1. Introduction
In [Reference Lindenstrauss, Richter and Zou20], we introduced equivariant Loday constructions. These generalize the non-equivariant Loday constructions, which include (topological) Hochschild homology, higher order Hochschild homology and torus homology.
In the equivariant case, we fix a finite group G. The starting point for a Loday construction is a G-commutative monoid in the sense of Hill and Hopkins [Reference Hill and Hopkins14]. In the setting of G-equivariant stable homotopy theory these are genuine G-commutative ring spectra whereas in the algebraic setting of Mackey functors G-commutative monoids are G-Tambara functors. Some equivariant homology theories such as the twisted cyclic nerve of Blumberg-Gerhardt-Hill-Lawson [Reference Blumberg, Gerhardt, Hill and Lawson3] and Hesselholt-Madsen’s Real topological Hochschild homology, 
$\mathsf{THR}$, [Reference Dotto, Moi, Patchkoria and Reeh6] can be identified with such equivariant Loday constructions [Reference Lindenstrauss, Richter and Zou20, § 7]. Here, 
$\mathsf{THR}$ is a homology theory for associative algebra spectra with anti-involution A and we identified this in the commutative case with the Loday construction over the one-point compactification of the sign-representation, 
$\mathsf{THR}(A) \simeq \mathcal{L}^{C_2}_{S^\sigma} (A)$. In the following we will often refer to S σ as the flip circle. In [Reference Lindenstrauss, Richter and Zou20, proposition 6.1], we show that for any G-simplicial set X, if we apply the functor 
$\underline{\pi}_0$ levelwise to the equivariant Loday construction of a connective genuine commutative G-algebra spectrum A to obtain a simplicial G-Tambara functor,
\begin{equation*}\underline{\pi}_0 (\mathcal{L}_X^G(A)) \cong \mathcal{L}^G_X(\underline{\pi}_0 (A)), \end{equation*} which relates 
$\mathcal{L}^{C_2}_{S^\sigma}$ of C 2-Tambara functors to 
$\mathsf{THR}$.
There is an algebraic version of 
$\mathsf{THR}$, called Real Hochschild homology [Reference Angelini-Knoll, Gerhardt and Hill1, definition 6.15] that takes associative algebras with anti-involution as input. These are associative k-algebras for some commutative ring k, such that 
$\tau(a) := \bar{a}$ satisfies 
$\overline{ab} = \bar{b} \bar{a}$ and such that the C 2-action is k-linear. In her thesis Chloe Lewis developed a Bökstedt-type spectral sequence for 
$\mathsf{THR}$ [Reference Lewis18] whose E 2-term consists of Real Hochschild homology groups. Other homology theories for associative algebras with anti-involution are reflexive homology [Reference Graves11] and involutive Hochschild homology [Reference Fernàndez-València and Giansiracusa9]. Reflexive homology is a homology theory associated to the crossed simplicial group that is the cyclic group of order two, 
$C_2 = \langle \tau\rangle$, in every simplicial degree, where we do not view C 2 as a constant simplicial group, but let τ interact with the category Δ by reversing the simplicial structure. Involutive Hochschild homology was defined in [Reference Fernàndez-València and Giansiracusa9]; the corresponding cohomology theory was developed by Braun [Reference Braun4], who defined a cohomology theory for involutive 
$A_\infty$-algebras, motivated by work of Costello on open Klein topological conformal field theories [Reference Costello5]. We slightly generalize the definition in [Reference Fernàndez-València and Giansiracusa9] and work over arbitrary commutative rings instead of fields.
We prove the identification of reflexive homology, 
$\mathsf{HR}_*$, with the homotopy groups of an equivariant Loday construction in section 6 and the one for involutive Hochschild homology, 
$\mathsf{iHH}_*$, in section 7:
Theorem (Theorems 6.4 and 7.2) Assume that R is a commutative ring with involution and that 2 is invertible in R. If the underlying abelian group of R is flat, then
\begin{equation*} \mathsf{iHH}_*^{\mathbb{Z}}(R) \cong \pi_*(\mathcal{L}^{C_2}_{S^\sigma}({\underline{R}}^\mathrm{fix} )(C_2/C_2)) \cong \mathsf{HR}^{+,\mathbb{Z}}_*(R,R). \end{equation*} Here, 
$\mathcal{L}^{C_2}_{S^\sigma}({\underline{R}}^\mathrm{fix} )$ is the C 2-equivariant Loday construction of the fixed point Tambara functor for R, 
$\underline R^\mathrm{fix}$, for the representation sphere of the real sign-representation, S σ. This is a simplicial Tambara functor and 
$\mathcal{L}^{C_2}_{S^\sigma}({\underline{R}}^\mathrm{fix} )(C_2/C_2)$ is its evaluation at the trivial orbit 
$C_2/C_2$. This yields a simplicial abelian group and we consider its homotopy groups.
If we work relative to a commutative ground ring k, then we obtain a corresponding result:
Theorem (Theorems 6.5 and 7.3) Assume that R is a commutative k-algebra with a k-linear involution and that 2 is invertible in R. If the underlying module of R is flat over k, then
\begin{equation*} \mathsf{iHH}_*^k(R) \cong \pi_*(\mathcal{L}^{C_2
,{\underline{k}}^c}_{S^\sigma}({\underline{R}}^\mathrm{fix})(C_2/C_2)) \cong
\mathsf{HR}^{+,k}_*(R,R). \end{equation*} In hindsight, this identifies the Loday construction over the C 2-Burnside Tambara functor with the Loday construction relative to 
${\underline{\mathbb{Z}}}^c$ under the above assumptions (see remark 5.3). We consider the examples of 
$\mathbb{F}_2$ and 
$\mathbb{Z}$ with the trivial C 2-action in section 8 in order to understand what happens if we drop these assumptions. There, the homotopy groups of the Loday constructions differ both from reflexive homology and from involutive Hochschild homology.
The relationship to the Real Hochschild homology of [Reference Angelini-Knoll, Gerhardt and Hill1] is more subtle: The latter takes all dihedral groups into account and for 
$D_2 = C_2$ their definition agrees with our equivariant Loday construction. We will establish a full comparison also for the higher 
$D_{2m}$ with equivariant Loday constructions in future work with Foling Zou (in preparation).
In section 9 we extend our results to the associative case, where we consider associative rings R and associative k-algebras with anti-involution where k is an arbitrary commutative ground ring. Usually, one cannot form Loday constructions without assuming commutativity, but the simplicial model of the one-point compactification of the sign-representation consists of two glued copies of the simplicial 1-simplex with its intrinsic ordering, so we can extend the definition to equivariant associative monoids in this case and we get results generalizing the above theorems:
Theorem (Theorem 9.4) Assume that R is an associative ring with anti-involution and that 2 is invertible in R. If the underlying abelian group of R is flat, then
\begin{equation*} \mathsf{iHH}_*^{\mathbb{Z}}(R) \cong \pi_*( \mathcal{L}^{C_2}_{S^\sigma}({\underline{R}}^\mathrm{fix} )(C_2/C_2)) \cong \mathsf{HR}^{+,\mathbb{Z}}_*(R,R). \end{equation*}If we work relative to a commutative ground ring k, then we obtain a corresponding result:
Theorem (Theorem 9.6) Assume that A is an associative k-algebra with a k-linear anti-involution and that 2 is invertible in A. If the underlying module of A is flat over k, then
\begin{equation*} \mathsf{iHH}_*^k(A) \cong \pi_*(\mathcal{L}^{C_2 ,{\underline{k}}^c}_{S^\sigma}({\underline{A}}^\mathrm{fix})(C_2/C_2))
\cong \mathsf{HR}^{+,k}_*(A,A). \end{equation*} The proofs, however, are different: In the case of an associative ring R with anti-involution the fixed point Mackey functor 
$\underline R^\mathrm{fix}$ is not an associative C 2-Green functor, so in particular it is not a C 2-Tambara functor. But it has the structure of a discrete Eσ-ring in the sense of [Reference Angelini-Knoll, Gerhardt and Hill1, § 6.3] and it is also a Hermitian Mackey functor in the sense of [Reference Dotto and Ogle7, definition 1.1.1]. Real Hochschild homology [Reference Angelini-Knoll, Gerhardt and Hill1, § 6.4] and Real topological Hochschild homology [Reference Dotto, Moi, Patchkoria and Reeh6, example 2.4] are defined for such objects, see also [Reference Horev15, proposition 7.1.1] for the analogous result for factorization homology of the flip circle S σ, so it is not surprising that one can extend the Loday construction for S σ to fixed point Mackey functors of rings with anti-involution. We endow the C 2-Mackey norm functor of 
$i_e^*\underline R^\mathrm{fix}$ with the structure of an associative Green functor so that 
$\underline R^\mathrm{fix}$ is a bimodule over it.
2. Equivariant Loday constructions
We recall the basic facts about equivariant Loday constructions for G-Tambara functors from [Reference Lindenstrauss, Richter and Zou20] for an arbitrary finite group G. We work with unital rings. We assume that ring maps preserve the unit, and that the unit acts as the identity on any module over the ring.
We consider simplicial G-sets X that are finite in every degree and call them finite simplicial G-sets. For every G-Tambara functor 
${\underline{T}}$ and every such X the simplicial G-Tambara functor 
$\mathcal{L}_X^G({\underline{T}})$ is the G-Loday construction for X and 
${\underline{T}}$. In simplicial degree n we define:
\begin{equation*} \mathcal{L}_X^G({\underline{T}})_n = X_n \otimes {\underline{T}},\end{equation*} where the formation of the tensor product with the finite G-set Xn uses the fact that G-Tambara functors are the G-commutative monoids in the setting of G-Mackey functors. This was proved by Mazur [Reference Mazur21] for cyclic p-groups for a prime p and by Hoyer [Reference Hoyer16] in the case of a general finite group G. As they show that the construction 
$X_n \otimes {\underline{T}}$ is functorial in Xn, the Loday construction is well-defined.
The above tensor can be made explicit. Every finite G-set is isomorphic to a finite disjoint union of orbits and Mazur and Hoyer show that for an orbit 
$G/H$ we obtain
\begin{equation*} G/H \otimes {\underline{T}} \cong N_H^Gi_H^*{\underline{T}}. \end{equation*} Here, 
$i_H^*$ restricts a G-Tambara functor to H, so for a finite H-set Y, 
$i_H^*{\underline{T}}(Y) := {\underline{T}}(G \times_H Y)$. The restriction functor has the norm functor 
$N_H^G$ as a left adjoint. A disjoint union of G-sets 
$X, X'$, 
$X \sqcup X'$ is sent to
\begin{equation*} (X \sqcup X') \otimes {\underline{T}} \cong (X \otimes {\underline{T}}) \Box
(X' \otimes {\underline{T}}), \end{equation*} so this determines every 
$X_n \otimes {\underline{T}}$ up to isomorphism.
3. Basic results about fixed point Tambara functors
In this section we study C 2-Mackey and Tambara functors. If L is an abelian group with involution 
$a\mapsto \bar a$, there is a C 2-Mackey functor 
${\underline{L}}^\mathrm{fix}$ given by
\begin{equation*} {\underline{L}}^\mathrm{fix} =
\begin{cases}
L^{C_2} & \text{at } C_2/C_2, \\
L & \text{at } C_2/e,
\end{cases}
\end{equation*} where 
$\mathsf{tr}(a)=a+\bar a$ for all 
$a\in L$ and 
$\mathsf{res}(a)=a$ for all 
$a\in L^{C_2}$. If R is a commutative ring whose multiplication is compatible with its involution, then we can define 
$\mathsf{norm}(a)=a\bar a$ and get a C 2-Tambara functor structure on 
${\underline{R}}^\mathrm{fix}$.
