ is also a right adjoint.

Definition 3.4 Given a model category with involution (M, τ), we define a weak equivalence (respectively fibration) of τ ⋉ M to be a map f such that ℙf is a weak equivalence (respectively fibration).

Proposition 3.5 These maps define a model structure on τ ⋉ M and we call this category and model structure the half-twisted model category.

Proof All that is required is to check that the lifting lemma applies in the case of

𝔻 : 𝒞 \overrightarrow{⟵}τ ⋉ 𝒞 : ℙ.

We have already shown that ℙ is a left adjoint and so preserves all colimits. Now we must check that every regular 𝔻J-cofibration is a weak equivalence, Take k: A → B a map in J, then

ℙ𝔻k = k ∨ τk: A ∨ τA → B ∨ τB

and since τ is a left Quillen functor (on 𝒞), it follows that ℙ𝔻k is an acyclic cofibration in M. A regular 𝔻J-cofibration in τ ⋉ M is a transfinite composition of pushouts of 𝔻J ([Hov99, Chapter 2]). Then since ℙ preserves these constructions (it is a left adjoint) and the set of acyclic cofibrations (of M) are closed under these operations ([Hov99, proof of 2.2.10]) the result follows.

The original reference for the lifting lemma is of course [Qui67, II.4], but the version we use is closest to that of [SS00, Lemma 2.3].

Lemma 3.6 (The Lifting Lemma) Let F : 𝒞 ⇄ 𝒟 : G be an adjoint pair of functors (F is the left adjoint) with 𝒞 a cofibrantly generated model category. Let I be the generating cofibrations and J the generating acyclic cofibrations for 𝒞.

Assume that 𝒟 has all small colimits and finite limits. Define a map f in 𝒟 to be a weak equivalence or a fibration if and only if Gf is so. Define the cofibrations to be those maps with in 𝒟 with the correct lifting property. Then this construction defines a cofibrantly generated model structure on 𝒟 provided:

(i). G preserves filtered colimits

(ii). Every regular F J-cofibration is a weak equivalence

Lemma 3.7 Let (M, τ) be a model category with involution, then ℙ: τ ⋉ M → M preserves cofibrations and hence is a left Quillen functor.

Proof The right adjoint 𝔻′ preserves fibrations and acyclic fibrations.

Definition 3.8 An involutary Quillen pair is an involutary adjunction that is also a Quillen pair.

Lemma 3.9 An involutary Quillen pair between model categories with involution, passes to a Quillen pair between the half-twisted model categories.

Proof If G is the right adjoint, then we have the equation ℙ ∘ (σ, τ) ⋉ G = G ∘ ℙ and the result follows by showing that (σ, τ) ⋉ G is a right Quillen functor.

Definition 3.10 An involutary Quillen equivalence is an involutary Quillen pair that is also a Quillen equivalence.

Proposition 3.11 An involutary Quillen equivalence passes to Quillen equivalence of the half-twisted model categories.

Proof Take an involutary Quillen equivalence (F, α) : (𝒞 , τ) ⇄ (𝒟, σ) : (G, β) then consider a cofibrant c → τc in τ ⋉ 𝒞 and a fibrant d → σd in σ ⋉ 𝒟. Then c is cofibrant in 𝒞 and d is fibrant in 𝒟 so a map F c → d is a weak equivalence if and only if c → Gd weak equivalence. But this is precisely the statement that a map (F c → σF c) → (d → σd) is a weak equivalence if and only if (c → τc) → (Gd → τGd) is a weak equivalence.

Definition 3.12 A monoidal model category with involution is a model category with involution that is also a monoidal category with involution and satisfies the pushout product axiom.

Lemma 3.13 If (M, τ) is a monoidal model category with involution then τ ⋉ M is a monoidal model category. Furthermore, if M satisfies the monoid axiom, so does τ ⋉ M.

