# CSS-T Codes From Reed Muller Codes

Emma Andrade  Department of Mathematics, California State University  Jessalyn
Bolkema  Department of Mathematics, Texas A&M University  Thomas Dexter
Department of Mathematics, California State University, Dominquez Hills
Harrison Eggers  School of Mathematical and Statistical Sciences, Clemson
University  Victoria Luongo  School of Mathematical and Statistical Sciences,
Clemson University  Felice Manganiello  School of Mathematical and Statistical
Sciences, Clemson University  Luke Szramowski  School of Mathematical and
Statistical Sciences, Clemson University

###### Abstract

CSS-T codes are a class of stabilizer codes introduced by Rengaswami et al
with desired properties for quantum fault-tolerance. In this work, we give a
comprehensive study of CSS-T codes built from Reed-Muller codes. These
classical codes allow for the construction of CSS-T code families with non-
vanishing asymptotic rate up to $\frac{1}{2}$ and possibly diverging minimum
distance. This desirable property enables constant overhead magic state
distillation.

##  1 Introduction

Quantum error correcting codes (QECC), together with methods to achieve fault-
tolerance, are essential to overcome the large number of errors that happen to
data and computations in a quantum computer. A set of quantum gates that are
universal is $\\{H,S,CX,T\\}$, where the Hadamard gate $H$, the phase gate
$S$, and the controlled-NOT gate $CX$ form a generating set of the Clifford
group and $T$ is a non-Clifford gate. Given a $\llbracket n,k,d\rrbracket$
stabilizer QECC, a challenging problem is to characterize the gates on the $k$
logical qubits encoded using transversal gates on the $n$ physical qubits.
Transversal gates are desirable for fault-tolerance as they prevent the spread
of errors between qubit levels.

Rengaswami _et al_ provides in [11, 10] algebraic conditions on stabilizer
QECC for their codespace to be preserved under physical transversal $T$ and
$T^{\dagger}$ gates. This is equivalent to saying that these physical gates
encode some logical gates. They also prove that, for any non-degenerate
stabilizer code that admits transversal $T$ and $T^{\dagger}$, there exists a
CSS code with the same parameters admitting the same gates. This implies that
the latter family of codes, denoted CSS-T, is an optimal family of stabilizer
codes admitting transversal $T$ and $T^{\dagger}$. Note that this is a unique
property of CSS-T codes since it is common knowledge that CSS codes are
generally not optimal in the family of stabilizer codes. CSS codes are a well-
studied family of stabilizer codes that rely on classical codes for their
definition. Many families of CSS codes have been constructed since their
definition in [4] and [12]; for an up-to-date list of CSS code constructions,
please visit [1].

In [10], the authors propose an open problem: find families of CSS-T codes
with non-vanishing asymptotic rate and asymptotic relative distance. These
properties would be desirable as they are a prerequisite for constant overhead
magic state distillation (MSD) [2], a construct that allows for transversal
encoding of a logical $T$ gate. The overhead is the ratio of noisy input
states to output $\epsilon$-noisy states, which is
$O(\log^{\gamma}(\frac{1}{\epsilon}))$, where
$\gamma=\frac{\log(n/k)}{\log(d)}$ for an $\llbracket n,k,d\rrbracket$ code
[2]. Examples of codes with constant overhead MSD appeared in [2] and [6].

In this work, we comprehensively study CSS-T codes from Reed-Muller codes.
These codes allow constructing CSS-T code families with non-vanishing
asymptotic rates up to $\frac{1}{2}$. Although the family does not have a non-
vanishing relative distance, the minimum distance diverges in specific cases,
satisfying the condition for constant overhead MSD given in [2].

This paper is organized as follows. In Section 2, we establish the notation
that we use throughout the manuscript. We recall the definition of CSS-T
codes, and we state the problem we address. In Section 3, we study the
asymptotic properties of families of Reed-Muller codes, focusing on the
asymptotic rate and relative distance. In this section, we establish
fundamental limitations of working only with Reed-Muller codes, meaning that
families of CSS codes defined only by Reed-Muller codes must have vanishing
asymptotic relative distance. In Section 4, we focus on constructing CSS-T
codes specifically using Reed-Muller codes. We translate the defining
properties of Reed-Muller codes to the classical operations of puncturing and
shortening linear codes and prove the main theorem, Theorem 14. We conclude in
Section 5 with some clarifying examples.

##  2 Definitions and Notation

This section states the definitions and the notation needed throughout the
manuscript. For definitions of stabilizer codes and other basic properties, we
refer the reader to [9, 5], whereas we refer to [7] for a reader interested in
classical coding theory.

The Calderbank-Shor-Steane (CSS) codes [4, 12] are a special class of
stabilizer codes constructed from classical error-correcting codes.
Specifically, if $C_{1}$ is $[n,k_{1},d_{1}]$, $C_{2}$ is $[n,k_{2},d_{2}]$,
and $C_{2}\subseteq C_{1}$, then $CSS(C_{1},C_{2})$ is a $\llbracket
n,k_{1}-k_{2},\geq\min\\{d_{1},d_{2}^{\perp}\\}\rrbracket$ quantum stabilizer
code. If $G_{2}$ and $G_{1}^{\perp}$ are generator matrices for the codes
$C_{2}$ and $C_{1}^{\perp}$, respectively, then a generator matrix for the
binary representation of stabilizers can be written as

$G_{S}=\left[\begin{array}[]{c|c}0&G_{1}^{\perp}\\\ \hline\cr
G_{2}&0\end{array}\right].$

Rengaswami _et al_ suggest in [11, 10] that a subfamily of CSS codes support a
physical transversal $T$ and $T^{\dagger}$ gates. This is equivalent to saying
that these physical gates encode some logical gates. The subfamily is defined
as follows.

######  Definition 1 (CSS-T codes [10]).

A CSS-T code is a CSS code associated with a pair $(C_{1},C_{2})$ of binary
linear codes satisfying

  1. 1.

$C_{2}$ is an even code, and

  2. 2.

for each codeword $x\in C_{2}$, there exists a self-dual code in
$C_{1}^{\perp}$ that is supported on $x$.

We elaborate on this definition and its equivalent formulations in Section 4.

Throughout, we consider Reed-Muller codes as candidates for both $C_{1}$ and
$C_{2}$ in the definition of a CSS-T code. We denote by
${\mathbb{F}}_{2}[x_{1},\dots,x_{m}]^{r}$ the set of all polynomials in $m$
variables with a total degree less than or equal to $r$. For
$p\in{\mathbb{F}}_{2}[x_{1},\dots,x_{m}]$, we denote by
$\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(p)$ the vector obtained by evaluation
of $p$ at ${\mathbb{F}}_{2}^{m}$; hence if
${\mathbb{F}}_{2}^{m}=\\{v_{1},\dots,v_{2^{m}}\\}$, then

$\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(p)=(p(v_{1}),\dots,p(v_{2^{m}})).$

######  Definition 2.

Given $r,m\in{\mathbb{N}}$ with $r\leq m$, the $r^{th}$ order Reed-Muller code
of length $2^{m}$, denoted $\operatorname{RM}(r,m)$, is the subspace of
${\mathbb{F}}_{2}^{m}$ give by

$\operatorname{RM}(r,m)=\\{\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(f)\mid
f\in{\mathbb{F}}_{2}[x_{1},x_{2},\ldots,x_{m}],\deg(f)\leq r\\}.$

In the following sections, we utilize the fact that $\operatorname{RM}(r,m)$
is a $[2^{m},\sum_{i=0}^{r}\binom{m}{i},2^{m-r}]$ linear code, as well as the
nested property of Reed-Muller codes, meaning that

$\operatorname{RM}(r_{2},m)\subseteq\operatorname{RM}(r_{1},m)$

if $r_{2}\leq r_{1}$. Finally, we recall that
$\operatorname{RM}(r,m)^{\perp}=\operatorname{RM}(m-r-1,m)$.

If $C$ is a linear code of length $n$ and $x\in C$, then
$\textnormal{supp}(x)=\\{i\in[n]\mid x_{i}\neq 0\\}$. If
$C=\operatorname{RM}(r,m)$ is a Reed-Muller code, then ${\mathbb{F}}_{2}^{m}$
is its index set, and if $x\in C$, then
$\textnormal{supp}(x)\subseteq{\mathbb{F}}_{2}^{m}$. If
$p\in{\mathbb{F}}_{2}[x_{1},\dots,x_{m}]^{r}$, then we denote with
$\textnormal{supp}(p)$ the support of
$x=\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(p)\in\operatorname{RM}(r,m)$,
meaning that $\textnormal{supp}(p)=\textnormal{supp}(x)$

##  3 Reed-Muller Codes’ Asymptotic Properties

We begin our analysis with the fundamental parameters of Reed-Muller codes,
establishing asymptotic minimum distance in Proposition 3 and asymptotic rate
in Proposition 4.

######  Proposition 3.

Let $C(s)=\operatorname{RM}(r(s),m(s))$ for $s\in{\mathbb{N}}$ with
$\lim_{s\to\infty}m(s)=\infty$. The sequence of codes
$(C(s))_{s\in{\mathbb{N}}}$ has non-vanishing minimum distance if and only if
$\lim_{s\to\infty}r(s)=c$.

###### Proof.

The code $\operatorname{RM}(r,m)$ has minimum distance $2^{m-r}$ and therefore
relative minimum distance $2^{m-r}/2^{m}=2^{-r}$. Thus the asymptotic relative
minimum distance of $C(s)$ is nonzero exactly when $\lim_{s\to\infty}r(s)=c$
for some $c\in{\mathbb{R}}$. ∎

######  Proposition 4.

Let $C(s)=\operatorname{RM}(r(s),m(s))$ for $s\in{\mathbb{N}}$ such that
$\lim_{s\to\infty}m(s)=\infty$. Let $R(C(s))$ be the rate of code $C(s)$. The
asymptotic rate of the sequence $(C(s))_{s\in{\mathbb{N}}}$ is

$\lim_{s\to\infty}R(C(s))=\Phi\left(\lim_{s\to\infty}\left(\frac{2r(s)-m(s)}{\sqrt{m(s)}}\right)\right)$

where $\Phi$ is the standard normal Gaussian CDF.

Such analysis of the asymptotic rate of Reed-Muller codes appears elsewhere in
the literature; see [8]. We include a proof in the interest of completeness.

###### Proof.

Consider a binomial random variable $X\sim B(m,\frac{1}{2})$. The rate of the
code $\operatorname{RM}(r,m)$ is given by

$\sum_{i=0}^{r}\binom{m}{i}\left(\frac{1}{2}\right)^{m}$

which coincides exactly with the cumulative probability $P(X\leq r)$. As
$m\to\infty$, by the Central Limit Theorem, $X$ tends towards a normal
distribution $\mathcal{N}(m/2,m/4)$ with associated cumulative density
function given by $P(X\leq r)=\Phi\left(\frac{2r-m}{\sqrt{m}}\right)$. Thus as
$m\to\infty$, the rate of the code $\operatorname{RM}(r,m)$ behaves as
$\Phi\left(\frac{2r-m}{\sqrt{m}}\right)$. That is,

$\displaystyle\lim_{s\to\infty}R(C(s))$
$\displaystyle=\lim_{s\to\infty}\Phi\left(\frac{2r(s)-m(s)}{\sqrt{m(s)}}\right)$
$\displaystyle=\Phi\left(\lim_{s\to\infty}\left(\frac{2r(s)-m(s)}{\sqrt{m(s)}}\right)\right)$

∎

We can now establish a necessary and sufficient condition for nonvanishing
rate.

######  Proposition 5.

Let $C(m)=\operatorname{RM}(\left\lfloor\frac{m-1}{2}\right\rfloor+t(m),m)$
for $m\in{\mathbb{N}}$ with $t(m)$ non-negative. Then the sequence
$(C(m))_{m\in{\mathbb{N}}}$ has nonvanishing asymptotic rate if and only if

$\lim_{m\to\infty}\frac{t(m)}{\sqrt{m}}\neq-\infty.$

###### Proof.

By previous proposition, the family ${C(m)}_{m\in{\mathbb{N}}}$ has asymptotic
rate

$\mathcal{R}=\Phi\left(\lim_{m\to\infty}\frac{2\left\lfloor\frac{m-1}{2}\right\rfloor+t(m)-m}{\sqrt{m}}\right).$

Thus $\mathcal{R}\neq 0$ if and only if

$\Phi\left(\lim_{m\to\infty}\frac{2\left\lfloor\frac{m-1}{2}\right\rfloor+t(m)-m}{\sqrt{m}}\right)\neq
0,$

or equivalently,

$\lim_{m\to\infty}\frac{2\left\lfloor\frac{m-1}{2}\right\rfloor+t(m)-m}{\sqrt{m}}\neq-\infty.$

Since

$\lim_{m\to\infty}\frac{2\left\lfloor\frac{m-1}{2}\right\rfloor+t(m)-m}{\sqrt{m}}=\lim_{m\to\infty}\frac{t(m)}{\sqrt{m}},$

the result follows.