Remark 3.1. Note that for an arbitrary finite group G, the functor that sends a commutative G-ring T to its fixed point G-Tambara functor is right adjoint to the functor that takes a G-Tambara functor 
${\underline{R}}$ to its underlying commutative G-ring 
${\underline{R}}(G/e)$ (see for instance [Reference Schuchardt, Spitz and Wisdom23, lemma 2.9]). This is completely analogous to the situation in Mackey functors, where the functor that sends an abelian group A with G-action to its fixed point G-Mackey functor 
$\;\underline A^{\mathrm{fix}}$ is right adjoint to the functor that takes a G-Mackey functor 
${\underline{M}}$ and sends it to the abelian group with G-action 
${\underline{M}}(G/e)$.
For 
$G=C_2$ a description of the counit map 
$\varrho_T
\colon {\underline{T}}^\mathrm{fix}(C_2/e) \rightarrow T$ and the unit map 
$\eta_{\underline{R}} \colon {\underline{R}} \rightarrow
{\underline{{\underline{R}}(C_2/e)}}^\mathrm{fix}$ of this adjunction are very straightforward (both in the Mackey case and in the Tambara case): The counit 
$\varrho_T \colon {\underline{T}}^\mathrm{fix}(C_2/e) = T \rightarrow T$ is the identity map.
At the free orbit
\begin{equation*} \eta_{\underline{R}}(C_2/e) \colon {\underline{R}}(C_2/e) \rightarrow
{\underline{{\underline{R}}(C_2/e)}}^\mathrm{fix}(C_2/e) = {\underline{R}}(C_2/e),\end{equation*} is the identity map and at the trivial orbit 
$C_2/C_2$
\begin{equation*} \eta_{\underline{R}}(C_2/C_2) \colon {\underline{R}}(C_2/C_2) \rightarrow
{\underline{{\underline{R}}(C_2/e)}}^\mathrm{fix}(C_2/C_2) = {\underline{R}}(C_2/e)^{C_2},\end{equation*}is the restriction map.
For an arbitrary finite group G this adjunction ensures that morphisms of commutative G-rings 
$f \colon R \rightarrow T$ are in bijective correspondence with morphisms of G-Tambara functors 
${\underline{f}} \colon \underline R^\mathrm{fix} \rightarrow {\underline{T}}^\mathrm{fix}$, because 
$\underline R^\mathrm{fix}(G/e) = R$.
In particular if k is a commutative ring and if 
$f \colon R \rightarrow T$ is a morphism of commutative k-algebras with involution, then we get a commutative diagram of C 2-Tambara functors
where iR and iT are the unit maps of R and T. Here 
${\underline{k}}^c$ denotes the fixed point Tambara for the trivial C 2-action, called the constant Tambara functor.
For a set Y we denote by 
$\mathbb{Z}\{Y\}$ the free abelian group generated by Y and for 
$y \in Y$ the corresponding generator in 
$\mathbb{Z}\{Y\}$ is 
$\{y\}$. When R is a commutative ring with involution the norm restriction of 
${\underline{R}}^\mathrm{fix}$ is given by
\begin{equation*} N^{C_2}_ei^{*}_e \underline R^\mathrm{fix} = N_e^{C_2}R =
\begin{cases}
( \mathbb{Z}\{R\} \oplus (R\otimes R)/C_2)/\mathrm{TR} & \text{at } C_2/C_2 \\
R\otimes R & \text{at } C_2/e,
\end{cases}
\end{equation*} where C 2 acts on 
$R\otimes R$ via 
$\tau(a\otimes b) = \bar
b\otimes \bar a$, 
$[a\otimes b]$ denotes the equivalence class of 
$a\otimes b$ in 
$ (R\otimes R)/C_2 $, and Tambara Reciprocity, TR, identifies 
$\{a+b\} \sim \{a\} +\{b\} + [a\otimes \bar b]$. Here 
$\mathsf{norm}(a\otimes b)=\{a\bar b\}$ and 
$\mathsf{tr}(a\otimes b)=[a\otimes b]$ for all 
$a\otimes b\in R\otimes R$, 
$\mathsf{res}(\{a\})=a\otimes \bar a$, and 
$\mathsf{res}([a\otimes b])=a\otimes b+\bar b\otimes \bar a$ (see [Reference Hill and Mazur13] for properties of the norm functor, especially Fact 4.4 in loc. cit.).
Lemma 3.2. Assume that 
$M$ and 
$N$ are two abelian groups with involution and assume that 2 is invertible in 
$M$ or in 
$N.$ Then there is an equivalence of 
$C_2$-Mackey functors
\begin{equation*}\underline M^\mathrm{fix}\Box\underline N^\mathrm{fix} \cong {\underline{(M\otimes N)}}^\mathrm{fix}\end{equation*} which is natural in 
$M$ and 
$N$. Here C2 acts on 
$M\otimes N$ by the diagonal action. If, in addition, 
$M$ and 
$N$ are both commutative rings with involution, then 
$\underline M^\mathrm{fix}$, 
$\underline N^\mathrm{fix}$, and 
${\underline{(M\otimes N)}}^\mathrm{fix}$ are C 2-Tambara functors, and the above equivalence is an equivalence of 
$C_2$-Tambara functors.
Proof. We have by the definition of the box product (see e.g. [Reference Hill and Mazur13, definition 3.1])
\begin{equation*}
\underline M^\mathrm{fix}\Box\underline N^\mathrm{fix} =
\begin{cases}
[M^{C_2} \otimes N^{C_2} \oplus (M\otimes N)/C_2]/\text{FR} &\text{at }C_2/C_2\\
M\otimes N &\text{at }C_2/e,
\end{cases}
\end{equation*} and we map it to 
${\underline{(M\otimes N)}}^\mathrm{fix}$ by using the Mackey or Tambara functor map corresponding to the identity map on the free orbit 
$C_2/e$ via the adjunction in remark 3.1. Then our map will consist of the identity on the free orbit and 
$\mathsf{res}$ on the trivial orbit, which will land in 
$(M\otimes N)^{C_2}$.
On the trivial orbit the map 
$\mathsf{res}$ includes 
$M^{C_2} \otimes
N^{C_2}$ into 
$(M\otimes N)^{C_2}$ and maps 
$(M\otimes N)/C_2$ to 
$(M\otimes N)^{C_2}$ by the map 
$[m\otimes n]\mapsto m\otimes n +\bar m\otimes\bar n$ (which is an equivalence since 2 is invertible in 
$M\otimes N$).
Then 
$\mathsf{res}$ maps 
$\underline M^\mathrm{fix}\Box\underline N^\mathrm{fix} (C_2/C_2)$ surjectively onto 
$ (M\otimes
N)^{C_2}$ because its restriction to the second summand does, and it is injective because its restriction to the second summand is injective and Frobenius Reciprocity allows us to identify the first summand into the second one: if 2 is invertible in M, any 
$m\in
M^{C_2}$ is equal to 
$\mathsf{tr} (m/2)$ so 
$m\otimes n$ is identified with 
$[m/2\otimes n]$, and if 2 is invertible in N similarly 
$m\otimes
n$ is identified with 
$[m\otimes n/2]$.
Lemma 3.3. If 
$R$ is a commutative ring with involution in which 2 is invertible and if 
$M$ is an abelian group with an involution, then there is an equivalence of C 2-Mackey functors
\begin{equation} (N^{C_2}_ei_e^*\underline R^\mathrm{fix}) \Box \underline M^\mathrm{fix} = (N^{C_2}_eR) \Box \underline M^\mathrm{fix} \cong {\underline{(R \otimes R \otimes
M)}}^\mathrm{fix}, \end{equation} which is natural in 
$M$ and 
$R$. Here, 
$C_2$ acts on 
$R \otimes R \otimes M$ by 
$\tau(a\otimes b \otimes m)=\bar b\otimes \bar a\otimes
\bar m$. If 
$M$ is also a commutative ring with involution, then (3.1) is an equivalence of 
$C_2$-Tambara functors.
Proof. Applying the formula for the box product ([Reference Hill and Mazur13, definition 3.1]) to the formula for the norm yields
\begin{align*}& (N^{C_2}_eR) \Box \underline M^\mathrm{fix}\\ &\quad =
\begin{cases}
\bigl( ( \mathbb{Z}\{R\} \oplus (R\otimes R)/C_2 ) /\mathrm{TR} \otimes M^{C_2} \oplus (R\otimes R \otimes M)/C_2 \bigr)/\mathrm{FR}\ \ \text{at } C_2/C_2 \\
R \otimes R \otimes M \ \text{at } C_2/e,
\end{cases}
\end{align*} and again we send it to 
${\underline{(R \otimes R \otimes
M)}}^\mathrm{fix}$ by the Mackey or Tambara functor map that corresponds via the adjunction of remark 3.1 to the identity on the free orbit. So the resulting map is the identity on the free orbit and 
$\mathsf{res}$ on the trivial orbit, which will land in 
$(R \otimes R \otimes M)^{C_2}$.
Again, since 2 is invertible the restriction of 
$\mathsf{res}$ to 
$(R\otimes R \otimes M)/C_2$, which sends 
$[a\otimes b\otimes m] \mapsto a\otimes b \otimes m + \bar b\otimes
\bar a\otimes \bar m $ is an isomorphism 
$(R\otimes R \otimes M)/C_2 \to (R\otimes R \otimes M)^{C_2} $. Frobenius Reciprocity identifies 
$\mathsf{tr}(a\otimes b)\otimes m$ with 
$[a\otimes b\otimes
\mathsf{res}(m)]\in (R\otimes R \otimes M)/C_2$ for all 
$a, b \in R$, 
$m\in M^{C_2}$. It also identifies 
$\{a\} \otimes \mathsf{tr}(m)$ with 
$[a\otimes \bar a \otimes m ] \in (R\otimes R \otimes M)/C_2$ for all 
$a\in R$, 
$m\in M$, and since 2 is invertible, 
$\mathsf{tr}(M) =M^{C_2}$ (any 
$m\in M^{C_2}$ is equal to 
$\mathsf{tr} (m/2)$). So in fact,
\begin{equation*}(N^{C_2}_eR) \Box \underline M^\mathrm{fix} (C_2/C_2) \cong (R \otimes R
\otimes M)/{C_2} \cong (R \otimes R \otimes M)^{C_2}, \end{equation*} and our map between 
$(N^{C_2}_eR)\Box \underline M^\mathrm{fix} $ and 
${\underline{(R \otimes
R\otimes M)}}^\mathrm{fix}$ is an isomorphism at both levels.
Note also that for a map of commutative C 2-rings 
$f\colon R \to C$ where 2 is invertible in both rings, the sequence of maps
\begin{equation*}N^{C_2}_ei^{*}_e \underline R^\mathrm{fix} \Box {\underline{C}}^\mathrm{fix} \to \underline R^\mathrm{fix}\Box {\underline{C}}^\mathrm{fix}
\cong {\underline{(R \otimes
C)}}^\mathrm{fix} \to {\underline{(C\otimes C)}}^\mathrm{fix} \to {\underline{C}}^\mathrm{fix} \end{equation*} that is given by the counit of the 
$(N_e^{C_2},i^*_e)$-adjunction, the identification of lemma 3.2, the map f, and multiplication, the corresponding map 
${\underline{(R \otimes R \otimes C)}}^\mathrm{fix}
\to {\underline{C}}^\mathrm{fix}$ is given by the multiplication in R and the R-module structure on C induced by the map f.
