# On analytic exponential functors on free groups

Minkyu Kim & Christine Vespa

###### Abstract.

This paper concerns exponential contravariant functors on free groups. We
obtain an equivalence of categories between analytic, exponential
contravariant functors on free groups and conilpotent cocommutative Hopf
algebras. This result explains how equivalences of categories obtained
previously by Pirashvili and by Powell interact. Moreover, we obtain an
equivalence between the categories of outer, exponential contravariant
functors on free groups and bicommutative Hopf algebras. We also go further by
introducing a subclass of analytic, contravariant functors on free groups,
called primitive functors; and prove an equivalence between primitive,
exponential contravariant functors and primitive cocommutative Hopf algebras.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. I The general case
    1. 3 Polynomial filtration of $\mathcal{C}$-modules
      1. 3.1 Cross-effects of $\mathcal{C}$-modules
      2. 3.2 Polynomial and analytic functors
      3. 3.3 Construction of the polynomial filtration
    2. 4 Coradical filtration of coaugmented coalgebras
      1. 4.1 Coradical filtration
      2. 4.2 Coradical filtration: the case of a field $\mathds{k}$
      3. 4.3 Examples of coradical filtrations
        1. 4.3.1 The polynomial bialgebra
        2. 4.3.2 The shuffle bialgebra
        3. 4.3.3 Finite-dimensional bialgebras
    3. 5 Isomorphism of polynomial and coradical filtrations
  4. II Application to $\mathsf{gr}^{\mathsf{op}}$
    1. 6 The operad $\mathfrak{Ass}^{u}$ and its associated category
      1. 6.1 The operad of unital associative algebras
      2. 6.2 The bimodule ${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$
      3. 6.3 Relation between $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ and ${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$
    2. 7 On a left ideal of $\mathds{kr}\mathsf{gr}^{\mathsf{op}}$
    3. 8 Application to exponential $\mathsf{gr}^{\mathsf{op}}$-modules
      1. 8.1 Analytic exponential $\mathsf{gr^{\mathsf{op}}}$-modules
      2. 8.2 Outer exponential $\mathsf{gr}^{\mathsf{op}}$-modules
      3. 8.3 Analytic $\mathsf{gr}^{\mathsf{op}}$-modules and Lie algebras
    4. 9 Primitive filtration of $\mathsf{gr}^{\mathsf{op}}$-modules
      1. 9.1 Primitive filtration
      2. 9.2 Representability of the induced $\mathsf{Cat}_{\mathfrak{Lie}}$-module
    5. 10 Application to primitive exponential $\mathsf{gr}^{\mathsf{op}}$-modules
      1. 10.1 Primitive filtration and coradical filtration
      2. 10.2 Primitive exponential $\mathsf{gr}^{\mathsf{op}}$-modules
      3. 10.3 Equivalence with restricted Lie algebras

###  1\. Introduction

Functors on the category $\mathsf{gr}$ of finitely generated free groups and
group homomorphisms appear naturally in different contexts. For example, they
give rise to natural examples of coefficients for the stable homology of the
family of automorphisms groups of free groups $\mathsf{Aut}(\mathsf{F_{n}})$
(see [2, 1, 10]), where $\mathsf{F}_{n}$ is the free group generated by $n$
elements. By [16], Hochschild-Pirashvili homology for a wedge of circle gives
rise to interesting functors on the category $\mathsf{gr}$. Since the
linearization of $\mathsf{gr}^{\mathsf{op}}$ is a subcategory of the linear
category of Jacobi diagrams in handlebodies introduced by Habiro and Massuyeau
in [5], a linear functor on this linear category gives rise, by restriction,
to a functor on $\mathsf{gr}^{\mathsf{op}}$. This is exploited in [9, 8, 19]
in order to study functors associated to (beaded) open Jacobi diagrams.

