# Complex cobordism modulo $c_{1}$-spherical cobordism and related genera

M. Bakuradze  Faculty of exact and natural sciences, A. Razmadze Math.
Institute, Iv. Javakhishvili Tbilisi State University, Georgia
<EMAIL_ADDRESS>
###### Abstract.

We prove that the ideal in complex cobordism ring ${\mathbf{M}\mathbf{U}}^{*}$
generated by the polynomial generators $S=(x_{1},x_{k},k\geq 3)$ of
$c_{1}$-spherical cobordism ring $W^{*}$, viewed as elements in
${\mathbf{M}\mathbf{U}}^{*}$ by forgetful map is prime. Using the Baas-
Sullivan theory of cobordism with singularities we define a commutative
complex oriented cohomology theory ${\mathbf{M}\mathbf{U}}^{*}_{S}(-)$,
complex cobordism modulo $c_{1}$-spherical cobordism, with the coefficient
ring ${\mathbf{M}\mathbf{U}}^{*}/S$. Then any $\Sigma\subseteq S$ is also
regular in ${\mathbf{M}\mathbf{U}}^{*}$ and therefore gives a multiplicative
complex oriented cohomology theory ${\mathbf{M}\mathbf{U}}^{*}_{\Sigma}(-)$.
The generators of $W^{*}[1/2]$ can be specified in such a way that for
$\Sigma=(x_{k},k\geq 3)$ the corresponding cohomology is identical to the Abel
cohomology, previously constructed in [14]. Another example corresponding to
$\Sigma=(x_{k},k\geq 5)$ gives the coefficient ring of the universal
Buchstaber formal group law after tensored by $\mathbb{Z}[1/2]$, i.e., is
identical to the scalar ring of the Krichever-Hoehn complex elliptic genus
[18], [17].

###### Key words and phrases:

Complex bordism, $SU$-bordism, Formal group law, Complex elliptic genus

###### 2010 Mathematics Subject Classification:

55N22; 55N35

The author was supported by EU fellowships for Georgian researchers, 2023
(57655523)

##  1\. Introduction and statements

The theory of complex cobordism ${\mathbf{M}\mathbf{U}}^{*}(-)$ and special
unitary cobordism ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}(-)$ play an important
role in cobordism theory [25]. The ring ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}$
is torsion free after localized away from 2

(1.1)
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]=\mathbb{Z}[1/2][y_{2},y_{3},\cdots],\,\,\,\,|y_{i}|=2i.$

With $SU$-structure forgetful inclusion in mind

${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]\subset{\mathbf{M}\mathbf{U}}^{*}[1/2]=\mathbb{Z}[1/2][a_{1},a_{2},a_{3},\cdots],\,\,\,\,|a_{i}|=2i,$

the generators $y_{i}$ can be treated as elements in
${\mathbf{M}\mathbf{U}}^{*}[1/2]$.

In particular $y_{i}$ is a $SU$-manifold if and only if all Chern numbers of
$y_{i}$ having factor $c_{1}$ are zero. Then we have to check the main Chern
number $s_{i}(y_{i})$ for Novikov’s criteria [23] (see (3.1) below) for the
membership of the set of polynomial generators in
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$.

In [16] Conner and floyd introduced $c_{1}$-spherical cobordism groups
$W^{2n}\subset{\mathbf{M}\mathbf{U}}^{2n}$. A complex cobordism class belongs
to $W^{2n}$ if and only if every Chern number involving $c_{1}^{2}$ vanishes.

$W^{*}$ is not a subring of ${\mathbf{M}\mathbf{U}}^{*}$. However, with
respect to some complex orientation and some $*$ multiplication (see Section
2), the ring $W^{*}$ is polynomial on generators in every positive even degree
except 4, [15], [20]:

(1.2) $W^{*}=Z[x_{1},x_{k}:k\geq
3],\,\,x_{1}={\mathbf{C}\mathbf{P}}_{1},\,x_{k}\in W_{2k}.$

The polynomial generators $x_{k}$ can be specified by the condition for the
main Chern numbers $s_{k}(x_{k})=d(k)d(k-1)$ for $k\geq 3$, where

$d(k)=\begin{cases}p,&\mbox{if $k+1=p^{s}$ for some prime $p$,}\\\
1,&\mbox{otherwise}.\end{cases}$

By [15] the cohomology theory $W^{*}(-)$ is complex oriented and (by
Proposition 3.12) the ring $W^{*}[1/2]$ is generated by the coefficients of
the corresponding formal group law $F_{W}$. In particular

(1.3) $\displaystyle W^{*}[1/2]\text{ is a free
}{\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]\text{ module generated by 1 and
}[{\mathbf{C}\mathbf{P}}_{1}].$

Our main observation is that the sequence of polynomial generators of $W^{*}$,
$(x_{k},k\geq 3)$ in (1.2) viewed as elements in ${\mathbf{M}\mathbf{U}}^{*}$
by forgetful map, generate the same ideal in ${\mathbf{M}\mathbf{U}}^{*}[1/2]$
as the coefficients $\alpha_{ij},i,j\geq 2$ of the universal formal group law.

We prove in Section 3 one of our main results

######  Theorem 1.1.

Let $S=(x_{1},x_{k},k\geq 3)$ be a sequence of polynomial generators of
$c_{1}$-spherical cobordism $W^{*}$. Then

i) $S$ is regular in ${\mathbf{M}\mathbf{U}}^{*}$.

ii) any subsequence $\Sigma$ of $S$ is regular in
${\mathbf{M}\mathbf{U}}^{*}$.

Then the use of Baas-Sullivan theory of cobordism with singularities defines a
multiplicative cohomology theory ${\mathbf{M}\mathbf{U}}^{*}_{\Sigma}(-)$,
with the scalar ring ${\mathbf{M}\mathbf{U}}^{*}/(\Sigma)$.

The obstruction to commutativity in ${\mathbf{M}\mathbf{U}}^{*}_{\Sigma}(-)$
studied in [21]. In some cases all obstructions vanish only for dimensional
reasons. So is for $\Sigma=S$ in Theorem 1.1.

In Section 3 we define ${\mathbf{M}\mathbf{U}}^{*}_{S}(-)$, the complex
cobordism modulo $c_{1}$-spherical cobordism.

######  Theorem 1.2.

Let $\Sigma=S$ in Theorem 1.1. There is a commutative complex oriented
cohomology theory ${\mathbf{M}\mathbf{U}}^{*}_{S}(-)$, with the coefficient
ring ${\mathbf{M}\mathbf{U}}^{*}/S$.

Let ${\mathbf{C}\mathbf{P}}_{i}$ be the cobordism class of complex projective
spae of dimension $2i$ and

$F_{U}=\sum\alpha_{ij}x^{i}y^{j}$

be the universal formal group law of complex cobordism. One can choose the
following polynomial generators $y_{2},y_{3}$ and $y_{4}$ in
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ viewed as elements in
${\mathbf{M}\mathbf{U}}^{*}[1/2]$ (see Section 3.5)

(1.4) $\displaystyle
y_{2}={\mathbf{C}\mathbf{P}}_{2}-\frac{9}{8}{\mathbf{C}\mathbf{P}}_{1}^{2},$
$\displaystyle y_{3}=-\alpha_{22},$ $\displaystyle
y_{4}=-\alpha_{23}+\frac{3}{2}\alpha_{22}{\mathbf{C}\mathbf{P}}_{1}.$

We need the following result on quasi-toric representatives in the
$SU$-bordism ring.

Theorem ([20], Thm 10.8.) There are quasi-toric $SU$-manifolds $M^{2i}$ ,
$i\geq 5$, with $s_{i}(M^{2i})=d(i)d(i-1)$ for odd $i$ and
$s_{i}(M^{2i})=2d(i)d(i-1)$ for even $i$, and they have the minimum possible
characteristic numbers $s_{i}$ and represent polynomial generators of
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$.

Let us specify the generators in (1.1) as

$y_{i}=[M^{2i}],i\geq 5.$

The motivation to this is that the Krichever complex elliptic genus (see [10],
§ E.5) vanishes on these generators.

We could also choose $y_{i},i\geq 5$ based on the following result

Theorem ([26], Thm 6.1.) The kernel of the complex elliptic genus on
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ is equal to the ideal of
$SU$-flops.

One interesting example of Theorem 1.1 is the Abel cohomology $h^{*}_{Ab}$
constructed in [14].

######  Theorem 1.3.

Let $\Sigma=(x_{i},i\geq 3)$, where $x_{i}=y_{i}$ is specified above.Then
${\mathbf{M}\mathbf{U}}^{*}_{\Sigma}(-)[1/2]$ is identical to the Abel
cohomology $h^{*}_{Ab}[1/2]$.

Another example of Theorem 1.1 is related to the Buchstaber formal group law
$F_{B}$ [6], [7]. After tensored with rationals the corresponding classifying
map $f_{B}$ is identical [3] to Krichver-Hoehn complex elliptic genus [18],
[17]. The scalar ring $\Lambda_{B}$ of $F_{B}$ is calculated in [11]. In
particular, it has only 2-torsion and the quotient

$\Lambda_{\mathcal{B}}=\Lambda_{B}/torsion$

is an integral domain. We will consider this example in Section 3 and prove
the following

######  Theorem 1.4.

Let $\Sigma=(x_{i},i\geq 5)$, where $x_{i}=y_{i}$ is specified above. Let
$\Lambda_{B}$ be the scalar ring of the universal Buchstaber formal group law.
Then cohomology theory ${\mathbf{M}\mathbf{U}}^{*}_{\Sigma}(-)[1/2]$ is
multiplicative, commutative, complex oriented and has the coefficient ring
identical to ${\mathbf{M}\mathbf{U}}^{*}[1/2]/\Sigma:=\Lambda_{B}[1/2]$.

