# Nonlinear conditions for ultradifferentiability: a uniform approach

David Nicolas Nenning, Armin Rainer, and Gerhard Schindl  Fakultät für
Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Recent work showed that a theorem of Joris (that a function $f$ is smooth if
two coprime powers of $f$ are smooth) is valid in a wide variety of
ultradifferentiable classes $\mathcal{C}$. The core of the proof was
essentially $1$-dimensional. In certain cases a multidimensional version
resulted from subtle reduction arguments, but general validity, notably in the
quasianalytic setting, remained open. In this paper we give a uniform proof
which works in all cases and dimensions. It yields the result even on infinite
dimensional Banach spaces and convenient vector spaces. We also consider more
general nonlinear conditions, namely general analytic germs $\Phi$ instead of
the powers, and characterize when $\Phi\circ f\in\mathcal{C}$ implies
$f\in\mathcal{C}$.

###### Key words and phrases:

Joris theorem, pseudo-immersion, ultradifferentiable classes, quasianalytic
and non-quasianalytic, holomorphic approximation, almost analytic extension

###### 2020 Mathematics Subject Classification:

26E10, 30E10, 32W05, 46E10, 46E25, 58C25

AR was supported by FWF-Project P 32905-N, DNN and GS by FWF-Project P 33417-N

##  1\. Introduction

A function $f$ is smooth provided that two relatively prime powers or,
equivalently, two consecutive powers of $f$ are smooth, by a theorem of Joris
[4]. In the recent paper [17] Thilliez showed that this result carries over to
Denjoy–Carleman classes of Roumieu type and, by refining Thilliez’s method, we
proved in [11] that it is valid in a wide variety of ultradifferentiable
classes. See the introduction in [11] for more on the historical development.

In this analysis the dimension of the domain of $f$ had some substantial
significance. The proof in dimension one was based on holomorphic
approximation and utilized tools of complex analysis in one variable.
Multidimensional versions could be obtained only by some subtle reduction
arguments (i.e., variants of Boman’s theorem [1] and Beurling-to-Roumieu
reduction). Consequently, general validity of the result in all dimensions
remained open in cases, most notably the quasianalytic case, where the
suitable reduction tools are not available. See the table in the introduction
of [11] for a summary of the known results (in that paper the result in
question was called property ($\mathscr{D}$)).

In the present paper we prove general validity in all dimensions and all
cases. In fact, the uniformity of our proof allows us to extend the result
even to infinite dimensional Banach spaces and convenient vector spaces. The
crucial observation is that, in arbitrary dimension, holomorphic approximation
in dimension one can be replaced by uniform unidirectional holomorphic
approximation combined with the polarization inequality.

Having proved that $\Phi\circ f\in\mathcal{C}$ implies $f\in\mathcal{C}$ for
$\Phi(t)=(t^{j},t^{j+1})$, $j\in\mathbb{N}_{\geq 1}$, and $\mathcal{C}$ a
suitable ultradifferentiable class, it is natural to ask what other $\Phi$
have this property. This question was studied by [2] and [5] in the
$C^{\infty}$-setting. By combining slightly adjusted arguments of [5] with our
result, we obtain a full characterization of the analytic germs
$\Phi:(\mathbb{K},0)\to(\mathbb{K}^{n},0)$ (where $\mathbb{K}$ is $\mathbb{R}$
or $\mathbb{C}$) having this property in terms of the support of the Taylor
series of $\Phi$. For example, it shows that $f\in\mathcal{C}$ provided that
$f^{2}+f^{3}\in\mathcal{C}$ and $f^{p}\in\mathcal{C}$, where $p$ is any
positive integer.

The outline of the paper is the following. Our main result is Theorem 1.1. For
its formulation we first need to define ultradifferentiable classes (Section
1.1) and specify the required assumptions (Section 1.2). The proof of Theorem
1.1 is given in Section 2, and in Section 3 we show that it extends to Banach
spaces and even to convenient vector spaces. Finally, in Section 4 we prove
the characterization alluded to above.

###  1.1. Ultradifferentiable classes

Let $\mathfrak{M}$ be a family of positive sequences $M=(M_{k})_{k\geq 0}$.
Let $U\subseteq\mathbb{R}^{d}$ be an open subset. The class
$\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$ consists, by definition, of all complex
valued functions $f\in C^{\infty}(U)$ with the following property: for all
compact subsets $K\subseteq U$ there exist $M\in\mathfrak{M}$ and $\rho>0$
such that

$\|f\|^{M}_{K,\rho}:=\sup_{x\in
K}\sup_{k\in\mathbb{N}}\frac{\|f^{(k)}(x)\|_{L^{k}(\mathbb{R}^{d},\mathbb{C})}}{\rho^{k}M_{k}}<\infty.$
(1)

Moreover, $\mathcal{E}^{(\mathfrak{M})}(U)$ consists of all complex valued
functions $f\in C^{\infty}(U)$ such that (1) holds for all compact $K\subseteq
U$, all $M\in\mathfrak{M}$, and all $\rho>0$. Here $f^{(k)}$ denotes the
Fréchet derivative of order $k$ of $f$ and
$\|f^{(k)}(x)\|_{L^{k}(\mathbb{R}^{d},\mathbb{C})}:=\sup\\{|f^{(k)}(x)(v_{1},\ldots,v_{k})|:\|v_{i}\|\leq
1\\}$ is the operator norm. In view of the polarization inequality (cf. [7,
7.13.1])

$\sup_{\|v\|\leq
1}|d_{v}^{k}f(x)|\leq\|f^{(k)}(x)\|_{L^{k}(\mathbb{R}^{d},\mathbb{C})}\leq(2e)^{k}\sup_{\|v\|\leq
1}|d_{v}^{k}f(x)|,$ (2)

where $d_{v}^{k}f(x):=\partial_{t}^{k}f(x+tv)|_{t=0}$, we can equivalently use

$\sup_{x\in K}\sup_{k\in\mathbb{N}}\sup_{\|v\|\leq
1}\frac{|d_{v}^{k}f(x)|}{\rho^{k}M_{k}}<\infty$ (3)

instead of (1) in the definition of $\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$ and
$\mathcal{E}^{(\mathfrak{M})}(U)$.

The global classes $\mathcal{B}^{\\{\mathfrak{M}\\}}(U)$ and
$\mathcal{B}^{(\mathfrak{M})}(U)$ are defined by taking the supremum over all
$x\in U$ in (1) (that is $\|f\|^{M}_{U,\rho}<\infty$) or equivalently (3). We
shall also need the Banach space $\mathcal{B}^{M}_{\rho}(U):=\\{f\in
C^{\infty}(U):\|f\|^{M}_{U,\rho}<\infty\\}$.

The classes $\mathcal{E}^{\\{\mathfrak{M}\\}}$ and
$\mathcal{B}^{\\{\mathfrak{M}\\}}$ are said to be of _Roumieu type_ ,
$\mathcal{E}^{(\mathfrak{M})}$ and $\mathcal{B}^{(\mathfrak{M})}$ of _Beurling
type_. By convention, we use $[\cdot]$ as a placeholder for $\\{\cdot\\}$ and
$(\cdot)$.

###  1.2. Admissible weight matrices

We need to impose some conditions on $\mathfrak{M}$.

######  Definition.

A family $\mathfrak{M}$ of positive sequences $M=(M_{k})_{k\geq 0}$ is said to
be a _weight matrix_ if $\mathfrak{M}$ is totally ordered with respect to the
pointwise order relation on sequences, and each $M\in\mathfrak{M}$ is log-
convex with $M_{0}=1\leq M_{1}$ and $M_{k}^{1/k}\to\infty$. A positive
sequence $M$ with these properties is called a _weight sequence_.

With $M$ we associate the sequence $m$ given by $k!\,m_{k}=M_{k}$. Assuming
that $m_{k}^{1/k}\to\infty$ we consider the function

$h_{m}(t):=\inf_{k\in\mathbb{N}}m_{k}t^{k},\quad\text{ for }t>0,\quad\text{
and }\quad h_{m}(0):=0,$

which is increasing, continuous on $[0,\infty)$, positive for $t>0$ and equals
$1$ for large $t$. For $t>0$ we put

$\displaystyle\overline{\Gamma}_{m}(t)$
$\displaystyle:=\min\\{k:h_{m}(t)=m_{k}t^{k}\\}$

and

$\displaystyle\underline{\Gamma}_{m}(t)$
$\displaystyle:=\min\Big{\\{}k:\frac{m_{k+1}}{m_{k}}\geq\frac{1}{t}\Big{\\}}.$

Trivially, $\underline{\Gamma}_{m}\leq\overline{\Gamma}_{m}$. Equality
$\overline{\Gamma}_{m}=\underline{\Gamma}_{m}$ holds, if $m$ is log-convex.

For positive sequences $M$, $N$ we set

$\operatorname{mg}(M,N):=\sup_{j,k\geq 0,\,j+k\geq
1}\left(\frac{M_{j+k}}{N_{j}N_{k}}\right)^{1/(j+k)}\in(0,\infty].$

Note that $\operatorname{mg}(M,M)<\infty$ is the condition (M.2) of Komatsu
[6] often called _moderate growth_.

######  Definition.

A weight matrix $\mathfrak{M}$ satisfying $m_{k}^{1/k}\to\infty$ for all
$M\in\mathfrak{M}$ is called

  * •

$\\{\text{\it admissible}\\}$ or _R-admissible_ if

    * –

$\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{M}\;\exists C\geq 1\;\forall
t>0:\overline{\Gamma}_{n}(Ct)\leq\underline{\Gamma}_{m}(t)$,

    * –

$\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{M}:\operatorname{mg}(M,N)<\infty.$

  * •

$(\text{\it admissible})$ or _B-admissible_ if

    * –

$\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{M}\;\exists C\geq 1\;\forall
t>0:\overline{\Gamma}_{m}(Ct)\leq\underline{\Gamma}_{n}(t)$,

    * –

$\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{M}:\operatorname{mg}(N,M)<\infty.$

By our convention, $[\text{admissible}]$ stands for $\\{\text{admissible}\\}$
(i.e. R-admissible) in the Roumieu case and $(\text{admissible})$ (i.e.
B-admissible) in the Beurling case.

######  Remark.

In the terminology of [11] a weight matrix is $[\text{admissible}]$ if and
only if it is $[\text{regular}]$ and has $[\text{moderate growth}]$.

These are the minimal requirements needed for the tools used in our proof; see
also Remark 4.3 in which the necessity of these assumptions is discussed. Note
that [admissibility] of the weight matrix $\mathfrak{M}$ neither implies nor
obstructs quasianalyticity of the corresponding class
$\mathcal{E}^{[\mathfrak{M}]}$. In fact, (non-)quasianalyticity plays no role
in our analysis.

###  1.3. Main results

######  Theorem 1.1.

Let $\mathfrak{M}$ be an [admissible] weight matrix. Let
$U\subseteq\mathbb{R}^{d}$ be an open subset. Then a function
$f:U\to\mathbb{C}$ belongs to $\mathcal{E}^{[\mathfrak{M}]}(U)$ if
$f^{j},f^{j+1}\in\mathcal{E}^{[\mathfrak{M}]}(U)$ for some positive integer
$j$.

If we apply the theorem to the weight matrix which consists but of a single
weight sequence $M$ (with $m_{k}^{1/k}\to\infty$,
$\overline{\Gamma}_{m}(Ct)\leq\underline{\Gamma}_{m}(t)$ for some $C\geq 1$,
and $\operatorname{mg}(M,M)<\infty$), then we immediately get a version for
Denjoy–Carleman classes.

As another special case we obtain the following result for Braun–Meise–Taylor
classes $\mathcal{E}^{[\omega]}$. For their definition and why it is a special
case of Theorem 1.1 we refer the reader to [11, Section 4.5] and the
references therein. By a _weight function_ we mean a continuous increasing
function $\omega:[0,\infty)\rightarrow[0,\infty)$ such that

  * •

$\omega(2t)=O(\omega(t))$ as $t\rightarrow\infty$,

  * •

$\omega(t)=o(t)$ as $t\rightarrow\infty$,

  * •

$\log(t)=o(\omega(t))$ as $t\rightarrow\infty$,

  * •

$t\mapsto\omega(e^{t})=:\varphi_{\omega}(t)$ is convex on $[0,\infty)$.

######  Corollary 1.2.

Let $\omega$ be a concave weight function. Let $U\subseteq\mathbb{R}^{d}$ be
an open subset. Then a function $f:U\to\mathbb{C}$ belongs to
$\mathcal{E}^{[\omega]}(U)$ if $f^{j},f^{j+1}\in\mathcal{E}^{[\omega]}(U)$ for
some positive integer $j$.

Actually, for a given weight function $\omega$, the conclusion of Corollary
1.2 holds _if and only if_ $\omega$ is equivalent to a concave weight
function, as we shall see in Remark 4.3. Two weight functions $\omega_{i}$,
$i=1,2$, are said to be _equivalent_ if $\omega_{1}(t)=O(\omega_{2}(t))$ and
$\omega_{2}(t)=O(\omega_{1}(t))$ as $t\to\infty$ which holds if and only if
$\mathcal{E}^{[\omega_{1}]}=\mathcal{E}^{[\omega_{2}]}$.

##  2\. Proof

###  2.1. Preliminaries

######  Lemma 2.1 ([11, Lemma 5.1]).

Let $M,N$ be weight sequences satisfying $m_{k}^{1/k}\to\infty$,
$n_{k}^{1/k}\to\infty$, and $C:=\operatorname{mg}(M,N)<\infty$. Then

$\displaystyle h_{m}(t)$ $\displaystyle\leq C^{j}n_{j}t^{j}h_{n}(Ct),\quad
t>0,~{}j\in\mathbb{N},$ (4) $\displaystyle h_{m}(t)$ $\displaystyle\leq
h_{n}\Big{(}\frac{eC}{2}t\Big{)}^{2},\quad t>0.$ (5)

For $\varepsilon>0$ let $\Omega_{\varepsilon}$ be the interior of the ellipse
in $\mathbb{C}$ with vertices $\pm\cosh(\varepsilon)$ and co-vertices $\pm
i\sinh(\varepsilon)$. By $\mathcal{H}(\Omega_{\varepsilon})$ we denote the
space of holomorphic functions on $\Omega_{\varepsilon}$. And
$\|g\|_{X}:=\sup_{z\in X}|g(z)|$ denotes the supremum norm.

The following lemma is a simple application of Hadamard’s three lines theorem.