To a functor from the category $\mathsf{gr}$ to the category of
$\mathds{k}$-modules (for $\mathds{k}$ a commutative ring), we can associate a
family of linear representations of the groups $\mathsf{Aut}(\mathsf{F}_{n})$,
for $n\in\mathbb{N}$. Hence, a functor from the category $\mathsf{gr}$ to the
category of $\mathds{k}$-modules is called a $\mathsf{gr}$-module and gives a
compatible family of representations of the groups
$\mathsf{Aut}(\mathsf{F_{n}})$, where the compatibilities are given by the
functoriality.

In the applications cited above, the functors on the category $\mathsf{gr}$
(resp. $\mathsf{gr}^{\mathsf{op}}$) satisfy further properties. One of the
advantages of considering functors instead of families of linear
representations lies in the exploitation of these properties.

This paper deals with the functors on the category $\mathsf{gr}^{\mathsf{op}}$
satisfying the exponential property (or, equivalently, the fact that the
functor is symmetric monoidal) and the analyticity property obtained from the
polynomial property in the sense of Eilenberg-Mac Lane.

On the one hand, by the work of Pirashvili [13] (see also [7, Remark 2.1,
Theorem 2.2] and [4]), the category of exponential functors on
$\mathsf{gr}^{\mathsf{op}}$ is equivalent to the category of cocommutative
Hopf algebras. On the other hand, Powell considers, in [14], the category of
analytic functors from $\mathsf{gr}^{\mathsf{op}}$ to the category of
$\mathds{k}$-vector spaces (for $\mathds{k}$ a field of characteristic $0$).
He proves an equivalence between this category and the category
$\mathcal{F}_{\mathfrak{Lie}}$ of linear functors on the linear PROP
associated to the Lie operad $\mathfrak{Lie}$.

This paper answers the following question: how do these two equivalences of
categories interact or more precisely can we describe the category of analytic
exponential functors on $\mathsf{gr}^{\mathsf{op}}$?

Let
$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$ be
the category of exponential and analytic functors on
$\mathsf{gr^{\mathsf{op}}}$ and
$\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}$ the category of cocommutative
and conilpotent Hopf algebras. We obtain the following result, which is valid
for any commutative ring $\mathds{k}$:

######  Theorem 1.

(Theorem 8.2) The evaluation on ${\mathsf{F}_{1}}$ induces an equivalence:

$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}.$

The proof of this result is independent of Powell’s result and is based on
Theorem 5.2 implying that the polynomial filtration of an exponential functor
on $\mathsf{gr^{\mathsf{op}}}$ and the coradical filtration coincide. Theorem
5.2 extends a result obtained by Touzé [18, Theorem 11.10] concerning functors
on an additive category. Recall that the category $\mathsf{gr^{\mathsf{op}}}$
is not additive.

For $\mathds{k}$ a field of characteristic zero, using the equivalence of
categories:

$P:\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathsf{Lie}_{\mathds{k}}$

given by the primitive elements of Hopf algebras, we obtain:

######  Corollary 2.

(Corollary 8.4) For $\mathds{k}$ a field of characteristic zero, the
composition of the evaluation on ${\mathsf{F}_{1}}$ with the functor $P$
induces an equivalence of categories:

$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{Lie}_{\mathds{k}}.$

Another important property which could be satisfied by a functor on the
category $\mathsf{gr}^{\mathsf{op}}$ is the outer property introduced in [16].
A functor $F:\mathsf{gr^{\mathsf{op}}}\to\mathsf{Mod}_{\mathds{k}}$ is an
outer functor if the inner automorphisms act trivially on $F(\mathsf{F}_{n})$
for all $n\in\mathbb{N}$. Outer functors appear naturally in [16] and [9]. For
$\mathds{k}$ a field of characteristic zero, Powell describes in [15] outer
analytic functors. More precisely he gives an equivalence of categories
between outer analytic functors on $\mathsf{gr^{\mathsf{op}}}$ and a
subcategory of $\mathcal{F}_{\mathfrak{Lie}}$ denoted by
$\mathcal{F}_{\mathfrak{Lie}}^{\mu}$.