In Section 4 we discuss a genus on $c_{1}$-spherical cobordism $W^{*}$

(1.5) $\phi_{W}:W^{*}[1/2]\to\mathbb{[}Z/2][x_{1},x_{3},x_{4}].$

The following questions remains open.

What about integral versions of Theorem 1.3 and Theorem 1.4?

Is the cohomology theory ${\mathbf{M}\mathbf{U}}^{*}/(x_{i},i\geq 5)$
commutative? The same question arises for other examples of Theorem 1.1 ii).

##  2\. Preliminaries

###  2.1. Krichever complex elliptic genus

The Krichever complex elliptic genus (also called Krichever-Höhn genus) has
been defined in [18] and [17]. Krichever wrote down its characteristic power
series $Q(x)$ using Baker-Akhiezer function. In [17], Höhn defined four
variable elliptic genus

(2.1)
$\phi_{KH}:{\mathbf{M}\mathbf{U}}^{*}\to\mathbb{Q}[p_{1},p_{2},p_{3},p_{4}],\,\,\,|p_{i}=2i|.$

determined by the following property: if one denotes by $f$ the exponent of
the corresponding formal group $F_{KH}$, then the series

$h(x):=\frac{f^{\prime}(x)}{f(x)}$

satisfies the differential equation

(2.2) $(h^{\prime}(x))^{2}=S(h(x)),$

where $S$ is the generic monic polynomial of degree 4:

$S(t)=p_{4}+p_{3}t+p_{2}t^{2}+p_{1}t^{3}+t^{4}.$

By definition [8] the universal Krichever formal group law $F_{K}$ is of the
form

(2.3)
$F_{K}(x,y)=xw(y)+yw(x)-w^{\prime}(0)xy+\frac{w(x)\beta(x)-w(y)\beta(y)}{xw(y)-yw(x)}x^{2}y^{2},$

where

$w=\frac{\partial F_{K}(x,y)}{\partial x}(x,0)$

is the invariant differential form of $F_{K}$ and
$\beta(x)=\frac{w^{\prime}(x)-w^{\prime}(0)}{2x}.$

The universal Krichever genus on $\phi_{K}$ on ${\mathbf{M}\mathbf{U}}^{*}$ is
the homomorphism classifying the formal group law $F_{K}$.

######  Lemma 1.

([3], Thm 2.3) The exponent of $F_{K}$ satisfies (2.3), i.e., $\phi_{K}$ is a
specialization of $\phi_{KH}$.

The universal Buchstaber formal group law $F_{B}$ is defined [5] as a
universal example among formal group laws of the form

$\frac{x^{2}A(y)-y^{2}A(x)}{xB(y)-yB(x)},$

where $A(0)=B(0)=1$.

If $A^{\prime}(0)=B^{\prime}(0)$ then $B(x)$ is identical to the image of
$\omega(x)$ under the classifying map $f_{B}$.

One can suggest kernel of $\phi_{K}$ directly in terms of the universal formal
group law $F_{U}$ as follows [4],[5]

######  Lemma 2.

Let $A(x,y)=\sum A_{ij}x^{i}y^{j}$ be defined by

$A(x,y)=F_{U}(x,y)(xw(y)-yw(x)),\,\,\text{\,\, where
\,\,\,}w(x)=\frac{\partial F_{U}(x,y)}{\partial x}(x,0)$

is the invariant differential form of $F_{U}$. Then $F_{k}$ is identical to
$F_{B}$ and the classifying map $f_{B}=f_{K}$ is the obvious quotient map

${\mathbf{M}\mathbf{U}}^{*}\to{\mathbf{M}\mathbf{U}}^{*}/(A_{ij},i,j\geq 3).$

######  Remark 1.

As shown in [11], all torsion elements of $\Lambda_{B}$, the scalar ring of
$F_{B}$ have order two. Furthermore, there is a set of multiplicative
generators for $\Lambda_{B}$ with one generator in every degree of the form
$p^{k}$, where $p$ is a prime and $k\geq 0$, and one generator in every degree
$2^{n}-2$, where $n\geq 3$; the latter generators generate the torsion
subring.

###  2.2. $c_{1}$-spherical cobordism

in [16] Conner and floyd introduced the groups
$W^{2n}\subset{\mathbf{M}\mathbf{U}}^{2n}$. A complex cobordism class belongs
to $W^{2n}$ if and only if every Chern number involving $c_{1}^{2}$ vanishes.

Recall the idempotent in [9], [15]

$\pi_{0}:{\mathbf{M}\mathbf{U}}^{*}\to{\mathbf{M}\mathbf{U}}^{*}:\,\,\,\pi_{0}=1+\sum_{k\geq
2}\alpha_{1k}\partial_{k}$

and the projection $\pi_{0}:{\mathbf{M}\mathbf{U}}^{*}\to W^{*}=Im\pi_{0}$.

By [15] Proposition 2.15 the multiplication $*$ in $W^{*}$ is given by

$a*b=\pi_{0}(a\cdot b)=ab+2[V]\partial a\partial b,$

where $[V]=\alpha_{12}\in{\mathbf{M}\mathbf{U}}^{4}$ is the cobordism class
${\mathbf{C}\mathbf{P}}_{1}^{2}-{\mathbf{C}\mathbf{P}}_{2}$.

Then $W^{*}$ is a subgroup, not a subring in ${\mathbf{M}\mathbf{U}}^{*}$.
With respect to the multiplication $*$ the ring $W^{*}$ is polynomial on
generators in every positive even degree except 4 as in (1.2).

###  2.3. The Abel formal group law

By definition [12] the universal Abel formal group law formal group is the
universal example of the formal group laws of the form

$F_{Ab}=x+y+\alpha_{11}xy+\sum_{i\geq 2}\alpha_{1i}(xy^{i}+x^{i}y).$

By Quillen [24] and Novikov [23] the scalar ring of the universal Lazard
formal group law is identical to ${\mathbf{M}\mathbf{U}}^{*}$, the ring of
complex cobordism. So to put it another way $F_{Ab}$ is classified from the
universal formal group law $F_{U}$ over ${\mathbf{M}\mathbf{U}}^{*}$

$F_{U}=x+y+\sum\alpha_{ij}x^{i}y^{j}$

by the quotient of ${\mathbf{M}\mathbf{U}}^{*}$ modulo the ideal $I_{Ab}$
generated by all coefficients $\alpha_{ij}\in{\mathbf{M}\mathbf{U}}^{*}$, but
only $\alpha_{1i}=\alpha_{i1}$

(2.4) $I_{Ab}=(\alpha_{ij}:i,j\geq 2).$

Buchstaber named the group law $F_{Ab}$ after Abel, motivated by Abel’s
article: ”Abel N. H. Methode generale pour trouver des fonctions d’une seule
quantite variable lorsqu’une propriete de ces fonctions est exprimee par une
equation entre deux variables oeuvres completes, I. Christiania, 1881. p.
1-10.”

In that article, Abel gave a general solution of a functional equation for the
exponent series of $F_{Ab}$.

######  Remark 2.

For our purposes, let us recall from [12] that the scalar ring $F_{Ab}$, i.e.,

$\Lambda_{Ab}={\mathbf{M}\mathbf{U}}^{*}/I_{Ab}$

is non polynomial infinitely generated ring. There is a set of multiplicative
generators for $\Lambda_{Ab}\otimes\mathbb{F}_{2}$ with one generator in every
complex degree $2^{k}$, $k\geq 0$.

$\Lambda_{Ab}$ is an integral domain. After tensored with rational numbers it
is a subring of the polynomial ring in two parameters of degree 2 and 4, i.e.,

(2.5)
$\Lambda_{Ab}\otimes\mathbb{Q}\subset\mathbb{Q}[\alpha,\beta],\,\,\,|\alpha|=2,|\beta|=4.$

in [14] proved that $F_{Ab}$ can be realized by commutative complex oriented
cohomology theory $h^{*}_{Ab}$ also called after Abel.

We will also need

######  Lemma 3.

The classifying map $f_{Ab}:{\mathbf{M}\mathbf{U}}^{*}\to\Lambda_{Ab}$
followed by $\Lambda_{Ab}\otimes\mathbb{Q}$ in is a specialization of the
Krichever-Hoehn complex elliptic genus $\phi_{KH}$.

###### Proof.

A direct proof of this fact could include a verification of the defining
property (2.3) of the Krichever-Höhn genus in case of $F_{Ab}$. Leaving this
to the reader, we derive the lemma from the fact that $F_{Ab}$ is obviously a
specialization of $F_{K}$ as $w(x)=1+\sum_{\geq 1}\alpha_{1i}x^{i}$ and
$\phi_{K}$ is a specialization of $\phi_{KH}$ by Lemma 1.

∎

###  2.4. Some auxiliary combinatorial definitions

Before we continue about the construction of the Abel cohomology, we need to
recall some combinatorial facts.