######  Lemma 2.2 ([11, Lemma 5.2] and [17, Lemma 3.2.4]).

Let $M,N$ be two weight sequences satisfying $m_{k}^{1/k}\to\infty$,
$n_{k}^{1/k}\to\infty$, and $C:=\operatorname{mg}(M,N)<\infty$. Let
$\varepsilon>0$. Let $g\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ and assume that there are constants
$L,a_{1},a_{2}>0$ such that

$\|g\|_{\Omega_{\varepsilon}}\leq L,\quad\|g\|_{[-1,1]}\leq
a_{1}h_{m}(a_{2}\varepsilon).$

Then with $a_{3}:=\max\\{a_{1},L\\}$ and $a_{4}:=eCa_{2}$ we have

$\|g\|_{\Omega_{\varepsilon/2}}\leq a_{3}h_{n}(a_{4}\varepsilon).$

###  2.2. Uniform unidirectional holomorphic approximation

Let $\mathbb{B}:=\\{x\in\mathbb{R}^{d}:\|x\|\leq 1\\}$ be the closed unit ball
in $\mathbb{R}^{d}$ and $\mathbb{S}:=\\{x\in\mathbb{R}^{d}:\|x\|=1\\}$ the
unit sphere. For any open subset $U\subseteq\mathbb{R}^{d}$ consider
$V_{U}:=\bigcup_{x\in U}(x+\mathbb{B})$ which is again open. If
$f:V_{U}\to\mathbb{C}$ is a smooth function, then the composites

$f_{x,v}(t):=f(x+tv),\quad t\in[-1,1],$

are well-defined and smooth for all $x\in U$ and $v\in\mathbb{S}$. For
notational convenience let $\Lambda_{U}$ be the collection of all line
segments $\lambda(t)=x+tv$, $t\in[-1,1]$, where $x\in U$ and $v\in\mathbb{S}$,
and $f_{\lambda}(t):=f_{x,v}(t)=f(x+tv)$.

The following theorem shows that ultradifferentiable bounds can be encoded by
specific uniform unidirectional holomorphic approximation. It is closely
related to [11, Theorem 5.3] (and [17, Proposition 3.3.2]), but special
attention has to be paid to the uniformity (in $\lambda$) of all the
estimates.

######  Theorem 2.3.

Let $U\subseteq\mathbb{R}^{d}$ be open and $f:V_{U}\to\mathbb{C}$ a smooth
function.

i Let $M^{(i)}$, $1\leq i\leq 3$, be weight sequences with
$(m^{(i)}_{k})^{1/k}\rightarrow\infty$ and

$\displaystyle\;\exists B_{1}\geq 1\;\forall
t>0:\overline{\Gamma}_{m^{(2)}}(B_{1}t)\leq\underline{\Gamma}_{m^{(1)}}(t),$
(6) $\displaystyle\;\exists B_{2}\geq 1\;\forall
j\in\mathbb{N}:m^{(2)}_{j+1}\leq B_{2}^{j+1}m^{(3)}_{j}.$ (7)

If $f\in\mathcal{B}^{M^{(1)}}_{B_{0}}(V_{U})$, then there exist constants
$K\geq 1$, $c_{1},c_{2}>0$ and $0<\varepsilon_{0}\leq\frac{1}{2}$ and
functions $f_{\lambda,\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that for all $0<\varepsilon\leq
2\varepsilon_{0}$ and all $\lambda\in\Lambda_{U}$,

$\|f_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon}}\leq
K,\quad\|f_{\lambda}-f_{\lambda,\varepsilon}\|_{[-1,1]}\leq
c_{1}h_{m^{(3)}}(c_{2}\varepsilon).$ (8)

The constants $K,c_{1},c_{2},\varepsilon_{0}$ are independent of $\lambda$ and
$\varepsilon$, in particular, $c_{2}=CB_{0}B_{1}$, where $C$ is an absolute
constant.

ii Let $N^{(i)}$, $1\leq i\leq 3$, be weight sequences with
$(n^{(i)}_{k})^{1/k}\rightarrow\infty$ and
$\operatorname{mg}(N^{(i)},N^{(i+1)})=D^{(i)}<\infty$. Assume that there exist
constants $K\geq 1$, $c_{1},c_{2}>0$ and $0<\varepsilon_{0}\leq\frac{1}{2}$
and functions $f_{\lambda,\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that for all $0<\varepsilon\leq
2\varepsilon_{0}$ and all $\lambda\in\Lambda_{U}$,

$\|f_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon}}\leq
K,\quad\|f_{\lambda}-f_{\lambda,\varepsilon}\|_{[-1,1]}\leq
c_{1}h_{n^{(1)}}(c_{2}\varepsilon),$ (9)

where the constants $K,c_{1},c_{2},\varepsilon_{0}$ are independent of
$\lambda$ and $\varepsilon$. Then
$f|_{U}\in\mathcal{B}^{N^{(3)}}_{\sigma}(U)$, where
$\sigma:=\frac{4e^{2}D^{(1)}D^{(2)}c_{2}}{a}$ and $a>0$ is an absolute
constant.

###### Proof.

(i) The assumption $f\in\mathcal{B}^{M^{(1)}}_{B_{0}}(V_{U})$ implies

$\sup_{\lambda\in\Lambda_{U}}\|f_{\lambda}^{(k)}\|_{[-1,1]}\leq
C_{0}B_{0}^{k}M^{(1)}_{k},\quad k\in\mathbb{N}.$

Then there are, by [3, Proposition 3.12] (thanks to (6) and (7)), constants
$c_{1},c_{2}>0$ and a function $F_{\lambda}\in C^{1}_{c}(\mathbb{C})$
extending $f_{\lambda}$ such that

$|\overline{\partial}F_{\lambda}(z)|\leq
c_{1}h_{m^{(3)}}(c_{2}d(z,[-1,1])),\quad z\in\mathbb{C}.$ (10)

Note that the constants $c_{1}=c_{1}(C_{0},B_{0},B_{1},B_{2})$ and
$c_{2}=12B_{0}B_{1}$ are independent of $\lambda$. By multiplication with a
suitable cut-off function, we may assume that the support of $F_{\lambda}$ is
contained in the disk $D$ centered at $0$ with radius $2$. Thus,

$F_{\lambda}(z)=\frac{1}{2\pi
i}\int_{D}\frac{\overline{\partial}F_{\lambda}(\zeta)}{\zeta-z}\,d\zeta\wedge
d\overline{\zeta},\quad z\in\mathbb{C},$

and hence, in view of (10), $\|F_{\lambda}\|_{\mathbb{C}}$ is uniformly
bounded in $\lambda$.

The function
$w_{\lambda,\varepsilon}:=\overline{\partial}F_{\lambda}\,\mathbf{1}_{\Omega_{\varepsilon}}$
satisfies

$\|w_{\lambda,\varepsilon}\|_{\mathbb{C}}\leq
c_{1}h_{m^{(3)}}(Cc_{2}\varepsilon),$

where $C>0$ is an absolute constant such that $d(z,[-1,1])\leq C\varepsilon$
for $z\in\Omega_{\varepsilon}$. Then the bounded continuous function
$v_{\lambda,\varepsilon}=\mathcal{K}*w_{\lambda,\varepsilon}$, where
$\mathcal{K}$ is the Cauchy kernel in $\mathbb{C}$, satisfies
$\overline{\partial}v_{\lambda,\varepsilon}=w_{\lambda,\varepsilon}$ in the
distributional sense in $\mathbb{C}$ and
$\|v_{\lambda,\varepsilon}\|_{\mathbb{C}}\leq
C\,\|w_{\lambda,\varepsilon}\|_{\mathbb{C}}$, where $C>0$ again is an absolute
constant. So $f_{\lambda,\varepsilon}:=F_{\lambda}-v_{\lambda,\varepsilon}$ is
holomorphic on $\Omega_{\varepsilon}$, continuous on
$\overline{\Omega}_{\varepsilon}$, and fulfills (8).

(ii) For $\lambda\in\Lambda_{U}$ and $0<\varepsilon\leq\varepsilon_{0}$
consider
$g_{\lambda,\varepsilon}:=f_{\lambda,\varepsilon}-f_{\lambda,2\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ which satisfy
$\|g_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon}}\leq 2K$ and
$\|g_{\lambda,\varepsilon}\|_{[-1,1]}\leq
2c_{1}h_{n^{(1)}}(2c_{2}\varepsilon)$, by (9). Then Lemma 2.2 implies, with
$c_{3}:=2\max\\{c_{1},K\\}$,

$\|g_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon/2}}\leq
c_{3}\,h_{n^{(2)}}(2eD^{(1)}c_{2}\varepsilon),$

for all $\lambda\in\Lambda_{U}$ and $0<\varepsilon\leq\varepsilon_{0}$. There
exists a (universal) constant $a>0$ such that the closed disk with radius
$a\varepsilon$ around any $t\in[-\frac{1}{2},\frac{1}{2}]$ is contained in
$\Omega_{\varepsilon/2}$. The Cauchy estimates and (4) yield

$\displaystyle\|g_{\lambda,\varepsilon}^{(j)}\|_{[-\frac{1}{2},\frac{1}{2}]}$
$\displaystyle\leq\frac{c_{3}\,j!\,h_{n^{(2)}}(2eD^{(1)}c_{2}\varepsilon)}{(a\varepsilon)^{j}}$
$\displaystyle\leq
c_{3}\Big{(}\frac{2eD^{(1)}D^{(2)}c_{2}}{a}\Big{)}^{j}N_{j}^{(3)}\,h_{n^{(3)}}(2eD^{(1)}D^{(2)}c_{2}\varepsilon),$

which means that

$\|g_{\lambda,\varepsilon}\|^{N^{(3)}}_{[-\frac{1}{2},\frac{1}{2}],\rho}\leq
c_{3}\,h_{n^{(3)}}(2eD^{(1)}D^{(2)}c_{2}\varepsilon)$

for all $\lambda\in\Lambda_{U}$ and $0<\varepsilon\leq\varepsilon_{0}$, where
$\rho:=\frac{2eD^{(1)}D^{(2)}c_{2}}{a}$. In an analogous way the first bound
in (9) gives

$\displaystyle\|f_{\lambda,\varepsilon_{0}}^{(j)}\|_{[-\frac{1}{2},\frac{1}{2}]}$
$\displaystyle\leq\frac{K}{(a\varepsilon_{0})^{j}}j!\leq
AK\,\rho^{j}N^{(3)}_{j}.$

Indeed, the assumption $(n^{(3)}_{k})^{1/k}\rightarrow\infty$ entails that for
all $\tau>0$ there exists $A$ such that $j!\leq A\,\tau^{j}N^{(3)}_{j}$ for
all $j$; so take $\tau:=2eD^{(1)}D^{(2)}c_{2}\varepsilon_{0}$. It follows that

$g_{\lambda}:=f_{\lambda,\varepsilon_{0}}+\sum_{j=1}^{\infty}g_{\lambda,\varepsilon_{0}2^{-j}}=f_{\lambda,\varepsilon_{0}}+\sum_{j=1}^{\infty}(f_{\lambda,\varepsilon_{0}2^{-j}}-f_{\lambda,\varepsilon_{0}2^{-j+1}})$

converges absolutely in the Banach space
$\mathcal{B}^{N^{(3)}}_{\rho}([-\frac{1}{2},\frac{1}{2}])$ for each
$\lambda\in\Lambda_{U}$, and

$\sup_{\lambda\in\Lambda_{U}}\|g_{\lambda}\|^{N^{(3)}}_{[-\frac{1}{2},\frac{1}{2}],\rho}<\infty.$
(11)

We have $f_{\lambda}=g_{\lambda}$ on $[-\frac{1}{2},\frac{1}{2}]$, since
$g_{\lambda}=f_{\lambda,\varepsilon_{0}2^{-k}}+\sum_{j=k+1}^{\infty}(f_{\lambda,\varepsilon_{0}2^{-j}}-f_{\lambda,\varepsilon_{0}2^{-j+1}})$
for every $k\in\mathbb{N}$, and so, for $t\in[-\frac{1}{2},\frac{1}{2}]$,

$|f_{\lambda}(t)-g_{\lambda}(t)|\leq|f_{\lambda}(t)-f_{\lambda,\varepsilon_{0}2^{-k}}(t)|+\Big{|}\sum_{j=k+1}^{\infty}(f_{\lambda,\varepsilon_{0}2^{-j}}(t)-f_{\lambda,\varepsilon_{0}2^{-j+1}}(t))\Big{|}$

which tends to $0$ as $k\to\infty$, by (9) and absolute convergence of the
sum. Now if $\lambda(t)=x+tv$ then
$d^{k}_{v}f(x)=f^{(k)}_{\lambda}(0)=g^{(k)}_{\lambda}(0)$. Thus, (11) and the
polarization inequality (2), imply
$f|_{U}\in\mathcal{B}^{N^{(3)}}_{\sigma}(U)$ with $\sigma:=2e\rho$. ∎

###  2.3. Proof of Theorem 1.1

The next proposition extends [11, Lemma 6.1]; again uniformity in $\lambda$ is
crucial.

######  Proposition 2.4.

Let $U\subseteq\mathbb{R}^{d}$ be open and $f:V_{U}\to\mathbb{C}$ any
function. Let $j$ be a positive integer. Let $M^{(i)}$, $1\leq i\leq k$ with
$k:=\lceil\log_{2}(j(j+1))\rceil+7$, be weight sequences satisfying
$(m^{(i)}_{\ell})^{1/\ell}\rightarrow\infty$ and

$\displaystyle\;\exists B\geq 1\;\forall
t>0:\overline{\Gamma}_{m^{(2)}}(Bt)\leq\underline{\Gamma}_{m^{(1)}}(t),$
$\displaystyle\operatorname{mg}(M^{(i)},M^{(i+1)})<\infty,\quad\text{ for
}2\leq i\leq k-1.$

There is a constant $D>0$ such that
$f^{j},f^{j+1}\in\mathcal{B}^{M^{(1)}}_{\rho}(V_{U})$ implies
$f\in\mathcal{B}^{M^{(k)}}_{D\rho}(U)$. As a consequence
$f^{j},f^{j+1}\in\mathcal{B}^{[M^{(1)}]}(V_{U})$ implies
$f\in\mathcal{B}^{[M^{(k)}]}(U)$.

###### Proof.

By the classical theorem of Joris, we may conclude that $f$ is smooth.