4. Working relative to a commutative ground ring
In [Reference Lindenstrauss, Richter and Zou20, § 8] we defined a G-equivariant Loday construction relative to a map of G-Tambara functors 
${\underline{k}} \rightarrow
{\underline{R}}$. In general, this construction is rather involved because its building blocks are relative norm-restriction terms: For an orbit 
$G/H$ we set
\begin{equation} G/H \otimes_{{\underline{k}}} {\underline{R}} := (G/H \otimes {\underline{R}} ) \Box_{(G/H \otimes{{\underline{k}}} )} {\underline{k}} =N_H^Gi_H^*({\underline{R}})
\Box_{N_H^Gi_H^*({\underline{k}})} {\underline{k}} =: N_H^{G,{\underline{k}}}i_H^*({\underline{R}}). \end{equation} This uses the naturality of 
$N_H^Gi_H^*(-)$ and the counit 
$N_H^Gi_H^*({\underline{k}}) \rightarrow {\underline{k}}$ of the norm-restriction adjunction.
In [Reference Lindenstrauss, Richter and Zou20] we define the relative equivariant Loday construction for any finite simplicial G-set X:
\begin{equation*} \mathcal{L}^{G, {\underline{k}}}_X({\underline{R}}) := \mathcal{L}^G_X ({\underline{R}})
\Box_{\mathcal{L}^G_X({\underline{k}})} {\underline{k}}.\end{equation*} If we consider fixed point C 2-Tambara functors 
${\underline{R}}^\mathrm{fix}$ and if we work relative to a constant Tambara functor 
${\underline{k}}^c$, then these terms simplify drastically. Recall that we denote by 
${\underline{k}}^c$ the constant Tambara functor which is the fixed point Tambara functor for the trivial C 2-action.
The purpose of this section is to relate the relative Loday construction of C 2-fixed point Tambara functors to the fixed point Tambara functor of the non-equivariant relative Loday construction. To that end we prove two crucial auxiliary results.
Proposition 4.1. Let 
$k \rightarrow R$ be a map of commutative C2-rings where C2 acts trivially on 
$k$ and 2 is invertible in R. Then
\begin{equation*} N_e^{C_2,{\underline{k}}^c}i_e^*({\underline{R}}^\mathrm{fix}) \cong {\underline{(R \otimes_k R)}}^\mathrm{fix}, \end{equation*} where C2 acts on 
$R \otimes_k R$ by 
$\tau(a\otimes b)=\bar b
\otimes \bar a$.
Proof. The relative box-product 
$ N_e^{C_2,{\underline{k}}^c}i_e^*({\underline{R}}^\mathrm{fix}) = N_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix}
)\Box_{N_e^{C_2}i_e^*({\underline{k}}^c)} {\underline{k}}^c$ is the coequalizer of the diagram
where ν is the composite of the map 
$N_e^{C_2}i_e^*{\underline{k}}^c \rightarrow
N_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})$ and the multiplication map of 
$N_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})$ and 
$\nu'$ uses the counit of the adjunction 
$\varepsilon \colon N_e^{C_2}i_e^*{\underline{k}}^c \rightarrow {\underline{k}}^c$ and the multiplication in 
${\underline{k}}^c$. As 
${\underline{k}}^c$ is the fixed point Tambara functor for the trivial action we can use the fact that 
$i_e^*$ and 
$N_e^{C_2}$ are strong symmetric monoidal and lemma 3.2 to get that
\begin{align*}N_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix}) \Box N_e^{C_2}i_e^*({\underline{k}}^c) & =
N_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix} \Box {\underline{k}}^c) =
N_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix} \Box {\underline{k}}^\mathrm{fix})\\
&\cong N_e^{C_2}i_e^*({\underline{(R\otimes k)}}^\mathrm{fix}).\end{align*}Then we can use lemma 3.3 to rewrite the diagram as
where now ν uses the map 
$k \rightarrow R$ and the induced k-module structure on R and 
$\nu'$ uses the multiplication in k.
Note that 
$R \otimes_k R \cong (R \otimes R) \otimes_{k \otimes k}
k$. We will show that taking the fixed point Tambara functor commutes with forming coequalizers.
To this end we consider the full subcategory of C 2-Tambara functors whose objects are fixed point Tambara functors and denote it by 
$C_2\text{-Tamb}^{\mathrm{fix}}$. But then the fact that the functor 
$T \mapsto
{\underline{T}}^\mathrm{fix}$ is right adjoint to evaluation at the free level implies that
\begin{align*} C_2\text{-Tamb}^{\mathrm{fix}}(\underline R^\mathrm{fix}, {\underline{T}}^\mathrm{fix}) & = C_2\text{-Tamb}(\underline R^\mathrm{fix}, {\underline{T}}^\mathrm{fix})\cong cC_2\text{-rings}(\underline R^\mathrm{fix}(C_2/e),
T) \\
& = cC_2\text{-rings}(R,
T) = cC_2\text{-rings}(R, {\underline{T}}^\mathrm{fix}(C_2/e)), \end{align*} where 
$cC_2\text{-rings}$ denotes the category of commutative C 2-rings. Therefore, if we restrict to the above full subcategory, taking the fixed point Tambara functor is left adjoint, hence preserves coequalizers.
Lemma 4.2. If 
$k$ is a commutative ring with trivial C2 action and 
$M$ and 
$N$ are two 
$k$-modules with a 
$k$-linear involution and 2 is invertible in 
$M$ or in 
$N$, then there is an equivalence of C2-Mackey functors
\begin{equation*}\underline M^\mathrm{fix}\Box_{{\underline{k}}^c} \underline N^\mathrm{fix} \cong {\underline{(M\otimes_k N)}}^\mathrm{fix}\end{equation*} which is natural in 
$M$ and 
$N$. Here C2 acts on 
$M\otimes N$ by the diagonal action. If 
$M$ and 
$N$ are both also commutative 
$k$-algebras, this is an equivalence of C2-Tambara functors.
Proof. Using lemma 3.2 we know that
\begin{equation*}\underline M^\mathrm{fix}\Box{{\underline{k}}^c} \Box \underline N^\mathrm{fix} = \underline M^\mathrm{fix}\Box{{\underline{k}}^\mathrm{fix}} \Box \underline N^\mathrm{fix} \cong
{\underline{(M\otimes k \otimes N)}}^\mathrm{fix}\end{equation*}and
\begin{equation*}\underline M^\mathrm{fix}\Box \underline N^\mathrm{fix} \cong {\underline{(M\otimes N)}}^\mathrm{fix}.\end{equation*}The result in the k-module case then follows from the fact that taking the fixed point Mackey functor commutes with forming coequalizers, which is completely analogous to the fact that the fixed point Tambara functor commutes with forming coequalizers which was shown in the proof of proposition 4.1 above. In the case of k-algebras, it follows directly by the argument in the proof there.
Theorem 4.3. Assume that 
$k \rightarrow R$ is a map of commutative 
$C_2$-rings where C2 acts trivially on 
$k$ and 2 is invertible in 
$R$, and let 
$X$ be a finite simplicial 
$C_2$-set. Then
\begin{equation*} \mathcal{L}^{C_2, {\underline{k}} ^c}_X({\underline{R}}^\mathrm{fix}) \cong {\underline{\mathcal{L}^k_X(R)}}^\mathrm{fix},\end{equation*} where C2 acts on each level 
$\mathcal{L}^k_{X_n}(R)$ by simultaneously using the action induced from the C2-action on Xn (exchanging copies of 
$R$ as needed) by naturality and acting on all copies of 
$R$.
Proof. Theorem 4.3 follows from the previous two results: proposition 4.1 says that for free orbits 
$C_2/e$,
\begin{equation*} \mathcal{L}^{C_2, {\underline{k}} ^c}_{C_2/e}({\underline{R}}^\mathrm{fix}) \cong {\underline{\mathcal{L}^k_{C_2/e}(R)}}^\mathrm{fix}.\end{equation*}Clearly for one-point orbits,
\begin{equation*} \mathcal{L}^{C_2, {\underline{k}} ^c}_{C_2/C_2}({\underline{R}}^\mathrm{fix}) = {\underline{R}}^\mathrm{fix}= {\underline{\mathcal{L}^k_{C_2/C_2}(R)}}^\mathrm{fix}.\end{equation*}If X and Y are disjoint C 2-sets
\begin{equation*} \mathcal{L}^{C_2, {\underline{k}} ^c}_{X\sqcup Y} ({\underline{R}}^\mathrm{fix}) \cong \mathcal{L}^{C_2, {\underline{k}} ^c}_{X} ({\underline{R}}^\mathrm{fix})
\Box_{{\underline{k}} ^c}
\mathcal{L}^{C_2, {\underline{k}} ^c}_{Y} ({\underline{R}}^\mathrm{fix}). \end{equation*}Then lemma 4.2 implies that the identification for the free and trivial orbits can be assembled into a statement about disjoint unions of orbits. This gives the desired identification in each fixed simplicial degree.
Face maps in X are surjective and the identifications above are compatible with fold maps, orbit surjections 
$C_2/e \rightarrow C_2/C_2$ and isomorphisms. As degeneracy maps just insert units, they are also compatible with the degreewise isomorphisms.
5. Identifying 
$\mathcal{L}_{S^\sigma }^{C_2}(\underline R^\mathrm{fix})$
In this section we will continue to work with the cyclic group of order 2, 
$C_2 = \langle \tau \mid \tau^2=e\rangle$, and we will consider the C 2-simplicial set S σ which models the one-point compactification of the real sign-representation,
where the C 2-action flips the two arcs. We will call S σ with this action the flip circle.
By [Reference Lindenstrauss, Richter and Zou20, (7.4)], for any C 2-Tambara functor 
${\underline{T}}$ we can express the C 2-Loday construction of 
${\underline{T}}$ with respect to S σ as a two-sided bar construction
\begin{equation}
\mathcal{L}^{C_2}_{S^{\sigma}}({\underline{T}}) \cong B({\underline{T}}, N^{C_2}_ei^{*}_e{\underline{T}}, {\underline{T}}).
\end{equation} We will simplify this for the C 2-Tambara functor 
${\underline{R}}^\mathrm{fix}$ associated to a commutative ring R with involution 
$a\mapsto \bar a$. We will repeatedly use the commutative C 2-ring 
$R \otimes R$, with
\begin{equation}
\tau(a\otimes b) = \bar b\otimes \bar a.
\end{equation}For a ring spectrum A with an anti-involution, Dotto, Moi, Patchkoria and Reeh observed [Reference Dotto, Moi, Patchkoria and Reeh6, p. 84], that
\begin{equation*} B(A, N_e^{C_2}i_e^*A, A) \simeq B(A, A \wedge A,
A), \end{equation*} where they use the flip-C 2-action on 
$A \wedge A$ (switching coordinates and acting on them, as in (5.2)). They identify 
$\mathsf{THR}(A)$ with 
$B(A, N_e^{C_2}i_e^*A, A)$ in [Reference Dotto, Moi, Patchkoria and Reeh6, theorem 2.23] under a flatness assumption on A.
The following result is an algebraic version of this result where we use the C 2-action on 
$R \otimes R$ that exchanges the coordinates and acts on both tensor factors.
Theorem 5.1. If R is a commutative ring with involution and 2 is invertible in R, then there is a natural equivalence of simplicial C2-Tambara functors
\begin{equation*}
\mathcal{L}^{C_2}_{S^{\sigma}}(\underline R^\mathrm{fix}) \cong
B (\underline R^\mathrm{fix}, N^{C_2}_ei^{*}_e\underline R^\mathrm{fix}, \underline R^\mathrm{fix}) \cong
{\underline{B(R, R\otimes R, R) }}^\mathrm{fix}
\end{equation*} where C2 acts on 
$R\otimes R$ as in (5.2).