∎

Alas, Proposition 3 and Proposition 5 together mean that a family of Reed-
Muller codes cannot have both nonvanishing rate and nonvanishing minimum
relative distance.

######  Proposition 6.

A family of Reed-Muller codes $\operatorname{RM}(r(m),m)$ cannot have both
nonvanishing rate and nonvanishing relative distance.

###### Proof.

Suppose the family $\\{\operatorname{RM}(r(m),m)\\}_{m\in{\mathbb{N}}}$ has
nonvanishing relative distance, and so $r(m)\to c$ for some $c\in{\mathbb{R}}$
as $m\to\infty$. It follows that

$\lim_{m\to\infty}\left(\frac{2r(m)-m}{\sqrt{m}}\right)=\lim_{m\to\infty}\left(\frac{2c-m}{\sqrt{m}}\right)=-\infty.$

The asymptotic rate of $\\{\operatorname{RM}(r(m),m)\\}_{m\in{\mathbb{N}}}$
can thus be evaluated

$\mathcal{R}=\Phi(-\infty)=0.$

∎

##  4 CSS-T codes from Reed-Muller Codes

Having established the asymptotic parameters of classical Reed-Muller codes,
we are now prepared to analyze the corresponding parameters in the quantum
case. First, we observe that a CSS code constructed from RM codes must
necessarily have vanishing (quantum) distance.

######  Corollary 7.

Let $C_{1}(s)=RM(r_{1}(s),m(s))$ and $C_{2}(s)=RM(r_{2}(s),m(s))$ for
$s\in{\mathbb{N}}$ with $\lim_{s\to\infty}m(s)=\infty$. The sequence of CSS
codes

$(\textsc{css}(C_{1}(s),C_{2}(s)))_{s\in{\mathbb{N}}}$

has vanishing minimum distance.

###### Proof.

Recall that $\textsc{css}(C_{1},C_{2})$ requires $C_{2}\subseteq C_{1}$, so
$r_{2}(s)\leq r_{1}(s)$ for all $s\in{\mathbb{N}}$. Suppose, seeking
contradiction, that the sequence of CSS codes has nonvanishing minimum
distance. Then, we have $r_{1}(s)\to c_{1}$, $r_{2}^{\perp}(s)\to c_{2}$ as
$s\to\infty$, for some $c_{1},c_{2}\in{\mathbb{R}}$. However,
$r_{2}^{\perp}(s)=m(s)-r_{2}(s)-1$, so $r_{2}^{\perp}(s)\to c_{2}$ and
$m(s)\to\infty$ implies that $r_{2}(s)\to\infty$. It follows from
$r_{2}(s)\leq r_{1}(s)$ that $r_{1}(s)\to\infty$, a contradiction. So such a
sequence of CSS codes has always vanishing minimum distance. ∎

This corollary leads to the following upper bound on the fare of CSS-T codes
when considering only RM codes.

######  Lemma 8.

The maximum rate of a CSS-T code defined by RM codes is $\frac{1}{2}$.

Now we focus on characterizing all CSS-T codes, specifically, that are defined
by Reed-Muller codes. Recall, from Definition 1, that the key property that
makes a CSS code a CSS-T code involves self-dual codes contained in
restrictions of dual codes.

A similar proposition to the one that follows is part of Remark 2 of [11]. We
provide here an algebraic proof of it.

######  Proposition 9.

An $[n,k]$ binary linear code $C$ contains a self-dual code if and only if $n$
is even and $C^{\perp}$ is self-orthogonal, meaning that $C^{\perp}\subseteq
C$.

###### Proof.

We begin by proving the forward direction. Let $C$ be an $[n,k]$ linear code
that contains a self-dual code $D=D^{\perp}$. Since $D\subseteq C$, we see
$C^{\perp}\subseteq D^{\perp}=D$. Combining these two relationships, we see
$C^{\perp}\subseteq D\subseteq C$. Thus, $C^{\perp}$ is self-orthogonal.

We now prove the backwards implication. Define

$S=\left\\{\overline{C}\mid C^{\perp}\subset\overline{C}\subset
C,\dim(\overline{C})=\frac{n}{2}\right\\}$

to be the set of subcodes of $C$ containing $C^{\perp}$ of dimension half of
their length. It holds that $\overline{C}\in S$ if and only if
$\overline{C}^{\perp}\in S$. Indeed, if $\overline{C}\in S$ we know that
$C^{\perp}\subseteq\overline{C}\subseteq C$. This implies that
$C^{\perp}\subseteq\overline{C}^{\perp}\subseteq(C^{\perp})^{\perp}=C$ and
$\dim(\overline{C}^{\perp})=\frac{n}{2}$, so $\overline{C}^{\perp}\in S$.

We use the Gaussian binomial coefficient to calculate the cardinality of $S$.
To account only for those codes which contain $C^{\perp}$, we consider the
chain quotient spaces

$C^{\perp}/C^{\perp}=\\{0\\}\subseteq\overline{C}/C^{\perp}\subseteq
C/C^{\perp}.$

Note that for every $\overline{C}\in S$, there is exactly one
$\overline{C}/C^{\perp}\subseteq C/C^{\perp}$, so by counting the number of
subspaces of the form $\overline{C}/C^{\perp}$ that are contained in
$C/C^{\perp}$, we find the cardinality of $S$. The dimension of
$\overline{C}/C^{\perp}$ is

$\dim\overline{C}/C^{\perp}=\dim(\overline{C})-\dim(C^{\perp})=\frac{n}{2}-(n-k)=k-\frac{n}{2},$

and the dimension of $C/C^{\perp}$ is $\dim(C)-\dim(C^{\perp})=k-(n-k)=2k-n$.

Since we know the dimension of each of these spaces, we can now use the
Gaussian binomial coefficient to compute the cardinality of $S$ as follows:

$|S|=\left[\begin{smallmatrix}2k-n\\\
k-\frac{n}{2}\end{smallmatrix}\right]_{2}=\prod_{i=0}^{k-\frac{n}{2}-1}\frac{2^{2k-n-i}-1}{2^{i+1}-1}.$

Note that $2^{2k-n-i}-1$ and $2^{i+1}-1$ are both odd for all
$i\in\mathbb{N}$. Therefore, $|S|$ is odd.

Recall that each code $\overline{C}$ in $S$ is paired off with exactly one
other code, its dual. Since $S$ is odd, then there must be at least one self-
dual code in $S$. Therefore, there exists a self-dual subcode in $C$. ∎

We note that the cardinality of the set $S$ can alternatively be deduced from
[3, Proposition 2.5].

######  Definition 10.

Given a code $C\subseteq{\mathbb{F}}_{2}^{n}$ and index set
$I\subseteq\\{1,\dots,n\\}$, produce the code $C$ punctured on $I$, denoted
$\textnormal{punct}(C,I)$, by removing from every $x\in C$ the coordinates in
$I$. The code $C$ shortened on $I$, denoted $\textnormal{short}(C,I)$,
consists of $x\in C$ such that $x_{i}=0$ for all $i\in I$ with coordinates in
$I$ removed.

The following classical result relating punctured and shortened codes is found
in Theorem 1.5.7 of [7].

######  Theorem 11.

Let $C$ be a $[n,k]$ binary linear code and $I\subseteq\\{1,\dots,n\\}$, then

$\textnormal{short}(C,I)^{\perp}=\textnormal{punct}(C^{\perp},I)$

It follows that the defining property of CSS-T codes can be expressed in terms
of punctured and shortened codes.

######  Corollary 12.

Let $C_{1}$ and $C_{2}$ be two linear codes and $C=\textsc{css}(C_{1},C_{2})$
a CSS code. $C$ is a CSS-T code if and only if for all $x\in C_{2}$ it holds
that

$\textnormal{punct}(C_{1},{\mathbb{F}}_{2}^{m}\setminus\textnormal{supp}(x))\subseteq\textnormal{short}(C_{1}^{\perp},{\mathbb{F}}_{2}^{m}\setminus\textnormal{supp}(x)).$

###### Proof.

We only need to focus on the self-dual containment condition. Let $x\in
C_{2}$, so by Proposition 9 we have that
$C_{1}^{\perp}|_{\textnormal{supp}(x)}$ contains a self-dual code if and only
if $(C_{1}^{\perp}|_{\textnormal{supp}(x)})^{\perp}$ is self-orthogonal,
meaning that

$(C_{1}^{\perp}|_{\textnormal{supp}(x)})^{\perp}\subseteq
C_{1}^{\perp}|_{\textnormal{supp}(x)}.$

Since
$C_{1}^{\perp}|_{\textnormal{supp}(x)}=\textnormal{short}(C_{1}^{\perp},{\mathbb{F}}_{2}^{m}\setminus{\textnormal{supp}(x)})$
and thanks to Theorem 11, the condition translates into

$\displaystyle\textnormal{punct}(C_{1},{\mathbb{F}}_{2}^{m}\setminus{\textnormal{supp}(x)})$
$\displaystyle=\textnormal{short}(C_{1}^{\perp},{\mathbb{F}}_{2}^{m}\setminus{\textnormal{supp}(x)})^{\perp}$
$\displaystyle\subseteq\textnormal{short}(C_{1}^{\perp},{\mathbb{F}}_{2}^{m}\setminus{\textnormal{supp}(x)}).$

∎

We can now completely characterize CSS-T codes from Reed-Muller codes.

######  Theorem 13.

Let $C_{1}=RM(\left\lfloor\frac{m-1}{2}\right\rfloor-t,m)$ and
$C_{2}=RM(r_{2},m)$. Then $\textsc{css}(C_{1},C_{2})$ is a CSS-T code if and
only if $r_{2}\leq 2t+1$ for $m$ even, or $r_{2}\leq 2t$ for $m$ odd.

###### Proof.

Let $x\in C_{2}$. Then there exists a polynomial
$p\in{\mathbb{F}}_{2}[x_{1},\dots,x_{m}]^{r_{2}}$ such that
$x=\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(p)$. For simplicity in notation, let
$r_{1}=\left\lfloor\frac{m-1}{2}\right\rfloor-t$, so
$C_{1}^{\perp}=\operatorname{RM}(m-r_{1}-1,m)$. Let $y\in C_{1}$, so there
exists a polynomial $q\in{\mathbb{F}}_{2}[x_{1},\dots,x_{m}]^{r_{1}}$ such
that $y=\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(q)$.

Consider the polynomials $p$ and $1-p$. Because we are working over the binary
field, it holds that $\textnormal{supp}(1-p)$ and $\textnormal{supp}(p)$
partition the set ${\mathbb{F}}_{2}^{m}$, meaning that
$\textnormal{supp}(1-p)\cup\textnormal{supp}(p)={\mathbb{F}}_{2}^{m}$ and
$\textnormal{supp}(1-p)\cap\textnormal{supp}(p)=\emptyset$. This gives us a
tool to split $\textnormal{supp}(q)$; specifically, given the polynomial
decomposition $q=(1-p)q+pq$, it holds that
$\textnormal{supp}((1-p)q)\subseteq\textnormal{supp}(1-p)$ and
$\textnormal{supp}(pq)\subseteq\textnormal{supp}(p)$. Moreover, by the
linearity of the evaluation map, it follows that
$\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(q)=\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}((1-p)q)+\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(pq)$.

Recall that $\textnormal{supp}(p)=\textnormal{supp}(x)$ whenever
$x=\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(p)$. Then

$\displaystyle\textnormal{punct}(y,{\mathbb{F}}_{2}^{m}\setminus\textnormal{supp}(x))$
$\displaystyle=\textnormal{punct}(\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(q),{\mathbb{F}}_{2}^{m}\setminus\textnormal{supp}(p))$
$\displaystyle=\textnormal{punct}(\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}((1-p)q)+\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(pq),\textnormal{supp}(1-p))$
$\displaystyle=\textnormal{punct}(\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(pq),\textnormal{supp}(1-p))$
$\displaystyle=\textnormal{short}(\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(pq),{\mathbb{F}}_{2}^{m}\setminus\textnormal{supp}(p))$

where the last equality holds because
$\textnormal{supp}(pq)\subseteq\textnormal{supp}(p)$.

To conclude, it is enough to prove that
$pq\in{\mathbb{F}}_{2}[x_{1},\dots,x_{m}]^{m-r_{1}-1}$, implying that
$\textnormal{ev}_{{\mathbb{F}}_{2}^{m}}(pq)\in C_{1}^{\perp}$. Indeed,

$\displaystyle\deg(pq)$ $\displaystyle\leq
r_{1}+r_{2}=\left\lfloor\frac{m-1}{2}\right\rfloor-t+r_{2}$
$\displaystyle=\begin{cases}h-1-t+2t+1,&m=2h\\\ h-t+2t,&m=2h+1\end{cases}$
$\displaystyle=h+t=m-r_{1}-1.$

Thus by Corollary 12, $\textsc{css}(C_{1},C_{2})$ is indeed CSS-T. ∎

We conclude this section with the final theorem joining all of our results.