Let
$\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
be the full subcategory of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ consisting
of outer $\mathsf{gr}^{\mathsf{op}}$-modules and
$\mathsf{Hopf}^{\mathsf{bc}}_{\mathds{k}}$ be the category of bicommutative
Hopf algebras. We obtain the following result:

######  Theorem 3.

(Theorem 8.3) The evaluation on ${\mathsf{F}_{1}}$ induces an equivalence of
categories:

$\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\simeq\mathsf{Hopf}^{\mathsf{bc}}_{\mathds{k}}.$

This result is deduced from the fact (see Theorem 8.3) that the composition by
the abelianization functor $\mathfrak{a}:\mathsf{gr}\to\mathsf{ab}$ (where
$\mathsf{ab}$ is the category consisting of the free abelian groups
$\mathds{Z}^{n}$ for $n\in\mathds{N}$ and group homomorphisms) induces an
equivalence of categories:

$\mathcal{F}^{\mathsf{exp}}(\mathsf{ab}^{\mathsf{op}};\mathds{k})\simeq\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k}).$

Let
$\mathcal{F}^{\mathsf{exp}}_{\omega,\mathsf{out}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
be the full subcategory of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ consisting
of analytic and outer $\mathsf{gr}^{\mathsf{op}}$-modules. For $\mathds{k}$ a
field of characteristic zero, we obtain:

######  Corollary 4.

(Corollary 8.4) For $\mathds{k}$ a field of characteristic zero, the
composition of the evaluation on ${\mathsf{F}_{1}}$ with the functor $P$
induces an equivalence of categories:

$\mathcal{F}^{\mathsf{exp}}_{\omega,\mathsf{out}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{Lie}^{\mathsf{ab}}_{\mathds{k}}.$

For $\mathds{k}$ a field of characteristic zero, our results and previous
results obtained by Pirashvili and Powell can be summarized in the following
commutative diagram:

${\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}})}$${\mathcal{F}_{\omega}(\mathsf{gr^{\mathsf{op}}})}$${\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}})}$${\mathcal{F}^{\mathsf{exp}}_{\omega,\mathsf{out}}(\mathsf{gr^{\mathsf{op}}})}$${\mathcal{F}_{\omega,\mathsf{out}}(\mathsf{gr^{\mathsf{op}}})}$${\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr^{\mathsf{op}}})}$${\mathcal{F}_{\mathsf{out}}(\mathsf{gr^{\mathsf{op}}})}$${\mathsf{Lie}_{\mathds{k}}}$${\mathcal{F}_{\mathfrak{Lie}}}$${\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}}$${\mathsf{Lie}^{\mathsf{ab}}_{\mathds{k}}}$${\mathcal{F}^{\mu}_{\mathfrak{Lie}}}$${\mathsf{Hopf}^{\mathsf{bc}}_{\mathds{k}}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where we omit the field $\mathds{k}$ in the notation of the categories of
functors. For example, we write
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}})$ instead of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$.

For $\mathds{k}$ a field of positive characteristic, in Corollary 10.19, we
obtain similar statements to Theorems 1, 3 by replacing the polynomial
filtration with the primitive filtration introduced in section 9.

The above results concern $\mathsf{gr}^{\mathsf{op}}$-modules. These are
consequences of more general results obtained in this paper and concerning
functors on a small category $\mathcal{C}$ with finite products and a null
object.

In part I, we study the polynomial filtration of functors on $\mathcal{C}$. In
section 3, we recall the polynomial filtration of functors. Section 4 concerns
the coradical filtration of coaugmented coalgebras. One of the main
ingredients of this paper is the description of the coradical filtration based
on idempotents and mimicking the notion of cross-effect for functors (see
Proposition 4.4). We also give examples of computations of the coradical
filtration using the description given in Proposition 4.4. The proof of this
result is based on Proposition 4.4. In section 5 (see Theorem 5.2) we prove
that the polynomial filtration of an exponential functor $F$ on $\mathcal{C}$,
evaluated on $X$, coincides with the coradical filtration of the bialgebra
$F(X)$ for any $X\in\mathcal{C}$.