Set $g:=f^{j}$ and $h:=f^{j+1}$ and consider $g_{\lambda}$ and $h_{\lambda}$
for $\lambda\in\Lambda_{U}$. (We may assume $j\geq 2$; otherwise the assertion
is trivial.) By Theorem 2.3(i), there exist constants $K\geq 1$,
$c_{1},c_{2}>0$ and $0<\varepsilon_{0}\leq\frac{1}{2}$ and functions
$g_{\lambda,\varepsilon},h_{\lambda,\varepsilon}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ such that for all $0<\varepsilon\leq
2\varepsilon_{0}$ and all $\lambda\in\Lambda_{U}$,

$\displaystyle\max\\{\|g_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon}},\|h_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon}}\\}\leq
K,$ (12)
$\displaystyle\max\\{\|g_{\lambda}-g_{\lambda,\varepsilon}\|_{[-1,1]},\|h_{\lambda}-h_{\lambda,\varepsilon}\|_{[-1,1]}\\}\leq
c_{1}h_{m^{(3)}}(c_{2}\varepsilon).$ (13)

Using $|a^{\ell}-b^{\ell}|\leq\ell\max\\{|a|,|b|\\}^{\ell-1}|a-b|$, we find
that
$g_{\lambda,\varepsilon}^{j+1}-h_{\lambda,\varepsilon}^{j}\in\mathcal{H}(\Omega_{\varepsilon})\cap
C^{0}(\overline{\Omega}_{\varepsilon})$ satisfies

$\displaystyle|g_{\lambda,\varepsilon}^{j+1}-h_{\lambda,\varepsilon}^{j}|$
$\displaystyle\leq|g_{\lambda,\varepsilon}^{j+1}-g_{\lambda}^{j+1}|+|h_{\lambda}^{j}-h_{\lambda,\varepsilon}^{j}|\leq
c_{3}h_{m^{(3)}}(c_{2}\varepsilon),\quad\text{ on }[-1,1].$

The positive constant $c_{3}$ is again independent of $\lambda$ and
$\varepsilon$ as are all further constants $c_{i}$ appearing below. Thus Lemma
2.2 implies

$\|h_{\lambda,\varepsilon}^{j}-g_{\lambda,\varepsilon}^{j+1}\|_{\Omega_{\varepsilon/2}}\leq
c_{4}h_{m^{(4)}}(Cec_{2}\varepsilon)=:\delta_{\varepsilon},$ (14)

where $C$ fulfills $C\geq\operatorname{mg}(M^{(i)},M^{(i+1)})$ for all $2\leq
i\leq k-1$. Set $r_{\varepsilon}:=\delta_{\varepsilon}^{\frac{1}{j+1}}$. We
may assume that $\varepsilon_{0}>0$ is chosen such that
$\delta_{\varepsilon}\leq r_{\varepsilon}\leq\frac{1}{2}$ for all
$0<\varepsilon\leq 2\varepsilon_{0}$.

Consider the continuous function

$u_{\lambda,\varepsilon}:=\varphi_{\varepsilon}\frac{\overline{g}_{\lambda,\varepsilon}h_{\lambda,\varepsilon}}{\max\\{|g_{\lambda,\varepsilon}|,r_{\varepsilon}\\}^{2}},$

where $\varphi_{\varepsilon}$ is a smooth function compactly supported in
$\Omega_{\varepsilon}$ and $1$ on $\Omega_{\varepsilon/2}$. Note that
$u_{\lambda,\varepsilon}=h_{\lambda,\varepsilon}/g_{\lambda,\varepsilon}$ in
$\Omega_{\varepsilon/2}\cap\\{|g_{\lambda,\varepsilon}|>r_{\varepsilon}\\}$.

The uniformity of (12), (13), and (14) allows us to apply Lemmas 4.2.1 to
4.2.4 in [17] and conclude the following: for all $\lambda\in\Lambda_{U}$ and
$0<\varepsilon\leq 2\varepsilon_{0}$,

$\displaystyle\|u_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon/2}}$
$\displaystyle\leq(2K)^{1/j},$ (15)
$\displaystyle\|f_{\lambda}-u_{\lambda,\varepsilon}\|_{[-1,1]}$
$\displaystyle\leq c_{5}r_{\varepsilon}^{1/j},$ (16)

where $c_{5}=c_{5}(K,j)$, cf. [17, Lemma 4.2.2]. The bounded continuous
function
$v_{\lambda,\varepsilon}=\mathcal{K}*(\overline{\partial}u_{\lambda,\varepsilon}\mathbf{1}_{\Omega_{\varepsilon/2}})$
satisfies
$\overline{\partial}v_{\lambda,\varepsilon}=\overline{\partial}u_{\lambda,\varepsilon}\mathbf{1}_{\Omega_{\varepsilon/2}}$
in the distributional sense in $\mathbb{C}$ and

$\|v_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon/2}}\leq
c_{6}\delta_{\varepsilon}^{1/s}$ (17)

where $s$ is any real number with $s>j(j+1)$ and $c_{6}=c_{6}(K,j,s)$, cf.
[17, Lemma 4.2.4] and also [17, Lemma 3.1.1].

Then
$f_{\lambda,\varepsilon}:=u_{\lambda,2\varepsilon}-v_{\lambda,2\varepsilon}$
is holomorphic in $\Omega_{\varepsilon}$ and continuous on $\mathbb{C}$. By
(15), (16), and (17), $\|f_{\lambda,\varepsilon}\|_{\Omega_{\varepsilon}}$ is
uniformly bounded and

$\|f_{\lambda}-f_{\lambda,\varepsilon}\|_{[-1,1]}\leq
c_{7}\delta_{2\varepsilon}^{1/s}=c_{7}\big{(}c_{4}h_{m^{(4)}}(2eCc_{2}\varepsilon)\big{)}^{1/s},$

for all $\lambda\in\Lambda_{U}$ and $0<\varepsilon\leq\varepsilon_{0}$.
Putting $s:=2^{k-6}=:2^{\ell}$ and applying (5) repeatedly, we find

$\displaystyle\|f_{\lambda}-f_{\lambda,\varepsilon}\|_{[-1,1]}\leq
c_{7}c_{4}^{1/s}h_{m^{(k-2)}}(2c_{2}(eC)^{\ell+1}\varepsilon).$

So, by Theorem 2.3(ii) (paying attention to the dependence of the constants),
we may conclude that there is a uniform constant $D>0$ such that
$f\in\mathcal{B}^{M^{(k)}}_{D\rho}(U)$ as claimed. ∎

###### Proof of Theorem 1.1.

It suffices to show that $f$ is locally of class
$\mathcal{E}^{[\mathfrak{M}]}$. Up to an affine transformation, we may assume
that $g:=f^{j}$ and $h:=f^{j+1}$ belong to
$\mathcal{B}^{[\mathfrak{M}]}(V_{U})$ and we will show that
$f\in\mathcal{B}^{[\mathfrak{M}]}(U)$, where $U$ is the open unit ball in
$\mathbb{R}^{d}$.

In the Roumieu case there exists $M^{(1)}\in\mathfrak{M}$ such that $g$,
$h\in\mathcal{B}^{\\{M^{(1)}\\}}(V_{U})$. By R-admissibility of
$\mathfrak{M}$, we find sequences $M^{(i)}\in\mathfrak{M}$ satisfying the
assumptions of Proposition 2.4 which implies that
$f\in\mathcal{B}^{\\{M^{(k)}\\}}(U)$.

In the Beurling case we fix any $M\in\mathfrak{M}$ and show that
$f\in\mathcal{B}^{(M)}(U)$. Now B-admissibility of $\mathfrak{M}$ provides
sequences $M^{(i)}\in\mathfrak{M}$ as required in Proposition 2.4 with
$M^{(k)}=M$. As $g$, $h\in\mathcal{B}^{(M^{(1)})}(V_{U})$, Proposition 2.4
gives $f\in\mathcal{B}^{(M)}(U)$. ∎

######  Remark 2.5.

Let $j$ be some positive integer. The map
$\varphi:\mathbb{C}^{U}\to\mathbb{C}^{U}\times\mathbb{C}^{U}$,
$f\mapsto(f^{j},f^{j+1})$ is injective. So there exists an inverse
$\psi:=\varphi^{-1}|_{\varphi(\mathbb{C}^{U})}$. Under the assumptions of
Theorem 1.1, the map $\psi$ takes bounded sets in
$\mathcal{E}^{[\mathfrak{M}]}(U)\times\mathcal{E}^{[\mathfrak{M}]}(U)$ to
bounded sets in $\mathcal{E}^{[\mathfrak{M}]}(U)$, if
$\mathcal{E}^{[\mathfrak{M}]}(U)$ is endowed with its natural locally convex
topology; cf. [12, Sec. 4.2].

##  3\. On Banach spaces and on convenient vector spaces

###  3.1. Ultradifferentiable classes of functions on Banach spaces

Let $E$ be a Banach space and $U\subseteq E$ an open subset. Let
$\mathfrak{M}$ be a weight matrix. For $M\in\mathfrak{M}$ and $\rho>0$ we
consider $\mathcal{B}^{M}_{\rho}(U)=\\{f\in
C^{\infty}(U,\mathbb{C}):\|f\|^{M}_{U,\rho}<\infty\\}$ as well as

$\displaystyle\mathcal{B}^{\\{\mathfrak{M}\\}}(U)$ $\displaystyle:=\\{f\in
C^{\infty}(U,\mathbb{C}):\;\exists
M\in\mathfrak{M}\;\exists\rho>0:\|f\|^{M}_{U,\rho}<\infty\\}\quad\text{ and}$
$\displaystyle\mathcal{B}^{(\mathfrak{M})}(U)$ $\displaystyle:=\\{f\in
C^{\infty}(U,\mathbb{C}):\;\forall
M\in\mathfrak{M}\;\forall\rho>0:\|f\|^{M}_{U,\rho}<\infty\\},$

where $\|f\|^{M}_{U,\rho}$ is defined by (1) (with $\mathbb{R}^{d}$ replaced
by $E$). Furthermore, we consider the local classes

$\displaystyle\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$ $\displaystyle:=\\{f\in
C^{\infty}(U,\mathbb{C}):\;\forall K\subseteq_{cp}U\;\exists
M\in\mathfrak{M}\;\exists\rho>0:\|f\|^{M}_{K,\rho}<\infty\\}\quad\text{ and}$
$\displaystyle\mathcal{E}^{(\mathfrak{M})}(U)$ $\displaystyle:=\\{f\in
C^{\infty}(U,\mathbb{C}):\;\forall K\subseteq_{cp}U\;\forall
M\in\mathfrak{M}\;\forall\rho>0:\|f\|^{M}_{K,\rho}<\infty\\}.$

Let $B(a,r):=\\{x\in E:\|x-a\|<r\\}$ be the open ball with center $a$ and
radius $r$.

######  Lemma 3.1.

Let $f:U\to\mathbb{C}$ be smooth. Then:

  1. (1)

$f\in\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$ if and only if

$\;\forall a\in U\;\exists M\in\mathfrak{M}\;\exists\rho>0\;\exists
r>0:f|_{B(a,r)}\in\mathcal{B}^{M}_{\rho}(B(a,r)).$ (18)

  2. (2)

$f\in\mathcal{E}^{(\mathfrak{M})}(U)$ if and only if

$\;\forall a\in U\;\forall M\in\mathfrak{M}\;\forall\rho>0\;\exists
r>0:f|_{B(a,r)}\in\mathcal{B}^{M}_{\rho}(B(a,r)).$ (19)

###### Proof.

(1) That (18) is sufficient for $f\in\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$ is
easily seen. Fix a compact subset $K\subseteq U$. For each $a\in K$ we get
$M_{a}\in\mathfrak{M}$, $\rho_{a}>0$, and a small ball $B(a,r_{a})$ such that
$f|_{B(a,r_{a})}\in\mathcal{B}^{M_{a}}_{\rho_{a}}(B(a,r_{a}))$. Now $K$ is
still covered by finitely many of the balls and we find
$\|f\|^{M}_{K,\rho}<\infty$ with $M:=\max M_{a}$ and $\rho:=\max\rho_{a}$,
where the maxima are taken over the corresponding $a$.

Let us show that $f\in\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$ implies (18). If
(18) does not hold, then there exists $a\in U$ such that

$\displaystyle\;\forall M\in\mathfrak{M}\;\forall\rho>0\;\forall r>0\;\forall
C>0\;\exists x\in B(a,r)\;\exists
k\in\mathbb{N}:\|f^{(k)}(x)\|_{L^{k}(E,\mathbb{C})}\geq C\rho^{k}M_{k}.$

By [3, Lemma 2.5], we may assume that $\mathfrak{M}$ is indexed by
$\mathbb{N}$ in the following way:
$\mathfrak{M}=\\{M^{(\ell)}:\ell\in\mathbb{N}\\}$ and $M^{(\ell_{1})}\leq
M^{(\ell_{2})}$ if and only if $\ell_{1}\leq\ell_{2}$. In particular, taking
$C=\rho=n$, $r=\tfrac{1}{n}$ and $M=M^{(n)}$, we find

$\displaystyle\;\forall n\in\mathbb{N}\;\exists x_{n}\text{ with
}\|x_{n}-a\|<\tfrac{1}{n}\;\exists
k_{n}\in\mathbb{N}:\|f^{(k_{n})}(x_{n})\|_{L^{k_{n}}(E,\mathbb{C})}\geq
n^{k_{n}+1}M^{(n)}_{k_{n}}.$

For the compact set $K:=\\{x_{n}\\}_{n\in\mathbb{N}}\cup\\{a\\}$ this means
that $\|f\|^{M}_{K,\rho}=\infty$ for all choices of $M\in\mathfrak{M}$ and
$\rho>0$, contradicting $f\in\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$.

(2) That (19) is sufficient for $f\in\mathcal{E}^{(\mathfrak{M})}(U)$ is again
easy to check. For the necessity assume that (19) does not hold. Then there
exist $a\in U$, $M\in\mathfrak{M}$, and $\rho>0$ such that

$\displaystyle\;\forall r>0\;\forall C>0\;\exists x\in B(a,r)\;\exists
k\in\mathbb{N}:\|f^{(k)}(x)\|_{L^{k}(E,\mathbb{C})}\geq C\rho^{k}M_{k}.$

Taking $C=n$ and $r=\tfrac{1}{n}$ we get

$\displaystyle\;\forall n\in\mathbb{N}\;\exists x_{n}\text{ with
}\|x_{n}-a\|<\tfrac{1}{n}\;\exists
k_{n}\in\mathbb{N}:\|f^{(k_{n})}(x_{n})\|_{L^{k_{n}}(E,\mathbb{C})}\geq
n\,\rho^{k_{n}}M_{k_{n}}.$

Thus $\|f\|^{M}_{K,\rho}=\infty$ for
$K:=\\{x_{n}\\}_{n\in\mathbb{N}}\cup\\{a\\}$, contradicting
$f\in\mathcal{E}^{(\mathfrak{M})}(U)$. ∎

###  3.2. Theorem 1.1 holds on Banach spaces

Let $E$ be a Banach space and $U\subseteq E$ an open subset. Let
$\mathbb{B}:=\\{x\in E:\|x\|\leq 1\\}$ be the closed unit ball in $E$ and
$\mathbb{S}:=\\{x\in E:\|x\|=1\\}$ the unit sphere. Consider
$V_{U}:=\bigcup_{x\in U}(x+\mathbb{B})$.