######  Theorem 14.

Let $t(m)$ be such that $\lim_{m\to\infty}\frac{t(m)}{\sqrt{m}}=c<\infty$
where $c$ is a non-negative constant. Let

$C_{1}(m)=\operatorname{RM}\left(\left\lfloor\frac{m-1}{2}\right\rfloor-t(m),m\right)\mbox{
and }C_{2}(m)=\operatorname{RM}(r_{2}(m),m)$

where $r_{2}(m)\leq 2t(m)+1$ for $m$ even, or $r_{2}(m)\leq 2t(m)$ for $m$
odd. Then, for $m\in{\mathbb{N}}$, the code $\textsc{css}(C_{1}(m),C_{2}(m))$
is a

$\left\llbracket
n,\left\lfloor\frac{m-1}{2}\right\rfloor-t(m)-r_{2}(m),2^{r_{2}(m)+1}\right\rrbracket$

CSS-T code with non-vanishing asymptotic rate

$\Phi(-2c)\leq\frac{1}{2}.$

If $\lim_{m\to\infty}\frac{t(m)}{\sqrt{m}}=\infty$, and

$C_{1}(m)=\operatorname{RM}\left(\left\lfloor\frac{m-1}{2}\right\rfloor-t(m),m\right)\mbox{
and }C_{2}(m)=\operatorname{RM}(r_{2}(m),m)$

where $r_{2}(m)\leq 2t(m)+1$ for $m$ even, or $r_{2}(m)\leq 2t(m)$ for $m$
odd, then for $m\in{\mathbb{N}}$, the code $\textsc{css}(C_{1}(m),C_{2}(m))$
is a

$\left\llbracket
n,\left\lfloor\frac{m-1}{2}\right\rfloor-t(m)-r_{2}(m),2^{r_{2}(m)+1}\right\rrbracket$

CSS-T code with vanishing rate.

###### Proof.

Theorem 13 implies that codes $\textsc{css}(C_{1}(m),C_{2}(m))$ are CSS-T
codes for any $m\in{\mathbb{N}}$.

The dimension of each CSS-T code is direct consequence of the formula of the
dimension of a CSS code based on codes $C_{1}(m)$ and $C_{2}(m)$. For the
minimum distance, for simplicity, denote with $r_{1}(m)$ maximum degree of the
multivariate polynomials defining the codewords of $C_{1}(m)$. Then, by
definition of minimum distance of a CSS code,

$\displaystyle d$
$\displaystyle\geq\min\\{d_{1},d_{2}^{\perp}\\}=\min\\{2^{m-r_{1}(m)},2^{m-(m-r_{2}(m)-1)}\\}=2^{r_{2}(m)+1}$

since $r_{2}(m)\leq r_{1}(m)\leq\left\lfloor\frac{m-1}{2}\right\rfloor$ by
Lemma 8.

Finally, the asymptotic rate of the family of codes
$(C_{2}(m))_{m\in{\mathbb{N}}}$ is vanishing since

$\displaystyle\lim_{m\to\infty}\frac{2r_{2}(m)-m}{\sqrt{m}}$
$\displaystyle=\lim_{m\to\infty}\frac{2(2t(m)+1)-m}{\sqrt{m}}$
$\displaystyle=4\lim_{m\to\infty}\frac{t(m)}{\sqrt{m}}+\lim_{m\to\infty}\frac{2}{\sqrt{m}}-\lim_{m\to\infty}\frac{m}{\sqrt{m}}=-\infty$

As a consequence, the asymptotic rate of the family of CSS-T codes defined by
$\textsc{css}(C_{1}(m),C_{2}(m))$ is the same as the family of Reed-Muller
codes $(C_{1}(m))_{m\in{\mathbb{N}}}$ given by

$\displaystyle\Phi\left(\lim_{m\to\infty}\frac{2\left(\left\lfloor\frac{m-1}{2}\right\rfloor-t(m)\right)-m)}{\sqrt{m}}\right)=\Phi\left(\lim_{m\to\infty}-2\frac{t(m)}{\sqrt{m}}\right)=\Phi(-2c).$

∎

##  5 Examples

Finally, we note that the complete characterization of Theorem 14 subsumes and
expands upon existing examples of CSS-T codes in the literature.

######  Example 15.

Let $t(m)=0$ and $r_{2}(m)=0$. Take $C_{1}(m)$ and $C_{2}(m)$ as defined in
Theorem 14. Then, for all $m\in{\mathbb{N}}$,
$\textsc{css}(C_{1}(m),C_{2}(m))$ forms a CSS-T code. Further, by Theorem 14,
the asymptotic rate of this family is

$\Phi(0)=\frac{1}{2}.$

This family of CSS-T codes achieves the maximum possible asymptotic rate.

######  Example 16.

In [11], a family of CSS-T codes from Reed-Muller codes is proposed. We
describe an equivalent family of codes using the techniques from Theorem 14.

Let $m$ be divisible by 3. Then, define $t(m)$ as follows:

$\displaystyle t(m)$
$\displaystyle=\frac{1}{3}\left\lfloor\frac{m-1}{2}\right\rfloor-\frac{2}{3}$
($m$ even) $\displaystyle t(m)$
$\displaystyle=\frac{1}{3}\left\lfloor\frac{m-1}{2}\right\rfloor-\frac{1}{3}$
$\displaystyle\textrm{($m$ odd)}.$

Let $r_{1}$ be defined as in Theorem 14, and let $r_{2}$ be equal to its upper
bound as stated in Theorem 14. That is,

$\displaystyle r_{2}(m)$ $\displaystyle=2t(m)+1$ ($m$ even) $\displaystyle
r_{2}(m)$ $\displaystyle=2t(m)$ $\displaystyle\textrm{($m$ odd)}.$

Then, $C_{1}=\operatorname{RM}(r_{1},m)$ and
$C_{2}=\operatorname{RM}(r_{2},m)$ define a CSS-T code by Theorem 14.

######  Example 17.

Let us consider functions of the form:

$t(m)=\lfloor c\sqrt{m}\rfloor$

for some positive $c\in\mathbb{Z}$. We can see, without difficulty that the
limit:

$\lim_{m\rightarrow\infty}\frac{\lfloor c\sqrt{m}\rfloor}{\sqrt{m}}=c$

By direct application of Theorem 14, we can notice that the CSS code
$CSS(C_{1},C_{2})$ defined by:

$C_{1}(m)=RM\left(\left\lfloor\frac{m-1}{2}\right\rfloor-\lfloor
c\sqrt{m}\rfloor,m\right)$ $C_{2}(m)=RM(r_{2}(m),m)$

is a CSS-T code with non-vanishing asymptotic rate $\Phi(-2c)$. Thus, by this
construction, it is possible to generate a CSS-T code with non-vanishing
asymptotic rate $\Phi(-2c)$, for any $c\in{\mathbb{N}}$.

######  Example 18.

Let us consider any linear function, $t(m)=\lfloor a\cdot x\rfloor$, with
$0<a<\frac{1}{8}$. We can see that the CSS code $CSS(C_{1},C_{2})$ defined by:

$C_{1}\left(\left\lfloor\frac{m-1}{2}\right\rfloor-\lfloor am\rfloor,m\right)$
$C_{2}\left(r_{2}(m),m\right)$

where:

$r_{2}(m)=\begin{cases}2t(m)+1&m\,\,\,\,even\\\ 2t(m)&m\,\,\,\,odd\end{cases}$

is a CSS-T code with vanishing rate, for any appropriately chosen $a$.

##  6 Conclusion

From Theorem 14, we have derived not only a meaningful classification for
CSS-T codes of this type, but furthermore, as a corollary, a way to generate
CSS-T codes which have their properties completely determined. Moreover, it
gives a useful classification of which type of Reed-Muller codes would yield
the most optimal results via this construction.

##  7 Acknowledgements

The authors were partially supported by the National Science Foundation under
grant DMS-1547399. Furthermore, the authors would like to thank Robert
Calderbank and Narayanan Rengaswamy for their fruitful comments about this
research.

## References

  * [1] Calderbank-Shor-Steane (CSS) stabilizer code.  In Victor V. Albert and Philippe Faist, editors, The Error Correction Zoo. 2022.
  * [2] Sergey Bravyi and Jeongwan Haah.  Magic-state distillation with low overhead.  Phys. Rev. A, 86:052329, Nov 2012.
  * [3] Eimear Byrne and Alberto Ravagnani.  Partition-balanced families of codes and asymptotic enumeration in coding theory.  Journal of Combinatorial Theory, Series A, 171:105169, 2020.
  * [4] A. R. Calderbank and Peter W. Shor.  Good quantum error-correcting codes exist.  Phys. Rev. A, 54:1098–1105, Aug 1996.
  * [5] Daniel Eric Gottesman.  Stabilizer Codes and Quantum Error Correction.  PhD thesis, California Institute of Technology, 1997.
  * [6] Jeongwan Haah, Matthew B. Hastings, D. Poulin, and D. Wecker.  Magic state distillation with low space overhead and optimal asymptotic input count.  Quantum, 1:31, oct 2017.
  * [7] W. Cary Huffman and Vera Pless.  Fundamentals of Error-Correcting Codes.  Cambridge University Press, 2003.
  * [8] Shrinivas Kudekar, Santhosh Kumar, Marco Mondelli, Henry D. Pfister, Eren Şaşoǧlu, and Rüdiger L. Urbanke.  Reed–Muller codes achieve capacity on erasure channels.  IEEE Transactions on Information Theory, 63(7):4298–4316, 2017\.
  * [9] Daniel A Lidar and Todd A Brun.  Quantum error correction.  Cambridge University Press, 2013.
  * [10] Narayanan Rengaswamy, Robert Calderbank, Michael Newman, and Henry D. Pfister.  Classical coding problem from transversal T gates.  In 2020 IEEE International Symposium on Information Theory (ISIT), pages 1891–1896, 2020.
  * [11] Narayanan Rengaswamy, Robert Calderbank, Michael Newman, and Henry D. Pfister.  On optimality of CSS codes for transversal T.  IEEE Journal on Selected Areas in Information Theory, 1(2):499–514, 2020.
  * [12] A. M. Steane.  Simple quantum error-correcting codes.  Phys. Rev. A, 54:4741–4751, Dec 1996.


$\iota=\eta_{\mathbb{1}}\colon\mathbb{1}\to i_{*}\mathbb{1}=A(i)$. One can
check that this structure transfers further onto the comparison functor
$E\colon\mathcal{M}(H)\overset{\sim}{\to}A(i)\,\text{-}\operatorname{Mod}$ by
taking the unique factorizations of $\lambda_{X,Y}$ and $\iota$ as
$A(i)$-linear morphisms:

$\textstyle{EX\otimes
EY\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{*}X\otimes
i_{*}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{X,Y}}$$\textstyle{i_{*}(X\otimes
Y)}$$\textstyle{EX\otimes_{A(i)}EY\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\lambda}_{X,Y}}$$\textstyle{E(X\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

As the induced maps $\overline{\lambda}_{X,Y}$ and $\overline{\iota}$ are
isomorphisms, this is actually a strong monoidal structure on $E$. It is
clearly compatible with the braidings.

The case of comodules is dual and left to the reader. ∎

######  9.4 Remark.

In [BDS15] we considered families of categories $\mathcal{M}(G)$ for quite
general topological groups $G$, with precise hypotheses depending on the
various examples; the tensor monadicity result, however, was (and can be)
proved only for subgroups $H\leq G$ with finite discrete quotient $G/H$. Our
formalism of Mackey and Green 2-functors is only meant to treat finite groups,
but can easily handle 2-categories of topological groupoids in order to
capture the full generality of these examples, as long as one finds suitable
spannable pairs $(\mathbb{G};\mathbb{J})$; we leave the latter task to
interested readers. However, we feel that a better generalization to
topological groupoids shouldn’t be such an “obvious” one but rather one which
also usefully describes (by new axioms) the observed behavior of subgroups
with _non-finite_ quotient (e.g. phenomena such as the Wirthmüller
isomorphisms of [BDS16] for the cases when $\omega_{f}\not\simeq\mathbb{1}$).
This is an interesting problem that remains to be solved.

##  10\. Examples of Green 2-functors

Many of the examples of Mackey 2-functors treated in [BD20, Ch. 4] are in fact
symmetric Green 2-functors. Let us stress that, in all cases, the
multiplicative structure is well-known, as indeed was the case for the
underlying structure of Mackey 2-functor: Our present contribution consists in
providing easily verifiable yet useful axioms for it. We now turn in this
section to the “easily verifiable” part. For simplicitly, we take
$\Bbbk=\mathbb{Z}$ and leave to the reader the straightforward adjustments
needed to work over other rings $\Bbbk$, when applicable.