In part II, we give applications of part I to
$\mathcal{C}=\mathsf{gr}^{\mathsf{op}}$. In section 6, we give an overview of
the operad $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ of unital associative algebras
and related constructions. The goal of this section is to recall Powell’s
assertion [14] that $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ induces a bilinear
functor
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}:\mathsf{Cat}_{\mathfrak{Lie}}^{\mathsf{op}}\times\mathds{k}\mathsf{gr}^{\mathsf{op}}\to\mathsf{Mod}_{\mathds{k}}$.
In section 7, we introduce a left ideal $\mathcal{I}$ of the linear category
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$ which is used to introduce the primitive
filtration of $\mathsf{gr}^{\mathsf{op}}$-modules. In section 8, we apply the
results of part I to study the polynomial filtration of exponential
$\mathsf{gr}^{\mathsf{op}}$-modules. In section 9, we introduce the primitive
filtration of $\mathsf{gr}^{\mathsf{op}}$-modules. In section 10, we study
primitive exponential $\mathsf{gr}^{\mathsf{op}}$-modules.

### Acknowledgements

We thank Geoffrey Powell for valuable comments on previous version of this
paper. We thank Kawazumi Sensei for introducing the authors to each other and
giving the opportunity to start this project. The first author appreciates
Leading Graduate Course for Frontiers of Mathematical Sciences and Physics;
and a KIAS Individual Grant MG093701 at Korea Institute for Advanced Study.
The second author would like to thank the JSPS (The Japan Society for the
Promotion of Science) for its support through the invitational fellowship for
research in Japan (long-term) ID L21510 and the university of Tokyo for its
hospitality.

###  2\. Notation

We denote by $\mathds{k}$ a commutative unital ring unless otherwise
specified.

We denote by $\mathds{N}$ the set of natural numbers $0,1,2,\cdots$.

We denote by $\mathcal{C}$ a small category with finite products $\times$ and
a null object $\ast$. We use the notation $\varepsilon_{X}:X\to\ast$ and
$\eta_{X}:\ast\to X$ to denote the unique morphisms and we use the notation
$p_{X}:=\eta_{X}\circ\varepsilon_{X}:X\to X$. We can view $\mathcal{C}$ as a
symmetric monoidal category of the form $(\mathcal{C},\times,\ast)$. The
example of main interest here is the category $\mathcal{C}_{H}$ generated by a
coaugmented cocommutative coalgebra $H$ (see section 5). The product is given
by the tensor product. Another example arises from the opposite category of
the category of finitely generated free groups and group homomorphisms. The
product is given by the free product $\ast$ of groups and the trivial group is
the null object. We choose a skeleton of the category which we denote by
$\mathsf{gr}^{\mathsf{op}}$. Its objects consist of natural numbers $n$ which
we identify with $\mathsf{F}_{n}$ the free group generated by $n$ elements
$x_{1},x_{2},\cdots,x_{n}$ (with the unit $e$). Using this notation, the free
product $\ast$ is given by the sum of natural numbers. A morphism from $n$ to
$m$ is given by a group homomorphism from $\mathsf{F}_{m}$ to
$\mathsf{F}_{n}$.

We introduce the notation of morphisms in the opposite category
$\mathsf{gr}^{\mathsf{op}}$. For a group homomorphism
$\rho:\mathsf{F}_{m}\to\mathsf{F}_{n}$, we denote by
$[\rho(x_{1})|\cdots|\rho(x_{m})]_{n}:n\to m$ the corresponding morphism in
$\mathsf{gr}^{\mathsf{op}}$. In particular, we set
$\Delta{:=}[x_{1}|x_{1}]_{1}$ for the fold map, $\nabla{:=}[x_{1}x_{2}]_{2}$
for the coproduct and $\gamma{:=}[x_{1}^{-1}]_{1}$ for the antipode
respectively. We denote by $\eta:0\to 1$, $\epsilon:1\to 0$ the trivial
morphisms.

We denote by $[X_{i}]^{n}_{i=1}$ a finite sequence of objects of
$\mathcal{C}$.