By Euclid’s algorithm for the natural numbers $m_{1},m_{2},\cdots,m_{k}$ one
can find integers $\lambda_{1},\lambda_{2},\cdots\lambda_{k}$ such that

(2.6)
$\lambda_{1}m_{1}+\lambda_{2}m_{2}+\cdots+\lambda_{k}m_{k}=gcd(m_{1},m_{2},\cdots,m_{k}).$

Let

(2.7)
$d(m)=gcd\bigg{\\{}\binom{m+1}{1},\binom{m+1}{2},\cdots,\binom{m+1}{m-1}\big{|}\,\,m\geq
1\bigg{\\}}.$

By [19] one has

$d(m)=\begin{cases}p,&\mbox{if $m+1=p^{s}$ for some prime $p$,}\\\
1,&\mbox{otherwise}\end{cases}$

therefore the elements

(2.8)
$e_{m}=\lambda_{1}\alpha_{1\,m}+\lambda_{2}\alpha_{2\,m-1}+\cdots+\lambda_{m}\alpha_{m\,1}$

may be chosen as the polynomial generators in ${\mathbf{M}\mathbf{U}}^{*}$.

Let $m\geq 4$ and let $\lambda_{2},\cdots,\lambda_{m-2}$ are such integers
that

(2.9)
$d_{2}(m):=\sum_{i=2}^{m-2}\lambda_{i}\binom{m+1}{i}=gcd\bigg{\\{}\binom{m+1}{2},\cdots,\binom{m+1}{m-2}\bigg{\\}}.$

Then by [11] Lemma 9.7 one has for $m\geq 3$

(2.10) $\displaystyle d_{2}(m)=d(m)d(m-1).$

Let us apply Euclid’s algorithm for the Chern numbers

$s_{m-1}(\alpha_{i,m-i})=\binom{m+1}{i}$

in (2.9) and let

$z_{k}=\sum_{i=2}^{k-1}\lambda_{i}\alpha_{i\,k+1-i},\,\,\,k\geq 3.$

Let us recall from [14] the constriction of abelian cohomology
$h_{Ab}^{*}(-)$.

######  Lemma 4.

Let $I_{Ab}$ be as in (2.4) and $I_{Ab}(l)\subset I_{Ab}$ be generated by
those elements of ${\mathbf{M}\mathbf{U}}^{*}$ whose degree is less or equal
$2l$. Let $I=(z_{3},\cdots)$ and $I(l)=(z_{3},\cdots,z_{l})$. Then

i) $I_{Ab}=I$;

ii) $I_{Ab}(l)=I(l)$.

Let’s put ${\mathbf{M}\mathbf{U}}^{*}=\mathbb{Z}[e_{1},e_{2},\cdots]$ for
$e_{m}$ in (2.8) and let

$A_{l}=\mathbb{Z}[e_{1},e_{2},\cdots,e_{l}]\text{ and
}A^{l+1}=\mathbb{Z}[e_{l+1},e_{l+2},\cdots],$

i.e.,

${\mathbf{M}\mathbf{U}}^{*}=A_{l}\otimes A^{l+1}.$

The preimage of $I(l)$ by obvious inclusion defines the ideal of $A_{l}$
denoted by same symbol so that

(2.11) ${\mathbf{M}\mathbf{U}}^{*}/I(l)=A_{l}/I(l)\otimes A^{l+1}.$

######  Remark 3.

We note here that by definition $I(l)$ is generated by those polynomials in
$\mathbb{Z}[e_{1},e_{2},\cdots,e_{l}]$ that are in the kernel of the quotient
map ${\mathbf{M}\mathbf{U}}^{*}\to{\mathbf{M}\mathbf{U}}_{*}/I_{Ab}$.

The construction of abelian cohomology is based on the following key result
[12], [14].

######  Lemma 5.

The ideals $I$ and $I(l)$ are prime, or equivalently
${\mathbf{M}\mathbf{U}}^{*}/I$ and ${\mathbf{M}\mathbf{U}}^{*}/I(l)$ are
integral domains.

Using Baas-Sullivan theory the commutative complex oriented cohomology
$h_{Ab}^{*}(-)$ with coefficient ring ${\mathbf{M}\mathbf{U}}^{*}/I$ was
defined in [14].

######  Remark 4.

We recall that for a commutative ring $R$, a regular sequence is a sequence
$(r_{1},\cdots,r_{n})$ in $R$ such that $r_{i}$ is not a zero-divisor on
$R/(r_{1},\cdots,r_{i-1}$, for $i=1,\cdots,n$.

If $R$ is a graded ring and the members of a regular sequence are homogeneous
of positive degree, then any permutation of a regular sequence is regular
[13]. Therefore we can regard our regular sequences as regular sets.

##  3\. Proofs

######  Lemma 6.

Let $J_{W}$ be the ideal in ${\mathbf{M}\mathbf{U}}^{*}$ generated by
$y_{2}=w_{1}*w_{1}=8{\mathbf{C}\mathbf{P}}_{2}-9{\mathbf{C}\mathbf{P}}_{1}^{2}$
and polynomial generators $(x_{i},i\geq 3)$ of $W^{*}$ as elements in
${\mathbf{M}\mathbf{U}}_{*}$ by group inclusion and $J_{W}(l)$ be
$(y_{2},x_{3},\cdots,x_{l})$. Let $I$ and $I(l)$ be the ideals in Lemma 4,
$\tilde{I}=(y_{2},I)$ and $\tilde{I}(l)=(y_{2},I(l))$. Then

i) $J_{W}(l)=\tilde{I}(l)$, $\forall l\geq 3$;

ii) The sequence $y_{2},x_{3},\cdots$ is regular in
${\mathbf{M}\mathbf{U}}^{*}$.

###### Proof.

i) First let us prove $\tilde{J}_{W}\subset\tilde{I}$. We have $I=I_{AB}$ by
Lemma 4. Also $\Lambda_{Ab}={\mathbf{M}\mathbf{U}}^{*}/I$ is an integral
domain by Lemma 5. Moreover ${\mathbf{M}\mathbf{U}}^{*}/I\otimes\mathbb{Q}$ is
polynomial ring in two parameters
${\mathbf{C}\mathbf{P}}_{1},{\mathbf{C}\mathbf{P}}_{2}$ by (3). Therefore it
suffices to check $x_{k}\in\tilde{I}$ viewed as elements in
${\mathbf{M}\mathbf{U}}^{*}[1/2]$.

For the latter we need two facts.

By $SU$ representatives in Introduction $J_{SU}\subset I$, where $J_{SU}$ is
the ideal in ${\mathbf{M}\mathbf{U}}^{*}[1/2]$ generated by
$y_{5},y_{6},\cdots$, special unitary flops in
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ of dimensions $\geq 10$. These
generators vanish under the Krichever genus and therefore vanish under its
specialization, the classifying map of the Abel formal group law. Then
$y_{3},y_{4}\in I$ by (1.4) therefore
$(y_{2},y_{3},y_{4},J_{SU})\subset\tilde{I}.$

Then by [15] $W^{*}[1/2]$ is a free
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ module generated by $1$ and
${\mathbf{C}\mathbf{P}}_{1}$. So after inverting 2 each generator $x_{k}$,
$k\leq l$, of $J_{W}(l)$ as element in ${\mathbf{M}\mathbf{U}}^{*}[1/2]$ is in
kernel of the quotient map to the integral domain
${\mathbf{M}\mathbf{U}}^{*}/(I(l),x_{2})$. Thus $x_{k}\in\tilde{I}(l)$ and
$J_{W}\subset\tilde{I}$ as the ideals in ${\mathbf{M}\mathbf{U}}^{*}$. Here we
use Remark 4 for $\tilde{I}(l)$.

Now let’s prove $I(l)\subset J_{W}(l)$, $\forall l\geq 2$ by induction on $l$.
For $l=2$ and $l=3$ this is true because
$y_{2}=8{\mathbf{C}\mathbf{P}}_{2}-9{\mathbf{C}\mathbf{P}}_{1}^{2}$ and
$x_{3}=y_{3}=z_{3}=-\alpha_{22}$. Let $I(l-1)\subset J_{W}(l-1)$ by induction
hypothesis. To prove $z_{l}\in(J_{W}(l-1),x_{l})$ recall that the main Chern
number $s_{l}(z_{l}-x_{l})=0$ by the definitions of $z_{l}$, $x_{l}$, i.e.,
(1.2) and (2.10). This means that $z_{l}-x_{l}=P(e_{1},e_{2},\cdots,e_{l-1})$
as decomposable element in ${\mathbf{M}\mathbf{U}}^{*}$. Here $e_{i}$ are as
in (2.8). On the other hand $x_{l}\in J_{W}(l)\subset\tilde{I}(l)$ hence
$P=x_{l}-z_{l}$ is in the kernel of the quotient map
${\mathbf{M}\mathbf{U}}^{*}\to{\mathbf{M}\mathbf{U}}^{*}/\tilde{I}(l-1)$, i.e.
$P\in\tilde{I}(l-1)=J_{W}(l-1)$. The induction hypothesis and Remark 3
completes the proof.

ii) By [14] again $I$ is prime ideal, therefore so is $\tilde{I}=J_{W}$ by
Remark 3.

∎

###  3.1. Proof of Theorem 1.1.

i) As $x_{3},\cdots,x_{l}\subset J_{W}(l)$ by Lemma 6 ii) and Remark 4 the
ring $R={\mathbf{M}\mathbf{U}}^{*}/(x_{3},\cdots,x_{l})$ is an integral
domain. In particular, the multiplication by
$x_{1}={\mathbf{C}\mathbf{P}}_{1}$ is mono in $R$. By Remark 4 again
$(x_{1},x_{3},\cdots,x_{l})$ is regular in ${\mathbf{M}\mathbf{U}}^{*}$
$\forall l$ and one has ii).