Then Theorem 2.3 and Proposition 2.4 remain valid for functions
$f:V_{U}\to\mathbb{C}$ (with literally the same proofs). Note that the
classical Joris theorem is valid on Banach spaces by Boman’s theorem.

######  Theorem 3.2.

Theorem 1.1 (and thus Corollary 1.2) hold if $U$ is an open subset of a Banach
space $E$.

###### Proof.

We treat Roumieu and Beurling case separately.

_Roumieu case._ Assume that
$f^{j},f^{j+1}\in\mathcal{E}^{\\{\mathfrak{M}\\}}(U)$. Fix $a\in U$; without
loss of generality we may assume that $a$ is the origin in $E$. By Lemma 3.1,
there exist $M\in\mathfrak{M}$, $\rho>0$, and $r>0$ such that
$g:=f^{j}|_{B(0,r)},h:=f^{j+1}|_{B(0,r)}\in\mathcal{B}^{M}_{\rho}(B(0,r))$.
Then $\tilde{g}(x):=g(\frac{r}{2}x)$ and $\tilde{h}(x):=h(\frac{r}{2}x)$ are
elements of $\mathcal{B}^{M}_{\frac{r}{2}\rho}(B(0,2))$; note that
$B(0,2)=V_{B(0,1)}$. By R-admissibility of $\mathfrak{M}$, we find sequences
$M^{(i)}\in\mathfrak{M}$ with $M^{(1)}=M$ satisfying the assumptions of
Proposition 2.4 which implies that the restriction of
$\tilde{f}(x):=f(\frac{r}{2}x)$ to $B(0,1)$ belongs to
$\mathcal{B}^{M^{(k)}}_{D\frac{r}{2}\rho}(B(0,1))$. Then
$f|_{B(0,\frac{r}{2})}\in\mathcal{B}^{M^{(k)}}_{D\rho}(B(0,\frac{r}{2}))$
which ends the proof in view of Lemma 3.1.

_Beurling case._ Assume that
$f^{j},f^{j+1}\in\mathcal{E}^{(\mathfrak{M})}(U)$. Fix $a\in U$,
$M\in\mathfrak{M}$, and $\rho>0$; we may again assume that $a$ is the origin.
By B-admissibility of $\mathfrak{M}$, there are sequences
$M^{(i)}\in\mathfrak{M}$ as required in Proposition 2.4 with $M^{(k)}=M$. Let
$D>0$ be the constant from Proposition 2.4. By Lemma 3.1, there exists $r>0$
such that $g,h\in\mathcal{B}^{M^{(1)}}_{\frac{\rho}{D}}(B(0,r))$. Then
$\tilde{g},\tilde{h}\in\mathcal{B}^{M^{(1)}}_{\frac{r\rho}{2D}}(B(0,2))$
(using the notation from the previous paragraph) and Proposition 2.4 implies
$\tilde{f}\in\mathcal{B}^{M^{(k)}}_{\frac{r\rho}{2}}(B(0,1))$ and,
consequently,
$f|_{B(0,\frac{r}{2})}\in\mathcal{B}^{M}_{\rho}(B(0,\frac{r}{2}))$. Invoking
Lemma 3.1 we are done. ∎

###  3.3. Theorem 1.1 holds on convenient vector spaces

In [8, 9, 10] and [16] the calculus of $\mathcal{E}^{[\mathfrak{M}]}$-maps has
been extended to maps between _convenient vector spaces_ , i.e., locally
convex spaces that are Mackey complete. We refer the reader to [7] for a
comprehensive treatment of convenient vector spaces and the
$c^{\infty}$-topology on them.

Let $E$ be a convenient vector space and $U\subseteq E$ a $c^{\infty}$-open
subset. Let $\mathfrak{M}$ be a weight matrix. Then
$\mathcal{E}^{[\mathfrak{M}]}(U)$ is by definition the set of all smooth
functions $f:U\to\mathbb{C}$ (i.e. smooth on smooth curves in $U$) such that
for each closed absolutely convex bounded subset $B$ of $E$ we have

$f\circ i_{B}\in\mathcal{E}^{[\mathfrak{M}]}(U_{B}),$

where $i_{B}:E_{B}\to E$ is the inclusion of the linear span $E_{B}$ of $B$ in
$E$ and $U_{B}:=U\cap E_{B}$. Note that $E_{B}$ equipped with the Minkowski
functional $p_{B}(x)=\inf\\{t>0:x\in tB\\}$ is a Banach space, since $E$ is
convenient.

######  Theorem 3.3.

Theorem 1.1 (and thus Corollary 1.2) hold if $U$ is an $c^{\infty}$-open
subset of a convenient vector space $E$.

###### Proof.

This is obvious by the definition of $\mathcal{E}^{[\mathfrak{M}]}(U)$ and
Theorem 3.2. ∎

##  4\. More general nonlinear conditions

As seen above the map $\Phi(t)=(t^{j},t^{j+1})$, where $j$ is some positive
integer, has the property

$\Phi\circ f\in\mathcal{E}^{[\mathfrak{M}]}\implies
f\in\mathcal{E}^{[\mathfrak{M}]},$ (20)

provided that $\mathfrak{M}$ is an [admissible] weight matrix. Here $t$ is a
complex or a real variable. Equivalently, the map $\Phi(t)=(t^{p},t^{q})$,
where $p$ and $q$ are positive coprime integers, has property (20); indeed,
all integers $j\geq pq$ have the form $j=ap+bq$ with $a,b\in\mathbb{N}$.

The question arises which maps $\Phi$ have the property (20) for all
continuous $f$. From now on we take continuity of $f$ as a basic assumption.
(The continuity of $f$ was not an issue in Theorem 1.1, since one of the
powers is necessarily odd and hence has a global continuous inverse.) This
problem was investigated in [2] and in [5] in the $C^{\infty}$-setting. In the
latter article the smooth germs $\Phi:(\mathbb{R},0)\to(\mathbb{R}^{n},0)$
(and $\Phi:(\mathbb{C},0)\to(\mathbb{C}^{n},0)$ which possess a complex Taylor
series, cf. [5, p.204]) with the property that $\Phi\circ f\in C^{\infty}$
implies $f\in C^{\infty}$ for all continuous germs $f$ with $f(0)=0$ were
characterized in terms of a condition on the support of the formal Taylor
series of $f$. They were called _pseudo-immersive_ germs.

The proof is based (similar to most proofs in [2]) on a reduction to the case
$\Phi(t)=(t^{p},t^{q})$ with $\operatorname{gcd}(p,q)=1$ which is mostly
algebraic and hence applies to our situation with marginal adjustments. We
thus obtain a characterization of the _analytic_ germs $\Phi$ satisfying (20)
for all continuous germs $f$ with $f(0)=0$ by the same condition on the
support of the Taylor series. Note that [admissibility] of the weight matrix
$\mathfrak{M}$ implies that $\mathcal{E}^{[\mathfrak{M}]}$ contains all
analytic maps and is stable under composition (by Corollaries 3.3 and 3.5 and
Proposition 1.1 in [3]). See also Remark 4.4.

###  4.1. A necessary condition

Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$. Let
$\Phi:(\mathbb{K},0)\to(\mathbb{K}^{n},0)$ be the germ of a non-zero analytic
map. Then $\Phi$ is represented by its convergent Taylor series

$\Phi(t)=\sum_{1\leq k<N}a_{k}t^{n_{k}},\quad\text{ where
}a_{k}\in\mathbb{K}^{n}\setminus\\{0\\}\text{ for all }k,$

$1\leq n_{1}<n_{2}<\cdots$, and $1\leq N\leq\infty$. That means that
$\\{n_{1},n_{2},\ldots\\}$ is the _support_ of the power series $\Phi(t)$.

If (20) holds for all continuous germs $f$ with $f(0)=0$, then
$\operatorname{gcd}(n_{1},n_{2},\ldots)=1$. Indeed, if there are integers
$p\geq 2$ and $\ell_{k}$ such that $n_{k}=p\ell_{k}$ for all $k$, then the
power series $\sum_{k}a_{k}x^{\ell_{k}}$ is convergent, defines an analytic
germ $\Psi$ so that $\Phi(t)=\Psi(t^{p})$, and it is easy to find a continuous
germ $f\not\in C^{1}$ with $f(0)=0$ such that $\Phi\circ f$ is analytic; cf.
[5, Theorem 2].

###  4.2. A characterization

We will see that the condition $\operatorname{gcd}(n_{1},n_{2},\ldots)=1$ is
also sufficient for (20) at least up to equivalence of analytic germs. Two
analytic germs $\Phi,\Psi:(\mathbb{K},0)\to(\mathbb{K}^{n},0)$ are said to be
_equivalent_ if there exist germs of analytic diffeomorphisms
$u:(\mathbb{K}^{n},0)\to(\mathbb{K}^{n},0)$ and
$v:(\mathbb{K},0)\to(\mathbb{K},0)$ such that $u\circ\Phi\circ v=\Psi$.
Equivalent germs either both satisfy or do not satisfy (20). Any non-zero
analytic germ $\Phi:(\mathbb{K},0)\to(\mathbb{K}^{n},0)$ is equivalent to a
germ whose first component is a positive power. So it is no restriction to
assume $\Phi_{1}(t)=t^{p}$.

######  Theorem 4.1.

Let $\Phi=(\Phi_{1},\ldots,\Phi_{n}):(\mathbb{K},0)\to(\mathbb{K}^{n},0)$ be
an analytic germ such that $\Phi_{1}(t)=t^{p}$ and let
$\\{n_{1},n_{2},\ldots\\}$ be the support of the power series $\Phi(t)$. Then
the following conditions are equivalent:

  1. (1)

Let $\mathfrak{M}$ be an [admissible] weight matrix. If
$f:(\mathbb{R}^{d},0)\to(\mathbb{K},0)$ is a function germ and $\Phi\circ f$
is of class $\mathcal{E}^{[\mathfrak{M}]}$, then $f$ is of class
$\mathcal{E}^{[\mathfrak{M}]}$.

  2. (2)

Let $\omega$ be a concave weight function. If
$f:(\mathbb{R}^{d},0)\to(\mathbb{K},0)$ is a function germ and $\Phi\circ f$
is of class $\mathcal{E}^{[\omega]}$, then $f$ is of class
$\mathcal{E}^{[\omega]}$.

  3. (3)

$\operatorname{gcd}(n_{1},n_{2},\ldots)=1$.

###### Proof.

Since (2) is a special case of (1) (cf. [11, Section 4.5]) and the necessity
of (3) for (2) follows from the discussion above, it remains to show that (3)
implies (1). This is achieved by adjusting the arguments in the proofs of
Theorems 3 and 3’ in [5]; actually the proof simplifies a bit, since all
series involved in the arguments are convergent.

We may assume that $p\geq 2$ and $n=2$. In fact, if $p=1$ there is nothing to
prove and if $n>2$ then one can find real constants $\gamma_{i}$ such that the
support of the power series
$(t^{p},\gamma_{2}\Phi_{2}(t)+\cdots+\gamma_{n}\Phi_{n}(t))$ satisfies (3).

Thus we may assume that $\Phi(t)=(t^{p},\varphi(t))$, where
$\varphi(t)=\sum_{1\leq k<N}a_{k}t^{n_{k}}$ with
$a_{k}\in\mathbb{K}\setminus\\{0\\}$ for all $k$. Then (3) takes the form
$\operatorname{gcd}(p,n_{1},n_{2},\ldots)=1$. The idea is to show that there
exist a positive integer $q$ and analytic germs $\alpha_{j}$ at $0$ such that

$t^{1+pq}=\sum_{j=0}^{p-1}\alpha_{j}(t^{p})\varphi(t)^{j}.$ (21)

Then (1) follows from Theorem 1.1 (by plugging $f$ into (21)).

We will sketch how to establish (21) following the steps in [5]; the main
difference is that in [5] all involved series are formal (power or Laurent)
series so that at the end an application of Borel’s lemma and some handling of
the flat terms is required.