A first vast family of examples comes from derivators. By an _additive
monoidal derivator_ we mean a monoidal derivator ([GPS14, §7]) whose
underlying derivator is additive (i.e. it lands in $\mathsf{ADD}$ as a
2-functor on $\mathsf{Cat}^{\mathrm{op}}$) and such that each (internal or
external) component of the tensor product is an additive functor of both
variables.

######  10.1 Theorem (Additive monoidal derivators).

If $\mathcal{D}\colon\mathsf{Cat}^{\mathrm{op}}\to\mathsf{ADD}$ is an additive
monoidal derivator, its restriction $\mathcal{M}:=\mathcal{D}|_{\mathsf{gpd}}$
is a Green 2-functor defined on
$(\mathbb{G};\mathbb{J})=(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$. If
$\mathcal{D}$ is braided or symmetric then so is the Green 2-functor
$\mathcal{M}$. Similarly, if $\mathcal{E}$ is an additive left or right
$\mathcal{D}$-module, its restriction
$\mathcal{N}:=\mathcal{E}|_{\mathsf{gpd}}$ is a left or right
$\mathcal{M}$-module.

###### Proof.

Since $\mathcal{D}$ is in particular an additive derivator, by [BD20, Theorem
4.1.1] its restriction $\mathcal{M}$ to finite groupoids is a Mackey 2-functor
for $(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$. By definition, the
multiplicative structure on the derivator $\mathcal{D}$ consists of a
pseudomonoid structure on $\mathcal{D}$ seen as an object of the Cartesian
2-category
$\operatorname{\mathsf{2Fun}}Fun(\mathsf{Cat}^{\mathrm{op}},\mathsf{ADD})$ of
additive prederivators, such that its multiplication morphism is cocontinuous
in both variables ([GPS14, Def. 7.15]). Restriction along the inclusion
$\mathsf{gpd}\hookrightarrow\mathsf{Cat}$ preserves products, hence
$\mathcal{M}$ is a pseudomonoid in the 2-category
$\operatorname{\mathsf{2Fun}}Fun(\mathsf{gpd}^{\mathrm{op}},\mathsf{ADD})$
equipped with the _Cartesian_ tensor structure. However, being biadditive, the
multiplication functor $\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ extends
along the (termwise) universal biadditive functor
$\mathcal{M}\times\mathcal{M}\to\mathcal{M}\otimes\mathcal{M}$, where
$\otimes$ is the diagonal tensor product of additive categories (3.10 and
4.3). The cocontinuity condition on $\mathcal{D}$ specializes to the condition
in 5.2 (2), so that we indeed obtain a Green 2-functor $\mathcal{M}$. The
discussion of braidings, symmetries and modules is similarly straightforward
and left to the reader. ∎

######  10.2 Example (Model categories).

Let $\mathcal{C}$ be a monoidal model category in the sense of [Hov99].
Suppose the homotopy category $\operatorname{Ho}(\mathcal{C})$ is additive,
for instance because the model structure is stable. Then there is an additive
monoidal derivator $\mathcal{D}$ whose value $\mathcal{D}(J)$ is the homotopy
category of $J$-shaped diagrams for the levelwise weak equivalences ([GPS14,
Ex. 7.19]). Hence 10.1 yields a Green 2-functor
$\mathcal{M}=\mathcal{D}|_{\mathsf{gpd}}$ for
$(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$. If the monoidal model category
$\mathcal{C}$ is braided or symmetric then so is the Green 2-functor
$\mathcal{M}$.

Among the many concrete examples of Green 2-functors that can be derived this
way, we should at least mention the following ones, for which there are well-
known symmetric monoidal stable Quillen model structures getting the job done.

######  10.3 Example (Linear representations).

The usual Mackey 2-functors of linear representations over a field (or
commutative ring) $\Bbbk$ are symmetric Green 2-functors thanks to the tensor
product over $\Bbbk$ equipped with diagonal $G$-action. Here $\mathcal{M}(G)$
could be e.g. the abelian category $\Bbbk G\,\text{-}\operatorname{Mod}$ of
all representations ([BD20, Ex. 4.1.4]), or its derived category
$\mathrm{D}(\Bbbk G)$ ([BD20, Ex. 4.1.5]). We can obtain more examples of
Green 2-functors by restricting attention to full subcategories of the
previous ones, as long as they are stable under restrictions, inductions and
tensor products (which may depend on $\Bbbk$). For exemple: the category
$\operatorname{mod}(\Bbbk G)$ of finitely generated representations, or
$\mathrm{perm}_{\Bbbk}(G)$ of finitely generated permutation representations,
or $\mathrm{proj}(\Bbbk G)$ of finite projective modules, or the idempotent
completion $\mathrm{perm}_{\Bbbk}(G)^{\natural}$ (that is the category of
trivial source modules), or categories of bounded complexes, etc. If $\Bbbk$
is not a field, we may want to consider the subcategory
$\mathcal{M}(G)=\operatorname{latt}_{\Bbbk}(G)$ of lattices, i.e. those $\Bbbk
G$-modules which are finitely generated projective over $\Bbbk$.

######  10.4 Example (Group ring spectra).

Let $\mathrm{Mod}_{R}$ be a symmetric monoidal stable model category of
modules over some commutative ring spectrum $R$ (for any of the highly
structured meanings of “commutative ring spectrum”). By specializing 10.2, we
get a Green 2-functor $\mathcal{M}$ such that
$\mathcal{M}(G)=\operatorname{Ho}((\mathrm{Mod}_{R})^{G})=\operatorname{Ho}(\mathrm{Mod}_{RG})$
is the homotopy category of left modules over the group ring spectrum $RG$.
When $R=H\Bbbk$ is the Eilenberg–Mac Lane spectrum of a (discrete) field
$\Bbbk$, we get back the derived category $\operatorname{D}(\Bbbk G)$ of 10.3.
For $R=\mathbb{S}$ the sphere spectrum, we get the “very naïve” $G$-spectra,
i.e. $G$-diagrams of spectra.

We now turn to the many Green 2-functors not arising this way from derivators.
In some situations, we can take quotients of Green 2-functors:

######  10.5 Proposition.

Let $\mathcal{M}$ be a Green 2-functor on any spannable pair
$(\mathbb{G};\mathbb{J})$ (6.13). Let $\mathcal{N}\subseteq\mathcal{M}$ be a
Mackey sub-2-functor, i.e. a collection of full additive subcategories
$\mathcal{N}(G)\subseteq\mathcal{M}(G)$ stable under restriction and
induction, and suppose each $\mathcal{N}(G)$ is a two-sided tensor ideal in
the monoidal category $\mathcal{M}(G)$. Then the additive quotient categories
$\mathcal{Q}(G):=\mathcal{M}(G)/\mathcal{N}(G)$ inherit a unique structure of
Green 2-functor $\mathcal{Q}$ on $(\mathbb{G};\mathbb{J})$ such that the
quotient functors $\mathcal{M}(G)\to\mathcal{Q}(G)$ are monoidal. If
$\mathcal{M}$ is braided or symmetric than so is $\mathcal{Q}$.

###### Proof.

A levelwise additive quotient by a Mackey sub-2-functor inherits a unique
structure of Mackey 2-functor by [BD20, Prop. 4.2.5]. Similarly, since each
$\mathcal{N}(G)$ is an ideal, the quotient categories and the induced
restriction functors between them inherit monoidal structures from
$\mathcal{M}$ and the induced 2-isomorphisms are still monoidal, so that we
again have a pseudomonoid in
$\operatorname{\mathsf{2Fun}}Fun(\mathbb{G}^{\mathrm{op}},\mathsf{ADD}_{\Bbbk}{}^{\\!\\!(\mathsf{ic})})$
as required. Since the adjunctions and monoidal structures on $\mathcal{Q}$
are induced by those of $\mathcal{M}$, we see that the projection formulas
similarly descend to $\mathcal{Q}$. The rest is as easy. ∎

######  10.6 Example (Stable module categories).

Consider the archetypical symmetric Green 2-functor $\mathcal{M}\colon
G\mapsto\Bbbk G\,\text{-}\operatorname{Mod}$ of linear representations over a
field $\Bbbk$ (10.3). It is defined on
$(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$, but we can restrict it to the
non-Cartesian pair $(\mathsf{gpd}{}^{\mathsf{f}};\mathsf{gpd}{}^{\mathsf{f}})$
for which it is a Green 2-functor in the general sense of 6.13. Now note that
projective modules form a tensor ideal $\mathcal{N}(G)=\Bbbk
G\,\text{-}\operatorname{Proj}\subseteq\mathcal{M}(G)$, by the projection
formula $U\otimes i_{*}(V)\cong i_{*}(i^{*}U\otimes V)$ for $i\colon
1\hookrightarrow G$ (and $G$ a group). Moreover, the family $\mathcal{N}(G)$
is stable under induction and restriction functors along all faithful $i$,
since $i_{*}$ and $i^{*}$ are adjoint on both sides hence are exact. By 10.5,
the stable module categories $\mathrm{Stab}(\Bbbk
G)=\mathcal{M}(G)/\mathcal{N}(G)$ inherit from $\mathcal{M}$ the structure of
a symmetric Green 2-functor for
$(\mathsf{gpd}{}^{\mathsf{f}};\mathsf{gpd}{}^{\mathsf{f}})$. (This example
cannot arise directly from 10.1 because $\mathcal{M}(1)\simeq 0$; see [BD20,
Rem. 4.2.7].)

######  10.7 Theorem (Presheaf Green 2-functors).

Let $\mathcal{A}$ be a cocomplete abelian category. Let $\mathcal{S}$ be a
Green 2-functor for $(\mathbb{G};\mathbb{J})$ whose values $\mathcal{S}(G)$
are essentially small additive categories. Then there is a Green 2-functor
$\mathcal{M}$ for $(\mathbb{G};\mathbb{J})$ such that
$\mathcal{M}(G)=\mathsf{Add}(\mathcal{S}(G)^{\mathrm{op}},\mathcal{A})$ is the
category of additive presheaves, and such that its 2-functoriality and its
tensor structure are extended from those of $\mathcal{S}$ along the Yoneda
embeddings $\mathcal{S}(G)\hookrightarrow\mathcal{M}(G)$. If $\mathcal{S}$ is
braided or symmetric then so is $\mathcal{M}$. Moreover, we have
$(i_{*}N)(X)\cong N(i^{*}X)$ for all $(i\colon H\to G)\in\mathbb{J}$,
$N\in\mathcal{M}(H)$ and $X\in\mathcal{S}(G)$.

###### Proof.

The underlying Mackey 2-functor of $\mathcal{M}$ and the formula for induction
$i_{*}$ are provided by [BD20, Prop. 7.3.2] and its proof (which are only
formulated for
$(\mathbb{G};\mathbb{J})=(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$, but
everything applies equally to the general case). The tensor structure
$\odot_{G}$ on each $\mathcal{S}(G)$ extends to $\mathcal{M}(G)$ by the
(additive) Day convolution product; the strong monoidal structure on each
restriction functor $u^{*}$ extends uniquely to $\mathcal{M}$ by the
(2-)universal property of the coend which defines Day convolution. The
corresponding external product $\boxdot$ on $\mathcal{M}$ is related to that
on $\mathcal{S}$ by a similar coend formula; in particular, again by the
universal property of coends, it suffices to check the bimorphism conditions
of 5.2 on $\mathcal{S}$, where it holds by hypothesis. We conclude that this
extended product on $\mathcal{M}$ defines a Green 2-functor. The remaining
details are straightforward and left to the reader. ∎

######  10.8 Example (Ordinary Mackey functors).

There exists a Green 2-functor $\mathcal{M}$ for
$(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$ such that $\mathcal{M}(G)$ is the
category $\operatorname{Mack}_{\Bbbk}(G)$ of ordinary $\Bbbk$-linear Mackey
functors for $G$, and another one such that $\mathcal{M}(G)$ is the category
$\operatorname{CohMack}_{\Bbbk}(G)$ of cohomological Mackey functors for $G$;
the so-called box product provides both tensor structures ([Bou97]). In both
cases, we obtain $\mathcal{M}$ by applying 10.7 to
$\mathcal{A}=\Bbbk\,\text{-}\operatorname{Mod}$ and a suitable Green 2-functor
$\mathcal{S}$. For the second $\mathcal{M}$, since
$\operatorname{CohMack}_{\Bbbk}(G)=\mathsf{Add}(\mathrm{perm}_{\mathbb{Z}}(G),\Bbbk\,\text{-}\operatorname{Mod})$
as tensor categories by Yoshida’s theorem (cf. [DH21]), we can take
$\mathcal{S}$ to be the (level-wise dual of the) Green 2-functor
$\mathcal{S}\colon G\mapsto\mathrm{perm}_{\mathbb{Z}}(G)^{\mathrm{op}}$ as in
10.3. For the first $\mathcal{M}$, we can take $\mathcal{S}(G)$ to be the
additive category of spans of finite $G$-sets, which defines a Mackey
2-functor $\mathcal{S}$ by [BD20, Thm. 7.2.3]; the cartesian product of
$G$-sets induces a tensor structure on $\mathcal{S}(G)$ extending (by
definition!) to the box product on $\mathcal{M}(G)$, and the preservation of
inductions is easily checked so that we get the required Green 2-functor
$\mathcal{S}$.