For a category $\mathcal{C}$ and objects $X,Y\in\mathcal{C}$, we denote by
$\mathcal{C}(X,Y)$ the set of morphisms from $X$ to $Y$.

For functors $F,G$ from $\mathcal{C}$ to $\mathsf{Mod}_{\mathds{k}}$, we
denote by $\mathrm{Nat}(F,G)$ the $\mathds{k}$-module consisting of natural
transformations from $F$ to $G$.

We denote by $\Sigma$ the category induced by the symmetric groups. In other
words, its objects are given by $n\in\mathds{N}$ and the morphism set
$\Sigma(n,m)$ is given by $\Sigma_{n}$ the $n$-th symmetric group if $n=m$;
and is empty if $n\neq m$.

We denote by $\underline{n}{:=}\\{1,\ldots,n\\}$ for $n>0$, and
$\underline{0}{:=}\emptyset$.

##  Part I The general case

###  3\. Polynomial filtration of $\mathcal{C}$-modules

In this section, we recall the notions of cross-effects and polynomial
filtration of $\mathcal{C}$-modules. These notions were introduced by
Eilenberg and Mac Lane in [3] for functors between categories of modules and
extended to functors on a monoidal category where the unit is a null object in
[6]. The aim of this section is to give an explicit way to construct the
polynomial approximations.

####  3.1. Cross-effects of $\mathcal{C}$-modules

Let $\mathds{k}$ be a commutative unital ring. A $\mathcal{C}$-module is a
functor from $\mathcal{C}$ to the category $\mathsf{Mod}_{\mathds{k}}$ of
$\mathds{k}$-modules and homomorphisms. We denote by
$\mathcal{F}(\mathcal{C};\mathds{k})$ the category of $\mathcal{C}$-modules
and natural transformations.

Let $\mathcal{F}^{(n)}_{\ast}(\mathcal{C};\mathds{k})$ be the full subcategory
of $\mathcal{F}(\mathcal{C}^{\times n};\mathds{k})$ consisting of
$F\in\mathcal{F}(\mathcal{C}^{\times n};\mathds{k})$ such that
$F(X_{1},\cdots,X_{n})\cong 0$ if $X_{i}=\ast$ for some $i$. We denote by
$\Delta_{n}:\mathcal{C}\to\mathcal{C}^{\times n}$ the diagonal functor.

######  Definition 3.1.

For $n\geq 1$, the $n$-th cross-effect functor is the left adjoint to the
precomposition with the diagonal functor
$\Delta_{n}^{\ast}:\mathcal{F}^{(n)}_{\ast}(\mathcal{C};\mathds{k})\to\mathcal{F}(\mathcal{C};\mathds{k})$.

In order to give an explicit description of the $n$-th cross-effect functor,
we introduce the following notation. Let $m,n$ be natural numbers such that
$1\leq m\leq n$. For $[X_{i}]^{n}_{i=1}$ a finite sequence of objects of
$\mathcal{C}$ and $p_{X_{m}}=\eta_{X_{m}}\circ\varepsilon_{X_{m}}:X_{m}\to
X_{m}$, we define the morphism:

$\tau_{m}([X_{i}]^{n}_{i=1}){:=}\left(\prod^{m-1}_{i=1}\mathrm{id}_{X_{i}}\right)\times
p_{X_{m}}\times\left(\prod^{n}_{i=m+1}\mathrm{id}_{X_{i}}\right):\prod^{n}_{i=1}{X_{i}}\to\prod^{n}_{i=1}{X_{i}}.$

The morphisms $\tau_{m}([X_{i}]^{n}_{i=1})$ are pairwise commuting
idempotents. Hence, for $F\in\mathcal{F}(\mathcal{C};\mathds{k})$ and
$X=\prod^{n}_{i=1}X_{i}$, the morphisms $e_{m}:F(X)\to F(X)$ defined by:

$e_{m}=\mathrm{id}_{F(X)}-F(\tau_{m}([X_{i}]^{n}_{i=1}))$

are pairwise commuting idempotents.