So in every simplicial degree n, 
$\mathcal{L}^{C_2}_{S^{\sigma}}(\underline R^\mathrm{fix})_n= \underline R^\mathrm{fix} \Box
(N^{C_2}_ei^{*}_e\underline R^\mathrm{fix} )^{\Box n} \Box \underline R^\mathrm{fix}$ is the fixed point Tambara functor of the C 2-ring 
$R \otimes (R\otimes R)^{\otimes n} \otimes
R$ with C 2-action given by
\begin{align*}
\tau& (a_0\otimes (a_1\otimes a_{2n+1})\otimes (a_2\otimes a_{2n})\otimes \cdots\otimes (a_n\otimes a_{n+2} )\otimes a_{n+1}) \\
& = \bar{a}_0\otimes (\bar{a}_{2n+1} \otimes \bar{a}_{1})\otimes (\bar{a}_{2n}\otimes \bar{a}_{2})\otimes \cdots\otimes (\bar{a}_{n+2} \otimes\bar{a}_{n} )\otimes \bar{a}_{n+1}.
\end{align*}One can visualize this C 2-action as
Remark 5.2. Note that in contrast to proposition 4.1 
$N^{C_2}_ei^{*}_e \underline R^\mathrm{fix}$ is not isomorphic to 
${\underline{(R\otimes R)}}^\mathrm{fix}$, even in very simple cases! For example, for 
$R=\mathbb{Z}$ with the trivial C 2-action, 
${\underline{(R\otimes R)}}^\mathrm{fix}$ is just 
$\underline{\mathbb{Z}}^c$ with respect to the constant action, while 
$N^{C_2}_ei^{*}_e \underline{\mathbb{Z}}^c$ is the C 2-Burnside Tambara functor (see for instance [Reference Lindenstrauss, Richter and Zou20, (5.1)]). We need an outer copy of 
${\underline{R}}^\mathrm{fix}$ in theorem 5.1 as a catalyst in order to achieve the desired simplification.
Proof. The proof follows by induction on n. The base case n = 0 is lemma 3.2 applied to 
$M=N=R$, and the inductive step can be done with the help of lemma 3.3 for 
$M= R \otimes
(R\otimes R)^{\otimes(n-1)} \otimes R$. Note that both lemmas proceed by identifying all the terms to the C 2-coinvariant (second) part of the box product on 
$C_2/C_2$, so these identifications of the term 
$\underline R^\mathrm{fix} \Box (N^{C_2}_ei^{*}_e\underline R^\mathrm{fix} )^{\Box n} \Box \underline R^\mathrm{fix}$ with 
${\underline{(R\otimes (R\otimes R)^{\otimes n} \otimes R)}} ^\mathrm{fix}$ behave as one would expect for internal multiplications and insertions of units. See also the comment below the proof of lemma 3.3. Therefore, these identifications are compatible with the simplicial structure maps.
Remark 5.3. It is important to remember that the equivariant Loday construction 
$\mathcal{L}_{X}^{C_2}(\underline R^\mathrm{fix})$ is not the Loday construction relative to 
${\underline{\mathbb{Z}}}^c$, but rather the Loday construction relative to the C 2-Burnside Tambara functor, and these are different. For example, taking the relative norm-restriction term from (4.1) 
$N_e^{C_2,
{\underline{\mathbb{Z}}}^c}i_e^*{\underline{\mathbb{Z}}}^c $ gives 
${\underline{\mathbb{Z}}}^c$, whereas taking 
$N_e^{C_2}i_e^*{\underline{\mathbb{Z}}}^c $ gives the C 2-Burnside Tambara functor.
However, theorem 5.1 shows that if 2 is invertible in R,
\begin{equation*}
\mathcal{L}^{C_2}_{S^{\sigma}}(\underline R^\mathrm{fix}) \cong
B(\underline R^\mathrm{fix}, N^{C_2}_ei^{*}_e\underline R^\mathrm{fix}, \underline R^\mathrm{fix}) \cong
{\underline{B(R, R\otimes R, R)}}^\mathrm{fix}
\end{equation*} where C 2 acts on 
$R\otimes R$ as in (5.2). Note that in the bar construction 
$B(R, R\otimes R, R)$ the ground ring is the ring of integers whereas for the Loday construction we work relative to the Burnside Tambara functor. Similarly, theorem 4.3 implies that for 
$k = \mathbb{Z}$ we also obtain that 
$\mathcal{L}^{C_2, {\underline{\mathbb{Z}}}^c}_{S^\sigma}({\underline{R}}^\mathrm{fix})$ can be identified with 
${\underline{B^{\mathbb{Z}}(R, R\otimes R, R)}}^\mathrm{fix}$ and therefore in hindsight we obtain that in this case the Loday construction relative to the Burnside Tambara functor agrees with the one relative to 
${\underline{\mathbb{Z}}}^c$.
Remark 5.4. If R is a commutative ring with involution and if M is an R-module with involution compatible with the involution on R in the sense that 
$\overline{rm}=\bar{r}\bar{m}$ for all 
$r \in R$, 
$m
\in M$, then the C 2-Mackey functor 
${\underline{M}}^\mathrm{fix}$ is a symmetric bimodule over the C 2-Tambara functor 
$\underline R^\mathrm{fix}$.
Equivariant Loday constructions on based G-simplicial sets X of a G-Tambara functor 
${\underline{T}}$ with coefficients in a G-Mackey functor 
${\underline{N}}$ which is a symmetric 
${\underline{T}}$-bimodule are defined analogously to those in the non-equivariant case. We place the coefficients at the basepoint in each simplicial degree. Then 
$\mathcal{L}_X^G({\underline{T}}; {\underline{N}})$ is a simplicial G-Mackey functor.
6. Relating
$\mathcal{L}_{S^\sigma}^{C_2}({\underline{R}})$ to reflexive homology
Let us for now consider a more general context: Let 
$k$ be a commutative ring and let 
$A$ be an associative 
$k$-algebra. We assume that A carries an anti-involution that we denote by 
$a\mapsto
\bar{a}$ and which we assume to be k-linear. Let M be an A-bimodule with an involution 
$m\mapsto \bar{m}$ that is compatible with the bimodule structure over A in the sense that 
$\overline{amb}= \bar{b} \bar{m} \bar{a}$ for all 
$a,b\in A$, 
$m\in M$. All tensor products will be over k in this section, unless otherwise indicated.
Graves [Reference Graves11, definition 1.8] defines an involution on every level of the Hochschild complex 
$\mathsf{CH}^k_n(A; M) =M\otimes A^{\otimes n}$ by
\begin{equation}
r_n(m\otimes a_1\otimes a_2\otimes\cdots\otimes a_n) =\bar m \otimes \bar a_n\otimes\cdots\otimes \bar a_2\otimes \bar a_1.
\end{equation} For the face maps of the Hochschild complex we get that 
$r_{n-1}\circ
d_i=d_{n-i} \circ r_n$, so these levelwise maps do not preserve the simplicial structure but they reverse it. Since this relation implies that 
$d\circ r_n=(-1)^n r_{n-1}\circ d$, applying rn at each level n does not induce a map on the associated chain complexes, unless we adjust the signs.
The C 2-actions given by the rn-maps together with the simplicial structure maps on 
$\mathsf{CH}^k_\cdot(A;M)$ turn 
$\mathsf{CH}^k_\cdot(A;M)$ into a functor from the crossed simplicial group 
$\Delta R^{op}$ in the sense of Fiedorowicz-Loday [Reference Fiedorowicz and Loday10] to the category of k-modules. In [Reference Graves11, definition 1.9], Graves defines reflexive homology as functor homology as follows:
\begin{equation*} \mathsf{HR}^{+,k}_*(A;M)=\mathsf{Tor}_*^{\Delta R^\mathrm{op}}(k^*, \mathsf{CH}^k_\cdot(A;M)).\end{equation*} Here 
$k^*$ is the constant right 
$\Delta R^\mathrm{op}$-module with value k at all objects. In [Reference Graves11, definition 2.1], he defines a bicomplex 
$C_{*,*}$ which is a bi-resolution of 
$k^*$. With its help he shows in [Reference Graves11, proposition 2.4] that 
$ \mathsf{HR}^{+,k}_*(A;M)$ is the homology of the complex 
$\mathsf{CH}^k_*(A;M)/(1-r)$, where r is obtained from the maps rn of (6.1) by
\begin{equation}
r(m\otimes a_1\otimes\cdots\otimes a_n)
=
(-1)^{\frac{n(n+1)}{2}} r_n(m\otimes a_1\otimes\cdots\otimes a_n)
= (-1)^{\frac{n(n+1)}{2}} \bar m \otimes \bar a_n\otimes\cdots\otimes \bar a_1.
\end{equation} With this choice of sign, the map r is a chain map, so the quotient by 
$1-r$ is still a chain complex. In the following we denote by 
$B^k_*(A, A\otimes A^\mathrm{op}, M)$ the chain complex associated to the simplicial k-module 
$B^k(A, A\otimes A^\mathrm{op}, M)$.
Theorem 6.1. Assume that 
$k$ is a commutative ring and that 
$A$ is an associative 
$k$-algebra with an anti-involution as above whose underlying 
$k$-module is flat. Let 
$M$ be an 
$A$-bimodule with a compatible involution as above, and assume that 2 is invertible in A. Then there is a 
$C_2$-equivariant quasi-isomorphism of chain complexes
\begin{equation}
B^k_*(A, A\otimes A^\mathrm{op}, M)\to \mathsf{CH}^k_*(A;M).
\end{equation} Here the generator τ of 
$C_2$ acts diagonally on 
$B^k_*(A, A\otimes
A^\mathrm{op}, M)$, where the action on 
$A\otimes A^\mathrm{op}$ is given by 
$\tau(a\otimes b)= \bar b\otimes \bar a$. On the Hochschild chain complex 
$C_2$ acts via 
$r$.
Corollary 6.2. Under the assumptions of theorem 6.1, we get homology isomorphisms
\begin{equation}
H_*(B^k_*(A, A\otimes A^\mathrm{op}, M)) \cong \mathsf{HH}^k_*(A;M)
\end{equation}
\begin{equation}
H_*(B^k_*(A, A\otimes A^\mathrm{op}, M)^{C_2} ) \cong \mathsf{HR}^{+,k}_*(A;M).
\end{equation}Proof. We get the first isomorphism because the map of theorem 6.1 is a quasi-isomorphism. It also follows from the fact that both complexes calculate 
$ \mathsf{Tor}_*^{A\otimes A^\mathrm{op} }(A,M)$ because of the assumption that A is flat over k. Note that in the case M = A the first isomorphism also follows from the fact that the bar construction on the left is isomorphic to the Segal-Quillen subdivision of the Hochschild complex [Reference Segal24].
The second isomorphism follows from the fact that 2 is invertible in both complexes: As 2 is invertible in A, the unit of A, 
$k \rightarrow
A$ factors through 
$k[\frac{1}{2}]$. We can express every level of each of the complexes as the direct sum of the +1-eigenspace and the −1-eigenspace of the action of the generator of C 2 on them. Since the actions commute with d, in fact each of the complexes breaks up as the direct sum of a positive subcomplex and a negative subcomplex.