As in [BD20, §4.4], we can consider Mackey 2-functors of “equivariant
objects”. They are quite often equipped with the structure of a Green
2-functor:

######  10.9 Theorem (Equivariant objects).

Let $G$ be a fixed finite groupoid, and let $\mathcal{S}\colon
G^{\mathrm{op}}\to\operatorname{\mathsf{PsMon}}(\mathsf{ADD})$ be a
pseudofunctor to the 2-category of additive monoidal categories. Then there
exists a Green 2-functor for
$\mathbb{G}=\mathbb{J}={\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}}$,
denoted $\Gamma(-;\mathcal{S})$, whose value at $(H,i_{H})$ is the category
$\Gamma((H,i_{H});\mathcal{S}):=\Gamma(H;\mathcal{S}\circ
i_{H}^{\mathrm{op}})$ of $H$-equivariant objects, i.e. of sections of the
Grothendieck fibration associated with $\mathcal{S}\circ
i_{H}^{\mathrm{op}}\colon H^{\mathrm{op}}\to\mathsf{ADD}$. If $\mathcal{S}$
takes values in braided or symmetric monoidal categories, the associated Green
2-functor inherits a braiding or symmetry accordingly.

###### Proof.

The underlying Mackey 2-functor for
${\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}}$ is given by
[BD20, Thm. 4.4.16], applied to $\mathcal{S}$ composed with the forgetful
2-functor $\operatorname{\mathsf{PsMon}}(\mathsf{ADD})\to\mathsf{ADD}$.

W must define the pairing and show that it preserves inductions. We only
sketch the construction, using the notations of _loc. cit._ , leaving all
(quite straightforward) verifications to the reader. The tensor product
$(X,\varphi)\otimes(Y,\psi)$ of two objects $(X,\varphi)$, $(Y,\psi)$ of
$\Gamma(H;\mathcal{S}\circ i_{H}^{\mathrm{op}})$ has object-components
$(X\otimes Y)_{p}:=X_{p}\otimes Y_{p}$ for $p\in\operatorname{Obj}H$, using
the given internal tensor of $\mathcal{S}(p)$; and morphism-components

$\textstyle{(\varphi\otimes\psi)_{g}\colon X_{p}\otimes
Y_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{g}\otimes\psi_{g}}$$\textstyle{g^{*}X_{q}\otimes
g^{*}Y_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{g^{*}(X_{q}\otimes
Y_{q})}$

for all $(g\colon p\to q)\in\operatorname{Mor}H$, using the given strong
monoidal structure of the functor $g^{*}=\mathcal{S}(i_{H}(g))$. The tensor
product of morphisms is similarly diagonal:
$(\xi\otimes\zeta)_{p}=\xi_{p}\otimes\zeta_{p}$. This defines a monoidal
structure on the category $\Gamma((H,i_{H});\mathcal{S})$ for each object
$(H,i_{H})\in{\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}}$.
For every 1-morphism $(i,\theta_{i})\colon(K,i_{K})\to(H,i_{H})$ in
${\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}}$, the
restriction functor
$(i,\theta_{i})^{*}\colon\Gamma((H,i_{H});\mathcal{S})\to\Gamma((K,i_{K});\mathcal{S})$
defined as in _loc. cit._ receives a strong monoidal structure from those
given on the pull-back functors
$(\theta^{-1}_{i,q})^{*}=\mathcal{S}(\theta^{-1}_{i,q})$ (for all
$q\in\operatorname{Obj}K$). Similarly, the natural isomorphism induced by a
2-cell $\alpha\colon(i,\theta_{i})\Rightarrow(j,\theta_{j})$ is monoidal,
which ultimately boils down to the natural transformation
$\alpha^{*}=\mathcal{S}(i_{H}\alpha)$ being monoidal by hypothesis. We thus
obtain a 2-functor
$\Gamma(-;\mathcal{S})\colon({\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}})^{{\mathrm{op}}}\to\operatorname{\mathsf{PsMon}}(\mathsf{ADD})$.

By 4.4, it only remains to see that the corresponding external pairing
preserves inductions in both variables. We leave this lengthy but
straightforward verification to the reader, who may compute using the explicit
Kan formulas for the adjunctions (cf. [BD20, Lemma 4.4.11]). ∎

We can apply the latter result to equivariant sheaves in geometry:

######  10.10 Example (Equivariant sheaves).

Suppose a finite group $G$ acts on a locally ringed space
$X=(X,\mathcal{O}_{X})$, and let $\operatorname{Mod}(X)$ be the symmetric
monoidal category of sheaves of $\mathcal{O}_{X}$-modules. Pulling back along
elements $g\in G$ defines a pseudoaction of $G$ on $\operatorname{Mod}(X)$,
i.e. a pseudofunctor $\mathcal{S}\colon
G^{\mathrm{op}}\to\operatorname{\mathsf{PsMon}}(\mathsf{ADD})$ which maps the
unique (unnamed) object of $G$ to $\operatorname{Mod}(X)$. By 10.9, we get a
symmetric Green functor $\mathcal{M}$ on
${\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}}$ whose value
at a subgroup $H\leq G$ is the category $\operatorname{Mod}(X/\\!\\!/H)$ of
$H$-equivariant sheaves of $\mathcal{O}_{X}$-modules. Many variations are
possible, by replacing $\operatorname{Mod}(X)$ with other monoidal categories.
For instance, we may consider constructible sheaves over some commutative
ring, or – if $X$ is a noetherian scheme – coherent or quasicoherent
$\mathcal{O}_{X}$-modules. We may also use chain complexes, or (variously
bounded) homotopy or derived categories. See [BD20, §4.4] for details.

Let us sketch some more examples, leaving the details to interested readers.

######  10.11 Example (Equivariant stable homotopy).

As in [BD20, Ex. 4.3.8], there is a Mackey 2-functor $\mathcal{M}$ for
$(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$ such that
$\mathcal{M}(G)=\operatorname{Ho}(\textrm{Sp}^{G})$ is the homotopy category
of genuine $G$-equivariant spectra. The latter is a symmetric monoidal
category by the smash product of $G$-spectra and the restriction functors
$i^{*}$ are strong symmetric monoidal. This structure extends to a bimorphism
$(\mathcal{M},\mathcal{M})\to\mathcal{M}$ turning $\mathcal{M}$ intro a
symmetric Green functor.

######  10.12 Example (Equivariant Kasparov theory).

As in [BD20, Ex. 4.3.9], there is a Mackey 2-functor $\mathcal{M}$ for
$(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$ (in fact for
$(\mathsf{gpd};\mathsf{gpd})$!) such that $\mathcal{M}(G)=\mathrm{KK}^{G}$ is
the $G$-equivariant Kasparov category of separable complex $G$-C*-algebras.
The latter is a symmetric monoidal category by the minimal tensor product of
C*-algebras equipped with the diagonal $G$-action. This structure extends to a
bimorphism which turns $\mathcal{M}$ into a symmetric monoidal Green functor.

######  10.13 Remark.

We suspect many Green 2-functors to arise from $G$-equivariant ring spectra
(cf. [LM06]) and/or from suitable $E_{\infty}$-Green functors in the sense of
Barwick _et al._[Bar17] [BGS20], by taking module categories of their values
at all $H\leq G$. In any case, it would be interesting to make precise the
way(s) that the categorification of Green functors due to Barwick _et al._ is
related to our own.

######  10.14 Remark.

All concrete examples of Green functors we have explicitly mentioned happen to
be symmetric. Nonetheless, it is easy to find Green functors which are braided
but not symmetric, or just “plain”, for instance those in 10.2 deriving from
suitably chosen monoidal model categories.

##  11\. Green 1-functors for general $(\mathbb{G};\mathbb{J})$

In this section we revisit the notion of (ordinary) _Green functor_ by
introducing a general definition which specializes to all variants in use for
finite groups (11.4). We base our definition (11.2) on the original view as
“levelwise ring with Frobenius formulas” and verify that, when the pair
$(\mathbb{G};\mathbb{J})$ is Cartesian, these are just monoids for the induced
monoidal structure (11.3).

We begin by recalling the following flexible definition of ordinary Mackey
functor. Let $(\mathbb{G};\mathbb{J})$ be a spannable pair (2.1) and fix a
commutative ring $\Bbbk$.

######  11.1 Definition ([BD20] [Del22]).

A _Mackey functor for $(\mathbb{G};\mathbb{J})$_ is an additive functor

$M\colon\tau_{1}\operatorname{{\sf
Span}}(\mathbb{G};\mathbb{J})\longrightarrow\Bbbk\,\text{-}\operatorname{Mod}$

taking values in $\Bbbk$-modules and defined on the semiadditive (truncated
ordinary) _span category_ of $(\mathbb{G};\mathbb{J})$ (see [BD20, Def. 2.5.1]
or [Del22, §2.3]). Concretely this means that $M$ assigns to every
$G\in\operatorname{Obj}\mathbb{G}$ a $\Bbbk$-module $M(G)$, to every
1-morphism $u\colon H\to G$ in $\mathbb{G}$ a $\Bbbk$-linear homomorphism
$u^{\scriptscriptstyle\bullet}\colon M(G)\to M(H)$ and, if $u\in\mathbb{J}$,
also a homomorphism $u_{\scriptscriptstyle\bullet}\colon M(H)\to M(G)$,
satisfying the following four axioms:

  1. (a)

_Functoriality:_ We have
$\mathrm{id}^{\scriptscriptstyle\bullet}=\mathrm{id}_{\scriptscriptstyle\bullet}=\mathrm{id}$,
and for composable $K\xrightarrow{v}H\xrightarrow{u}G$ we have $(u\circ
v)^{\scriptscriptstyle\bullet}=v^{\scriptscriptstyle\bullet}\circ
u^{\scriptscriptstyle\bullet}$ and when $u,v\in\mathbb{J}$ also $(u\circ
v)_{\scriptscriptstyle\bullet}=u_{\scriptscriptstyle\bullet}\circ
v_{\scriptscriptstyle\bullet}$.

  2. (b)

_Isomorphism invariance:_ Whenever $u\simeq v$ are isomorphic in $\mathbb{G}$,
we have $u^{\scriptscriptstyle\bullet}=v^{\scriptscriptstyle\bullet}$ and when
$u,v\in\mathbb{J}$ also
$u_{\scriptscriptstyle\bullet}=v_{\scriptscriptstyle\bullet}$.

  3. (c)

_Additivity:_ every coproduct $G\overset{i}{\to}G\sqcup
H\overset{\;j}{\leftarrow}H$ in $\mathbb{G}$ induces an isomorphism
$(i^{\scriptscriptstyle\bullet},j^{\scriptscriptstyle\bullet})^{t}\colon
M(G\sqcup H)\overset{\sim}{\to}M(G)\oplus M(H)$ with inverse
$(i_{\scriptscriptstyle\bullet},j_{\scriptscriptstyle\bullet})$ (hence
$M(\emptyset)\cong 0$).

  4. (d)

_Mackey formula:_ For every Mackey square in $\mathbb{G}$ with $i$ and $q$ in
$\mathbb{J}$

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{\underset{\scriptstyle\gamma}{\overset{\sim}{\rotatebox[origin={c}]{90.0}{$\Downarrow$}}}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{G}$

we have the equality $u^{\scriptscriptstyle\bullet}\circ
i_{\scriptscriptstyle\bullet}=q_{\scriptscriptstyle\bullet}\circ
p^{\scriptscriptstyle\bullet}$ of maps $M(H)\to M(K)$.

As in [Del22], we denote by

$\operatorname{Mack}_{\Bbbk}(\mathbb{G};\mathbb{J}):=\operatorname{\mathrm{Fun}}_{\mathrm{add}}(\tau_{1}\operatorname{{\sf
Span}}(\mathbb{G};\mathbb{J}),\Bbbk\,\text{-}\operatorname{Mod})$

the $\Bbbk$-linear category of Mackey functors and natural transformations.
Note that a natural transformation $\alpha\colon M\to N$ is the same thing as
a family of maps $\alpha_{G}\colon M(G)\to M(G)$
($G\in\operatorname{Obj}\mathbb{G}$) which is compatible with respect to the
restriction maps $u^{\scriptscriptstyle\bullet}$ for all $u\colon H\to G$ and,
for $u\in\mathbb{J}$, also with the induction maps
$u_{\scriptscriptstyle\bullet}$.