###  3.2. Proof of Theorem 1.3

We use Lemma 6 and apply the specified generators $x_{3},\cdots$ in
$W^{*}[1/2]$ such that $(x_{3},\cdots)=(z_{3},\cdots)$ in
${\mathbf{M}\mathbf{U}}^{*}[1/2]$.

∎

###  3.3. Proof of Theorem 1.2.

We have to prove that all obstructions to commutativity vanish. This follows
by dimension argument and definition of obstruction elements in [21]. In
particular, the obstruction elements to commutativity
$\tilde{\mathcal{P}}_{2}(x_{i})$ defined by certain cohomology operation [21]
Theorem 3.4

$\tilde{\mathcal{P}}_{2}:{\mathbf{M}\mathbf{U}}^{-2n}\to{\mathbf{M}\mathbf{U}}^{-4n-2}\otimes\mathbb{F}_{2}.$

So that all obstruction elements are of odd complex dimension $-2i-1$.
Therefore they vanish in the scalar ring
${\mathbf{M}\mathbf{U}}^{*}/(x_{1},x_{3},\cdots)\otimes\mathbb{F}_{2}$ which
is evenly generated as a quotient of $\Lambda_{Ab}$ by Remark 2.

Alternatively, we can use Busato’s long proof [14] of commutativity of
$h^{*}_{Ab}(-)$. The proof is that $\tilde{\mathcal{P}}_{2}(\alpha_{ij})$
projects to $0$ in $\Lambda_{Ab}\otimes\mathbb{F}_{2}$, for
$\alpha_{ij},\,\,i,j\geq 2$. In our case, we have to consider one more element
$\alpha_{11}=x_{1}$. By [21] Theorem 3.4 we have
$\tilde{\mathcal{P}}_{2}(\alpha_{11})=\alpha_{22}=x_{3}$ modulo $\alpha_{11}$.
So that the obstruction element $\tilde{\mathcal{P}}_{2}(x_{1})$ vanishes.

###  3.4. Proof of Theorem 1.4.

One can use the fact that ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ is a
subring in $W^{*}[1/2]$ and follow [2] where we proved the statement for
generators $(y_{i},i\geq 5)$ in ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$.
The scalar ring $\Lambda_{B}$ of $F_{B}$ and its classifying map $f_{B}$ is
calculated in [11] (see also Remark 1). In particular, $\Lambda_{B}$ has the
ideal of elements of order two generated by $f_{B}(e_{2^{n}-2})$, $n\geq 3$.
Here $e_{i}$ are as in (2.8). The quotient by torsion

$\Lambda_{\mathcal{B}}=\Lambda_{B}/(f_{B}(e_{2^{n}-2}),\,\,n\geq 3).$

is an integral domain.

∎

######  Remark 5.

Using the arguments in the proof of Theorem 1.2 one can prove that for
$\Sigma$ containing all generators $x_{n}$ of $W^{*}$ with odd indices
$n=2i-1\geq 1$, the cohomology theory $h^{*}_{\Sigma}$ is commutative.

###  3.5. Proof of (1.4)

We have to check that $y_{2},y_{3},y_{4}$ are represented by
$\mathop{\mathrm{SU}}$-manifolds i.e., all Chern numbers of $y_{i}$ having
factor $c_{1}$ are zero. Then we have to check the main Chern number
$s_{i}(y_{i})$ for Novikov’s criteria [23] for the membership of the set of
polynomial generators in ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$.

In particular, $\mathop{\mathrm{SU}}$ manifold $x_{n}$ is a polynomial
generator if and only if

(3.1) $\displaystyle s_{n}(y_{n})=\begin{cases}\pm 2^{k}p&\mbox{if
}n=p^{l},\,\,\,p\,\,\,\mbox{is odd prime, }\\\ \pm 2^{k}p&\mbox{if
}n+1=p^{l},\,\,\,p\,\,\,\mbox{is odd prime, }\\\ \pm
2^{k}&\mbox{otherwise}.\end{cases}$

Rewrite the elements $y_{i}$ in ${\mathbf{M}\mathbf{U}}^{*}$ as follows

$\displaystyle
y_{2}={\mathbf{C}\mathbf{P}}_{2}-\frac{9}{8}{\mathbf{C}\mathbf{P}}_{1}^{2};$
$\displaystyle
2y_{3}=-3{\mathbf{C}\mathbf{P}}_{3}+8{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{2}-5{\mathbf{C}\mathbf{P}}_{1}^{3};$
$\displaystyle
y_{4}=-2{\mathbf{C}\mathbf{P}}_{1}^{4}+7{\mathbf{C}\mathbf{P}}_{1}^{2}{\mathbf{C}\mathbf{P}}_{2}-3{\mathbf{C}\mathbf{P}}_{2}^{2}-4{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{3}+2{\mathbf{C}\mathbf{P}}_{4}.$

The proof follows from the following calculations for the main Chern numbers.

$\displaystyle c_{1}c_{1}[{\mathbf{C}\mathbf{P}}_{1}^{2}]=8,$ $\displaystyle
c_{1}c_{1}[{\mathbf{C}\mathbf{P}}_{2}]=9,$ $\displaystyle
c_{2}[{\mathbf{C}\mathbf{P}}_{1}^{2}]=4,$ $\displaystyle
c_{2}[{\mathbf{C}\mathbf{P}}_{2}]=3.$

This implies that that $c_{1}c_{1}(y_{2})=0$. There are no more Chern numbers
involving $c_{1}$ and $s_{2}(y_{2})=3$. Therefore $y_{2}$ forms a generator of
${\mathbf{M}\mathbf{S}\mathbf{U}}^{4}[1/2]$.

For $y_{3}$:

$\displaystyle X$ $\displaystyle c_{3}(X)$ $\displaystyle c_{1}c_{2}(X)$
$\displaystyle c_{1}c_{1}c_{1}(X)$ $\displaystyle{\mathbf{C}\mathbf{P}}_{3}$
$\displaystyle 4$ $\displaystyle 24$ $\displaystyle 64$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{2}$
$\displaystyle 6$ $\displaystyle 24$ $\displaystyle 54$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}^{3}$ $\displaystyle 8$ $\displaystyle
24$ $\displaystyle 48.$

It follows all Chern numbers of $y_{3}$ involving $c_{1}$ are zero. Then
$y_{3}$ forms a generator in ${\mathbf{M}\mathbf{S}\mathbf{U}}^{6}[1/2]$ as
$s_{3}(y_{3})=-2\cdot 3$ satisfies the Novikov criteria.

Similarly for $y_{4}$: we have $s_{4}(y_{4})=2\cdot 5$ and

$\displaystyle X$ $\displaystyle c_{1}c_{1}c_{1}c_{1}(X)$ $\displaystyle
c_{1}c_{1}c_{2}(X)$ $\displaystyle c_{1}c_{3}(X)$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}^{4}$ $\displaystyle 384$
$\displaystyle 192$ $\displaystyle 64$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}^{2}{\mathbf{C}\mathbf{P}}_{2}$
$\displaystyle 432$ $\displaystyle 204$ $\displaystyle 60$
$\displaystyle{\mathbf{C}\mathbf{P}}_{2}^{2}$ $\displaystyle 486$
$\displaystyle 216$ $\displaystyle 54$
$\displaystyle{\mathbf{C}\mathbf{P}}_{1}{\mathbf{C}\mathbf{P}}_{3}$
$\displaystyle 512$ $\displaystyle 224$ $\displaystyle 56$
$\displaystyle{\mathbf{C}\mathbf{P}}_{4}$ $\displaystyle 625$ $\displaystyle
250$ $\displaystyle 50.$

∎

##  4\. $c_{1}$-spherical cobordism modulo quasi-toric manifolds

Recall again from [26] that in special unitary cobordism
${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ one can construct polynomial
generators $y_{n},\,\,\,n\geq 5$ using Euclid’s algorithm and $SU$-flops or
equivalently generators specified in Introduction.

The restriction of the Krichever complex elliptic genus

${\mathbf{M}\mathbf{U}}^{*}\to\mathbb{Q}[p_{1},p_{2},p_{3},p_{4}]$

on ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ gives the genus

${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]\to\mathbb{Z}[1/2][y_{2},y_{3},y_{4}],$

as it is zero on all $y_{i},i\geq 5$.

As $W^{*}[1/2]$ is a free ${\mathbf{M}\mathbf{S}\mathbf{U}}^{*}[1/2]$ module
generated by 1 and $x_{1}={\mathbf{C}\mathbf{P}}_{1}$ one has the following

######  Theorem 4.1.

$SU$-flops of dimension $\geq 10$ as elements in the ring $W^{*}[1/2]$ with
$*$ product generate the kernel ideal of the genus

$\phi_{W}:W^{*}[1/2]\to\mathbb{Z}[1/2][x_{1},x_{3},x_{4}].$

## References

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x_{2})]\cdot[\mathtt{cp}_{(n,m+o)}\,x\,\mathtt{to}\,a,c.\>a\otimes c]$

The other properties in Definition 2 are proved by similar mechanisms and we
thus obtain a graded exponential comonad.