We define an endomorphism of $F(X)$ by:

$\chi_{F}([X_{i}]^{n}_{i=1}){:=}(e_{1}\circ e_{2}\circ\cdots\circ
e_{n}):F(X)\to F(X).$

For $n\geq 1$, we define
$\mathrm{cr}^{n}(F)\in\mathcal{F}^{(n)}_{\ast}(\mathcal{C};\mathds{k})$ by

(1)
$(\mathrm{cr}^{n}(F))([X_{i}]^{n}_{i=1}){=}\mathrm{Im}\left(\chi_{F}([X_{i}]^{n}_{i=1})\right).$

This induces a functor

$\mathrm{cr}^{n}:\mathcal{F}(\mathcal{C};\mathds{k})\to\mathcal{F}^{(n)}_{\ast}(\mathcal{C};\mathds{k}).$

######  Proposition 3.2.

The functor
$\mathrm{cr}^{n}:\mathcal{F}(\mathcal{C};\mathds{k})\to\mathcal{F}^{(n)}_{\ast}(\mathcal{C};\mathds{k})$
is the $n$-th cross-effect functor.

###### Proof.

Let $F\in\mathcal{F}(\mathcal{C};\mathds{k})$ and
$G\in\mathcal{F}^{(n)}_{\ast}(\mathcal{C};\mathds{k})$. We give a direct
construction of a natural isomorphism
$\Phi:\mathrm{Nat}(F,\Delta_{n}^{\ast}(G))\to\mathrm{Nat}(\mathrm{cr}^{n}(F),G)$.
Let $\beta\in\mathrm{Nat}(F,\Delta_{n}^{\ast}(G))$. For an $n$-tuple
$[X_{i}]^{n}_{i=1}$ in $\mathcal{C}$, let $X=\prod^{n}_{i=1}X_{i}$ and
$\pi_{i}:X\to X_{i}$ be the $i$-th projection. We define
$\Phi(\beta):(\mathrm{cr}^{n}(F))\left([X_{i}]^{n}_{i=1}\right)\to
G\left([X_{i}]^{n}_{i=1}\right)$ to be the composition:

$(\mathrm{cr}^{n}(F))\left([X_{i}]^{n}_{i=1}\right)\hookrightarrow
F(X)\stackrel{{\scriptstyle\beta_{X}}}{{\to}}(\Delta_{n}^{\ast}(G))(X)=G\left([X]^{n}_{i=1}\right)\stackrel{{\scriptstyle
G([\pi_{i}]^{n}_{i=1})}}{{\longrightarrow}}G\left([X_{i}]^{n}_{i=1}\right).$

The inverse of $\Phi$ is given by
$\Psi:\mathrm{Nat}(\mathrm{cr}^{n}(F),G)\to\mathrm{Nat}(F,\Delta_{n}^{\ast}(G))$
defined as follows. Let $\beta^{\prime}\in\mathrm{Nat}(\mathrm{cr}^{n}(F),G)$.
For $Z\in\mathcal{C}$ with the $n$-times iterated diagonal map
$\Delta^{(n)}_{Z}:Z\to Z^{\times n}$, we define
$\Psi(\beta^{\prime}):F(Z)\to(\Delta_{n}^{\ast}(G))(Z)$ as the composition:

$F(Z)\stackrel{{\scriptstyle
F(\Delta^{(n)}_{Z})}}{{\longrightarrow}}F(Z^{\times
n})\stackrel{{\scriptstyle\chi_{F}([Z]^{n}_{i=1})}}{{\longrightarrow}}(\mathrm{cr}^{n}(F))([Z]^{n}_{i=1})\stackrel{{\scriptstyle\beta^{\prime}_{[Z]^{n}_{i=1}}}}{{\longrightarrow}}(\Delta_{n}^{\ast}(G))(Z).$

∎

######  Proposition 3.3.

Let $F\in\mathcal{F}(\mathcal{C};\mathds{k})$. Then we have a natural
isomorphism:

$F(\prod^{n}_{i=1}X_{i})\cong
F(\ast)\oplus\left(\bigoplus^{n}_{k=1}\bigoplus_{1\leq i_{1}<\cdots<i_{k}\leq
n}(\mathrm{cr}^{k}(F))(X_{i_{1}},\cdots,X_{i_{k}})\right).$