Since the quasi-isomorphism is a C 2-map, it preserves this decomposition, and as it is a quasi-isomorphism, it must be a quasi-isomorphism on the positive and negative subcomplexes, respectively. That means that we get an isomorphism
\begin{equation*}H_*( B^k_*(A, A\otimes A^\mathrm{op}, M)^{C_2}) \to H_*(\mathsf{CH}^k_*(A;M) ^{C_2}),\end{equation*}but since 2 is invertible, we have a chain isomorphism
\begin{equation*} \mathsf{CH}^k_*(A;M) ^{C_2} \to \mathsf{CH}^k_*(A;M) _{C_2}= \mathsf{CH}^k_*(A;M)/(1-r),\end{equation*}and hence the claim follows with [Reference Graves11, proposition 2.4].
Proof of theorem 6.1
We consider two 
$A\otimes A^\mathrm{op}$-flat resolutions of A: We use 
$B^k_*(A, A\otimes A^\mathrm{op}, A\otimes A^\mathrm{op})$ with 
$A\otimes A^\mathrm{op}$ acting on the rightmost coordinate and 
$B^k_*(A,A,A)$ where 
$A^\mathrm{op}$ acts on the left and A on the right, as in the 
$\mathsf{Tor}$-identification of Hochschild homology. We let C 2 act on 
$B_*(A, A\otimes A^\mathrm{op},
A\otimes A^\mathrm{op})$ by acting diagonally on all the coordinates, and denote the action of the generator on it by τ. This action is simplicial, and therefore commutes with d. We let C 2 act on 
$B^k_*(A,A,A)$ by setting
\begin{equation*}r(a_0\otimes a_1\otimes \cdots \otimes a_n\otimes a_{n+1})
=(-1)^\frac{n(n+1)}{2} \bar a_{n+1}\otimes \bar a_n \otimes
\cdots\otimes \bar a_1 \otimes \bar a_0.\end{equation*}Because of the sign adjustment, r is a chain map. We only know that the two resolutions are flat, not that they are projective. But any chain map between them that covers the identity on A induces an isomorphism on H 0, which is the only nontrivial homology group for both complexes, and therefore is a quasi-isomorphism.
We define 
$f_n \colon B^k_n(A, A\otimes A^\mathrm{op}, A\otimes A^\mathrm{op}) \to B^k_{n}(A,A,A)$ as
\begin{align*} f_n(a_0
\otimes &(a_1\otimes a_{2n+2})
\otimes (a_2\otimes a_{2n+1}) \otimes \cdots \otimes (a_{n+1}\otimes a_{n+2}))\\
= & a_{n+2}a_{n+3}\cdots a_{2n+1} a_{2n+2} a_0\otimes a_1\otimes a_2\otimes\cdots\otimes a_{n+1}.
\end{align*} This is a simplicial 
$A\otimes A^\mathrm{op}$-module map, and covers the identity on the A being resolved since in level 0 it sends 
$a_0\otimes
(a_1\otimes a_2)$ to 
$a_2 a_0 \otimes a_1$ and both of these map down to 
$a_2 a_0 a_1\in A$. This map is not C 2-equivariant, but if we define 
$g := r\circ
f\circ\tau$, we get 
$g_n \colon B^k_n(A, A\otimes A^\mathrm{op}, A\otimes
A^\mathrm{op}) \to B^k_{n}(A,A,A)$ with
\begin{align*} g_n(a_0
\otimes &(a_1\otimes a_{2n+2})
\otimes (a_2\otimes a_{2n+1}) \otimes \cdots \otimes (a_{n+1}\otimes a_{n+2}))\\
= & (-1)^\frac{n(n+1)}{2} a_{n+2}\otimes a_{n+3}\otimes \cdots \otimes a_{2n+1} \otimes a_{2n+2}\otimes a_0 a_1 a_2 \cdots a_{n+1}.
\end{align*} This is not a simplicial map but it is an 
$A\otimes A^\mathrm{op}$-module map and it is a chain map since r, f, and τ are chain maps. Again, it covers the identity on A since on level 0, 
$a_0\otimes (a_1\otimes
a_2)\mapsto a_2 \otimes a_0 a_1$ and both of these map down to 
$a_2
a_0 a_1\in A$.
We now use the fact that 2 is invertible in A and consider the map
\begin{equation*}\frac {f+g}{2} \colon B^k_*(A, A\otimes A^\mathrm{op}, A\otimes A^\mathrm{op})\to B^k_*(A,A,A),\end{equation*} which is a map of 
$A\otimes A^\mathrm{op}$-chain complexes and covers the identity on A since f and g are such maps. This map is also equivariant because
\begin{equation*} r\circ\frac {f+g}{2}
= r\circ\frac {f+r\circ f\circ\tau}{2}
=\frac {r\circ f + f\circ\tau}{2}
=\frac {r\circ f \circ\tau + f}{2}\circ\tau
=\frac {f+g}{2}\circ\tau.
\end{equation*} So 
$\frac{f+g}{2}$ is a quasi-isomorphism of flat 
$A\otimes A^\mathrm{op}$-complexes. By lemma 6.3 below, if we tensor it over 
$A\otimes A^\mathrm{op}$ with the 
$A\otimes A^\mathrm{op}$-module M, we get a quasi-isomorphism
\begin{equation*} \frac{f+g}{2}\otimes\mathrm{id}_M \colon B^k_*(A, A\otimes A^\mathrm{op}, M)\to
\mathsf{CH}^k_*(A;M). \end{equation*}This map is equivariant because it is the tensor product of two equivariant maps.
Lemma 6.3. Let 
$R$ be an associative ring and let 
$\phi\colon C_*\to D_*$ be a quasi-isomorphism between two bounded below chain complexes of flat right 
$R$-modules. Let 
$M$ be a left 
$R$-module. Then 
$\phi\otimes \mathrm{id}_M \colon C_*\otimes_R M \to D_*\otimes_R M$ is a quasi-isomorphism as well.
Proof. Since ϕ is a quasi-isomorphism, its mapping cone, 
$\mathrm{cone}(\phi)$, is acyclic. The mapping cone is also a bounded-below chain complex of flat right R-modules, so it can be viewed as a flat resolution of the 0-module, possibly with a shift. We suspend it, so that 
$\Sigma^a \mathrm{cone}(\phi)$ is a non-negative chain complex whose bottom chain group is in degree zero. Since flat resolutions can be used to calculate 
$\mathsf{Tor}$,
\begin{equation*}H_*(\Sigma^a\mathrm{cone}(\phi)\otimes_R M) \cong\mathsf{Tor}_{*}^R(0, M) =0\end{equation*} for all 
$*$. So 
$\Sigma^a\mathrm{cone}(\phi)\otimes_R M$ and hence 
$\mathrm{cone}(\phi)\otimes_R M =\mathrm{cone}(\phi\otimes_R \mathrm{id}_M)$ is acyclic. But that forces 
$\phi\otimes_R \mathrm{id}_M$ to be a quasi-isomorphism.
Taking our identification of 
$\mathcal{L}_{S^\sigma}^{C_2}({\underline{R}})$ with 
${\underline{B(R, R\otimes R,R) }}^\mathrm{fix}$ from theorem 5.1 together with corollary 6.2 we obtain the following comparison result between the homology groups of the C 2-Loday construction for the flip circle S σ and 
${\underline{R}}^\mathrm{fix}$ on the one hand and the reflexive homology groups on the other hand:
Theorem 6.4. Assume that 
$R$ is a commutative ring with involution and that 2 is invertible in 
$R$. If the underlying abelian group of 
$R$ is flat over 
$\mathbb{Z}$, then
\begin{equation*} \pi_*( \mathcal{L}^{C_2}_{S^\sigma}({\underline{R}}^\mathrm{fix} )(C_2/C_2)) \cong \mathsf{HR}^{+,\mathbb{Z}}_*(R,R). \end{equation*}The relative version follows directly from corollary 6.2, theorem 4.3, and the identification in (5.1):
Theorem 6.5. Assume that 
$R$ is a commutative 
$k$-algebra with a 
$k$-linear involution and that 2 is invertible in 
$R$. If the underlying module of 
$R$ is flat over 
$k$, then
\begin{equation*} \pi_*(\mathcal{L}^{C_2 ,{\underline{k}}^c}_{S^\sigma}({\underline{R}}^\mathrm{fix})(C_2/C_2)) \cong
\mathsf{HR}^{+,k}_*(R,R). \end{equation*}7. Involutive Hochschild homology as a Loday construction
Involutive Hochschild cohomology was defined in [Reference Braun4]. Fernàndez-València and Giansiracusa extended the definition to involutive Hochschild homology. The input is an associative algebra with anti-involution and in [Reference Fernàndez-València and Giansiracusa9] the authors work relative to a field k.
A straightforward generalization of their definition [Reference Fernàndez-València and Giansiracusa9, definition 3.3.1] to arbitrary commutative ground rings is as follows:
Definition 7.1. Let 
$k$ be a commutative ring, let 
$A$ be an associative algebra with anti-involution and let 
$M$ be an involutive 
$A$-bimodule. The involutive Hochschild homology groups of 
$A$ with coefficients in 
$M$ are
\begin{equation*} \mathsf{iHH}_*^k(A;M) = \mathsf{Tor}_*^{A^{ie}}(A;M). \end{equation*}Here A ie is the involutive enveloping algebra. As in the classical case its role is to describe (involutive) A-bimodules: There is an equivalence of categories between the category of involutive A-bimodules and the category of modules over A ie [Reference Fernàndez-València and Giansiracusa9, proposition 2.2.1]. As a k-module
\begin{equation*} A^{ie} = A \otimes_k A \otimes_k k[C_2]\end{equation*}and the multiplication on A ie is determined by
\begin{equation*} (a \otimes b \otimes \tau^i) \cdot (c \otimes d \otimes \tau^j) =
(a \otimes b) \cdot \tau^i(c \otimes d) \otimes \tau^{i+j}.\end{equation*} Here, 
$\tau(c \otimes d)$ is again 
$\bar{d} \otimes \bar{c}$, so
\begin{equation*} (a \otimes b \otimes \tau) \cdot (c \otimes d \otimes \tau^j) =
(a\bar{d} \otimes \bar{c}b) \otimes \tau^{1+j}.\end{equation*} Hence we can view A ie as a twisted group algebra 
$(A \otimes
A^{\mathrm{op}})[C_2]$. As before, every involutive algebra A is an involutive A-bimodule.
Of course we know from the classical setting of Hochschild homology that the above definition does not yield what you want if A is not flat as a k-module.
We obtain a comparison theorem between involutive Hochschild homology and the homology of the C 2-Loday construction of the circle S σ for 
${\underline{R}}^\mathrm{fix}$.
Theorem 7.2. Let 
$R$ be a commutative ring with a 
$C_2$-action. Assume that 2 is invertible in 
$R$ and that the underlying abelian group of 
$R$ is flat. Then
\begin{equation*} \pi_*(\mathcal{L}^{C_2}_{S^{\sigma}}(\underline R^\mathrm{fix})(C_2/C_2)) \cong \mathsf{iHH}^{\mathbb{Z}}_*(R). \end{equation*}And again over a general commutative k, there is a relative version:
Theorem 7.3. Let 
$k$ be a commutative ring and let 
$R$ be a commutative 
$k$-algebra with a 
$k$-linear 
$C_2$-action. Assume that 2 is invertible in 
$R$ and that the underlying 
$k$-module of 
$R$ is flat. Then
\begin{equation*} \pi_*(\mathcal{L}^{C_2,{\underline{k}}^c}_{S^{\sigma}}(\underline R^\mathrm{fix})(C_2/C_2)) \cong \mathsf{iHH}^k_*(R). \end{equation*} We prove theorem 7.2 by comparing 
$\mathsf{iHH}_*^{\mathbb{Z}}(R;M)$ for an involutive R-bimodule M to the homotopy groups at the 
$C_2/C_2$-level of the simplicial Mackey functor 
${\underline{B(R,R\otimes R,
M)}}^\mathrm{fix}$ where C 2 acts on 
$R\otimes R$ by 
$\tau(a\otimes b)
=\bar b \otimes \bar a$. The lemmata below should be used for 
$k=\mathbb{Z}$. The proof of theorem 7.3 is similar, just over a general commutative ground ring k.