Let $z$ be an indeterminate. Recall that $\mathbb{K}\\{z\\}$ denotes the ring
of convergent power series in $z$ with coefficients in $\mathbb{K}$. The
convergent power series $\varphi(z)=\sum_{1\leq k<N}a_{k}z^{n_{k}}$ has the
form

$\varphi(z)=\varphi_{0}(z^{p})+z\varphi_{1}(z^{p})+\cdots+z^{p-1}\varphi_{p-1}(z^{p})$

where $\varphi_{j}(z^{p})\in\mathbb{K}\\{z^{p}\\}$ for all $j=0,\ldots,p-1$.
Introducing another independent indeterminate $x$, set

$S(x):=\varphi_{0}(z^{p})+x\varphi_{1}(z^{p})+\cdots+x^{p-1}\varphi_{p-1}(z^{p})\in\mathbb{K}\\{z^{p}\\}[x].$

Furthermore, define

$\displaystyle G_{kj}(x):=S(x)^{j}x^{kp},\quad
j=0,1,\ldots,p-1,~{}k=0,1,\ldots,(p-1)^{2}.$

Let $\mathbb{K}\\{\\!\\{z\\}\\!\\}$ denote the field of fractions of
$\mathbb{K}\\{z\\}$. It is easy to see that
$\mathbb{K}\\{\\!\\{z\\}\\!\\}=\\{z^{n}g(z):n\in\mathbb{Z},\,g(z)\in\mathbb{K}\\{z\\}\\}$.
We look for nontrivial $a_{kj}\in\mathbb{K}\\{\\!\\{z^{p}\\}\\!\\}$ such that

$\sum_{j=0}^{p-1}\sum_{k=0}^{(p-1)^{2}}a_{kj}G_{kj}(x)=xH(x^{p})$

with $H(x^{p})\in\mathbb{K}\\{\\!\\{z^{p}\\}\\!\\}[x^{p}]$. This amounts to an
underdetermined homogeneous system of linear equations over the field
$\mathbb{K}\\{\\!\\{z^{p}\\}\\!\\}$; see [5, p.201] for details. So we find
$a_{kj}\in\mathbb{K}\\{\\!\\{z^{p}\\}\\!\\}$ for $j=0,1,\ldots,p-1$,
$k=0,1,\ldots,r$, and some $r\geq 0$, not all zero, and
$H(x^{p})\in\mathbb{K}\\{\\!\\{z^{p}\\}\\!\\}[x^{p}]$ such that

$\sum_{j=0}^{p-1}\sum_{k=0}^{r}a_{kj}G_{kj}(x)=xH(x^{p}).$

Regrouping the terms on the left-hand side, we obtain
$A_{j}(x^{p})\in\mathbb{K}\\{\\!\\{z^{p}\\}\\!\\}[x^{p}]$, $j=0,1,\ldots,p-1$,
not all zero, such that

$\sum_{j=0}^{p-1}A_{j}(x^{p})S(x)^{j}=xH(x^{p}).$ (22)

Taking $r$ as small as possible, at least one of the series $A_{0}(z^{p})$, …
, $A_{p-1}(z^{p})$, which are obtained by replacing $x^{p}$ with $z^{p}$, is
non-zero and also $H(z^{p})$ is non-zero. Indeed, if $H(z^{p})=0$ then (22)
shows that $\varphi(z)=S(z)$ is algebraic over
$\mathbb{K}\\{\\!\\{z^{p}\\}\\!\\}\subseteq\mathbb{K}(\\!(z^{p})\\!)\subseteq\mathbb{C}(\\!(z^{p})\\!)$
of degree at most $p-1$; here $\mathbb{K}(\\!(z)\\!)$ is the field of formal
Laurent series in $z$ with coefficients in $\mathbb{K}$. The automorphism
$\sigma$ of $\mathbb{C}(\\!(z)\\!)$ induced by $z\mapsto\xi z$, where
$\xi=e^{2\pi i/p}$, leaves invariant $\mathbb{C}(\\!(z^{p})\\!)$ and generates
the cyclic group $\langle\sigma\rangle$. Thus $\varphi(z)$ (as an element of
$\mathbb{C}(\\!(z)\\!)$) is invariant under a non-trivial subgroup of
$\langle\sigma\rangle$, say $\langle\sigma^{b}\rangle$ with $1\leq b\leq p-1$
and $b|p$. Then

$\sum_{1\leq k<N}a_{k}z^{n_{k}}=\sum_{1\leq k<N}a_{k}\xi^{bn_{k}}z^{n_{k}}$

which implies $\xi^{bn_{k}}=1$ for all $k$, since all $a_{k}\neq 0$. But that
means that $p/b$ divides all $n_{k}$, contradicting
$\operatorname{gcd}(p,n_{1},n_{2},\ldots)=1$. Cf. [5, p.202].

Multiplying (22) by a suitable power of $z^{p}$, we may suppose that the
coefficients of $A_{j}$ and $H$ belong to $z^{p}\mathbb{K}\\{z^{p}\\}$.
Substituting $z$ for $x$, we find

$\sum_{j=0}^{p-1}A_{j}(z^{p})\varphi(z)^{j}=zH(z^{p})$

with $A_{j}(z^{p})$ and $H(z^{p})$ belonging to $z^{p}\mathbb{K}\\{z^{p}\\}$.
Since $H(z^{p})\neq 0$, there exist $q\in\mathbb{N}_{\geq 1}$,
$c\in\mathbb{K}\setminus\\{0\\}$, and $H_{1}(z^{p})\in\mathbb{K}\\{z^{p}\\}$
such that $H(z^{p})=z^{pq}(c+z^{p}H_{1}(z^{p}))$ from which it is easy to
conclude (21). ∎

######  Remark 4.2.

Let $\Phi$ be as in Theorem 4.1 with
$\operatorname{gcd}(n_{1},n_{2},\ldots)=1$ and assume that $\mathfrak{M}$ is
an [admissible] weight matrix. Then the map
$\mathcal{E}^{[\mathfrak{M}]}\ni\Phi\circ f\mapsto
f\in\mathcal{E}^{[\mathfrak{M}]}$ in (1) takes bounded sets to bounded sets.
This follows from Remark 2.5 and (21), since the superposition operator
$g\mapsto h\circ g$ is bounded, even continuous, among
$\mathcal{E}^{[\mathfrak{M}]}$ (in finite dimensions), provided that
$\mathcal{E}^{[\mathfrak{M}]}$ is stable under composition; see [12, Theorem
4.13].

######  Remark 4.3.

Let $g$ be a continuous germ at $0\in\mathbb{R}$ such that $g(0)=1$. Then
$f:=1/g$ and $h:=f-1$ are continuous germs and $h(0)=0$. Consider the analytic
germ $\varphi(t)=\frac{1}{1+t}-1=\sum_{k\geq 1}(-1)^{k}t^{k}$. Then
$\varphi\circ h=g-1$. Any analytic germ
$\Phi:(\mathbb{R},0)\to(\mathbb{R}^{n},0)$ that has $\varphi$ as a component
evidently satisfies the condition 4.1(3). Now if $g$ is of class
$\mathcal{E}^{[\mathfrak{M}]}$ and (20) holds, then $h$ is of class
$\mathcal{E}^{[\mathfrak{M}]}$ and consequently also $f=1/g$. That means that
(20) for analytic $\Phi$ implies that $\mathcal{E}^{[\mathfrak{M}]}$ is
_inverse-closed_.

In view of [3, Theorem 4.8] we infer that concavity of the weight function
$\omega$ (up to equivalence) is a necessary condition for the conclusions of
Corollary 1.2 and Theorem 4.1(2).

Next we want to discuss implications of the conclusions of Theorem 1.1 and
Theorem 4.1(1) and compare them with the assumption of admissibility for the
weight matrix $\mathfrak{M}$; our goal is to convince the reader that
admissibility is not only sufficient but also “close to” necessary. Suppose
that

$m_{k}^{1/k}\to\infty\quad\text{ for all }M\in\mathfrak{M};$ (23)

this guarantees that $\mathcal{E}^{[\mathfrak{M}]}$ contains the real analytic
class. Consider also the conditions

$\displaystyle\begin{split}\;\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{N}&:\sup_{k}\Big{(}\frac{M_{k+1}}{N_{k}}\Big{)}^{1/k}<\infty\qquad\text{
in the Roumieu case,}\\\ \;\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{N}&:\sup_{k}\Big{(}\frac{N_{k+1}}{M_{k}}\Big{)}^{1/k}<\infty\qquad\text{
in the Beurling case,}\end{split}$ (24)

which characterize stability under derivation of
$\mathcal{E}^{[\mathfrak{M}]}$.

Fact i:

Assuming (24), the conclusions of Theorem 1.1 and Theorem 4.1(1) imply that
$\mathcal{E}^{[\mathfrak{M}]}$ is stable under composition.

Indeed, we saw that (20) for analytic $\Phi$ entails that
$\mathcal{E}^{[\mathfrak{M}]}$ is inverse-closed which in turn implies (see
[12, Theorem 4.9 and 4.11] and [13])

$\displaystyle\begin{split}\;\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{N}&:\sup_{j\leq
k}\frac{m_{j}^{1/j}}{n_{k}^{1/k}}<\infty\qquad\text{ in the Roumieu case,}\\\
\;\forall M\in\mathfrak{M}\;\exists N\in\mathfrak{N}&:\sup_{j\leq
k}\frac{n_{j}^{1/j}}{m_{k}^{1/k}}<\infty\qquad\text{ in the Beurling
case.}\end{split}$ (25)

Now (25), in conjunction with (24), yields that $\mathcal{E}^{[\mathfrak{M}]}$
is stable under composition (_loc. cit._). Conversely, since
$\mathcal{E}^{[\mathfrak{M}]}$ contains the real analytic class (thanks to
(23)), stability under composition entails inverse-closedness and hence (25).
Note that (24) follows from the moderate growth part in the definition of an
[admissible] weight matrix.

Fact ii:

Assuming (23) and (24), the other part in the definition of admissibility,
i.e. the control of $\overline{\Gamma}_{n}$ by $\underline{\Gamma}_{m}$ (resp.
$\overline{\Gamma}_{m}$ by $\underline{\Gamma}_{n}$), also implies stability
under composition.

This follows from almost analytic extension (see [3, Remark 2.7]). But the
property “control of $\overline{\Gamma}_{n}$ by $\underline{\Gamma}_{m}$”
seems not to be _substantially_ stronger than stability under composition.
This is evidenced by Fact iii and Fact iv.

Fact iii:

For any weight function $\omega$, the class $\mathcal{E}^{[\omega]}$ being
stable under composition implies the existence of a weight matrix
$\mathfrak{M}$ with $\mathcal{E}^{[\omega]}=\mathcal{E}^{[\mathfrak{M}]}$ such
that $m$ is log-convex and thus $\overline{\Gamma}_{m}=\underline{\Gamma}_{m}$
for all $M\in\mathfrak{M}$.

See [3, Theorem 4.8] and [15, Proposition 3]); in this case (23) and moderate
growth (hence (24)) are automatic.

Fact iv:

If a general weight matrix $\mathfrak{M}$ fulfills (23), (25), and

$\displaystyle\begin{split}\;\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{N}&:\sup_{k}\frac{m_{k}}{m_{k-1}}n_{k}^{-1/k}<\infty\qquad\text{
in the Roumieu case,}\\\ \;\forall M\in\mathfrak{M}\;\exists
N\in\mathfrak{N}&:\sup_{k}\frac{n_{k}}{n_{k-1}}m_{k}^{-1/k}<\infty\qquad\text{
in the Beurling case,}\end{split}$ (26)

then $\mathfrak{M}$ admits the desired control of $\overline{\Gamma}_{n}$ by
$\underline{\Gamma}_{m}$.

Indeed, in the Roumieu case, for given $M\in\mathfrak{M}$ there exist
$L,N\in\mathfrak{M}$ and constants $C_{i}>0$ such that for all $j\leq k$,

$\frac{m_{j}}{m_{j-1}}\stackrel{{\scriptstyle\eqref{eq:qr}}}{{\leq}}C_{1}\ell_{j}^{1/j}\stackrel{{\scriptstyle\eqref{eq:almostincreasing}}}{{\leq}}C_{2}n_{k}^{1/k}\leq
C_{3}\frac{n_{k}}{n_{k-1}};$

the last inequality it true for every weight sequence. It follows that
$\overline{\Gamma}_{n}(C_{3}t)\leq\underline{\Gamma}_{m}(t)$ for all $t>0$
(see [14, Lemma 3.3]). The Beurling case is analogous.

######  Remark 4.4.

In principle, one can apply the above method also to the more general case
that $\Phi$ is the germ of a $\mathcal{E}^{[\mathfrak{M}]}$-mapping, but the
ramification $z^{p}$ causes a loss of regularity and, since the series are
formal, an ultradifferentiable Borel lemma must be used which introduces flat
terms whose handling again entails a loss of regularity.

## References

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# Pirashvili–Richter-type theorems for the reflexive and dihedral homology
theories

Daniel Graves  Lifelong Learning Centre, University of Leeds, Woodhouse,
Leeds, LS2 9JT, UK<EMAIL_ADDRESS>
###### Abstract.

Reflexive homology and dihedral homology are the homology theories associated
to the reflexive and dihedral crossed simplicial groups respectively. The
former has recently been shown to capture interesting information about
$C_{2}$-equivariant homotopy theory and its structure is related to the study
of “real” objects in algebraic topology. The latter has long been of interest
for its applications in $O(2)$-equivariant homotopy theory and connections to
Hermitian algebraic $K$-theory. In this paper, we show that the reflexive and
dihedral homology theories can be interpreted as functor homology over
categories of non-commutative sets, after the fashion of Pirashvili and
Richter’s result for the Hochschild and cyclic homology theories.

###### Key words and phrases:

reflexive homology, dihedral homology, functor homology, crossed simplicial
group, involutive non-commutative sets, involutive algebra

###### 1991 Mathematics Subject Classification:

18G15, 18G90, 16E40, 16E30

## Introduction

_Functor homology_ plays an important role in modern homological algebra. The
subject was pioneered by Connes ([Con83]) who described the cyclic homology
theory as derived functors over the cyclic category. This was extended to
incorporate dihedral structure by Loday ([Lod87]) and Krasauskas, Lapin and
Solov’ev ([KLS87]). Collectively, this work led to the notion of a crossed
simplicial group (see [FL91] and [Kra87]). Crossed simplicial groups extended
the notion of cyclic homology to detect other structure that is compatible
with a unital, associative multiplication, such as an order-reversing
involution.

A great many homology theories for algebras have interpretations in terms of
functor homology: Hochschild homology ([Lod98, 6.2.2]); cyclic homology
([Con83], [Lod98, 6.2.8]); dihedral homology ([Lod87], [KLS87]); reflexive
homology ([Gra23b]); symmetric homology ([Aul10], [Fie]); hyperoctahedral
homology ([Gra22], [Fie]); higher-order Hochschild homology ([Pir00]); gamma
homology ([PR00]); André-Quillen homology ([Pir03]); $E_{n}$-homology
([LR11]); and Leibniz homology ([HV15]).

The _reflexive crossed simplicial group_ and the _dihedral crossed simplicial
group_ are two of the _fundamental crossed simplicial groups_. The associated
homology theories, _reflexive homology_ and _dihedral homology_ , are defined
as functor homology over their respective indexing categories $\Delta R^{op}$
and $\Delta D^{op}$.

The dihedral homology theory has long been of interest for its applications to
$O(2)$-equivariant homotopy theory and Hermitian algebraic $K$-theory (see
[Lod87], [KLS87], [Dun89], [Lod90], [Lod93], [Cor93a], [Cor93b], [Lod96]). By
comparison, the study of the reflexive homology theory is a modern venture.
Recent work of the author has studied the applications of the reflexive
crossed simplicial group to $C_{2}$-equivariant homotopy theory and notes that
the structure of the reflexive crossed simplicial group is used in the study
of _real_ objects in algebraic topology ([Gra23b]).

From one point of view, reflexive homology and dihedral homology can be viewed
as extensions of Hochschild homology and cyclic homology respectively,
detecting the extra structure of an order-reversing involution.