Proof Take f: (c → τc) → (d → τd) a cofibration in τ ⋉ M and g: (x → τx) → (y → τy) a fibration. Then we consider the following pullback diagram in M, which is ℙ applied to the corresponding diagram in τ ⋉ M.

\xymatrix{F (d, x) \ar[dr]^{ℙα} \ar[drr] \ar[ddr] \\  & P \ar[d] \ar[r] & F (c, x) \ar[d] \\  & F (d, y) \ar[r] & F (c, y)}

Since ℙf is a cofibration and ℙg is a fibration it follows that α is a fibration that is trivial if either of f or g is.

The second part of the pushout product axiom concerns the cofibrant replacements of 𝕀. The diagram

[figure: ∗; Q𝕀; Qi; ∼; 𝕀; i; ∗; τQ𝕀; ∼; τ𝕀]

gives a cofibrant replacement of the unit in τ ⋉M and a cofibrant replacement of the unit in M: ∗ → Q𝕀 → 𝕀. Then we see that for cofibrant X in M Q𝕀 ⊗ X → 𝕀 ⊗ X is a weak equivalence (by the pushout product axiom in M) hence the corresponding statement in τ ⋉ M holds. The statement regarding the monoid axiom holds by an equally straightforward argument using the fact that ℙ preserves cofibrations.

4 The half-twisted model category of rational SO(2)- spectra

I will construct a Quillen equivalence between rational O(2)-spectra with the cyclic model structure and W ⋉ S_{ℚ}^{SO(2)} –mod

Proposition 4.1 The category SO(2)-equivariant S-modules, S_{ℚ}^{SO(2)} –mod, is a model category with involution.

Proof We define the twist on this category to be h^{†}j^{∗}, and note that h^{†}j^{∗} ∘h^{†}j^{∗} = 1 we use equality rather than natural isomorphism since it actually preserves the underlying sets of these spectra. It is the composite of the two left Quillen functors and hence is a left Quillen functor.

Thus we have a model category W ⋉ S_{ℚ}^{SO(2)} –mod and the next few results should be expected.

Remark 4.2 Let ε_{1}^{SO(2)} denote the monoidal inflation functor from spectra to SO(2)-spectra (including the relevant change of universe functor) then 1: ε_{1}^{SO(2)}X → ε_{1}^{SO(2)}X is an object of the half-twisted category, and maps to ε_{1}^{O(2)}X under C. There appear to be no non-trivial natural maps X → h^{†}j^{∗}X for general X, which is to be expected since the group of homomorphisms O(2) → SO(2) is ℤ/2, that is there are no interesting group homomorphisms to give a change of groups functor from SO(2)-spectra to O(2)-spectra.

Theorem 4.3 There is a monoidal Quillen equivalence

C : W ⋉ S_{ℚ}^{SO(2)} –mod \overrightarrow{⟵}S_{ℚ}^{𝒞} –mod : I

where IX = h: i^{∗}X → h^{†}j^{∗}i^{∗}X, If = i^{∗}f and C(w: A → h^{†}j^{∗}A) is the O(2)- spectrum associated to that half-twisted object, C acts on maps in a similar way described below.

Proof Almost all of the work has been done in 1.2 we are merely going to repeat the arguments to construct a map of O(2)-spectra from a map in the half-twisted category and then check that we have a well-behaved adjunction.

But first, let us recap the construction of an O(2)-spectrum from an object of the half-twisted category. Take h: X → h^{†}j^{∗}X, a map of order two then CX(V) = (X ∘ i^{∗})(V) = X(i^{∗}V) as a topological space. We give this an O(2)-action by letting h ∈ O(2) our chosen reflection act as h: X(i^{∗}V) → j^{∗}(X(i^{∗}V)). Now we must add the structure of a prespectrum: σ: S^{W −V} ∧ CX(V) → CX(W) is defined by the corresponding structure map of X, that this is an O(2)-map is encoded in the fact that h is a map of prespectra. It is clear that this gives a spectrum (i.e. the adjoints of the structure maps are equivariant homeomorphisms). As we did above for the spheres, we define the 𝕃-spectrum action using the underlying set of of the structure map for X, and then checking that this gives an O(2)-map by considering the map h. Repeat this process twice more, for the ring spectra S and S_{ℚ}^{SO(2)} and we are done.