We proceed by showing that the induced comonad over
$\mathrm{Syn}(\mathscr{T})$ is Lipschitz. This is entailed by the inequation
$r\mathbin{\bullet}a([v],[w])\leq a([v]_{\ast_{r}},[w]_{\ast_{r}})$ which is
deduced from the following chain of (in)equalities,

$\displaystyle\>r\mathbin{\bullet}a([v],[w])$ $\displaystyle\triangleq
r\mathbin{\bullet}a(v,w)$ $\displaystyle\triangleq
r\mathbin{\bullet}\bigvee\Big{\\{}q\mid v=_{q}w[z/y][x/z]\text{ is a theorem
of }\mathscr{T}\Big{\\}}$
$\displaystyle=\bigvee\Big{\\{}r\mathbin{\bullet}q\mid v=_{q}w[z/y][x/z]\text{
is a theorem of }\mathscr{T}\Big{\\}}$ {$(r\mathbin{\bullet}-)$ preserves
joins}
$\displaystyle\leq\bigvee\Big{\\{}q\mid\mathtt{pr}_{(r,[s])}\,a\,\mathtt{fr}\,x.\,v=_{q}\mathtt{pr}_{(r,[s])}\,a\,\mathtt{fr}\,x.\,w[z/y][x/z]\text{
is a theorem of }\mathscr{T}\Big{\\}}$
$\displaystyle=\bigvee\Big{\\{}q\mid\mathtt{pr}_{(r,[s])}\,a\,\mathtt{fr}\,x.\,v=_{q}\mathtt{pr}_{(r,[s])}\,a\,\mathtt{fr}\,y.\,w\text{
is a theorem of }\mathscr{T}\Big{\\}}$ {$\alpha$-equivalence}
$\displaystyle\triangleq
a(\mathtt{pr}_{(r,[s])}\,a\,\mathtt{fr}\,x.\,v,\mathtt{pr}_{(r,[s])}\,a\,\mathtt{fr}\,y.\,w)$
$\displaystyle=a([v]_{\ast_{r}},[w]_{\ast_{r}})$

The final step is to show that if the interpretation of an equation
$\Gamma\rhd v=_{q}w:\mathbb{A}$ with $q\in B$ is satisfied in
$\mathrm{Syn}(\mathscr{T})$ then it is a theorem of the graded
$\mathcal{V}\lambda$-theory $\mathscr{T}$. This is proved via a similar
reasoning to the one used in characterising $\sim$ (in the beginning of the
proof). ∎

Proof of Theorem 7

1\. $D$ is well-defined. It is straightforward to check that each $D_{n}$ is
symmetric monoidal when equipped with the reordering isomorphism

$\displaystyle\phi^{D_{n}}_{X,Y}:(X\otimes Y)\otimes\ldots\otimes(X\otimes
Y)\to(X\otimes\ldots\otimes X)\otimes(Y\otimes\ldots\otimes Y)$ (27)

obtained by successive application of the symmetry and associator
transformations in $\mathsf{C}$. It follows that
$D:\mathbb{N}\to(\mathsf{SymMon}[\mathsf{C},\mathsf{C}],\mathrm{Id},\circ)$ is
well-defined.

2\. $D$ is an $\mathbb{N}$-graded comonad. To see this we define,

$\epsilon:D_{1}\to\mathrm{Id}\qquad\text{and}\qquad\delta^{m,n}:D_{m\cdot
n}\to D_{m}D_{n}$

as the identity transformation and the iterated re-bracketing transformation
$(-)^{\otimes(n\cdot m)}\cong((-)^{\otimes n})^{\otimes m}$ constructed using
the associator of $\mathsf{C}$ when $m,n\neq 0$, the isomorphisms $I\cong
I\otimes\ldots\otimes I$ when $n=0$ and the identity on $I$ when $m=0$. It is
easy to check that this defines an $\mathbb{N}$-graded comonad (Definition 1):
the first diagram of (10) commutes trivially since all arrows are identities;
the second diagram of (10) commutes because the two re-bracketing schemes
coincide.

3\. $D$ is a non-symmetric oplax monoidal functor
$\mathbb{N}\to(\mathsf{SymMon}[\mathsf{C},\mathsf{C}],I,\otimes)$. Next we
show that, apart from the symmetry condition, part (i) of Definition 2 holds.
To see that this is the case define,

$e:D_{0}\to I\qquad\text{and}\qquad d^{m,n}:D_{m+n}\to D_{m}\otimes D_{n}$

as the identity on $I$ and the re-bracketing transformation
$(-)^{\otimes(m+n)}\cong(-)^{\otimes m}\otimes(-)^{\otimes n}$ built from the
associator if $m,n\neq 0$, and the isomorphisms
$(-)^{\otimes(m+0)}\cong(-)^{\otimes m}\otimes I$ and $(-)^{\otimes(0+n)}\cong
I\otimes(-)^{\otimes n}$ given by the unitors of $\mathsf{C}$ otherwise. Again
it is easy to check that the analogues of the diagrams (10) commute. $D$ is
not a _symmetric_ oplax monoidal functor
$\mathbb{N}\to(\mathsf{SymMon}[\mathsf{C},\mathsf{C}],I,\otimes)$ since it
would mean, for example, that
$\gamma_{X,X}=\operatorname{\mathsf{id}}_{X\otimes X}$, which in general is
not the case.

4\. $D$ satisfies diagrams (1-3) in condition (ii) of Definition 2. This is
straightforward given the definitions. However, $D$ does not satisfy diagram
(4) in condition (ii) of Definition 2. This is because, moving counter-
clockwise along the diagram, we encounter $\phi^{D_{s}}_{D_{n},D_{m}}$ which,
as defined in (27) performs some permutations using the symmetry
transformation $\gamma$ of $\mathsf{C}$. Moving clockwise on the other hand,
$D_{s}d^{n,m}$ and $\delta^{s,(n+m)}$ only perform re-bracketing operations. ∎

Proof of Theorem 8

1\. $E$ is well-defined. Let us start by showing that
$E:\mathbb{N}\to\mathsf{SymMon}[\mathsf{C},\mathsf{C}]$ is well-defined, i.e.
that each $E_{n}$ is a symmetric monoidal endofunctor on $\mathsf{C}$. Each
$E_{n}$ inherits the monoidal structure of $D_{n}=(-)^{\otimes n}$ through its
defining universal property as the following diagrams describe.

(36)

In order to obtain a symmetric monoidal structure on each $E_{n}$, we will
prove that the morphism
$\phi^{D_{n}}_{X,Y}\cdot(\epsilon^{n}_{X}\otimes\epsilon^{n}_{Y}):E_{n}X\otimes
E_{n}Y\to D_{n}(X\otimes Y)$ is a cone for the diagram defining
$E_{n}(X\otimes Y)$. Note that the morphisms $\sigma\cdot\phi^{D_{n}}_{X,Y}$
in the diagram above are precisely those of the shape
$\phi_{X,Y}^{D_{n}}\cdot(\sigma\otimes\sigma)$. Since
$\epsilon^{n}_{X}\otimes\epsilon^{n}_{Y}$ equalises all the maps
$\sigma_{1}\otimes\sigma_{2}$ with
$\sigma_{1},\sigma_{2}\in\operatorname{\mathrm{Sym}}\left(n\right)$, it
equalises in particular those of the shape $\sigma\otimes\sigma$ and it
follows that the right-hand squares commute. This proves that
$\phi^{D_{n}}_{X,Y}\cdot(\epsilon^{n}_{X}\otimes\epsilon^{n}_{Y})$ is indeed a
cone. Therefore there exists a unique morphism
$\phi^{E_{n}}_{X,Y}:E_{n}X\otimes E_{n}Y\to E_{n}(X\otimes Y)$, which is just
the restriction of $\phi^{D_{n}}$ to the symmetric elements of $D_{n}\otimes
D_{n}$. The argument for the diagram on the right is similar and based on the
fact that the equation $\sigma\cdot\phi^{D_{n}}=\tau\cdot\phi^{D_{n}}$ follows
immediately from the coherence theorem of symmetric monoidal categories.
Lastly, we need to show that each $E_{n}$ is symmetric. This follows from the
commutativity of the front, top, bottom and sides of the diagram below which,
when combined with the fact that $\epsilon^{n}$ is component-wise mono, yields
the commutativity of the back of the diagram.