The original functor homology interpretations of Hochschild homology and
cyclic homology were given in terms of $\Delta^{op}$ and $\Delta C^{op}$, the
categories indexing simplicial objects and cyclic objects respectively.
Pirashvili and Richter ([PR02]) gave an alternative functor homology
interpretation of Hochschild homology and cyclic homology. They showed that
Hochschild homology and cyclic homology can be expressed as functor homology
over the _category of non-commutative sets_ , $\mathcal{F}(as)$. The category
of non-commutative sets was introduced by Feĭgin and Tsygan ([FT87, A10]). It
is the category that encapsulates the structure of a monoid in a symmetric
monoidal category ([Pir02]) and it is isomorphic to $\Delta S$, the category
associated to the symmetric crossed simplicial group ([FL91], [Aul10], [Fie]).

The author introduced an extension of $\mathcal{F}(as)$, called the _category
of involutive non-commutative sets_ and denoted $\mathcal{IF}(as)$. This
category encapsulates the structure of an involutive monoid in a symmetric
monoidal category ([Gra20]) and is isomorphic to $\Delta H$, the category
associated to the hyperoctahedral crossed simplicial group ([FL91], [Gra20],
[Gra22], [Fie]).

In this paper we prove that, just as Hochschild and cyclic homology can be
interpreted as functor homology over the category of non-commutative sets
(equivalently, over $\Delta S$), reflexive homology and dihedral homology can
be expressed as functor homology over the category of involutive non-
commutative sets (equivalently, over $\Delta H$).

The paper is structured as follows.

In Sections 1 and 2 we recall the necessary background on functor homology,
crossed simplicial groups and their homology theories. In Section 3 we recall
the category of involutive non-commutative sets, $\mathcal{IF}(as)$, and the
subcategory of based involutive non-commutative sets, $\mathcal{I}\Gamma(as)$.
In Section 4, we recall the special cases of functor homology for involutive
algebras in terms of _Loday functors_ and _bar constructions_.

In Sections 5 and 6 we recall Pirashvili and Richter’s identification of the
Hochschild and cyclic homology theories as functor homology over the category
of non-commutative sets. We also recall the category-theoretic framework
introduced independently by Słomińska and Zimmermann, which allows for an
alternative proof of Pirashvili and Richter’s result. This will be the
framework we use to prove our results.

In Sections 7 and 8 we prove the technical results that we require for our
theorems.

In Section 9 we prove our first main theorem (Theorem 9.2) which, in the
specific case of an involutive algebra (Corollary 9.3), can be stated as
follows:

######  Theorem.

Let $A$ be an involutive, associative $k$-algebra. There exist isomorphisms of
graded $k$-modules

$HD_{\star}(A)=\mathrm{Tor}_{\star}^{\Delta
D^{op}}\left(k^{\ast},\mathcal{L}(A)\right)\cong\mathrm{Tor}_{\star}^{\mathcal{IF}(as)}\left(B,\mathsf{H}_{A}\right)\cong\mathrm{Tor}_{\star}^{\Delta
H}\left(B,\mathsf{H}_{A}\right)$

where $\mathcal{L}(A)$ is the dihedral Loday functor (also known as the
dihedral bar construction); $\mathsf{H}_{A}$ is the hyperoctahedral bar
construction; and the functor $B$ can be found in Definition 9.1.

In Section 10 we prove our second main theorem (Theorem 10.1) which, in the
specific case of an involutive algebra (Corollary 10.2), can be stated as
follows:

######  Theorem.

Let $A$ be an involutive, associative $k$-algebra and let $M$ be an involutive
$A$-bimodule. There is an isomorphism of graded $k$-modules

$HR_{\star}\left(A,M\right)=\mathrm{Tor}_{\star}^{\Delta
R^{op}}\left(k^{\ast},\mathcal{L}(A,M)\right)\cong\mathrm{Tor}_{\star}^{\mathcal{I}\Gamma(as)}\left(B^{\prime},\mathsf{R}(A,M)\right)$

where $\mathcal{L}(A,M)$ is the reflexive Loday functor; $\mathsf{R}(A,M)$ is
a restriction of the hyperoctahedral bar construction to the subcategory
$\mathcal{I}\Gamma(as)$; and $B^{\prime}$ is the restriction of the functor
$B$ to the subcategory $\mathcal{I}\Gamma(as)$.

### Acknowledgements

I would like to thank Jake Saunders, Ieke Moerdijk and Sarah Whitehouse for
interesting and helpful conversations whilst writing this paper. I would like
to thank Andrew Fisher, Jack Davidson and James Brotherston for their thoughts
and comments on an earlier version of this document.

### Conventions

Throughout the paper, we will let $k$ be a unital, commutative ring,
$\mathbf{Mod}_{k}$ be the category of $k$-modules and $\otimes$ denote the
tensor product of $k$-modules. All $k$-algebras in this paper will be assumed
to be unital. For $n\geqslant 0$, we will let $[n]$ denote the set
$\left\\{0,1\dotsc,n\right\\}$.

##  1\. Functor homology

We begin by recalling the categories of left and right modules over a small
category. We also recall the tensor product of $\mathbf{C}$-modules and its
left-derived functors (see [Mac70, Section 3], [PR02, Section 1.6] for
instance).

######  Definition 1.1.

For a small category $\mathbf{C}$ we define
$\mathbf{CMod}=\mathrm{Fun}\left(\mathbf{C},\mathbf{Mod}_{k}\right)$ and
$\mathbf{ModC}=\mathrm{Fun}\left(\mathbf{C}^{op},\mathbf{Mod}_{k}\right)$.

######  Definition 1.2.

For a small category $\mathbf{C}$, we define the $k$-constant functor,
$k^{\ast}$ in $\mathbf{CMod}$ to be the functor that sends every object of
$\mathbf{C}$ to $k$ and every morphism in $\mathbf{C}$ to the identity map on
$k$.

######  Definition 1.3.

There is a tensor product

$-\otimes_{\mathbf{C}}-\colon\mathbf{ModC}\times\mathbf{CMod}\rightarrow\mathbf{Mod}_{k}$

defined as the coend

$G\otimes_{\mathbf{C}}F=\int^{C\in\mathrm{Ob}(\mathbf{C})}G(C)\otimes F(C).$

The left derived functors of the tensor product $-\otimes_{\mathbf{C}}-$ are
denoted by $\mathrm{Tor}_{\star}^{\mathbf{C}}\left(-,-\right)$.

##  2\. Crossed simplicial groups

We begin by recalling some notation for certain groups that we will use
throughout the paper. We will recall the notion of crossed simplicial group
together with examples. We will recall the functor homology definitions of
reflexive homology, dihedral homology and hyperoctahedral homology.

###  2.1. A gathering of groups

######  Definition 2.1.

We define the groups that we will use throughout the paper. Let $n\geqslant
0$.

  * •

The _reflexive group_ is defined by

$R_{n+1}=\left\langle r_{n+1}\mid r_{n+1}^{2}=1\right\rangle.$

The generator $r_{n+1}$ acts on $[n]$ by $r_{n+1}(i)=n-i$.

  * •

Let

$C_{n+1}=\left\langle t_{n+1}\mid t_{n+1}^{n+1}=1\right\rangle$

denote the _cyclic group of order $n+1$_. We note that $C_{n+1}$ acts on the
set $[n]$ as follows: $t_{n+1}(i)=i+1$ for $0\leqslant i\leqslant n-1$ and
$t_{n+1}(n)=0$.

  * •

Let

$D_{n+1}=\left\langle r_{n+1},\,t_{n+1}\mid
r_{n+1}^{2}=t_{n+1}^{n+1}=1,\,r_{n+1}t_{n+1}r_{n+1}=t_{n+1}^{-1}\right\rangle.$

denote the _dihedral group_. We note that $D_{n+1}$ acts on the set $[n]$. The
generator $t_{n+1}$ acts as for the cyclic group $C_{n+1}$. The generator
$r_{n+1}$ acts as for $R_{n+1}$.

  * •

We denote by $\Sigma_{n+1}$ the _symmetric group_ on the set $[n]$.

  * •

Let $\Sigma_{n+1}^{+}$ be the subgroup of permutations which fix $0$.

  * •

The _hyperoctahedral group_ $H_{n+1}$ is the semi-direct product
$C_{2}^{n+1}\rtimes\Sigma_{n+1}$ where $\Sigma_{n+1}$ acts on $C_{2}^{n+1}$ by
permuting the factors.

  * •

Let $H_{n+1}^{+}$ denote the subgroup of $H_{n+1}$ consisting of elements
$(z_{0},\,z_{1},\dotsc,z_{n};\sigma)$ such that $z_{0}=1\in C_{2}$ and
$\sigma\in\Sigma_{n+1}^{+}$.

The following lemma gives us a specific instance of the dihedral group inside
the hyperoctahedral group. This will be of use for our technical results in
Section 7.

######  Lemma 2.2.

The elements $R=\left(t_{2},\dotsc,t_{2};r_{n+1}\right)$ and
$T=\left(1,\dotsc,1;t_{n+1}\right)$ generate a subgroup of $H_{n+1}$
isomorphic to $D_{n+1}$.

###### Proof.

We see that
$R^{2}=\left(t_{2},\dotsc,t_{2};r_{n+1}\right)^{2}=1=\left(1,\dotsc,1;t_{n+1}\right)^{n+1}=T^{n+1}$.
We also see that

$RTR=\left(t_{2}^{2},\dotsc,t_{2}^{2};r_{n+1}t_{n+1}r_{n+1}\right)=\left(1,\dotsc,1;t_{n+1}^{-1}\right)=T^{-1}$

which completes the proof. ∎

###  2.2. Crossed simplicial groups

The theory of crossed simplicial groups was developed independently by
Fiedorowicz and Loday ([FL91]) and Krasauskas ([Kra87]), motivated by Connes’
construction of cyclic homology ([Con83]).

######  Definition 2.3.

The category $\Delta$ has as objects the sets $[n]$ for $n\geqslant 0$, with
order-preserving maps as morphisms.

We recall, from [FL91, 1.1], that a family of groups $\left\\{G_{n}\right\\}$
for $n\geqslant 0$ is called a _crossed simplicial group_ if there is a
category $\Delta G$ such that the objects are the sets $[n]$; $\Delta$ is a
subcategory of $\Delta G$; $\mathrm{Aut}_{\Delta G}([n])=G_{n}$; and any
morphism $[n]\rightarrow[m]$ in $\Delta G$ can be uniquely written as a pair
$\left(\varphi,g\right)$ with $g\in G_{n}$ and
$\varphi\colon[n]\rightarrow[m]$ in $\Delta$.

######  Example 2.4.

We recall the _fundamental crossed simplicial groups_.

  * •

If we take the trivial group for each $n$, we recover the category $\Delta$.

  * •

If we take the family of reflexive groups, $\left\\{R_{n+1}\right\\}$ (see
[FL91, Example 2] and [Gra23b, 1.1]), we obtain the _reflexive category_
$\Delta R$, (see [Gra23b, 1.2]).

  * •

If we take the family of cyclic groups, $\left\\{C_{n+1}\right\\}$ (see [FL91,
Example 4]), we obtain the _cyclic category_ $\Delta C$, which is the same as
Connes’ category $\Lambda$ ([Con83]).

  * •

If we take the family of dihedral groups, $\left\\{D_{n+1}\right\\}$ (see
[FL91, Example 4]), we obtain the _dihedral category_ $\Delta D$, (see [Lod87,
Section 3], and also [KLS87, 1.1], where this category is denoted by $\Xi$).

  * •

If we take the family of symmetric groups, $\left\\{\Sigma_{n+1}\right\\}$
(see [FL91, Example 6]), we obtain the _symmetric category_ $\Delta S$ (see
[Aul10, 1.1]).

  * •

If we take the family of hyperoctahedral groups, $\left\\{H_{n+1}\right\\}$
(see [FL91, Example 6]), we obtain the _hyperoctahedral category_ $\Delta H$
(see [Gra22, 1.2]).

###  2.3. Functor homology theories

Given a crossed simplicial group, we have an associated functor homology
theory. In this subsection we recall the definition of reflexive homology from
[Gra23b]; the definition of dihedral homology from [Lod87] and [KLS87]; and
the definition of hyperoctahedral homology from [Fie] and [Gra22].

######  Definition 2.5.

Let $F\colon\Delta R^{op}\rightarrow\mathbf{Mod}_{k}$. The _reflexive homology
of $F$_ is defined to be

$HR_{\star}(F)=\mathrm{Tor}_{\star}^{\Delta R^{op}}\left(k^{\ast},F\right).$

######  Definition 2.6.

Let $F\colon\Delta D^{op}\rightarrow\mathbf{Mod}_{k}$. The _dihedral homology
of $F$_ is defined to be

$HD_{\star}(F)=\mathrm{Tor}_{\star}^{\Delta D^{op}}\left(k^{\ast},F\right).$

It was shown by Fiedorowicz and Loday [FL91, 6.16] that the functor homology
theory for $\Delta H^{op}$ is isomorphic to Hochschild homology. However, the
functor homology theory associated to $\Delta H$ is very different. As shown
in previous work of the author ([Gra22], [Gra23a]), inspired by Fiedorowicz’s
preprint ([Fie]), the functor homology theory for $\Delta H$ has applications
in $C_{2}$-equivariant stable homotopy theory. Whilst hyperoctahedral homology
does not appear directly in this paper, functor homology over the category
$\Delta H$ does, so we include the definition for reference.

######  Definition 2.7.

Let $F\colon\Delta H\rightarrow\mathbf{Mod}_{k}$. The _hyperoctahedral
homology of $F$_ is defined to be

$HO_{\star}(F)=\mathrm{Tor}_{\star}^{\Delta H}(k^{\ast},F).$

###  2.4. A gathering of groupoids

We will want to consider crossed simplicial groups not only as a family of
groups but also as a groupoid. We do this in the natural way: the objects are
the sets $[n]$ for $n\geqslant 0$ and the automorphisms are the group elements
in degree $n$. Recall the groups defined in Subsection 2.1.

######  Definition 2.8.

We will make use of the following groupoids.

  * •

Let $\mathbf{H}$ denote the _groupoid of hyperoctahedral groups_. The objects
are the sets $[n]$ and
$\mathrm{Hom}_{\mathbf{H}}\left([n],[n]\right)=H_{n+1}$.

  * •

Let $\mathbf{H}^{+}$ denote the sub-groupoid of $\mathbf{H}$ such that
$\mathrm{Hom}_{\mathbf{H}^{+}}\left([n],[n]\right)=H_{n+1}^{+}$.