Now for the maps, with the above construction we can simply define Cf_{V} = f_{V} (as a set map) for f a map as below.

\xymatrix{X \ar[d]^{h_{X}} \ar[r]^{f} & Y \ar[d]^{h_{Y}} \\ h^{†}j^{∗}X \ar[r]^{h^{†}j^{∗}f} & h^{†}j^{∗}Y}

It is now routine that Cf gives an O(2)-equivariant map of 𝕃-spectra, and similarly that is gives an O(2)-equivariant map of S-modules and S_{ℚ}^{SO(2)}-modules.

That these are adjoint functors is immediate, the pair C, I are actually an equivalence of categories, (I would hesitate from saying more though X = ICX). Now we consider the model structures, for which we need a diagram (left adjoints will be on the top and left).

\xymatrix{ & W ⋉ S_{ℚ}^{SO(2)} –mod \ar[dl]_{C} \ar[dr]_{ℙ} \\ S_{ℚ}^{𝒞} –mod \ar[ur]_{I} \ar[rr]_{i^{∗}} \ar[d] &  & S_{ℚ}^{SO(2)} –mod \ar[ul]_{𝔻} \ar[ll]_{0(2)_{+}∧_{SO(2)}} \ar[d] \\ 𝒞 M𝒮 \ar[u] \ar[rr]_{i^{∗}} &  & SO(2)M𝒮 \ar[u] \ar[ll]_{0(2)^{+}∧_{SO(2)}}}

We show that I is a right Quillen functor. Take a fibration or trivial fibration f in S_{ℚ}^{𝒞} –mod, then If is a fibration or trivial fibration exactly when ℙIf = i^{∗}f is. Now i^{∗}f is a fibration or trivial fibration in S_{ℚ}^{SO(2)} –mod, precisely when it is so in SO(2)M𝒮. Hence the result follows since i^{∗}: 𝒞 M𝒮 → SO(2)M𝒮 is a right Quillen functor.

Now we must show that (C, I) are a Quillen equivalence, a repeat of the above argument shows that I detects and preserves all weak equivalences (since i^{∗}: 𝒞 M𝒮 → SO(2)M𝒮 does so). Now we show that the composite X → ICX → If̂CX is a weak equivalence in W ⋉ S_{ℚ}^{SO(2)} –mod, the first arrow is an isomorphism, the second is I applied to a π_{∗}^{ℚ}-isomorphism over the groups in 𝒞 (i.e. the subgroups of SO(2)) and hence a weak equivalence. Since the functors i^{∗}, j^{∗} and h^{†} = I_{j^{∗}i^{∗}U}^{i^{∗}U} are all strong monoidal, so is I.

The following remark is obvious from our constructions and is used in the above proof.

Remark 4.4 The following diagram is a commutative diagram of Quillen pairs (with left adjoints on top or on the left).

\xymatrix{S_{ℚ}^{𝒞} –mod \ar[r]_{I} \ar[dr]_{i^{∗}} & W ⋉ S_{ℚ}^{SO(2)} –mod \ar[l]_{C} \ar[d]^{ℙ} \\  & S_{ℚ}^{SO(2)} –mod \ar[ul]_{0(2)_{+}∧_{SO(2)}} \ar[u]^{𝔻}}

References

[Hov99] Mark Hovey. Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999.

[Mac71] Saunders MacLane. Categories for the Working Mathematician. Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.

[May98] J. P. May. Equivariant and nonequivariant module spectra. J. Pure Appl. Algebra, 127(1):83–97, 1998.

[Qui67] Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin, 1967.

[SS00] Stefan Schwede and Brooke E. Shipley. Algebras and modules in monoidal model categories. Proc. London Math. Soc. (3), 80(2):491–511, 2000.