We can now introduce a corresponding general notion of Green functor:

######  11.2 Definition (Green 1-functor for $(\mathbb{G};\mathbb{J})$).

Let $M,N,L$ be Mackey functors for the spannable pair
$(\mathbb{G};\mathbb{J})$. We define a _Dress 333Compare with the definition
of “pairing” given in [Dre73, p. 195]. pairing_ $\beta\colon(M,N)\to L$ to be
a family of $\Bbbk$-linear morphisms $\beta=\beta_{G}\colon
M(G)\otimes_{\Bbbk}N(G)\to L(G)$ for $G\in\operatorname{Obj}\mathbb{G}$
satisfying:

  1. (a)

Naturality for restrictions:
$u^{\scriptscriptstyle\bullet}(\beta(x,y))=\beta(u^{\scriptscriptstyle\bullet}(x),u^{\scriptscriptstyle\bullet}(y))$
for all $(H\overset{u}{\to}G)\in\mathbb{G}$.

  2. (b)

The _Frobenius formulas_ (or _projection formulas_)
$\beta(x,u_{\scriptscriptstyle\bullet}(y))=u_{\scriptscriptstyle\bullet}(\beta(u^{\scriptscriptstyle\bullet}(x),y))$
and
$\beta(u_{\scriptscriptstyle\bullet}(y),x)=u_{\scriptscriptstyle\bullet}\beta(y,u^{\scriptscriptstyle\bullet}(x))$
for all $x\in M(G)$ and $y\in M(H)$, when $u\in\mathbb{J}$.

A _Green functor for $(\mathbb{G};\mathbb{J})$_ is then a Mackey functor $M$
for $(\mathbb{G};\mathbb{J})$ together with a Dress pairing
$\smash{(M,M)\overset{\cdot}{\to}M}$ which is associative and unital, i.e. one
which turns each $M(G)$ into an associative unital $\Bbbk$-algebra. Similarly,
a left (or right) module $N$ over the Green functor $M$ is a Mackey functor
$N$ together with a pairing $(M,N)\to N$ (resp. $(N,M)\to N$) which turns each
$N(G)$ into a left (or right) $M(G)$-module.

The following reassuring result is well-known at least in some cases (cf.
[Lew80, Prop. 1.4] or [Bou97, §2]), but we provide the general proof for
completeness.

######  11.3 Proposition.

Suppose the pair $(\mathbb{G};\mathbb{J})$ is Cartesian. Then:

  1. (1)

The products of $\mathbb{G}$ induce an additive symmetric monoidal structure
on the span category $\tau_{1}\operatorname{{\sf
Span}}(\mathbb{G};\mathbb{J})$ and thus, by Day convolution with respect to
$\otimes_{\Bbbk}$, a $\Bbbk$-linear closed symmetric monoidal structure on
$\operatorname{Mack}_{\Bbbk}(\mathbb{G};\mathbb{J})$.

  2. (2)

The data of a Dress pairing $\beta\colon(M,N)\to L$ for
$(\mathbb{G};\mathbb{J})$ (as in 11.2) is equivalent to that of a morphism
$\overline{\beta}\colon M\otimes N\to L$ in the tensor category of (1).

  3. (3)

Under the correspondence of (2), a (commutative) Green functor corresponds to
a (commutative) monoid in
$\operatorname{Mack}_{\Bbbk}(\mathbb{G};\mathbb{J})$, and similarly for
modules.

###### Proof.

It is straightforward to check that the products $G\times H$ of $\mathbb{G}$
extend to spans to define a symmetric monoidal structure on
$\tau_{1}\operatorname{{\sf Span}}(\mathbb{G};\mathbb{J})$; it is additive in
both variables since finite products distribute over finite coproducts
(because $\mathbb{G}$ is extensive by hypothesis, and reasoning as in [CLW93,
Prop. 2.5]; cf. [PS07]). We can then apply $\Bbbk$-linear Day convolution (see
e.g. [Lor21, §6.2]) to get the tensor structure $\otimes$ on
$\operatorname{Mack}_{\Bbbk}(\mathbb{G};\mathbb{J})$ as claimed in part (1).

Note that, by the universal property of Day convolution as a coend, a morphism
$\overline{\beta}\colon M\otimes N\to L$ in
$\operatorname{Mack}_{\Bbbk}(\mathbb{G};\mathbb{J})$ is uniquely determined by
the $\Bbbk$-linear maps

$\tilde{\beta}_{G,H}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.78539pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M(G)\otimes_{\Bbbk}N(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.56316pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\textrm{can.}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 59.78539pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.78539pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{(M\otimes N)(G\times
H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
144.2363pt\raise 6.68887pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.98889pt\hbox{$\scriptstyle{\overline{\beta}_{G\times
H}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 166.68935pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
166.68935pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{L(G\times H)}$}}}}}}}}}}}\ignorespaces$

natural in $G,H\in\tau_{1}\operatorname{{\sf Span}}(\mathbb{G};\mathbb{J})$,
obtained by precomposing $\overline{\beta}$ with the coend structure map.
Moreover, $\overline{\beta}$ is the associative and unital structure map of a
monoid or module precisely when the family
$\tilde{\beta}=\\{\tilde{\beta}_{G,H}\\}$ is associative and unital in an
evident corresponding sense (e.g. the case of a monoid $M=N=L$ yields
precisely a lax monoidal structure on the functor $M$, and the general case is
very similar).

To prove part (2), it will therefore suffice to see that the data of such a
natural family $\tilde{\beta}$ is equivalent to that of a Dress pairing
$\beta$. Indeed, we get a bijective correspondence
$\tilde{\beta}\leftrightarrow\beta$ just as in 4.12, by restricting along
diagonals and projections: Given $\tilde{\beta}$, we construct
$\beta=\\{\beta_{G}\\}_{G}$ as the composite

$\beta_{G}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.15414pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M(G)\otimes_{\Bbbk}N(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.67581pt\raise 7.68333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.5389pt\hbox{$\scriptstyle{\tilde{\beta}_{G,G}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 59.15414pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.15414pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{L(G\times
G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
114.12909pt\raise 6.55833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.19167pt\hbox{$\scriptstyle{\delta^{\scriptscriptstyle\bullet}_{G}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 131.68463pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
131.68463pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{L(G)}$}}}}}}}}}}}\ignorespaces,$

whereas if we are given $\beta$, we get the following family of maps:

$\tilde{\beta}_{G,H}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.78539pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M(G)\otimes_{\Bbbk}N(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
35.43259pt\raise 5.72221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.36111pt\hbox{$\scriptstyle{\operatorname{pr}^{\scriptscriptstyle\bullet}_{1}\otimes\operatorname{pr}^{\scriptscriptstyle\bullet}_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 63.78539pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
63.78539pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{L(G\times H)\otimes_{\Bbbk}N(G\times
H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
176.86906pt\raise 6.28055pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.58057pt\hbox{$\scriptstyle{\beta_{G\times H}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 200.8018pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
200.8018pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{L(G\times H)}$}}}}}}}}}}}\ignorespaces.$

Given a natural family $\tilde{\beta}$, we must check that the family $\beta$
forms a Dress pairing. And indeed, $\beta$ is clearly natural with respect to
restrictions (though usually not for inductions!), and it satisfies the two
projection formulas by a simpler version of the proof of 6.4. (Just draw in
$\Bbbk\,\text{-}\operatorname{Mod}$ the analog of that proof’s last diagram,
which will commute by the naturality of $\tilde{\beta}$ for both restrictions
and inductions together with the Mackey formula for the square (6.8).)

Conversely, if we are given a Dress pairing $\beta$ we must show that the
corresponding $\tilde{\beta}$ is natural in both variables and with respect to
both restriction and induction maps. The naturality for restrictions follows
immediately from that of $\beta$. The naturality with respect to inductions,
say for two 1-cells $i\colon G^{\prime}\to G$ and $j\colon H^{\prime}\to H$ in
$\mathbb{J}$, follows by a simpler version of the proof of 6.11. (Just draw in
$\Bbbk\,\text{-}\operatorname{Mod}$ the analog of that proof’s large diagram,
which will commute by the two Frobenius formulas for $\beta$, the
functoriality of induction and restriction, and the Mackey formula for the two
squares (6.12).) This concludes the proof of part (2).

For part (3), in view of the remarks made at the beginning, it suffices to
check that a family $\tilde{\beta}$ is associative and unital precisely when
the corresponding Dress pairing is. This is a straightforward verification we
leave to the reader. ∎

######  11.4 Examples (See [Del22] for more examples and details).

If $(\mathbb{G};\mathbb{J})$ is
$(\mathsf{gpd}{}^{\mathsf{f}};\mathsf{gpd}{}^{\mathsf{f}})$,
$(\mathsf{gpd};\mathsf{gpd}{}^{\mathsf{f}})$ or
$({\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}};{\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}})$,
as in 2.3, the associated Mackey and Green functors in our sense are the
classical notions of, respectively: globally defined Mackey and Green functors
(a.k.a. bifree biset functors [Bou10]), globally defined inflation functors
(a.k.a. right-free biset functors), and Mackey and Green functors for a fixed
group $G$. With $\mathbb{G}=\mathbb{J}=\mathsf{gpd}$ we get the variant of
global Mackey and Green functors studied by Ganter [Gan13] and Nakaoka
[Nak16b] [Nak16a], which also allows “induction” along non-faithful functors
(e.g. deflations). The category of the latter Mackey functors contains that of
all biset functors as a reflective (tensor-ideal) subcategory; see [DH21].

######  11.5 Remark.

Let us review the analogies between Green 1-functors and Green 2-functors. The
more general definition of a Green 2-functor via the projection formulas
(6.13) categorifies the original point of view on Green 1-functors (11.2),
while the definition via bimorphisms (5.4) valid in the common situation of
Cartesian pairs corresponds to viewing a Green 1-functor as a lax monoidal
functor, i.e. as the data $\tilde{\beta}$ in the proof of 11.3.

In order to complete this picture, we should also be able to view Green
2-functors (for a fixed Cartesian pair) as pseudomonoids in a suitable tensor
2-category $\operatorname{\operatorname{\mathsf{Mack}}^{\mathsf{ic}}_{\Bbbk}}$
of Mackey 2-functors. For the Cartesian pairs in 2.3 where $\mathbb{J}$
consists of _faithful_ functors, at least (which unfortunately excludes the
pair $(\mathsf{gpd};\mathsf{gpd})$), we provided in [BD21b] a biequivalence
$\operatorname{\operatorname{\mathsf{Mack}}^{\mathsf{ic}}_{\Bbbk}}\simeq\operatorname{\mathsf{PsFun}}_{\sqcup}(\Bbbk\textrm{\sf
S}\widehat{\textrm{\sf pan}},\mathsf{ADD}^{\mathsf{ic}}_{\Bbbk})$ for a
certain bicategory $\Bbbk\textrm{\sf S}\widehat{\textrm{\sf pan}}$ of _Mackey
2-motives_ , which contains the “usual” span bicategory $\operatorname{{\sf
Span}}$ but has more 2-cells. We could thus use Day convolution for monoidal
bicategories ([Cor19]) to put a tensor structure on
$\operatorname{\operatorname{\mathsf{Mack}}^{\mathsf{ic}}_{\Bbbk}}$, but first
we need to check that the Cartesian product of $\mathbb{G}$ extends to a
symmetric monoidal structure on $\Bbbk\textrm{\sf S}\widehat{\textrm{\sf
pan}}$. We conjecture this to be possible by extending the arguments in
[Hof11]. We do not pursue here this motivic approach, though elegant, as it
would require yet another layer of machinery for a (so far) small return in
applications.

##  12\. The origins of classical Green functors

In this final section we explain how ordinary Green functors can be obtained
from Green 2-functors by several procedures deserving the name of
“decategorification”. We then show how the classical Green functors (as can be
found e.g. in [Web00]) arise this way from the natural examples of Green
2-functors discussed in Section 10. As before, $(\mathbb{G};\mathbb{J})$
denotes a (almost always) Cartesian spannable pair (Hypotheses 2.1 and 2.10)
and $\Bbbk$ the chosen commutative ring of coefficients.

######  12.1 Construction (K-decategorification; see [BD20, Ch. 2.5]).

The ordinary $\mathrm{K}_{0}$ of additive categories defines a functor
$\operatorname{K_{0}^{add}}\colon\tau_{1}\mathsf{Add}\to\operatorname{Ab}$,
which to a small additive category $\mathcal{A}\in\mathsf{Add}$ associates the
Grothendieck group
$\operatorname{K_{0}^{add}}(\mathcal{A})=\mathrm{K}_{0}(\mathcal{A},\oplus,0)$
and to an additive functor $F\colon\mathcal{A}\to\mathcal{B}$ the induced
homomorphism
$\operatorname{K_{0}^{add}}(\mathcal{A})\to\operatorname{K_{0}^{add}}(\mathcal{B})$;
with isomorphic functors inducing the same homomorphism. By [BD20, §5.2],
every Mackey 2-functor
$\mathcal{M}\colon\mathbb{G}^{\mathrm{op}}\to\mathsf{Add}$ for
$(\mathbb{G};\mathbb{J})$ and $\Bbbk=\mathbb{Z}$ admits a canonical additive
extension $\widehat{\mathcal{M}}\colon\operatorname{{\sf
Span}}(\mathbb{G};\mathbb{J})\to\mathsf{Add}$ to the span bicategory. Then the
composite map

$M:=\operatorname{K_{0}^{add}}\circ\tau_{1}(\widehat{\mathcal{M}})\colon\tau_{1}\operatorname{{\sf
Span}}(\mathbb{G};\mathbb{J})\to\tau_{1}\mathsf{Add}\to\operatorname{Ab}$

is additive, i.e. is an ordinary Mackey functor for $(\mathbb{G};\mathbb{J})$
as in 11.1.