  * •

Let $\mathbf{D}$ denote the sub-groupoid of $\mathbf{H}$ such that
$\mathrm{Hom}_{\mathbf{D}}\left([n],[n]\right)$ is the dihedral subgroup of
$H_{n+1}$ generated by the elements $R$ and $T$ of Lemma 2.2.

  * •

Let $\mathbf{R}$ denote the sub-groupoid of $\mathbf{H}$ such that
$\mathrm{Hom}_{\mathbf{R}}\left([n],[n]\right)$ is the reflexive subgroup of
$H_{n+1}$ generated by the element $R$ of Lemma 2.2.

##  3\. Involutive non-commutative sets

We recall the categories of non-commutative sets and involutive non-
commutative sets together with the subcategories of based non-commutative sets
and based, involutive, non-commutative sets.

###  3.1. The category of non-commutative sets

######  Definition 3.1.

The _category of non-commutative sets_ , $\mathcal{F}(as)$, has as objects the
sets $[n]$ for $n\geqslant 0$. An element
$f\in\mathrm{Hom}_{\mathcal{F}(as)}([n],[m])$ is a map of finite sets such
that $f^{-1}(i)$ is a totally ordered set for each $i\in[m]$. Given composable
morphisms $f\in\mathrm{Hom}_{\mathcal{F}(as)}([n],[m])$ and
$g\in\mathrm{Hom}_{\mathcal{F}(as)}([m],[k])$, the composite is given by the
underlying map of finite sets $g\circ f$, together with the preimage data
given by the ordered disjoint union:

$\left(g\circ f\right)^{-1}(i)=\coprod_{j\in g^{-1}(i)}f^{-1}(j).$

######  Definition 3.2.

The _category of based non-commutative sets_ , $\Gamma(as)$, is the
subcategory of $\mathcal{F}(as)$ with the same set of objects and whose
morphisms satisfy $f(0)=0$.

######  Remark 3.3.

We note that the version of the category $\Gamma(as)$ that we use here has a
total ordering on the preimage of the basepoint. There are versions of this
category that forgo a total-ordering on the preimage of the basepoint. See the
paper of Hartl, Pirashvili and Vespa [HPV13, Example 2.11] for example.

###  3.2. The category of involutive non-commutative sets

######  Definition 3.4.

A _finite, totally ordered $C_{2}$-set_ consists of a finite set whose
elements have a total ordering together with a superscript label from the
group $C_{2}$ for each element.

######  Definition 3.5.

Let $t$ denote the generator of $C_{2}$. We define an action of $C_{2}$ on
finite, totally-ordered $C_{2}$-sets by

$t\ast\left\\{j_{1}^{\alpha_{j_{1}}}<\cdots<j_{r}^{\alpha_{j_{r}}}\right\\}=\left\\{j_{r}^{t\alpha_{j_{r}}}<\cdots<j_{1}^{t\alpha_{j_{1}}}\right\\}.$

Explicitly, we invert the ordering and multiply each label by $t\in C_{2}$.

######  Definition 3.6.

The _category of involutive, non-commutative sets_ , $\mathcal{IF}(as)$, has
as objects the sets $[n]$ for $n\geqslant 0$. An element
$f\in\mathrm{Hom}_{\mathcal{IF}(as)}([n],[m])$ is a map of finite sets such
that $f^{-1}(i)$ is a totally ordered $C_{2}$-set for each $i\in[m]$. Given
composable morphisms $f\in\mathrm{Hom}_{\mathcal{IF}(as)}([n],[m])$ and
$g\in\mathrm{Hom}_{\mathcal{IF}(as)}([m],[k])$, the composite is given by the
underlying map of finite sets $g\circ f$, together with the preimage data
given by the ordered disjoint union:

$\left(g\circ f\right)^{-1}(i)=\coprod_{j^{\alpha}\in g^{-1}(i)}\alpha\ast
f^{-1}(j)$

with the action from Definition 3.5.

######  Remark 3.7.

We recall from [Gra22, 1.7] (see also [Gra20, 3.11]) that there is an
isomorphism of categories $\Delta H\cong\mathcal{IF}(as)$.

######  Definition 3.8.

The _category of based, involutive, non-commutative sets_ ,
$\mathcal{I}\Gamma(as)$, is the subcategory of $\mathcal{IF}(as)$ with the
same objects and whose morphisms satisfy $f(0)=0$.

##  4\. Involutive algebras, Loday functors and bar constructions

We recall the definition of an involutive $k$-algebra. We also recall the
definition of an involutive bimodule (see [Lod98, 5.2.1] for instance). We
recall the Loday functors for the reflexive homology and dihedral homology of
an involutive $k$-algebra. We also recall the hyperoctahedral bar
construction, used to define the hyperoctahedral homology of an involutive
$k$-algebra.

###  4.1. Involutive algebras

######  Definition 4.1.

An _involutive $k$-algebra_ consists of an associative $k$-algebra $A$
together with an order-reversing $k$-algebra automorphism $i\colon
A\rightarrow A$ which squares to the identity. This is commonly written as a
map $a\mapsto\overline{a}$ satisfying the following conditions for $a,\,b\in
A$ and $\lambda\in k$:

  * •

$\overline{a+b}=\overline{a}+\overline{b}$;

  * •

$\overline{ab}=\overline{b}\,\overline{a}$;

  * •

$\overline{\overline{a}}=a$;

  * •

$\overline{\lambda}=\lambda$.

######  Definition 4.2.

Let $A$ be an involutive $k$-algebra. An _involutive $A$-bimodule_ is an
$A$-bimodule $M$ equipped with a map $m\mapsto\overline{m}$ such that
$\overline{a_{1}ma_{2}}=\overline{a_{2}}\,\overline{m}\,\overline{a_{1}}$ for
$a_{1}$, $a_{2}\in A$.

###  4.2. Loday functors and homology theories for involutive algebras

######  Definition 4.3.

For an associative $k$-algebra $A$ and an $A$-bimodule $M$, the _Loday
functor_ is the simplicial $k$-module

$\mathcal{L}(A,M)\colon\Delta^{op}\rightarrow\mathbf{Mod}_{k}$

given on objects by $[n]\mapsto M\otimes A^{\otimes n}$ and determined on
morphisms by

$\partial_{i}\left(m\otimes a_{1}\otimes\cdots\otimes
a_{n}\right)=\begin{cases}\left(ma_{1}\otimes a_{2}\otimes\cdots\otimes
a_{n}\right)&i=0\\\ \left(m\otimes a_{1}\otimes\cdots\otimes
a_{i}a_{i+1}\otimes\cdots\otimes a_{n}\right)&1\leqslant i\leqslant n-1\\\
\left(a_{n}m\otimes a_{1}\otimes\cdots\otimes a_{n-1}\right)&i=n\end{cases}$

and

$s_{j}\left(m\otimes a_{1}\otimes\cdots\otimes
a_{n}\right)=\begin{cases}\left(m\otimes 1_{A}\otimes
a_{1}\otimes\cdots\otimes a_{n}\right)&j=0\\\ \left(m\otimes
a_{1}\otimes\cdots\otimes a_{j}\otimes 1_{A}\otimes
a_{j+1}\otimes\cdots\otimes a_{n}\right)&j\geqslant 1.\end{cases}$

Following [Gra23b, 1.8, 1.9] we can extend the Loday functor to a functor on
$\Delta R^{op}$ and therefore define the reflexive homology of an involutive
algebra with coefficients in an involutive bimodule. (We note that this
functor was denoted by $\mathcal{L}^{+}(A,M)$ and the homology was denoted by
$HR_{\star}^{+}(A,M)$ in that paper.)

######  Definition 4.4.

Let $A$ be an involutive $k$-algebra and let $M$ be an involutive
$A$-bimodule. We extend the Loday functor $\mathcal{L}(A,M)$ to a functor

$\mathcal{L}\left(A,M\right)\colon\Delta R^{op}\rightarrow\mathbf{Mod}_{k}$

by defining

$r_{n+1}\left(m\otimes a_{1}\otimes\cdots\otimes
a_{n}\right)=\left(\overline{m}\otimes\overline{a_{n}}\otimes\cdots\otimes\overline{a_{1}}\right).$

We denote the _reflexive homology of $A$ with coefficients in $M$_ by

$HR_{\star}(A,M)=\mathrm{Tor}_{\star}^{\Delta
R^{op}}\left(k^{\ast},\mathcal{L}(A,M)\right).$

We can extend the Loday functor for $\Delta R^{op}$ to a functor on $\Delta
D^{op}$. This first appeared in [Lod87, Section 2] and [KLS87, Section 1] (see
also [Lod90, 1.4], [Dun89, Section 2] for example) and is also sometimes
called the _dihedral bar construction_. We note that in this case we have
introduced cycles so we must have $M=A$ and we write $\mathcal{L}(A)$ rather
than $\mathcal{L}(A,A)$.

######  Definition 4.5.

Let $A$ be an involutive $k$-algebra. We extend the Loday functor
$\mathcal{L}(A)$ from $\Delta R^{op}$ to a functor

$\mathcal{L}\left(A\right)\colon\Delta D^{op}\rightarrow\mathbf{Mod}_{k}$

by defining

$t_{n+1}\left(a_{0}\otimes a_{1}\otimes\cdots\otimes
a_{n}\right)=\left(a_{n}\otimes a_{0}\otimes\cdots\otimes a_{n-1}\right).$

We denote the _dihedral homology of $A$_ by

$HD_{\star}(A)=\mathrm{Tor}_{\star}^{\Delta
D^{op}}\left(k^{\ast},\mathcal{L}(A)\right).$

###  4.3. The hyperoctahedral bar construction

######  Definition 4.6.

Let $A$ be an involutive $k$-algebra. The _hyperoctahedral bar construction_

$\mathsf{H}_{A}\colon\Delta H\rightarrow\mathbf{Mod}_{k}$

is given on objects by $[n]\mapsto A^{\otimes n+1}$. Let
$(\varphi,g)\in\mathrm{Hom}_{\Delta H}\left([n],[m]\right)$ (so
$\varphi\in\mathrm{Hom}_{\Delta}\left([n],[m]\right)$ and
$g=\left(z_{0},\dotsc,z_{n};\sigma\right)\in H_{n+1}$). We define
$\mathsf{H}_{A}\left(\varphi,g\right)$ to be determined $k$-linearly by

$\mathsf{H}_{A}\left(\varphi,g\right)(a_{0}\otimes\cdots\otimes
a_{n})=b_{0}\otimes\cdots\otimes b_{m},$

where

$b_{i}=\prod_{j\in(\varphi\circ\sigma)^{-1}(i)}a_{j}^{z_{j}}$

where the product is ordered according to the map $\varphi$ and

$a_{j}^{z_{j}}=\begin{cases}a_{j}&z_{j}=1\\\
\overline{a_{j}}&z_{j}=t_{2}.\end{cases}$

Note that an empty product is defined to be the multiplicative unit $1_{A}$.

###  4.4. Bar constructions on categories of non-commutative sets

######  Definition 4.7.

Let $A$ be an involutive, associative $k$-algebra. By abuse of notation, we
will write $\mathsf{H}_{A}\colon\mathcal{IF}(as)\rightarrow\mathbf{Mod}_{k}$
for the functor obtained by pre-composing the hyperoctahedral bar construction
with the isomorphism of [Gra22, 1.7]. In this setting, the functor
$\mathsf{H}_{A}$ is defined as follows. On objects we define
$\mathsf{H}_{A}([n])=A^{\otimes n+1}$.

Let $a_{0}\otimes\cdots\otimes a_{n}\in A^{\otimes n+1}$. Let
$f\colon[n]\rightarrow[m]$ in $\mathcal{IF}(as)$. We define
$\mathsf{H}_{A}(f)$ to be determined $k$-linearly by

$\mathsf{H}_{A}(f)\left(a_{0}\otimes\cdots\otimes
a_{n}\right)=b_{0}\otimes\cdots\otimes b_{m},$

where

$b_{i}=\prod_{j\in f^{-1}(i)}a_{j}^{z_{j}}$

with the product ordered according to the total ordering on $f^{-1}(i)$ and
$a_{j}^{z_{j}}$ given as in Definition 4.6.

######  Definition 4.8.

Let $A$ be an involutive, associative $k$-algebra and let $M$ be an involutive
$A$-bimodule. We define a functor
$\mathsf{R}(A,M)\colon\mathcal{I}\Gamma(as)\rightarrow\mathbf{Mod}_{k}$ as
follows. On objects we define $\mathsf{R}(A,M)([n])=M\otimes A^{\otimes n}$.

Let $a_{0}\otimes\cdots\otimes a_{n}\in M\otimes A^{\otimes n}$. In
particular, note that $a_{0}\in M$. Let $f\colon[n]\rightarrow[m]$ in
$\mathcal{I}\Gamma(as)$. We define $\mathsf{R}(A,M)(f)$ to be determined
$k$-linearly by

$\mathsf{R}(A,M)(f)\left(a_{0}\otimes\cdots\otimes
a_{n}\right)=b_{0}\otimes\cdots\otimes b_{m},$

where

$b_{i}=\prod_{f^{-1}(i)}a_{j}^{z_{i}}$

with the product ordered according to the total ordering on $f^{-1}(i)$ and
$a_{j}^{z_{j}}$ given as in Definition 4.6.

######  Remark 4.9.

We observe that if we take $M=A$, then the functor $\mathsf{R}(A,A)$ is the
restriction of the hyperoctahedral bar construction $\mathsf{H}_{A}$ to the
subcategory $\mathcal{I}\Gamma(as)\subset\mathcal{IF}(as)$.

##  5\. The results of Pirashvili and Richter

We give a slight reformulation of Pirashvili and Richter’s theorem.

######  Theorem 5.1 ([PR02, 1.3]).

For functors $F\colon\Gamma(as)\rightarrow\mathbf{Mod}_{k}$ and
$T\colon\mathcal{F}(as)\rightarrow\mathbf{Mod}_{k}$, there are isomorphisms of
graded $k$-modules

$\mathrm{Tor}_{\star}^{\Delta C^{op}}\left(k^{\ast},T\circ
i_{1}\right)\cong\mathrm{Tor}_{\star}^{\mathcal{F}(as)}\left(b,T\right)\quad\text{and}\quad\mathrm{Tor}_{\star}^{\Delta^{op}}\left(k^{\ast},F\circ
i_{2}\right)\cong\mathrm{Tor}_{\star}^{\Gamma(as)}\left(\overline{b},F\right),$

where $i_{1}\colon\Delta C^{op}\rightarrow\mathcal{F}(as)$ and
$i_{2}\colon\Delta^{op}\rightarrow\Gamma(as)$ are inclusions of subcategories;
the functor $b$ was defined in [PR02, 1.7] and $\overline{b}$ is the
restriction of $b$ to the subcategory $\Gamma(as)$.