---
$\textstyle{E_{n}X\otimes
E_{n}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{E_{n}X,E_{n}Y}}$$\scriptstyle{\phi^{E_{n}}_{X,Y}}$$\scriptstyle{\epsilon^{n}_{X}\otimes\epsilon^{n}_{Y}}$$\textstyle{E_{n}Y\otimes
E_{n}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{E_{n}}_{Y,X}}$$\scriptstyle{\epsilon^{n}_{Y}\otimes\epsilon^{n}_{X}}$$\textstyle{D_{n}X\otimes
D_{n}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{D_{n}X,D_{n}Y}}$$\scriptstyle{\phi^{D_{n}}_{X,Y}}$$\textstyle{D_{n}Y\otimes
D_{n}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{D_{n}}_{Y,X}}$$\textstyle{E_{n}(X\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{n}\gamma_{X,Y}}$$\scriptstyle{\epsilon^{n}_{X\otimes
Y}}$$\textstyle{E_{n}(Y\otimes
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon^{n}_{Y,X}}$$\textstyle{D_{n}(X\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{n}\gamma_{X,Y}}$$\textstyle{D_{n}(Y\otimes
X)}$

2\. $E$ is an $\mathbb{N}$-graded comonad. Since $E_{1}=D_{1}$ by definition,
$E$ inherit the trivial counit $\mathrm{Id}=\epsilon:E_{1}\to\mathrm{Id}$ of
$D$. For the comultiplication, we again rely of the universal property
defining $E$, but we will also need the assumption that $(X\otimes-)$
preserves the limit defining $E_{n}$ for every $\mathsf{C}$-object $X$. First,
we claim that $D_{m}\epsilon^{n}:D_{m}E_{n}\to D_{m}D_{n}$ is the equaliser of
all permutations of the type
$\sigma_{1}\otimes\ldots\otimes\sigma_{m}:D_{m}D_{n}\to
D_{m}D_{n},\sigma_{i}\in\operatorname{\mathrm{Sym}}\left(n\right)$. It is
clearly a cone, so we need to show that it is a limiting cone. For this,
consider another cone $\phi:C\to D_{m}D_{n}$ which equalises all permutations
of the shape
$\sigma_{1}\otimes\ldots\otimes\sigma_{m},\sigma_{i}\in\operatorname{\mathrm{Sym}}\left(n\right)$.
We can iteratively construct a unique morphism $\psi:C\to D_{m}E_{n}$ such
that $D_{m}\epsilon^{n}\cdot\psi=\phi$. To do this, consider the following
diagram.

$\textstyle{D_{m}E_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes\epsilon^{n}}$$\textstyle{D_{m-1}E_{n}\otimes
D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes\sigma_{m}}$$\textstyle{D_{m-1}E_{n}\otimes
D_{n}}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{E_{n}\otimes
E_{n}\otimes
D_{m-2}D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes\epsilon^{n}\otimes\operatorname{\mathsf{id}}}$$\textstyle{E_{n}\otimes
D_{m-1}D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes(\sigma_{2}\otimes\operatorname{\mathsf{id}})}$$\textstyle{E_{n}\otimes
D_{m-1}D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon^{n}\otimes\operatorname{\mathsf{id}}}$$\textstyle{E_{n}\otimes
D_{m-1}D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon^{n}\otimes\operatorname{\mathsf{id}}}$$\textstyle{D_{m}D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}\otimes\operatorname{\mathsf{id}}}$$\textstyle{D_{m}D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\psi_{2}}$$\scriptstyle{\psi}$$\textstyle{D_{m}D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigotimes_{i=1}^{m}\sigma_{i}}$$\textstyle{D_{m}D_{n}}$

By assumption, $\epsilon^{n}\otimes\operatorname{\mathsf{id}}$ on the bottom
row of squares is an equaliser. Moreover, since $\phi$ equalises all
permutations of the shape $\bigotimes_{i=1}^{m}\sigma_{i}$, it in particular
equalises those of the shape $\sigma_{1}\otimes\operatorname{\mathsf{id}}$ and
therefore forms a cone for the diagram defining the equaliser
$\epsilon^{n}\otimes\operatorname{\mathsf{id}}$. We can therefore find a
unique mediating morphism $\psi_{1}$ such that
$\phi=(\epsilon^{n}\otimes\operatorname{\mathsf{id}})\cdot\psi_{1}$. Next,
note that $\psi_{1}$ equalises all permutations of the shape
$\operatorname{\mathsf{id}}\otimes\sigma_{2}\otimes\operatorname{\mathsf{id}}$:

$\displaystyle\,(\epsilon^{n}\otimes\operatorname{\mathsf{id}})\cdot(\operatorname{\mathsf{id}}\otimes(\sigma_{2}\otimes\operatorname{\mathsf{id}}))\cdot\psi_{1}$
$\displaystyle=(\operatorname{\mathsf{id}}\otimes(\sigma_{2}\otimes\operatorname{\mathsf{id}}))\cdot(\epsilon^{n}\otimes\operatorname{\mathsf{id}})\cdot\psi_{1}$
$\displaystyle=(\operatorname{\mathsf{id}}\otimes(\sigma_{2}\otimes\operatorname{\mathsf{id}}))\cdot\phi$
$\displaystyle\\{\text{Definition\ of }\psi_{1}\\}$
$\displaystyle=(\operatorname{\mathsf{id}}\otimes(\sigma_{2}^{\prime}\otimes\operatorname{\mathsf{id}}))\cdot\phi$
$\displaystyle\\{\sigma_{2}^{\prime}\in\operatorname{\mathrm{Sym}}\left(n\right),\text{
Definition of }\phi\\}$
$\displaystyle=(\epsilon^{n}\otimes\operatorname{\mathsf{id}})\cdot(\operatorname{\mathsf{id}}\otimes(\sigma_{2}^{\prime}\otimes\operatorname{\mathsf{id}}))\cdot\psi_{1}$

and thus
$(\operatorname{\mathsf{id}}\otimes(\sigma_{2}\otimes\operatorname{\mathsf{id}}))\cdot\psi_{1}=(\operatorname{\mathsf{id}}\otimes(\sigma_{2}^{\prime}\otimes\operatorname{\mathsf{id}}))\cdot\psi_{1}$
since $\epsilon^{n}\otimes\operatorname{\mathsf{id}}$ is mono. By using the
symmetry transformation $X\otimes E_{n}\otimes X\to X\otimes X\otimes E_{n}$
and the assumption that the tensor preserves the equalisers defining $E_{n}$,
it is not hard to see that
$\operatorname{\mathsf{id}}\otimes(\epsilon^{n}\otimes\operatorname{\mathsf{id}})$
is the equaliser of all these permutations. It follows that there exists a
unique mediating $\psi_{2}:C\to E_{n}\otimes E_{n}\otimes D_{m-2}D_{n}$. By
repeating this argument, using the fact that $\phi$ equalises the product of
symmetries where all but one component are identities, we can iteratively
eliminate each $D_{n}$ until we produce the mediating morphism $\psi:C\to
D_{m}E_{n}$ satisfying $D_{m}\epsilon^{n}\cdot\psi=\phi$.

Having shown that $D_{m}\epsilon^{n}$ is an equaliser, we use its universal
property and the definition of $E_{mn}$ to build unique natural
transformations $\chi^{m,n}:E_{mn}\to D_{m}E_{n}$ and then
$\eta^{m,n}:E_{mn}\to E_{m}E_{n}$. The following diagram commutes because
$\epsilon^{m,n}$ equalises all permutations in
$\operatorname{\mathrm{Sym}}\left(mn\right)$, and thus in particular those of
the shape
$\sigma_{1}\otimes\ldots\otimes\sigma_{m},\sigma_{i}\in\operatorname{\mathrm{Sym}}\left(n\right)$.
This, combined with the fact proved above that
$D_{m}\epsilon^{n}:D_{m}E_{n}\to D_{m}D_{n}$ is the equalizer of these
permutations, provides the unique natural transformation $\chi^{m,n}:E_{mn}\to
D_{m}E_{n}$.

(41)

Using this diagram, we build the comultiplication $\eta^{m,n}:E_{mn}\to
E_{m}E_{n}$ as follows.

(48)

The squares commute by definition, and since $\epsilon^{mn}$ equalises all
permutations of $\operatorname{\mathrm{Sym}}\left(mn\right)$, it equalises
those permuting $m$ blocks of $n$-tensors. The fact that $\chi^{m,n}:E_{mn}\to
D_{m}E_{n}$ also equalises those permutations follows from the fact that
$D_{m}\epsilon^{n}\cdot\chi^{m,n}=\delta^{m,n}\cdot\epsilon^{mn}$ (by
definition of $\chi^{m,n}$, see Diagram 41) and the fact that
$D_{m}\epsilon^{n}$ is component-wise mono. This gives us the comultiplication
$\eta^{m,n}:E_{mn}\to E_{m}E_{n}$ by the universal property of $E_{m}$ w.r.t.
the permutations of $\operatorname{\mathrm{Sym}}\left(m\right)$.

Finally, we need to check that the counit and comultiplication satisfy (10).
The first diagram is trivial because $E_{1}=D_{1}=\mathrm{Id}$. For the second
diagram notice that the commutativity of the left-hand-side trapeze in the
diagram above shows proves that $E$ is a sub- $\mathbb{N}$-graded comonad of
$D$. We can therefore reduce the commutativity of the second diagram in (10)
to its counterpart for $D$ by post-composing with a mono.

3\. $E$ is an $\mathbb{N}$-graded exponential comonad. It remains to define
monoidal natural transformations $e:E_{0}\to I$ and $d^{m,n}:E_{m+n}\to
E_{m}\otimes E_{n}$. The former is again inherited from $D$ since
$E_{0}=D_{0}=I$. We construct the latter in a similar fashion to
comultiplication. Specifically we use the universal property defining $E$ and
the assumption that the tensor product preserves the limits defining $E_{n}$.
Since $\epsilon^{m+n}$ equalises all permutations of
$\operatorname{\mathrm{Sym}}\left(m+n\right)$, it equalises those that leave
the first $m$ component unchanged. This guarantees the commutativity of the
following diagram and the existence of the unique morphism $E_{m+n}\to
D_{m}\otimes E_{n}$.