###### Proof.

The previous idempotents $e_{i}$’s are simultaneously diagonalizable since
they commute with each other. For $1\leq i_{1}<\cdots<i_{k}\leq n$, the
simultaneous eigenspace consisting of $v\in F(X)$ such that $e_{i}v=v$ only
for $i=i_{1},\cdots i_{k}$ coincides with
$(\mathrm{cr}^{k}(F))(X_{i_{1}},\cdots,X_{i_{k}})$. The eigenspace consisting
of $v\in F(X)$ such that $e_{i}v=0$ for any $i$ coincides with $F(\ast)$. This
observation leads to the natural isomorphism. ∎

####  3.2. Polynomial and analytic functors

In this section, we give an overview of polynomial functors by using cross-
effects. We define the polynomial functor approximation as being the right
adjoint of the inclusion.

######  Definition 3.4.

A functor $F\in\mathcal{F}(\mathcal{C};\mathds{k})$ has degree less than or
equal to $n$, written $\mathrm{deg}(F)\leq n$, if $\mathrm{cr}^{n+1}(F)\cong
0$; and $F\in\mathcal{F}(\mathcal{C};\mathds{k})$ has degree $n$ if
$\mathrm{deg}(F)\leq n$ and $\mathrm{deg}(F)\not\leq(n-1)$. A
$\mathcal{C}$-module $F$ is a polynomial functor if there exists
$n\in\mathbb{N}$ such that $\mathrm{deg}(F)\leq n$. Denote by
$\mathcal{F}_{\leq n}(\mathcal{C};\mathds{k})$ the full subcategory of
$\mathcal{F}(\mathcal{C};\mathds{k})$ consisting of functors with degree less
than or equal to $n$.

Note that, if $\mathrm{deg}(F)\leq n$, then $\mathrm{deg}(F)\leq(n+1)$.

######  Definition 3.5.

Let $i_{n}:\mathcal{F}_{\leq
n}(\mathcal{C};\mathds{k})\to\mathcal{F}(\mathcal{C};\mathds{k})$ be the
inclusion. The $n$-th polynomial approximation
$P_{n}:\mathcal{F}(\mathcal{C};\mathds{k})\to\mathcal{F}_{\leq
n}(\mathcal{C};\mathds{k})$ is the right adjoint to the inclusion $i_{n}$.

######  Proposition 3.6.

  * •

For $F\in\mathcal{F}(\mathcal{C};\mathds{k})$, the counits of the adjunctions
induce the following commutative diagram:

${F}$${\cdots}$${i_{n}(P_{n}(F))}$${i_{n+1}(P_{n+1}(F))}$${\cdots}$

  * •

For $G\in\mathcal{F}_{\leq n}(\mathcal{C};\mathds{k})$, the unit $G\to
P_{n}(i_{n}(G))$ is an isomorphism.

###### Proof.

Let $j:\mathcal{F}_{\leq
n}(\mathcal{C};\mathds{k})\to\mathcal{F}_{\leq(n+1)}(\mathcal{C};\mathds{k})$
be the inclusion. The unit $i_{n}(P_{n}(F))\to F$ induces a natural
transformation $j(P_{n}(F))\to P_{n+1}(F)$ by the adjunction with respect to
$i_{n+1}$. We apply $i_{n+1}$ to this to obtain $i_{n}(P_{n}(F))\to
i_{n+1}(P_{n+1}(F))$. The commutativity in the first part follows from
adjunction properties.