Pirashvili and Richter proved their theorem by using the axiomatic
characterization of $\mathrm{Tor}$ functors, which required detailed analysis
of certain cyclic sets. However, it was shown by Słomińska and Zimmermann that
there theorems can be proved using category-theoretic methods.

##  6\. The framework of Słomińska and Zimmermann

Słomińska ([Sło03]) and Zimmermann ([Zim04]) independently developed a
categorical framework that generalizes the theorems of Pirashvili and Richter
([PR02]).

Let $\mathbf{C}$ be a small category and let $\mathbf{A}$ and $\mathbf{B}$ be
subcategories of $\mathbf{C}$, such that all three categories have the same
set of objects. Following Słomińska, we will write
$\mathbf{C}=\mathbf{A}\circ\mathbf{B}$ if every morphism $f$ in $\mathbf{C}$
can be uniquely written as a composite, $f_{\mathbf{A}}\circ f_{\mathbf{B}}$,
of a morphism in $\mathbf{B}$ followed by a morphism in $\mathbf{A}$. We note
that we would write $\mathbf{C}=\mathbf{A}\Join\mathbf{B}$ in the notation of
Zimmermann.

######  Example 6.1.

The following important examples follow from the very definition of a crossed
simplicial group. Recall the crossed simplicial group categories from Example
2.4 and the groupoids of Definition 2.8. We have decompositions $\Delta
R=\Delta\circ\mathbf{R}$; $\Delta C=\Delta\circ\mathbf{C}$; $\Delta
D=\Delta\circ\mathbf{D}$; $\Delta S=\Delta\circ\mathbf{S}$; and $\Delta
H=\Delta\circ\mathbf{H}$.

##  7\. Decomposing the category of involutive non-commutative sets

In this section we prove the technical results required to prove Theorem 9.2.
This involves decomposing the category $\mathcal{IF}(as)$ into the dihedral
category $\Delta D^{op}$ and the groupoid $\mathbf{H}^{+}$.

######  Lemma 7.1.

There is a decomposition $\mathcal{IF}(as)=\Delta\circ\mathbf{H}$.

###### Proof.

As shown in [Gra22, 1.7] (see also [Gra20, 3.11]), there is an isomorphism of
categories $\mathcal{IF}(as)\cong\Delta H$ from which the result follows. ∎

######  Lemma 7.2.

There is a decomposition $\mathbf{H}=\mathbf{D}\circ\mathbf{H}^{+}$.

###### Proof.

Let $\left(z_{0},\dotsc,z_{n};\sigma\right)$ be an element of $H_{n+1}$. For
ease of notation, we will write $\tau$ for $t_{n+1}$; $t$ for $t_{2}$; and $r$
for $r_{n+1}$. Suppose $\sigma(0)=k$. We can therefore write
$\sigma=\tau^{k}\tau^{-k}\sigma$, where $\tau^{-k}\sigma$ is a permutation in
$\Sigma_{n+1}^{+}$. We now have two cases. On the one hand, suppose that
$z_{k}=1\in C_{2}$. In this case, we can write

$\left(z_{0},\dotsc,z_{n};\sigma\right)=\left(1,\dotsc,1;\tau^{k}\right)\circ\left(1,z_{k+1},\dotsc,z_{k-1};\tau^{-k}\sigma\right)$

and we see that $\left(1,\dotsc,1;\tau^{k}\right)\in D_{n+1}$ and
$\left(1,z_{k+1},\dotsc,z_{k-1};\tau^{-k}\sigma\right)\in H_{n+1}^{+}$ as
required.

On the other hand, suppose $z_{k}=t_{2}=t\in C_{2}$. In this case we can write

$\left(z_{0},\dotsc,z_{n};\sigma\right)=\left(t,\dotsc,t;r\right)\circ\left(tz_{n},\dotsc,tz_{0};r\sigma\right).$

Now, $tz_{k}=1$ and this occurs in position $n-k$, so we can apply a similar
argument to the previous case to write

$\displaystyle\left(z_{0},\dotsc,z_{n};\sigma\right)$
$\displaystyle=\left(t,\dotsc,t;r\right)\circ\left(tz_{n},\dotsc,tz_{0};r\sigma\right)$
$\displaystyle=\left(t,\dotsc,t;r\right)\circ\left(1,\dotsc,1;\tau^{n-k}\right)\circ\left(1,tz_{k-1},\dotsc,tz_{k+1};\tau^{-(n-k)}r\sigma\right).$

We observe that

$\tau^{-(n-k)}r\sigma(0)=\tau^{-(n-k)}r(k)=\tau^{-(n-k)}(n-k)=0$

so $\left(1,tz_{k-1},\dotsc,tz_{k+1};\tau^{-(n-k)}r\sigma\right)\in
H_{n+1}^{+}$ and
$\left(t,\dotsc,t;r\right)\circ\left(1,\dotsc,1;\tau^{n-k}\right)\in D_{n+1}$
as required. ∎

##  8\. Decomposing the category of based, involutive non-commutative sets

In this section we prove the technical results required to prove Theorem 10.1.
This involves decomposing the category $\mathcal{I}\Gamma(as)$ into the
reflexive category $\Delta R^{op}$ and the groupoid $\mathbf{H}^{+}$.

######  Lemma 8.1.

The category $\Delta R^{op}$ is a subcategory of $\mathcal{I}\Gamma(as)$.

###### Proof.

We begin by noting that $\Delta^{op}$ is a subcategory of
$\mathcal{I}\Gamma(as)$. In [PR02, 1.4], it is shown that $\Delta^{op}$ is a
subcategory of $\Gamma(as)$ by explaining how to include face and degeneracy
maps. We can include the category $\Delta^{op}$ into $\mathcal{I}\Gamma(as)$
as a subcategory by adding identity labels to all preimages of singletons.

We include the reflections $r_{n+1}$ for $n\geqslant 0$ from $\Delta R^{op}$
into $\mathcal{I}\Gamma(as)$ as follows. For the reflection

$r_{n+1}\colon[n]\rightarrow[n]$

we write

$r_{n+1}^{-1}(i)=\begin{cases}\left\\{0^{t}\right\\}&i=0\\\
\left\\{(n-i+1)^{t}\right\\}&i>0\end{cases}$

in $\mathcal{I}\Gamma(as)$. A straightforward check shows that the generators
of $\Delta^{op}$ and the maps $r_{n+1}$ generate the category $\Delta R^{op}$
as a subcategory of $\mathcal{I}\Gamma(as)$. ∎

######  Lemma 8.2.

There is a decomposition $\mathcal{I}\Gamma(as)=\Delta
R^{op}\circ\mathbf{H}^{+}$.

###### Proof.

By Lemmata 7.1 and 7.2 we have a decomposition $\mathcal{IF}(as)=\left(\Delta
D\right)\circ\mathbf{H}^{+}$. The groupoid $\mathbf{H}^{+}$ is a subcategory
of $\mathcal{I}\Gamma(as)$, since the elements of this groupoid preserve $0$.
We also know that $\mathcal{I}\Gamma(as)$ is a subcategory of
$\mathcal{IF}(as)$. Combining these facts, together with the duality
isomorphism $\Delta D\cong\Delta D^{op}$ of [Dun89, 1.4], we obtain

$\displaystyle\mathcal{I}\Gamma(as)$
$\displaystyle=\mathcal{IF}(as)\cap\mathcal{I}\Gamma(as)$
$\displaystyle=\left(\Delta
D\circ\mathbf{H}^{+}\right)\cap\mathcal{I}\Gamma(as)$
$\displaystyle=\left(\Delta
D\cap\mathcal{I}\Gamma(as)\right)\circ\mathbf{H}^{+}$
$\displaystyle\cong\left(\Delta
D^{op}\cap\mathcal{I}\Gamma(as)\right)\circ\mathbf{H}^{+}.$

By Lemma 8.1, $\Delta R^{op}$ is a subcategory of $\mathcal{I}\Gamma(as)$ and
so $\Delta R^{op}$ is a subcategory of the intersection $\Delta
D^{op}\cap\mathcal{I}\Gamma(as)$. Now, consider a morphism in $\Delta D^{op}$
which is not in $\Delta R^{op}$. Since $\Delta R^{op}$ contains all the order-
preserving maps and all the reflections, such a morphism must contain a non-
identity cycle. However, such a map does not preserve $0$ and so cannot be an
morphism in $\mathcal{I}\Gamma(as)$. Therefore, we have $\Delta
R^{op}=\left(\Delta D^{op}\cap\mathcal{I}\Gamma(as)\right)$ and so
$\mathcal{I}\Gamma(as)=\Delta R^{op}\circ\mathbf{H}^{+}$ as required. ∎

##  9\. Dihedral homology

We use the decompositions of Section 7 to prove our first main theorem.

######  Definition 9.1.

Let $B\colon\mathcal{IF}(as)^{op}\rightarrow\mathbf{Mod}_{k}$ be the functor
defined on objects by

$B([n])=k\left[\coprod_{m\geqslant
0}\mathrm{Hom}_{\mathbf{H}^{+}}\left([m],[n]\right)\right]$

and determined on morphisms by pre-composition. By abuse of notation we will
also let $B$ denote the functor $\Delta H^{op}\rightarrow\mathbf{Mod}_{k}$
obtained by pre-composing with the opposite of the isomorphism of [Gra22,
1.7].

######  Theorem 9.2.

Let $F\colon\mathcal{IF}(as)\rightarrow\mathbf{Mod}_{k}$. There exist
isomorphisms of graded $k$-modules

$HD_{\star}(F\circ i\circ D)=\mathrm{Tor}_{\star}^{\Delta
D^{op}}\left(k^{\ast},F\circ i\circ
D\right)\cong\mathrm{Tor}_{\star}^{\mathcal{IF}(as)}\left(B,F\right)\cong\mathrm{Tor}_{\star}^{\Delta
H}\left(B,F\right),$

where $D\colon\Delta D^{op}\rightarrow\Delta D$ is the dihedral duality
isomorphism of [Dun89, 1.4] and $i\colon\Delta D\rightarrow\mathcal{IF}(as)$
is the inclusion of the subcategory.

###### Proof.

By Lemmata 7.1 and 7.2, we have decompositions

$\mathcal{IF}(as)=\Delta\circ\mathbf{H}=\Delta\circ\mathbf{D}\circ\mathbf{H}^{+}=\left(\Delta\circ\mathbf{D}\right)\circ\mathbf{H}^{+},$

where $\Delta\circ\mathbf{D}$ is precisely the dihedral category $\Delta D$.

By [Sło03, 1.1] or [Zim04, 2.7], we have an isomorphism of graded $k$-modules

$\mathrm{Tor}_{\star}^{\mathcal{IF}(as)}\left(B,F\right)\cong\mathrm{Tor}_{\star}^{\Delta
D}\left(k^{\ast},F\circ i\right),$

where $k^{\ast}$ is the $k$-constant right $\Delta D$-module. The first
isomorphism follows from the fact that $k^{\ast}\circ D$ is the $k$-constant
right $\Delta D^{op}$-module, which we also call $k^{\ast}$. The second
isomorphism follows from the isomorphism of [Gra22, 1.7]. ∎

######  Corollary 9.3.

Let $A$ be an involutive $k$-algebra. There exist isomorphisms of graded
$k$-modules

$HD_{\star}(A)=\mathrm{Tor}_{\star}^{\Delta
D^{op}}\left(k^{\ast},\mathcal{L}(A)\right)\cong\mathrm{Tor}_{\star}^{\mathcal{IF}(as)}\left(B,\mathsf{H}_{A}\right)\cong\mathrm{Tor}_{\star}^{\Delta
H}\left(B,\mathsf{H}_{A}\right).$

###### Proof.

By [Fie, Lemma 2.2], the composite $\mathsf{H}_{A}\circ i\circ D\colon\Delta
D^{op}\rightarrow\mathbf{Mod}_{k}$ is the Loday functor $\mathcal{L}(A)$ of
Definition 4.5 and the result follows from Theorem 9.2. ∎

######  Remark 9.4.

Pirashvili and Richter’s result was the first to draw a link between the
category of non-commutative sets and cyclic homology. This connection has been
shown to go further, notably in papers of Angeltveit ([Ang08], [Ang09]), where
the category of non-commutative sets plays an important role in forming a
cyclic bar construction for $A_{\infty}$ $H$-spaces. In future work, the
author hopes to investigate further connections between the category of
involutive, non-commutative sets and dihedral homology.

##  10\. Reflexive homology

We use the decomposition of Section 8 to prove our second main theorem.

######  Theorem 10.1.

Let $F\colon\mathcal{I}\Gamma(as)\rightarrow\mathbf{Mod}_{k}$. There is an
isomorphism of graded $k$-modules

$HR_{\star}(F\circ i)=\mathrm{Tor}_{\star}^{\Delta
R^{op}}\left(k^{\ast},F\circ
i\right)\cong\mathrm{Tor}_{\star}^{\mathcal{I}\Gamma(as)}\left(B^{\prime},F\right)$

where $B^{\prime}$ is the restriction of the functor $B$ from Definition 9.1
to the subcategory $\mathcal{I}\Gamma(as)$ and $i\colon\Delta
R^{op}\rightarrow\mathcal{I}\Gamma(as)$ is the inclusion of the subcategory.

###### Proof.

By Lemma 8.2, we have a decomposition $\mathcal{I}\Gamma(as)=\left(\Delta
R^{op}\right)\circ\mathbf{H}^{+}$. The theorem now follows from [Sło03, 1.1]
or [Zim04, 2.7]. ∎

######  Corollary 10.2.

Let $A$ be an involutive $k$-algebra and let $M$ be an involutive
$A$-bimodule. There is an isomorphism of graded $k$-modules

$HR_{\star}\left(A,M\right)=\mathrm{Tor}_{\star}^{\Delta
R^{op}}\left(k^{\ast},\mathcal{L}(A,M)\right)\cong\mathrm{Tor}_{\star}^{\mathcal{I}\Gamma(as)}\left(B^{\prime},\mathsf{R}(A,M)\right).$

###### Proof.

The corollary follows from Theorem 10.1, upon noting that the composite

$\mathsf{R}(A,M)\circ i\colon\Delta R^{op}\rightarrow\mathbf{Mod}_{k}$

coincides with the Loday functor $\mathcal{L}(A,M)$ of Definition 4.4. ∎

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