$\textstyle{E_{m+n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon^{m+n}}$$\textstyle{D_{m+n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{m,n}}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes\sigma,\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes\tau,\tau\in\operatorname{\mathrm{Sym}}\left(n\right)}$$\scriptstyle{\dots}$$\textstyle{D_{m+n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{m,n}}$$\textstyle{D_{m}\otimes
E_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\mathsf{id}}_{D_{m}}\otimes\epsilon^{n}}$$\textstyle{D_{m}\otimes
D_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes\sigma,\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}$$\scriptstyle{\operatorname{\mathsf{id}}\otimes\tau,\tau\in\operatorname{\mathrm{Sym}}\left(n\right)}$$\scriptstyle{\dots}$$\textstyle{D_{m}\otimes
D_{n}}$

Using this diagram we build the following analogue of the comultiplication
case.

(55)

The squares on the right-hand-side commute by definition, moreover
$\epsilon^{m+n}$ equalises the permutations leaving the last $n$ components
unchanged since it equalises all permutations in
$\operatorname{\mathrm{Sym}}\left(m+n\right)$. Since
$\operatorname{\mathsf{id}}\otimes\,\epsilon^{n}$ is mono, we conclude that
the arrow $E_{m+n}\to D_{m}\otimes E_{n}$ also equalises these permutations,
which in turn defines the unique transformation $E_{m+n}\to E_{m}\otimes
E_{n}$. Let us call it $d^{m,n}$ as well. Next, to check that the following
diagram commutes

$\textstyle{E_{m+n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{n,m}}$$\scriptstyle{d^{m,n}}$$\textstyle{E_{m}\otimes
E_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{E_{m},E_{n}}}$$\textstyle{E_{n}\otimes
E_{m}}$

we can again resort to the universal property defining $E$. We just sketch the
argument as it is similar to those used above. Since $d^{n,m}$ is the unique
morphism from the cone $E_{m+n}$ to the limit $E_{n}\otimes E_{m}$ following
the construction above, and since $\gamma_{E_{m},E_{n}}\cdot d^{m,n}$ also
makes $E_{m+n}$ a cone for the same diagram, $\gamma_{E_{m},E_{n}}\cdot
d^{m,n}=d^{n,m}$ by unicity.

Finally we need to check the diagrams (ii) of definition 2. The first three
can reduced to the commutativity of their counterparts for $D$ by post-
composition with a mono via (36), (48) and (55). The last diagram we need to
check is:

(60)

We reason by post-composing with monos:

$\displaystyle\,D_{s}(\epsilon^{n}\otimes\epsilon^{m})\cdot\epsilon^{s}_{E_{n}\otimes
E_{m}}\cdot E_{s}d^{n,m}\cdot\eta^{s,n+m}$
$\displaystyle=D_{s}(\epsilon^{n}\otimes\epsilon^{m})\cdot
D_{s}d^{n,m}\cdot\epsilon^{s}_{E_{n+m}}\cdot\eta^{s,n+m}$
$\displaystyle\\{\text{Nat. of $\epsilon^{s}$}\\}$
$\displaystyle=D_{s}((\epsilon^{n}\otimes\epsilon^{m})\cdot{d^{n,m}})\cdot\epsilon^{s}{E_{n+m}}\cdot\eta^{s,n+m}$
$\displaystyle=D_{s}(d^{n,m}\cdot\epsilon^{m+n})\cdot\epsilon^{s}_{E_{n+m}}\cdot\eta^{s,n+m}$
$\displaystyle\\{\text{\eqref{eq:sub_exponential}}\\}$
$\displaystyle=D_{s}d^{n,m}\cdot
D_{s}\epsilon^{m+n}\cdot\epsilon^{s}_{E_{n+m}}\cdot\eta^{s,n+m}$
$\displaystyle=D_{s}d^{n,m}\cdot\delta^{s,n+m}\cdot\epsilon^{s\cdot(n+m)}$
$\displaystyle\\{\text{\eqref{eq:sub_comonad}}\\}$
$\displaystyle=D_{s}d^{n,m}\cdot\delta^{s,n+m}\cdot\sigma\cdot\epsilon^{s\cdot(n+m)}$
$\displaystyle\\{\text{Definition of $\epsilon^{s\cdot(n+m)}$}\\}$
$\displaystyle=\phi_{D_{n},D_{m}}^{D_{s}}\cdot(\delta^{s,n}\otimes\delta^{s,m})\cdot
d^{s\cdot n,s\cdot m}\cdot\epsilon^{s\cdot(n+m)}$ {$\star$}
$\displaystyle=\phi_{D_{n},D_{m}}^{D_{s}}\cdot(\delta^{s,n}\otimes\delta^{s,m})\cdot(\epsilon^{s\cdot
n}\otimes\epsilon^{s\cdot m})\cdot d^{s\cdot n,s\cdot m}$
$\displaystyle\\{\text{\eqref{eq:sub_exponential}}\\}$
$\displaystyle=\phi_{D_{n},D_{m}}^{D_{s}}\cdot(D_{s}\epsilon^{n}\cdot\epsilon^{s}_{E_{n}}\cdot\eta^{s,n})\otimes(D_{s}\epsilon^{m}\cdot\epsilon^{s}_{E_{m}}\cdot\eta^{s,m})\cdot
d^{s\cdot n,s\cdot m}$ $\displaystyle\\{\text{\eqref{eq:sub_comonad}}\\}$
$\displaystyle=\phi_{D_{n},D_{m}}^{D_{s}}\cdot(D_{s}\epsilon^{n}\otimes
D_{s}\epsilon^{m})\cdot((\epsilon^{s}_{E_{n}}\cdot\eta^{s,n})\otimes(\epsilon^{s}_{E_{m}}\cdot\eta^{s,m}))\cdot
d^{s\cdot n,s\cdot m}$
$\displaystyle=D_{s}(\epsilon^{n}\otimes\epsilon^{m})\cdot\phi_{E_{n},E_{n}}^{D_{s}}\cdot((\epsilon^{s}_{E_{n}}\cdot\eta^{s,n})\otimes(\epsilon^{s}_{E_{m}}\cdot\eta^{s,m}))\cdot
d^{s\cdot n,s\cdot m}$ $\displaystyle\\{\text{Nat. of $\phi^{D_{s}}$}\\}$
$\displaystyle=D_{s}(\epsilon^{n}\otimes\epsilon^{m})\cdot\epsilon^{s}_{E_{n}\otimes
E_{m}}\cdot\phi_{E_{n},E_{n}}^{E_{s}}\cdot(\eta^{s,n}\otimes\eta^{s,m})\cdot
d^{s\cdot n,s\cdot m}$ $\displaystyle\\{\text{\eqref{eq:sub_monoidal}}\\}$

Then because
$D_{s}(\epsilon^{n}\otimes\epsilon^{m})\cdot\epsilon^{s}_{E_{n}\otimes E_{m}}$
is mono we prove our claim. The step marked with $\\{\star\\}$ needs further
explanation. The expressions $D_{s}d^{n,m}\cdot\delta^{s,n+m}$ and
$\phi_{D_{n},D_{m}}^{D_{s}}\cdot(\delta^{s,n}\otimes\delta^{s,m})\cdot
d^{s\cdot n,s\cdot m}$ may differ. However by the coherence theorem of
symmetric monoidal categories they may only differ by some permutation
$\sigma$, i.e.  we can always find a permutation $\sigma$ such that
$D_{s}d^{n,m}\cdot\delta^{s,n+m}\cdot\sigma=\phi_{D_{n},D_{m}}^{D_{s}}\cdot(\delta^{s,n}\otimes\delta^{s,m})\cdot
d^{s\cdot n,s\cdot m}$. Also by the definition of $E_{s\cdot(n+m)}$ we know
that $\epsilon^{s\cdot(n+m)}=\sigma\cdot\epsilon^{s\cdot(n+m)}$. ∎

Proof of Proposition 1 To see that $n\mathbin{\bullet}-$ preserves arbitrary
joins we compute,

$\displaystyle\,n\mathbin{\bullet}\left(\bigvee X\right)$
$\displaystyle\triangleq\left(\bigvee X\right)\otimes\dots\otimes\left(\bigvee
X\right)$ $\displaystyle=\bigvee\left(X\otimes\dots\otimes X\right)$
{$\otimes$ preserves joins} $\displaystyle=\bigvee\\{x_{1}\otimes\dots\otimes
x_{n}\mid x_{1},\dots,x_{n}\in X\\}$
$\displaystyle=\bigvee\\{x\otimes\dots\otimes x\mid x\in X\\}$ $\\{\star\\}$
$\displaystyle\triangleq\bigvee n\mathbin{\bullet}X$

where the step marked with $(\star)$ follows from the fact that the inequation
below holds.