######  12.2 Remark (Variant K-decategorifications).

If the value-categories of $\mathcal{M}$ are equipped with further structure,
we may prefer to use a more pertinent version of $\mathrm{K}_{0}$-groups. For
instance, if $\mathcal{M}$ lands in exact (e.g. abelian) categories and exact
functors, or in triangulated categories and triangulated functors, we could
use $\operatorname{K_{0}^{exa}}$ or $\operatorname{K_{0}^{tri}}$ respectively,
the $\mathrm{K}_{0}$-group of exact or triangulated categories.

######  12.3 Remark (Higher K-theory).

If $\mathcal{M}$ takes values in exact categories, we may consider Quillen’s
higher algebraic K-theory of exact categories
$\mathcal{E}\mapsto\operatorname{K^{exa}_{*}}(\mathcal{E})=\pi_{*}\Omega
BQ\mathcal{E}$. Since the $Q$-construction $Q\mathcal{E}$ is functorial for
exact functors, and since isomorphic functors induce the same map on K-theory
groups, precisely the same argument as in 12.1 yields a Mackey functor
$G\mapsto\operatorname{K^{exa}_{*}}(\mathcal{M}(G))$.

######  12.4 Construction (Hom-decategorification; see [BD21a, App. A]).

Let $\mathcal{M}$ be a Mackey 2-functor for $(\mathbb{G};\mathbb{J})$, and
suppose $\\{X_{G},Y_{G},\lambda_{u},\rho_{u}\\}_{G,u}$ is a given _coherent
family of pairs of objects in $\mathcal{M}$_. It consists of objects
$X_{G},Y_{G}\in\mathcal{M}(G)$ for $G\in\operatorname{Obj}\mathbb{G}$ and
identifications $\lambda_{u}\colon X_{H}\overset{\sim}{\to}u^{*}X_{G}$ and
$\rho_{u}\colon u^{*}Y_{G}\overset{\sim}{\to}Y_{H}$ for every 1-morphism
$u\colon H\to G$ of $\mathbb{G}$, suitably compatible with the 2-morphisms of
$\mathbb{G}$. Then there exists an ordinary Mackey functor
$M=M(\\{X_{G},Y_{G},\lambda_{u},\rho_{u}\\}_{G,u})$ for
$(\mathbb{G};\mathbb{J})$ with value

$M(G)=\operatorname{Hom}_{\mathcal{M}(G)}(X_{G},Y_{G})$

at $G\in\operatorname{Obj}\mathbb{G}$. Restriction maps
$u^{\scriptscriptstyle\bullet}=M([G\overset{u}{\leftarrow}H=H])\colon M(G)\to
M(H)$ and transfer maps
$u_{\scriptscriptstyle\bullet}=M([H=H\overset{u}{\to}G])\colon M(H)\to M(G)$
(if $u\in\mathbb{J}$) are defined by

$u^{\scriptscriptstyle\bullet}(\varphi):=\rho_{u}\circ
u^{*}(\varphi)\circ\lambda_{u}\quad\textrm{and}\quad
u_{\scriptscriptstyle\bullet}(\psi):={}^{\ell}\varepsilon\circ
u_{*}(\rho^{-1}_{u}\psi\lambda_{u}^{-1})\circ{}^{r\\!}\eta$

respectively, for all $\varphi\in\mathcal{M}(G)(X_{G},Y_{G})$ and
$\psi\in\mathcal{M}(H)(X_{H},Y_{H})$.

We are now ready for the last three abstract results of our theory, which
specify three independent (but occasionally overlapping) methods of squeezing
ordinary Green functors from a Mackey or Green 2-functor.

######  12.5 Theorem (Green functors via K-decategorification).

Let $\mathcal{M}$ be a Green 2-functor for $(\mathbb{G};\mathbb{J})$ and
suppose its value categories $\mathcal{M}(G)$ are all small. Then:

  1. (1)

The K-decategorification
$M=\operatorname{K_{0}^{add}}\circ\tau_{1}(\widehat{M})$ of $\mathcal{M}$
(12.1) is an ordinary Green functor for $(\mathbb{G};\mathbb{J})$, commutative
if $\mathcal{M}$ is braided. As a ring, $M(G)$ is just the
$\mathrm{K}_{0}$-ring of $\mathcal{M}(G)$ with multiplication induced by its
internal tensor product.

  2. (2)

Similarly, if $\mathcal{N}$ is a left (or right) module over $\mathcal{M}$
(5.9), its K-decategorification $N$ is a left (or right) module over the Green
functor $M$ of (1).

Moreover, if $\mathcal{M}$ takes values in exact or triangulated categories
and if the tensor products of $\mathcal{M}$ preserve exact sequences or
triangles in both variables, the same conclusion holds for the
K-decategorifications using $\operatorname{K_{0}^{exa}}$,
$\operatorname{K^{exa}_{*}}$ or $\operatorname{K_{0}^{tri}}$ instead (as in
Remarks 12.2 and 12.3).

###### Proof.

We know from [BD20, Prop. 2.5.5] that the K-decategorification of a Mackey
2-functor is an ordinary Mackey functor (the proof—essentially recalled in
12.1—immediately adapts to the exact and triangulated variations, as well as
to higher K-theory as in 12.3). By hypothesis the tensor structure of a Green
2-functor preserves directs sums, respectively exact sequences or triangles,
in both variables. It follows that it turns each K-group into a ring and each
restriction map into a ring morphism; and similarly for the actions on
modules. Evidently the rings are commutative if $\mathcal{M}$ is braided.
Finally, the Frobenius formulas (b) of 11.2 follow immediately from 6.4. Hence
we get ordinary Green functors as claimed. (See [Wei13, IV §6] on how to
induce parings in higher K-theory of exact categories.) This settles part (1),
and part (2) is similar. ∎

######  12.6 Theorem (Green functors via Hom-decategorification, I).

Let $\mathcal{M}$ be any Mackey (!) 2-functor for $(\mathbb{G};\mathbb{J})$.
Then:

  1. (1)

The Hom-decategorification
$M=M(\\{X_{G},X_{G},\lambda_{u},\lambda^{-1}_{u}\\}_{G,u})$ of $\mathcal{M}$
for any coherent choice of objects such that $X_{G}=Y_{G}$ and
$\rho_{u}=\lambda_{u}^{-1}$ (12.4) is a Green 2-functor for
$(\mathbb{G};\mathbb{J})$. As a ring, each $M(G)$ is just
$\operatorname{End}_{\mathcal{M}(G)}(X_{G})$. The resulting Green functors are
commutative if the endomorphism ring of each $X_{G}$ is commutative.

  2. (2)

Similarly, the Hom-decategorification for any coherent choice
$\\{X_{G},Y_{G},\lambda_{u},\rho_{u}\\}$ of pairs of objects in $\mathcal{M}$
is a left module over $M(\\{Y_{G},Y_{G},\rho^{-1}_{u},\rho_{u}\\}_{G,u})$ and
a right module over $M(\\{X_{G},X_{G},\lambda_{u},\lambda^{-1}_{u}\\}_{G,u})$,
the latter two being the endomorphism Green functors as in (1).

###### Proof.

We know from [BD21a, Theorem A.7] that the Hom-decategorification of any
Mackey 2-functor $\mathcal{M}$ at any coherent family of pairs is a Mackey
functor. The product and action pairings of the claims, which are simply given
by composition of morphisms, are clearly preserved by the restriction maps
$u^{\scriptscriptstyle\bullet}$ of these Mackey functors because we suppose
$\lambda$ and $\rho$ to be mutual inverses. Hence the only remaining doubts
concern the Frobenius formulas.

Let $u\colon H\to G$ be in $\mathbb{J}$. We only verify the first formula
$x\cdot
u_{\scriptscriptstyle\bullet}(y)=u_{\scriptscriptstyle\bullet}(u^{\scriptscriptstyle\bullet}(x)\cdot
y)$ in the ring case (1), as the proofs of the second formula or for modules
(2) are similar. Thus let $x\in
M(G)=\operatorname{End}_{\mathcal{M}(G)}(X_{G})$ and $y\in
M(H)=\operatorname{End}_{\mathcal{M}(H)}(X_{H})$, and consider the following
diagram in $\mathcal{M}(G)$ (where we write
$\rho:=\rho_{u}=\lambda_{u}^{-1}$):

$\textstyle{X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{r\\!}\eta}$$\scriptstyle{u_{\scriptscriptstyle\bullet}(y)}$$\scriptstyle{u_{\scriptscriptstyle\bullet}(u^{\scriptscriptstyle\bullet}(x)\cdot
y)}$$\textstyle{X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{X_{G}}$$\textstyle{u_{*}u^{*}X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{*}(\rho^{-1}y\rho)}$$\scriptstyle{u_{*}(u^{*}(x)\cdot\rho^{-1}y\rho)}$$\textstyle{u_{*}u^{*}X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{\ell}\varepsilon}$$\scriptstyle{u_{*}u^{*}(x)}$$\textstyle{u_{*}u^{*}X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{\ell}\varepsilon}$

The right square commutes by naturality of ${}^{\ell}\varepsilon$, the left
square by definition of $u_{\scriptscriptstyle\bullet}(y)$, and the lower
triangle by functoriality of $u_{*}$. Since the bottom curved arrow is also
equal to
$u_{*}(\rho^{-1}\rho\,u^{*}(x)\,\rho^{-1}y\rho)=u_{*}(\rho^{-1}(u^{\scriptscriptstyle\bullet}(x)\,y)\rho)$,
the outer square commutes by definition of
$u_{\scriptscriptstyle\bullet}(u^{\scriptscriptstyle\bullet}(x)\cdot y)$. We
deduce from all this that the top triangle also commutes, proving the claimed
formula.

All remaining details are straightforward and are left to the reader. ∎

######  12.7 Theorem (Green functors via Hom-decategorification, II).

Let $\mathcal{M}$ be a Green 2-functor for $(\mathbb{G};\mathbb{J})$. Then:

  1. (1)

Suppose $\\{X_{G},Y_{G},\lambda_{u},\rho_{u}\\}_{G,u})$ is a coherent family
of pairs in $\mathcal{M}$ such that: Each $X_{G}$ is a comonoid in
$\mathcal{M}(G)$, each $Y_{G}$ a monoid, each $\lambda_{u}$ a comonoid map and
each $\rho_{u}$ a monoid map. Then its Hom-decategorification $M$ (as in 12.4)
is a Green functor for $(\mathbb{G};\mathbb{J})$ where the multiplication on
$M(G)=\operatorname{Hom}_{\mathcal{M}(G)}(X_{G},Y_{G})$ is the associated
convolution product.

  2. (2)

The Mackey functor induced by a family whose objects are left (resp. right)
modules and comodules over the monoids and comonoids of part (1) and whose
coherence maps are morphisms of such, is a left (resp. right) module over the
Green functor of (1).

###### Proof.

As before we verify (1) and leave the similar (2) to the reader.