In the following we will always assume that R is a commutative k-algebra with a k-linear C 2-action, that 2 is invertible in R and that the underlying k-module of R is flat over k.
Lemma 7.4. The zeroth homotopy group of the simplicial 
$k$-module 
${\underline{B^k(R,
R\otimes_k R, M)}}^\mathrm{fix}(C_2/C_2)$ is isomorphic to the zeroth involutive Hochschild homology group of 
$R$ with coefficients in M:
\begin{equation*} \pi_0({\underline{B^k(R, R\otimes_k R, M)}}^\mathrm{fix}(C_2/C_2)) \cong R
\otimes_{R^{ie}} M = \mathsf{iHH}_0^k(R;M). \end{equation*}Proof. As 2 is invertible, taking C 2-fixed points is isomorphic to taking C 2-coinvariants and both functors are exact. Thus we have to identify the quotient of 
$(R \otimes M)_{C_2}$ by the bimodule action and this yields 
$(R \otimes_{R \otimes_k R} M)_{C_2}$ which is isomorphic to 
$(M/\{am
-ma, a \in R, m \in M\})_{C_2}$. By [Reference Fernàndez-València and Giansiracusa9, proposition 2.4.1], 
$R \otimes_{R^{ie}} M$ is isomorphic to the pushout of
and this proves the claim.
Lemma 7.5. Assume that 
$0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$ is a short exact sequence of Rie-modules and abbreviate the simplicial k-module 
${\underline{B^k(R, R\otimes_k R, M_i)}}^\mathrm{fix}(C_2/C_2)$ by BMi. Then we get an induced long exact sequence on homotopy groups
Proof. As we assume that R is flat over k, tensoring with R is exact, and as 2 is invertible, taking fixed points is exact. Therefore, in every simplicial degree n, the sequence
\begin{equation*} 0 \rightarrow (BM_1)_n \rightarrow (BM_2)_n \rightarrow
(BM_3)_n \rightarrow 0\end{equation*}is short exact and hence we obtain a short exact sequence of simplicial k-modules
\begin{equation*} 0 \rightarrow BM_1 \rightarrow BM_2 \rightarrow BM_3 \rightarrow 0\end{equation*}which yields a long exact sequence on homotopy groups.
Lemma 7.6. Assume that 
$P$ is a projective 
$R^{ie}$-module. Then 
$\pi_n{\underline{B^k(R, R\otimes_k R, P)}}^\mathrm{fix}(C_2/C_2) \cong 0$ for all positive n.
Proof. In the category of 
$R^{ie}$-modules, 
$R^{ie}$ is a projective generator and every module can be written as a quotient of a direct sum of copies of R ie. Our construction sends a direct sum of modules to a direct sum of simplicial objects, yielding a direct sum of associated chain complexes. Retracts of modules give retracts of the associated chain complexes. It therefore suffices to check the claim for 
$P=R^{ie}$.
If D is any k-module with a C 2-action such that 2 acts invertibly on D, then there is an isomorphism
\begin{equation*} (D \otimes_k k[C_2])^{C_2} \cong D \end{equation*} where on the left hand side we consider the diagonal C 2-action: First note that 
$D \otimes_k k[C_2]$ with the diagonal action is isomorphic to 
$D \otimes_k k[C_2]$ where the C 2-action is only on the right-hand factor. The isomorphism 
$\psi \colon D \otimes_k k[C_2]
\rightarrow D \otimes_k k[C_2]$ sends a generator 
$d \otimes \tau^i$ to 
$\tau^{-i}d \otimes \tau^i$. Then, as 2 acts invertibly, we have
\begin{equation*} (D \otimes_k k[C_2])^{C_2} \cong (D \otimes_k k[C_2])_{C_2} = (D
\otimes_k k[C_2]) \otimes_{k[C_2]} k. \end{equation*} So in total, 
$(D \otimes_k k[C_2])^{C_2} \cong D$.
Therefore, in every simplicial degree n we can identify
\begin{equation*} {\underline{B^k_n(R, R\otimes_k R, R^{ie})}}^\mathrm{fix}(C_2/C_2) = (R \otimes_k (R
\otimes_k R)^{\otimes_k n} \otimes_k (R \otimes_k R \otimes_k k[C_2]))^{C_2}\end{equation*} with 
$R \otimes_k (R
\otimes_k R)^{\otimes_k n} \otimes_k (R \otimes_k R)$. But then we are left with the bar construction 
$B^k(R, R \otimes_k R, R \otimes_k R)$ and this has trivial homotopy groups in positive degrees.
Proposition 7.7. Assume that 
$R$ is a commutative 
$k$-algebra with a 
$k$-linear involution such that 2 is invertible in 
$R$ and assume that 
$M$ is an involutive 
$R$-bimodule. Then
\begin{equation*} \pi_*{\underline{B^k(R, R\otimes_k R, M)}}^\mathrm{fix}(C_2/C_2) \cong
\mathsf{iHH}^k_*(R;M). \end{equation*}Proof. Lemmata 7.4–7.6 imply that 
$\pi_*{\underline{B^k(R, R\otimes_k R, -)}}^\mathrm{fix}(C_2/C_2)$ has the same axiomatic description as 
$\mathsf{Tor}_*^{R^{ie}}(R;-)$.
Proof of theorems 7.2 and 7.3
Theorem 7.2 is a special case of proposition 7.7 working with 
$k=\mathbb{Z}$ (although we are working over the C 2-Burnside Tambara functor, not over 
$\mathbb{Z}^c$) and with M = R. Theorem 7.3 is the relative version.
Remark 7.8. Graves states a comparison result in [Reference Graves11, theorem 9.1] between reflexive homology, 
$\mathsf{HR}^{+,k}_*(A;M)$, and involutive Hochschild homology, 
$\mathsf{iHH}_*^k(A;M)$. The assumptions are slightly too restrictive there: Fernàndez-València and Giansiracusa prove in [Reference Fernàndez-València and Giansiracusa9, proposition 3.3.3] that 
$\mathsf{iHH}^k_*(A;M) \cong \mathsf{HH}^k_*(A;M)_{C_2}$ if the characteristic of the ground field is different from 2 and Graves shows in [Reference Graves11, proposition 2.4], that 
$\mathsf{HH}^k_*(A;M)_{C_2} \cong
\mathsf{HR}^{+,k}_*(A;M)$ if 2 is invertible in the ground ring. The assumption on A being projective as a 
$k[C_2]$-module comes for free if we work over a field of characteristic different from 2 thanks to Maschke’s theorem. For an arbitrary ring R, we also get that an arbitrary 
$R[G]$-module M is projective if M is projective as an R-module and if 
$|G|$ is invertible in R [Reference Merling22, proposition 4.4].
Remark 7.9. If a finite group G carries a homomorphism 
$\varepsilon \colon G \rightarrow C_2$, then one can consider an associated crossed simplicial group and the corresponding (co)homology theory, see for instance [Reference Angelini-Knoll, Merling and Péroux2, Reference Koam and Pirashvili17].
For an arbitrary finite group 
$C_2 \neq G \neq \{e\}$ without an interesting homomorphism to C 2, there is only the version of an associated crossed simplicial group by viewing G as a constant simplicial group because there is no meaningful way in which G can act on the simplicial category, as the automorphisms of Δ are isomorphic to C 2 (see for instance [Reference Dyckerhoff and Kapranov8, proposition 1.13]).
On the other hand, if G is a group with n elements 
$\{g_1, \ldots,
g_n\}$, we can consider the unreduced suspension of G, SG. This is the graph
and the group G acts by sending an element 
$g \in G$ and an edge labelled by gi to the edge ggi. We can model this graph by a finite simplicial G-set.
Thus if R is a commutative algebra with a G-action, then
\begin{equation}
\pi_*\mathcal{L}^G_{SG}({\underline{R}}^\mathrm{fix})(G/G)\end{equation} is a perfectly fine homology theory. We propose (7.1) as a generalization of reflexive homology to arbitrary finite groups, at least if 
$|G|$ is invertible, and will investigate its properties in future work.
We are grateful to the referee who pointed out that Hahn and Wilson [Reference Hahn and Wilson12, question 6.3] suggest that 
$H{\underline{\mathbb{F}_p}} \wedge^L_{N_e^{C_p}H\mathbb{F}_p} H{\underline{\mathbb{F}_p}}$ might be relevant for the Segal conjecture for the group Cp. Here, 
$H\mathbb{F}_p$ is the Eilenberg-MacLane spectrum of 
$\mathbb{F}_p$ and 
$H{\underline{\mathbb{F}_p}}$ denotes the Cp-equivariant Eilenberg-MacLane spectrum associated to the constant Tambara functor on 
$\mathbb{F}_p$. The spectral analogue of our Loday construction, 
$\mathcal{L}^{C_p}_{SC_p}(H{\underline{\mathbb{F}_p}})$, can be identified with 
$H{\underline{\mathbb{F}_p}} \wedge^L_{N_e^{C_p}H\mathbb{F}_p} H{\underline{\mathbb{F}_p}} = B(H{\underline{\mathbb{F}_p}},
{N_e^{C_p}H\mathbb{F}_p}, H{\underline{\mathbb{F}_p}}^)$, similar to [Reference Lindenstrauss, Richter and Zou20, § 7.5].
8. The cases
${\underline{\mathbb{F}_2}}^c$ and
${\underline{\mathbb{Z}}}^c$
For our results we had to assume that 2 is invertible in our commutative ring and that the underlying abelian group is flat. So it is a natural question to ask what happens if we drop these assumptions. We first study the simplest and most extreme case.
8.1. Comparison for
${\underline{\mathbb{F}_2}}^c$
We consider 
$\mathbb{F}_2$ with the trivial C 2-action, so the fixed point Tambara functor is the constant Tambara functor: 
${\underline{\mathbb{F}_2}}^c =
{\underline{\mathbb{F}_2}}^\mathrm{fix}$. Graves calculates reflexive homology of the ground ring in [Reference Graves11, proposition 5.1] and in the case of 
$\mathbb{F}_2$ we obtain
\begin{equation*} \mathsf{HR}^{+,\mathbb{F}_2}_*(\mathbb{F}_2) \cong H_*(BC_2, \mathbb{F}_2) \end{equation*} and this is 
$\mathbb{F}_2$ in all non-negative degrees. Note that here it does not matter whether we view 
$\mathbb{F}_2$ as a commutative 
$\mathbb{F}_2$-algebra or as a commutative ring (a commutative 
$\mathbb{Z}$-algebra). Similarly, we can calculate the involutive Hochschild homology of 
$\mathbb{F}_2$ as an involutive 
$\mathbb{F}_2$-algebra (or as a commutative 
$\mathbb{Z}$-algebra) and obtain
\begin{equation*} \mathsf{iHH}^{\mathbb{F}_2}_*(\mathbb{F}_2; \mathbb{F}_2) = \mathsf{Tor}_n^{\mathbb{F}_2[C_2]}(\mathbb{F}_2, \mathbb{F}_2) \cong
H_*(BC_2; \mathbb{F}_2). \end{equation*}Hence, involutive Hochschild homology and reflexive homology agree in this case.