$x_{1}\otimes\dots\otimes
x_{n}\leq\left(\bigvee\\{x_{1},\dots,x_{n}\\}\right)\otimes\dots\otimes\left(\bigvee\\{x_{1},\dots,x_{n}\\}\right)\hskip
42.67912pt(x_{1},\dots,x_{n}\in X)$

∎

Proof of Theorem 9 Consider two $\mathsf{C}$-morphisms $f,g:X\to Y$. We
reason,

$\displaystyle\,n\mathbin{\bullet}a(f,g)$
$\displaystyle\triangleq\underbrace{a(f,g)\otimes\ldots\otimes
a(f,g)}_{n\text{ times}}$ $\displaystyle\leq a(f^{\otimes n},g^{\otimes n})$
$\displaystyle\\{\text{$\otimes$ in $\mathsf{C}$ is
$\mathcal{V}\text{-}\mathsf{Cat}$-enriched}\\}$ $\displaystyle\triangleq
a(D_{n}f,D_{n}g)$ $\displaystyle\leq
a(D_{n}f\cdot\epsilon^{n}_{X},D_{n}g\cdot\epsilon^{n}_{X})$
$\displaystyle\\{\text{$\mathsf{C}$ is
$\mathcal{V}\text{-}\mathsf{Cat}$-enriched}\\}$
$\displaystyle=a(E_{n}f,E_{n}g)$ {limit of $\mathscr{D}$ is
$\mathcal{V}\text{-}\mathsf{Cat}$-enriched}

∎

Proof of Proposition 2 Consider a diagram
$\mathscr{D}:I\to\mathcal{V}\text{-}\mathsf{Cat}$ and the respective limit
cone $(p_{i}:\lim\mathscr{D}\to\mathscr{D}(i))_{i\in I}$. We need to show that
the cone $(p_{i}\otimes\operatorname{\mathsf{id}}:\lim\mathscr{D}\otimes
X\to\mathscr{D}(i)\otimes X)_{i\in I}$ for the diagram $\mathscr{D}\otimes X$
is a limit. A key component of our proof is the existence of a right adjoint
$\mathrm{U}:\mathcal{V}\text{-}\mathsf{Cat}\to\mathsf{Set}$ that makes the
following diagram commute [26, page 181].

$\textstyle{\mathcal{V}\text{-}\mathsf{Cat}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-\otimes
X)}$$\scriptstyle{\mathrm{U}}$$\textstyle{\mathcal{V}\text{-}\mathsf{Cat}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{U}}$$\textstyle{\mathsf{Set}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-\times\mathrm{U}X)}$$\textstyle{\mathsf{Set}}$

The functor $\mathrm{U}$ merely forgets the $\mathcal{V}$-categorical
structure and keeps the underlying set. It preserves limits and therefore
$(p_{i}:\mathrm{U}\lim\mathscr{D}\to\mathrm{U}\mathscr{D}(i))_{i\in I}$ is the
limit cone of $\mathrm{U}\mathscr{D}$. Then because limits commute with limits
we also know that
$(p_{i}\times\operatorname{\mathsf{id}}:\mathrm{U}\lim\mathscr{D}\times\mathrm{U}X\to\mathrm{U}\mathscr{D}(i)\times\mathrm{U}X)_{i\in
I}$ is the limit cone of $\mathrm{U}\mathscr{D}\times\mathrm{U}X$. Observe
that this last cone is the $\mathrm{U}$-image of
$(p_{i}\otimes\operatorname{\mathsf{id}}:\lim\mathscr{D}\otimes
X\to\mathscr{D}(i)\otimes X)_{i\in I}$. Next, every cone
$(f_{i}:A\to\mathscr{D}(i)\otimes X)_{i\in I}$ for $\mathscr{D}\otimes X$ is
also a cone for $\mathrm{U}\mathscr{D}\times\mathrm{U}X$ which yields a
_unique_ mediating function
$h:\mathrm{U}A\to\mathrm{U}\lim\mathscr{D}\times\mathrm{U}X$. The last step of
our proof is to lift $h$ to a $(\mathcal{V}\text{-}\mathsf{Cat})$-morphism
$h:A\to\lim\mathscr{D}\otimes X$.

Let $a$ be the underlying function of $A$ and $x$ be the underlying function
of $X$. As described in [26, page 181], the limit of $\mathscr{D}$ is
generally characterised as $\lim\mathscr{D}\cong(\lim\mathrm{U}\mathscr{D},l)$
where,

$l(y,z)=\bigwedge_{i\in I}d_{i}(p_{i}(y),p_{i}(z))$

Then observe that $h$ can be decomposed into $\langle h^{1},h^{2}\rangle$ and
analogously for all the $f_{i}$ with $i\in I$. Observe as well that for all
$i\in I$ the equation $h^{2}=f^{2}_{i}$ holds. Then for all $i\in I$, $y,z\in
A$ we obtain,

$\displaystyle\,a(y,z)$ $\displaystyle\leq(d_{i}\otimes x)(f_{i}(y),f_{i}(z))$
$\displaystyle=d_{i}(f^{1}_{i}(y),f^{1}_{i}(z))\otimes
x(f^{2}_{i}(y),f^{2}_{i}(z))$
$\displaystyle=d_{i}(f^{1}_{i}(y),f^{1}_{i}(z))\otimes x(h^{2}(y),h^{2}(z))$
$\big{\\{}h^{2}=f^{2}_{i}$ for all $i\in I\big{\\}}$
$\displaystyle=d_{i}(p_{i}(h^{1}(y)),p_{i}(h^{1}(z)))\otimes
x(h^{2}(y),h^{2}(z))$

Thus the inequation $a(x,y)\leq\bigwedge_{i\in
I}d_{i}(p_{i}(h^{1}(y)),p_{i}(h^{1}(z)))\otimes x(h^{2}(y),h^{2}(z))$ holds.
Finally,

$\displaystyle\,a(y,z)$ $\displaystyle\leq\bigwedge_{i\in
I}d_{i}(p_{i}(h^{1}(y)),p_{i}(h^{1}(z)))\otimes x(h^{2}(y),h^{2}(z))$
$\displaystyle=\left(\bigwedge_{i\in
I}d_{i}(p_{i}(h^{1}(y)),p_{i}(h^{1}(z)))\right)\otimes x(h^{2}(y),h^{2}(z))$
{$\otimes$ preserves arbitrary meets}
$\displaystyle=l(h^{1}(y),h^{1}(z))\otimes x(h^{2}(y),h^{2}(z))$
$\displaystyle=(l\otimes x)(h(y),h(z))$

Since the inequation $a(y,z)\leq(l\otimes x)(h(y),h(z))$ holds for all $y,z\in
A$ the function $h:A\to\lim\mathscr{D}\otimes X$ must be a
$\mathcal{V}\text{-}\mathsf{Cat}$-morphism, which proves our claim. ∎

Proof of Proposition 3 The proof is inspired by an analogous one in [10] and
hinges on the fact that the contraction $\epsilon^{n}:E_{n}(V)\to V^{\otimes
n}$ is split mono. To prove the latter, let us consider the symmetrisation
operator
$\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\sigma:V^{\otimes
n}\to V^{\otimes n}$ [4, 6] – it is a contraction because the properties of
norms entail,

$\left\lVert\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\sigma\right\rVert\leq\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\lVert\sigma\rVert=\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}1=1$

It is then straightforward to show that this operator restricts on the
codomain to a linear map $\partial^{n}:V^{\otimes n}\to E_{n}(V)$ by taking
advantage of the fact that $\operatorname{\mathrm{Sym}}\left(n\right)$ is a
group. Moreover $E_{n}(V)$ inherits its norm from $V^{\otimes n}$ which yields
$\lVert\partial^{n}\rVert=\lVert\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\sigma\rVert$.
Thus $\partial^{n}$ is a linear contraction as well. Next, in order to prove
that $\partial^{n}$ is a retraction of $\epsilon^{n}$ consider the following
facts. By construction we have $\sigma\cdot\epsilon^{n}=\epsilon^{n}$ for all
symmetries $\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)$ which gives
rise to the equation
$\left(\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\sigma\right)\cdot\epsilon^{n}=\partial^{n}\cdot\epsilon^{n}=\operatorname{\mathsf{id}}\cdot\,\epsilon^{n}$.
Moreover $\epsilon^{n}$ is an inclusion. Therefore for every vector $v\in
E_{n}(V)$ we obtain,

$\displaystyle\,\partial^{n}(\epsilon^{n}(v))=\textstyle{\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\sigma}(\epsilon^{n}(v))=\epsilon^{n}(v)=v$

The final step is to prove that every cone $f:U\to V^{\otimes
n}\,\hat{\otimes}_{\pi}\,W$ factorises uniquely through
$\epsilon^{n}\otimes\operatorname{\mathsf{id}}$. By composition we obtain a
linear contraction $(\partial^{n}\otimes\operatorname{\mathsf{id}})\cdot
f:U\to E_{n}(V)\,\hat{\otimes}_{\pi}\,W$. Let us show that it factorises $f$
through $\epsilon^{n}\otimes\operatorname{\mathsf{id}}$. Consider a vector
$u\in U$. By construction we know that
$f(u)=\sigma\otimes\operatorname{\mathsf{id}}\,(f(u))$ for all permutations
$\sigma$ on $n$. This entails,

$\displaystyle\,f(u)$
$\displaystyle=\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\,\sigma\otimes\operatorname{\mathsf{id}}\,(f(u))$
$\displaystyle=\frac{1}{n!}\left(\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\,\sigma\right)\otimes\operatorname{\mathsf{id}}\,(f(u))$
{Addition distributes over $(-\otimes\operatorname{\mathsf{id}})$}
$\displaystyle=\left(\frac{1}{n!}\sum_{\sigma\in\operatorname{\mathrm{Sym}}\left(n\right)}\,\sigma\right)\otimes\operatorname{\mathsf{id}}\,(f(u))$
{Scaling distributes over $(-\otimes\operatorname{\mathsf{id}})$}
$\displaystyle=\partial^{n}\otimes\operatorname{\mathsf{id}}\,(f(u))$

We thus obtain the chain of equalities
$(\epsilon^{n}\otimes\operatorname{\mathsf{id}})\cdot(\partial^{n}\otimes\operatorname{\mathsf{id}})(f(u))=(\epsilon^{n}\otimes\operatorname{\mathsf{id}})(f(u))=f(u)$.
Finally unicity follows from the fact that $\epsilon^{n}$ is split mono. ∎