Write $(X_{G},\delta_{G},\epsilon_{G})$ and $(Y_{G},\mu_{G},\iota_{G})$ for
the given comonoid and monoid structures. By definition, the convolution
product of two elements $x,y\in M(G)$ is $x\cdot y=\mu_{G}\circ(x\otimes
y)\circ\delta_{G}$. For any $u\colon H\to G$, the restriction map
$u^{\scriptscriptstyle\bullet}\colon M(G)\to M(H)$ preserves the convolution
product by the commutativity of the following diagram:

$\textstyle{u^{*}X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}(x\cdot
y)}$$\scriptstyle{u^{*}\delta_{G}}$$\textstyle{u^{*}(X_{G}\otimes
X_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}(x\otimes
y)}$$\textstyle{u^{*}(Y_{G}\otimes
Y_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}\mu_{G}}$$\textstyle{u^{*}Y_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{u}}$$\textstyle{X_{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{u}}$$\scriptstyle{\delta_{H}}$$\textstyle{Y_{H}}$$\textstyle{X_{H}\otimes
X_{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{\scriptscriptstyle\bullet}(x)\otimes
u^{\scriptscriptstyle\bullet}(y)}$$\scriptstyle{\lambda_{u}\otimes\lambda_{u}}$$\textstyle{u^{*}X_{G}\otimes
u^{*}X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{str}}$$\scriptstyle{\simeq}$$\scriptstyle{u^{*}x\otimes
u^{*}y}$$\textstyle{u^{*}Y_{G}\otimes
u^{*}Y_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{str}}$$\scriptstyle{\simeq}$$\scriptstyle{\rho_{u}\otimes\rho_{u}}$$\textstyle{u^{*}Y_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{H}}$

Note that the left and right pentagons commute because $\lambda_{u}$ is a
comonoid morphism and $\rho_{u}$ is a monoid morphism. A similar diagram shows
that $u^{\scriptscriptstyle\bullet}$ preserves the unit, that is
$u^{\scriptscriptstyle\bullet}(\iota_{G}\epsilon_{G})=\iota_{H}\epsilon_{H}$.
It remains to verify the Frobenius formulas; we only check $x\cdot
u_{\scriptscriptstyle\bullet}(y)=u_{\scriptscriptstyle\bullet}(u^{\scriptscriptstyle\bullet}(x)\cdot
y)$ and omit the analogous verification of the other one. For any two $x\in
M(G),y\in M(H)$, observe the following diagram, where $\mathrm{Rp}^{1}$ is the
first right projection map of 6.1:

(12.8) $\textstyle{X_{G}\otimes
X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{r\\!}\eta}$$\scriptstyle{\mathrm{id}\otimes{}^{r\\!}\eta}$$\textstyle{X_{G}\otimes
u_{*}u^{*}X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Rp}^{1}}$$\scriptstyle{\simeq}$$\scriptstyle{x\otimes
u_{*}(\rho^{-1}_{u}y\lambda^{-1}_{u})}$$\textstyle{Y_{G}\otimes
u_{*}u^{*}Y_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Rp}^{1}}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{id}\otimes{}^{\ell}\varepsilon}$$\textstyle{Y_{G}\otimes
Y_{G}}$$\textstyle{u_{*}u^{*}(X_{G}\otimes
X_{G})}$$\textstyle{u_{*}(u^{*}X_{G}\otimes
u^{*}X_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{*}(\mathrm{str})}$$\scriptstyle{\simeq}$$\scriptstyle{u_{*}(u^{*}x\otimes\rho_{u}^{-1}y\lambda_{u}^{-1})}$$\textstyle{u_{*}(u^{*}Y_{G}\otimes
u^{*}Y_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{u_{*}(\mathrm{str})}$$\scriptstyle{\mathrm{Lp}^{1}}$$\textstyle{u_{*}u^{*}(Y_{G}\otimes
Y_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{\ell}\varepsilon}$

It is commutative: The central square commutes by the naturality of
$\mathrm{Rp}^{1}$. The left square commutes because it is the perimeter of the
following diagram,

$\textstyle{X_{G}\otimes
X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{r\\!}\eta}$$\scriptstyle{\mathrm{id}\otimes{}^{r\\!}\eta}$$\textstyle{X_{G}\otimes
u_{*}u^{*}X_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{r\\!}\eta}$def.$\scriptstyle{\mathrm{Rp}^{1}_{X_{G},u^{*}X_{G}}}$$\textstyle{u_{*}u^{*}(X_{G}\otimes
X_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{*}u^{*}(\mathrm{id}\otimes{}^{r\\!}\eta)}$$\textstyle{u_{*}u^{*}(X_{G}\otimes
u_{*}u^{*}X_{G})}$$\textstyle{u_{*}(u^{*}X_{G}\otimes
u^{*}X_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{*}(\mathrm{str})}$$\scriptstyle{\simeq}$$\scriptstyle{u_{*}(\mathrm{id}\otimes
u^{*}{}^{r\\!}\eta)}$$\textstyle{u_{*}(u^{*}X_{G}\otimes
u^{*}u_{*}u^{*}X_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{*}(\mathrm{str})}$$\scriptstyle{\simeq}$$\scriptstyle{u_{*}(\mathrm{id}\otimes{}^{r\\!}\varepsilon)}$$\textstyle{u_{*}(u^{*}X_{G}\otimes
u^{*}X_{G})}$

which commutes by: the naturality of ${}^{r\\!}\eta$ (for the top square), the
naturality of the strong monoidal isomorphism of $u^{*}$ (the middle square),
and a unit-counit relation for $i^{*}\dashv i_{*}$ (the bottom triangle). The
right square in (12.8) commutes by a similar diagram, after replacing
$\mathrm{Rp}^{1}$ with $(\mathrm{Lp}^{1})^{-1}$ (see 6.4). Thus (12.8) is
commutative, as claimed, and we are now ready to contemplate our last diagram:

As before, the two pentagons commute (even before $u_{*}$) because
$\lambda=\lambda_{u}$ and $\rho=\rho_{u}$ are morphisms of (co)monoids. The
two rhombuses commute by the naturality of ${}^{r\\!}\eta$ and
${}^{\ell}\varepsilon$, respectively, and the middle square (even before
$u_{*}$) by the functoriality of $\otimes=\otimes_{H}$. Thus the whole diagram
commutes, completing the proof. ∎

######  12.9 Remark.

If $\mathcal{M}$ takes values in $\mathbb{Z}$-graded categories and grading-
preserving functors, we have graded versions of all the Green functors and
modules in Theorems 12.6 and 12.7, with values given instead by the graded
Homs: $G\mapsto M(G)=\operatorname{Hom}^{*}_{\mathcal{M}(G)}(X_{G},Y_{G})$.
For Green 2-functors with values in tensor triangulated categories one
typically uses the grading induced by the suspension functor.

######  12.10 Example.

In the local case
$\mathbb{G}=\mathbb{J}={\mathsf{gpd}{}^{\mathsf{f}}_{\\!{}^{\scriptscriptstyle/}\\!G}}$
for a fixed $G$, any pair of objects $X,Y\in\mathcal{M}(G)$ in the “top”
category gives rise to a coherent family of pairs simply by restriction:
$X_{H}=u^{*}X_{G}$ and $Y_{H}=u^{*}X_{H}$ for all $u\colon H\to G$, with
$\lambda=\rho=\mathrm{id}$.

######  12.11 Example.

For any Green 2-functor $\mathcal{M}$, the tensor unit objects
$X_{G}=Y_{G}=\mathbb{1}$ form a coherent family of pairs thanks to the unit
isomorphisms $u^{*}X_{G}\cong X_{H}$ of the strong monoidal functors $u^{*}$.
The resulting Green functor
$G\mapsto\operatorname{End}_{\mathcal{M}(G)}(\mathbb{1})$ is commutative by
the Eckmann–Hilton argument. In case the categories carry tensor-compatible
gradings as in [SA04] (e.g. when $\mathcal{M}$ takes values in tensor
triangulated categories), we obtain a _graded_ commutative version
$G\mapsto\operatorname{End}_{\mathcal{M}(G)}^{*}(\mathbb{1})$.

$*\;\;*\;\;*$

We conclude by explaining how the classical Green functors used in algebra and
topology arise by K- or Hom-decategorification, or sometimes both. All Hom-
examples occur by specializing either 12.10 or 12.11, occasionally in graded
form (12.9). The relevant pair $(\mathbb{G};\mathbb{J})$ is always the same as
for the used Green 2-functor. Further variations are possible and are left to
the interested reader.

######  12.12 Example (Group cohomology).

The cohomology ring Green functor $G\mapsto\mathrm{H}^{*}(G;\Bbbk)$ arises by
Hom-decategorification at the tensor units of the derived category Green
2-functor $G\mapsto\operatorname{D}(\Bbbk G)$ (10.3). For every
$V\in\operatorname{D}(\Bbbk G)$, (hyper-) cohomology with twisted coefficients
$H\mapsto\mathrm{H}^{*}(H;V|_{H})$ yields a Mackey functor for $G$ which is a
module over the cohomology ring Green functor (for $G$).

######  12.13 Example (Fixed points).

For a fixed group $G$, suppose $A$ is a $G$-algebra, i.e. a monoid in
$\operatorname{Mod}(\Bbbk G)$. Then $H\mapsto\mathrm{H}^{0}(H;A)=A^{H}$ yields
a Green functor for $G$ whose value at $H\leq G$ is the subalgebra of
$H$-fixed points in $A$. Indeed, this arises by Hom-decategorification as in
12.7 of the ($G$-local version of the) Green 2-functor
$H\mapsto\operatorname{Mod}(\Bbbk H)$ at the family of pairs induced (as in
12.10) by the comonoid $X_{G}=\mathbb{1}=\Bbbk$ and the monoid $Y_{G}=A$.
Passing to the derived category, we get a Green functor for $G$ whose value at
$H$ is the whole graded cohomology algebra $\mathrm{H}^{*}(H;A)$.

######  12.14 Example (Tate cohomology).

If in 12.12 we replace the derived category with the stable category Green
2-functor $G\mapsto\operatorname{Stab}(\Bbbk G)$ (10.6), we similarly obtain
the Tate cohomology Green functor $G\mapsto\widehat{\mathrm{H}}^{*}(G;\Bbbk)$
and Tate cohomology with any twisted coefficients as a ($G$-local) module over
it.

######  12.15 Example (The $\mathrm{G}_{0}$-ring).

If $\Bbbk$ is a field (so that $-\otimes_{\Bbbk}-$ is exact), consider the
Green 2-functor $G\mapsto\mathrm{mod}(\Bbbk G)$ of finite dimensional modules
(10.3). By K-decategorification via $\operatorname{K_{0}^{exa}}$ (see 12.2),
we get the Green 2-functor whose values are $\mathrm{G}_{0}(\Bbbk G)$, the
G-theory group of the group algebras.

######  12.16 Example (The trivial source ring).

The trivial source ring Green functor is the
$\operatorname{K_{0}^{add}}$-decategorification of the Green 2-functor
$G\mapsto\mathrm{perm}_{\Bbbk}(G)^{\natural}$ of trivial source modules
(10.3). This is usually defined over a field $\Bbbk$ of positive
characteristic $p>0$, in which case such modules are known as $p$-permutation
modules.

######  12.17 Example (The representation ring).

For a general $\Bbbk$, the representation ring (or Green ring) Green functor
$G\mapsto\mathrm{R}_{\Bbbk}(G)$ is the
$\operatorname{K_{0}^{add}}$-decategorification of the Green 2-functor
$G\mapsto\operatorname{latt}_{\Bbbk}(G)$ of $\Bbbk G$-lattices (10.3).

######  12.18 Example (Algebraic K-theory).

For a field $\Bbbk$ of characteristic zero, consider the Green 2-functor
$G\mapsto\mathrm{proj}(\Bbbk G)=\mathrm{mod}(\Bbbk G)$ of finitely generated
(necessarily projective) modules (10.3; note that $-\otimes_{\Bbbk}-$ is exact
on $\Bbbk G$-projectives even if $\Bbbk$ is not a field, but we also want a
tensor unit!). Via K-decategorification as in 12.3, we obtain from it a
(graded) Green functor $G\mapsto\mathrm{K}_{*}^{\mathrm{alg}}(\Bbbk G)$ whose
values are the algebraic K-theory groups of the group algebra, with
multiplication induced by the Hopf algebra structure of $\Bbbk G$.

######  12.19 Example (The Burnside ring).

The Burnside ring Green functor
$G\mapsto\mathrm{B}(G)=\mathrm{K}_{0}(G\textrm{-}\mathsf{set},\sqcup,\times)$
(denoted $A(G)$ by topologists) arises in two ways. Firstly as the
$\operatorname{K_{0}^{add}}$-decategorification of the Green 2-functor of
“representable” ordinary Mackey functors, i.e. the additive category of spans
of finite $G$-sets $G\mapsto\mathbb{Z}\tau_{1}\operatorname{{\sf
Span}}(G\textrm{-}\mathsf{set})$ (see 10.8), since its K-ring is the same as
for $G$-sets. Secondly as the Hom-decategorification at the tensor units of
the stable homotopy Green 2-functor $G\mapsto\operatorname{SH}(G)$ (10.11).
For any genuine $G$-spectrum $X\in\operatorname{SH}(G)$, the homotopy groups
$H\mapsto\pi_{*}^{H}(X)=\operatorname{SH}(H)_{*}(\mathbb{1},X|_{H})$ yield a
graded Mackey functor for $G$, i.e. a graded module over the Burnside ring
(for $G$). The latter can all be realized this way by the existence of
equivariant Eilenberg–Mac Lane spectra.

######  12.20 Example (The complex representation ring).

Similarly, the complex representation ring Green functor
$G\mapsto\mathrm{R}_{\mathbb{C}}(G)$ arises both as the K-decategorification
of the Green 2-functor $G\mapsto\operatorname{mod}(\mathbb{C}G)$ (as in
Examples 12.15 or 12.17 with $\Bbbk=\mathbb{C}$), and as the Hom-
decategorification at the tensor units of the Kasparov theory Green 2-functor
$G\mapsto\mathrm{KK}(G)$ (10.12). For any separable $G$-C*-algebra
$A\in\mathrm{KK}(G)$, the equivariant K-theory groups
$H\mapsto\mathrm{K}_{*}^{H}(A)=\mathrm{KK}(H)_{*}(\mathbb{1},A|_{H})$ yield a
graded Mackey functor for $G$ which is a module over the representation ring.
This recovers results of [Del14, §4].

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