If we compare this to the 2-sided bar construction 
$B(\mathbb{F}_2,
\mathbb{F}_2\otimes \mathbb{F}_2, \mathbb{F}_2) \cong B^{\mathbb{F}_2}(\mathbb{F}_2,
\mathbb{F}_2\otimes \mathbb{F}_2, \mathbb{F}_2)$, then this bar construction is isomorphic to the constant simplicial object with value 
$\mathbb{F}_2$ and therefore here we obtain
\begin{equation*} \pi_n{\underline{B(\mathbb{F}_2, \mathbb{F}_2\otimes \mathbb{F}_2, \mathbb{F}_2)}}^\mathrm{fix}(C_2/C_2) = \pi_nB(\mathbb{F}_2, \mathbb{F}_2\otimes \mathbb{F}_2,
\mathbb{F}_2) = \begin{cases} \mathbb{F}_2, & n = 0, \\ 0, &
\text{otherwise.} \end{cases} \end{equation*} Hence in this case 
$\pi_*{\underline{B(\mathbb{F}_2, \mathbb{F}_2\otimes \mathbb{F}_2,
\mathbb{F}_2)}}^\mathrm{fix}(C_2/C_2)$ agrees neither with reflexive homology nor with involutive Hochschild homology.
What about 
$\pi_*B({\underline{\mathbb{F}_2}}^c, N_e^{C_2}(\mathbb{F}_2),
{\underline{\mathbb{F}_2}}^c)(C_2/C_2)$? Note that
\begin{equation*}N_e^{C_2}(\mathbb{F}_2)(C_2/C_2) \cong \mathbb{Z}/4\mathbb{Z} \quad \text{ and }
N_e^{C_2}(\mathbb{F}_2)(C_2/e) \cong \mathbb{F}_2. \end{equation*} In 
${\underline{\mathbb{F}_2}}^c \Box N_e^{C_2}(\mathbb{F}_2)$ we obtain:
\begin{align*}
C_2/C_2: & \quad \left(\mathbb{F}_2 \otimes \mathbb{Z}/4\mathbb{Z} \oplus (\mathbb{F}_2 \otimes \mathbb{F}_2 \otimes
\mathbb{F}_2)/C_2\right)/\mathrm{FR} \\
C_2/e: & \quad \mathbb{F}_2 \otimes \mathbb{F}_2 \otimes
\mathbb{F}_2 \cong \mathbb{F}_2
\end{align*} The C 2-Weyl action is trivial on 
$\mathbb{F}_2 \otimes \mathbb{F}_2 \otimes \mathbb{F}_2 \cong
\mathbb{F}_2$. Frobenius reciprocity yields
\begin{equation*} [1 \otimes 1 \otimes 1] = [\mathsf{res}(1) \otimes 1 \otimes 1] \sim 1
\otimes \mathsf{tr}(1 \otimes 1) = 2 \cdot 1 \otimes 1 \otimes 1 = 0\end{equation*} so at the 
$C_2/C_2$-level we are left with one copy of 
$\mathbb{F}_2$ and we obtain 
${\underline{\mathbb{F}_2}}^c \Box N_e^{C_2}(\mathbb{F}_2) \cong {\underline{\mathbb{F}_2}}^c$.
This identifies 
$B({\underline{\mathbb{F}_2}}^c, N_e^{C_2}(\mathbb{F}_2),
{\underline{\mathbb{F}_2}}^c)(C_2/C_2)$ with the constant simplicial object with value 
$\mathbb{F}_2$, and therefore
\begin{equation*} \pi_n\mathcal{L}^{C_2}_{S^\sigma}({\underline{\mathbb{F}_2}}^c)(C_2/C_2) \cong \begin{cases} \mathbb{F}_2, & n = 0, \\ 0, &
\text{otherwise.} \end{cases} \end{equation*} At the free orbit, we also get the constant simplicial object with value 
$\mathbb{F}_2$, and in total we get an isomorphism of simplicial Tambara functors between 
$\mathcal{L}^{C_2}_{S^\sigma}({\underline{\mathbb{F}_2}}^c)$ and 
${\underline{B(\mathbb{F}_2, \mathbb{F}_2\otimes
\mathbb{F}_2, \mathbb{F}_2)}}^c$.
8.2. Comparison for
${\underline{\mathbb{Z}}}^c$
We consider the ring of integers and this only carries a trivial C 2-action. We know that norm restriction of 
${\underline{\mathbb{Z}}}^c$ gives the C 2-Burnside Tambara functor, 
$N_e^{C_2}i_e^*{\underline{\mathbb{Z}}}^c \cong
{\underline{A}}$. This is the monoidal unit for the 
$\Box$-product. We showed in [Reference Lindenstrauss, Richter and Zou19, lemma 5.1] that for two arbitrary commutative rings A and B, 
${\underline{A}}^c \Box {\underline{B}}^c \cong {\underline{(A \otimes
B)}}^c$ and hence
\begin{equation*} {\underline{\mathbb{Z}}}^c \Box {\underline{\mathbb{Z}}}^c \cong {\underline{(\mathbb{Z} \otimes \mathbb{Z})}}^c \cong
{\underline{\mathbb{Z}}}^c. \end{equation*}Proposition 8.1. There is an isomorphism of simplicial C2-Tambara functors
\begin{equation*} \mathcal{L}_{S^\sigma}^{C_2}({\underline{\mathbb{Z}}}^c) \cong {\underline{\mathbb{Z}}}^c\end{equation*} where the right-hand side denotes the constant simplicial C 2-Tambara functor with value 
${\underline{\mathbb{Z}}}^c$.
Proof. By the above arguments we get for an arbitrary simplicial degree n:
\begin{align*}
\mathcal{L}_{S^\sigma}^{C_2}({\underline{\mathbb{Z}}}^c)_n & = {\underline{\mathbb{Z}}}^c \Box
(N_e^{C_2}i_e^*{\underline{\mathbb{Z}}}^c)^{\Box n}
\Box {\underline{\mathbb{Z}}}^c \\
& \cong {\underline{\mathbb{Z}}}^c \Box {\underline{\mathbb{Z}}}^c \\
& \cong {\underline{\mathbb{Z}}}^c.
\end{align*}The simplicial structure maps induce the identity maps under these isomorphisms.
Corollary 8.2. The homotopy groups of 
$\mathcal{L}_{S^\sigma}^{C_2}({\underline{\mathbb{Z}}}^c)$ are
\begin{equation*} \pi_*(\mathcal{L}_{S^\sigma}^{C_2}({\underline{\mathbb{Z}}}^c)) \cong \begin{cases}
{\underline{\mathbb{Z}}}^c, & *=0, \\
0, & * \gt 0.\end{cases}\end{equation*}Corollary 8.3. For the C2-Tambara functor 
${\underline{\mathbb{Z}}}^c$ the homotopy groups
\begin{equation*} \pi_*(\mathcal{L}_{S^\sigma}^{C_2}({\underline{\mathbb{Z}}}^c))(C_2/C_2) \end{equation*} are neither isomorphic to 
$\mathsf{HR}^{+,\mathbb{Z}}_*(\mathbb{Z})$ nor to 
$\mathsf{iHH}^{\mathbb{Z}}_*(\mathbb{Z})$.
Proof. We saw above that 
$\pi_*(\mathcal{L}_{S^\sigma}^{C_2}({\underline{\mathbb{Z}}}^c))(C_2/C_2)$ is concentrated in degree 
$*=0$ with value 
$\mathbb{Z}$ whereas 
$\mathsf{HR}^{+,\mathbb{Z}}_*(\mathbb{Z})$ and 
$\mathsf{iHH}^{\mathbb{Z}}_*(\mathbb{Z})$ both give 
$H_*(C_2; \mathbb{Z})$.
9. The case of rings and algebras with anti-involution
We assume now that R is an associative ring with anti-involution. In this case the fixed point Mackey functor 
${\underline{R}}^\mathrm{fix}$ is not an associative Green functor: For 
$a, b \in R^{C_2}$ we get that 
$\tau(ab) = \bar{b}\bar{a} = ba$ and as a and b do not necessarily commute, ab is not, in general, a fixed point. But 
$\underline R^\mathrm{fix}$ does carry the structure of a discrete Eσ-ring [Reference Angelini-Knoll, Gerhardt and Hill1, example 6.12] and it is also a Hermitian Mackey functor in the sense of [Reference Dotto and Ogle7, definition 1.1.1]
We can still define a replacement of the norm-restriction object that we call 
$\tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})$ in order to avoid confusion with the commutative case. We claim that this can be done in the setting of associative C 2-Green functors.
Definition 9.1. We define 
$\tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})$ at the free level as
\begin{equation*} \tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})(C_2/e) := R \otimes R^\mathrm{op} \end{equation*} and at the trivial orbit 
$C_2/C_2$ we define
\begin{equation*} \tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})(C_2/C_2) := (\mathbb{Z}\{R\} \oplus (R \otimes
R^\mathrm{op})/C_2)/\mathrm{TR}, \end{equation*} where the Tambara reciprocity relation, 
$\mathrm{TR}$, identifies 
$\{a+b\} \sim \{a\} + \{b\} +[a\otimes \bar{b}]$ for all 
$a,b\in R$, just as in the norm-restriction construction in the commutative case.
The restriction map is
\begin{equation*} \mathsf{res}\{a\} := a \otimes \bar{a}, \qquad \mathsf{res}[a \otimes b] := a
\otimes b + \bar{b} \otimes \bar{a} \end{equation*} and the transfer sends 
$a \otimes b$ to
\begin{equation*} \mathsf{tr}(a \otimes b) := [a \otimes b]. \end{equation*}The definition above is completely analogous to the commutative case, so indeed, this does define a C 2-Mackey functor.
Note that this definition of the norm agrees with the norm defined in [Reference Hill and Mazur13, definition 3.13] as a C 2-Mackey functor. In [Reference Hill and Mazur13] Hill and Mazur use the notation 
$N^G_H$ for the norm functor for Mackey functors and they use 
$\mathcal{N}_H^G$ for the norm functor for Tambara functors.
The norm functor for Mackey functors and the restriction functor are both strong symmetric monoidal. In our case we start with a ring with an anti-involution and we consider the norm functor with a non-trivial Weyl action on the tensor factors of R. However, it is straightforward to check that this additional Weyl action does not interfere with the product:
Lemma 9.2. We can endow 
$\tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix}) =
\tilde{N}_e^{C_2}R$ with the structure of an associative 
$C_2$-Green functor. 
$\square$
Proposition 9.3. The 
$C_2$-Mackey functor 
${\underline{R}}^\mathrm{fix}$ is an 
$\tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})$-bimodule.
Proof. We know that
\begin{align*}& (\tilde{N}^{C_2}_ei^{*}_e \underline R^\mathrm{fix}) \Box \underline R^\mathrm{fix} \\ &\quad=
\begin{cases}
\bigl( ( \mathbb{Z}\{R\} \oplus (R\otimes R^\mathrm{op})/C_2 ) /\mathrm{TR} \otimes R^{C_2} \oplus (R\otimes R^\mathrm{op} \otimes R)/C_2 \bigr)/\mathrm{FR}& \text{at } C_2/C_2 \\
R \otimes R^\mathrm{op} \otimes R& \text{at } C_2/e.
\end{cases}
\end{align*} We define the left 
$\tilde{N}^{C_2}_ei^{*}_e \underline R^\mathrm{fix}$-module structure of 
$\underline R^\mathrm{fix}$ by
\begin{equation*} (a \otimes b) \otimes c \mapsto acb\end{equation*}at the free level. At the trivial level we have three types of terms:
(1) For
$a \in R$ and
$x \in R^{C_2}$ we send
$\{a\} \otimes x$ to
$ax\bar{a}$.(2) Whereas for
$a, b \in R$ and
$x \in R^{C_2}$ we define
\begin{equation*} [a \otimes b] \otimes x \mapsto axb +\bar{b}x\bar{a}. \end{equation*}The resulting elements are fixed points under the anti-involution.
(3) The C 2-action on
$R \otimes R^\mathrm{op} \otimes R$ sends a generator
$a
\otimes b \otimes y$ to
$\bar{b} \otimes \bar{a} \otimes
\bar{y}$. We send a C 2-equivalence class
$[a \otimes b \otimes
y]$ to
$ayb + \bar{b}\bar{y}\bar{a}$.
We have to check that this action is well-defined and satisfies associativity and a unit condition.
A direct inspection shows that 
$[a \otimes b] \otimes x$ and 
$[\bar{b} \otimes \bar{a}] \otimes x$ map to the same element. Similarly, the value on 
$[a \otimes b \otimes y]$ and 
$[\bar{b} \otimes \bar{a} \otimes \bar{y}]$ agrees. It is also straightforward to see that the module structure respects Tambara reciprocity. For Frobenius reciprocity we have to compare three expressions:
•
$[a \otimes b] \otimes x$ is
$\mathsf{tr}(a \otimes b) \otimes x$ and this is identified with
$[a \otimes b \otimes \mathsf{res}(x)] = [a \otimes b \otimes
x]$. Both terms are mapped to
$axb + \bar{b}x\bar{a}$ because x is a fixed point.• A term
$\{a\} \otimes \mathsf{tr}(y)$ is sent to
$a(y + \bar{y})\bar{a}$. It is identified with
$[\mathsf{res}\{a\} \otimes y] = [a \otimes \bar{a}
\otimes y]$ and this goes to
$ay\bar{a}+ a\bar{y}\bar{a}$.• We have
\begin{equation*} [a \otimes b] \otimes (y + \bar{y}) = [a \otimes b] \otimes
\mathsf{tr}(y) = \mathsf{res}([a \otimes b]) \otimes y. \end{equation*}All these terms are mapped to
$ayb + \bar{b}y\bar{a} + a\bar{y}b +
\bar{b}\bar{y}\bar{a}$.
As 
$\{1\}$ acts neutrally at the trivial level and as 
$1 \otimes 1$ acts neutrally at the free level, the unit condition is satisfied. Associativity can be proven with a very tedious calculation.
This shows that the map 
$\tilde{N}_e^{C_2}i_e^*\underline R^\mathrm{fix} \Box \underline R^\mathrm{fix} \rightarrow \underline R^\mathrm{fix}$ defined above yields a left 
$\tilde{N}_e^{C_2}i_e^*\underline R^\mathrm{fix}$-module structure on 
$\underline R^\mathrm{fix}$.
The following is a sketch of the construction of the right 
$\tilde{N}_e^{C_2}i_e^*\underline R^\mathrm{fix}$-module structure on 
$\underline R^\mathrm{fix}$: We have to define 
$\underline R^\mathrm{fix} \Box \tilde{N}_e^{C_2}i_e^*\underline R^\mathrm{fix}
\rightarrow \underline R^\mathrm{fix}$, that is: a map from
\begin{align*} &\underline R^\mathrm{fix} \Box (\tilde{N}^{C_2}_ei^{*}_e \underline R^\mathrm{fix})\\ &\quad =
\begin{cases}
R^{C_2} \otimes \bigl( ( \mathbb{Z}\{R\} \oplus (R\otimes R^\mathrm{op})/C_2 )
/\mathrm{TR} \oplus (R \otimes R\otimes R^\mathrm{op})/C_2 \bigr)/\mathrm{FR}& \text{at } C_2/C_2 \\
R \otimes R \otimes R^\mathrm{op} & \text{at } C_2/e
\end{cases}
\end{align*} to 
$\underline R^\mathrm{fix}$. At the free level we send 
$a \otimes (b \otimes c)$ to cab and this propagates to the trivial level where we map 
$[y \otimes a \otimes b]$ to 
$bya + \bar{a}\bar{y}\bar{b}$ and 
$x \otimes [a \otimes b]$ to 
$bxa
+ \bar{a}x\bar{b}$. A term 
$x \otimes \{a\}$ goes to 
$\bar{a}xa$. Then a proof dual to the above shows that this indeed gives a well-defined right module structure and that this right-module structure is compatible with the left-module structure so that we actually obtain a bimodule structure.
We use this bimodule structure of 
$\underline R^\mathrm{fix}$ over 
$\tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})$ for the definition of 
$\mathcal{L}_{S^\sigma}^{C_2}(\underline R^\mathrm{fix})$ by declaring 
$C_2/e \otimes \underline R^\mathrm{fix}$ to be 
$\tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix})$ and of course 
$C_2/C_2 \otimes
\underline R^\mathrm{fix}$ is just 
$\underline R^\mathrm{fix}$. As the simplices in S σ are lined up on two copies of 
$\Delta(-,[1])$, that are just glued at the endpoints, the associativity of R suffices to obtain well-defined face and degeneracy maps and therefore a well-defined Loday construction 
$\mathcal{L}_{S^\sigma}^{C_2}(\underline R^\mathrm{fix})$. As a simplicial C 2-Mackey functor, 
$\mathcal{L}_{S^\sigma}^{C_2}(\underline R^\mathrm{fix})$ is isomorphic to 
$B(\underline R^\mathrm{fix}, \tilde{N}_e^{C_2}i_e^*({\underline{R}}^\mathrm{fix}), \underline R^\mathrm{fix})$.
We now state and prove the analogue of theorems 6.4 and 7.2:
Theorem 9.4. Assume that 
$R$ is an associative ring with anti-involution and that 2 is invertible in 
$R$. If the underlying abelian group of 
$R$ is flat, then
\begin{equation*} \mathsf{iHH}_*^{\mathbb{Z}}(R) \cong \pi_*( \mathcal{L}^{C_2}_{S^\sigma}({\underline{R}}^\mathrm{fix}
)(C_2/C_2)) \cong \mathsf{HR}^{+,\mathbb{Z}}_*(R,R). \end{equation*}Proof. We only point out where the differences to the proof in the commutative case are. As in lemma 3.3, we can show (by literally using the same proof) that there is an isomorphism of C 2-Mackey functors
\begin{equation*} \tilde{N}_e^{C_2}i_e^* {\underline{R}}^{\mathrm{fix}} \Box {\underline{M}}^{\mathrm{fix}} \cong {\underline{(R \otimes R^{op} \otimes
M)}}^{\mathrm{fix}}, \end{equation*}if 2 is invertible in R and if R is an associative ring with anti-involution.
The arguments in § 5 go through with the difference that we have to replace 
$R \otimes R$ by 
$R \otimes R^{op}$ and theorem 5.1 gives an isomorphism of C 2-Mackey functors
\begin{equation*} \mathcal{L}_{S^\sigma}^{C_2}(\underline R^\mathrm{fix}) \cong B(\underline R^\mathrm{fix}, \tilde{N}_e^{C_2}i_e^*\underline R^\mathrm{fix}, \underline R^\mathrm{fix}) \cong
{\underline{B(R, R \otimes R^{op}, R)}}^\mathrm{fix}. \end{equation*} Section 6 is already formulated for associative algebras and also the homological algebra arguments in section 7 go through but we have to replace 
$R
\otimes R$ by the enveloping algebra 
$R \otimes R^{op}$.
In the setting where we choose a commutative ring k and A is an associative k-algebra with an anti-involution that fixes k, we first have to define a relative analogue of the norm.
Note that the unit map 
$k \rightarrow A$ induces a map of C 2-Green functors 
$N_e^{C_2}i_e^*{{\underline{k}}^c} \rightarrow \tilde{N}_e^{C_2}i_e^*{{\underline{A}}^\mathrm{fix}}$.
Definition 9.5. We define 
$\tilde{N}_e^{C_2, {\underline{k}}^c}({\underline{A}}^\mathrm{fix})$ as
\begin{equation*} \tilde{N}_e^{C_2, {\underline{k}}^c}({\underline{A}}^\mathrm{fix}) :=
\tilde{N}_e^{C_2}i_e^*{{\underline{A}}^\mathrm{fix}}
\Box_{N_e^{C_2}i_e^*{{\underline{k}}^c}} {\underline{k}}^c. \end{equation*} With this we can define 
$\mathcal{L}_{S^\sigma}^{C_2,{\underline{k}}^c}({\underline{A}}^\mathrm{fix})$ for an associative k-algebra A with anti-involution and obtain an isomorphism of simplicial C 2-Mackey functors
\begin{equation*} \mathcal{L}_{S^\sigma}^{C_2,{\underline{k}}^c}({\underline{A}}^\mathrm{fix}) \cong B({\underline{A}}^\mathrm{fix}, \tilde{N}_e^{C_2, {\underline{k}}^c}i_e^*({\underline{A}}^\mathrm{fix}), {\underline{A}}^\mathrm{fix}).\end{equation*}We get an analogue of theorems 6.5 and 7.3.
Theorem 9.6. Assume that A is an associative 
$k$-algebra with a 
$k$-linear anti-involution and that 2 is invertible in 
$A$. If the underlying module of 
$A$ is flat over 
$k$, then
\begin{equation*} \mathsf{iHH}_*^k(A) \cong \pi_*(\mathcal{L}^{C_2, {\underline{k}}^c}_{S^\sigma}({\underline{A}}^\mathrm{fix})(C_2/C_2))
\cong \mathsf{HR}^{+,k}_*(A,A). \end{equation*}Proof. We have to adapt the statement and the proof of proposition 4.1 and claim that for an associative k-algebra A with anti-involution we obtain that
\begin{equation*} \tilde{N}_e^{C_2,{\underline{k}}^c}i_e^*({\underline{A}}^\mathrm{fix}) \cong {\underline{(A
\otimes_k A^{op})}}^\mathrm{fix}, \end{equation*} where C 2 acts on 
$A \otimes_k A^{op}$ by 
$\tau(a\otimes b)=\bar b
\otimes \bar a$.
The proof goes through, when we consider the adjunction between the full subcategory of C 2-fixed point Mackey functors of associative rings with anti-involution and the category of rings with anti-involution. Then the proof of the adjunction can be copied. This yields an analogue of theorem 4.3 in the associative setting.
\begin{equation*} \mathcal{L}_{S^\sigma}^{C_2,{\underline{k}}^c}({\underline{A}}^\mathrm{fix}) \cong
{\underline{\mathcal{L}_{S^\sigma}^k(A)}}^\mathrm{fix} \end{equation*} The other changes are similar to the absolute case of an associative ring with anti-involution but of course now we have to replace 
$A \otimes_k A$ by the enveloping algebra 
$A \otimes_k A^{op}$.
Acknowledgements
We thank Sarah Whitehouse and Jack Davidson for helpful comments.
The first author was supported by NSF grant DMS-2004300 at the beginning of working on this paper, and by a grant from the Simons Foundation (#359565, Ayelet Lindenstrauss) afterwards. The second author thanks the Department of Mathematics of the Indiana University Bloomington for its hospitality and support. An anonymous referee helped us to improve the paper with many insightful comments.
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