For the second part, it suffices to verify that the postcomposition
$\mathrm{Nat}(G^{\prime},G)\to\mathrm{Nat}(G^{\prime},P_{n}(i_{n}(G)))$ gives
a bijection. Note that $i_{n}:\mathcal{F}_{\leq
n}(\mathcal{C};\mathds{k})\to\mathcal{F}(\mathcal{C};\mathds{k})$ is a fully
faithful functor by definition. Equivalently, we have a bijection
$\mathrm{Nat}(G^{\prime},G)\to\mathrm{Nat}(i_{n}(G^{\prime}),i_{n}(G))$. The
result follows from the adjunction
$\mathrm{Nat}(G^{\prime},P_{n}(i_{n}(G)))\cong\mathrm{Nat}(i_{n}(G^{\prime}),i_{n}(G))$.
∎

######  Remark 3.7.

The existence of a right adjoint to the inclusion $i_{n}$ is proved in Section
3.3. It turns out that the natural transformations in the first part of
Proposition 3.6 are inclusions. In particular, the sequence of
$i_{n}(P_{n}(F))$’s gives a filtration of $F$ called the polynomial filtration
of $F$.

######  Definition 3.8.

A $\mathcal{C}$-module $F$ is analytic if $F$ is the colimit of its polynomial
filtration. Denote by $\mathcal{F}_{\omega}(\mathcal{C};\mathds{k})$ the full
subcategory of $\mathcal{F}(\mathcal{C};\mathds{k})$ consisting of analytic
$\mathcal{C}$-module.

####  3.3. Construction of the polynomial filtration

In this section, we show that there exists a polynomial approximation as in
Definition 3.5. We give an explicit description of the functor $P_{n}(F)$ and
we prove that it corresponds to the right adjoint of the inclusion.

######  Definition 3.9.

For $X$ an object of $\mathcal{C}$, we define $(P_{n}(F))(X)$ to be the kernel
of the composite,

$F(X)\stackrel{{\scriptstyle
F(\Delta^{(n+1)}_{X})}}{{\to}}F(X^{\times(n+1)})\stackrel{{\scriptstyle\chi}}{{\longrightarrow}}F(X^{\times(n+1)})$

where $\Delta^{(n+1)}_{X}$ is the $(n+1)$-times iterated diagonal map of $X$
and $\chi=\chi_{F}([X]^{n+1}_{i=1})$. Denote by
$P_{n}(F)\in\mathcal{F}(\mathcal{C};\mathds{k})$ the induced functor.

######  Lemma 3.10.

For $F\in\mathcal{F}(\mathcal{C};\mathds{k})$, we have
$\mathrm{cr}^{n+1}(P_{n}(F))\cong 0$. In particular,
$\mathrm{deg}(P_{n}(F))\leq n$.

###### Proof.

We prove that $\mathrm{cr}^{n+1}(P_{n}(F))([X_{i}]^{n+1}_{i=1})=0$ for a
sequence $[X_{i}]^{n+1}_{i=1}$ of objects of $\mathcal{C}$. It suffices to
show that $\chi_{P_{n}(F)}([X_{i}]^{n+1}_{i=1})=0$. Let
$X=\prod^{n+1}_{i=1}X_{i}$. Let $\pi:X^{\times(n+1)}\to\prod^{n+1}_{i=1}X_{i}$
be $\prod^{n+1}_{i=1}\pi_{i}$ where $\pi_{i}=\prod_{j\neq
i}\varepsilon_{j}:X=X_{1}\times\cdots\times X_{n+1}\to X_{i}$. It is easy to
check that the diagram below commutes for any $j$ where $\tau_{j}$ is
introduced in section 3.1:

${X^{\times(n+1)}}$${X^{\times(n+1)}}$${\prod^{n+1}_{i=1}X_{i}}$${\prod^{n+1}_{i=1}X_{i}}$$\scriptstyle{\tau_{j}({[X]}^{n+1}_{i=1})}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\tau_{j}({[X_{i}]}^{n+1}_{i=1})}$

By combining the above commutative diagrams iteratively, we obtain

$F(\pi)\circ\chi_{F}([X]^{n+1}_{i=1})=\chi_{F}([X_{i}]^{n+1}_{i=1})\circ
F(\pi).$

Moreover, $\pi\circ\Delta^{(n+1)}_{X}=\mathrm{id}_{X}$ implies
$F(\pi)\circ\chi_{F}([X]^{n+1}_{i=1})\circ
F(\Delta^{(n+1)}_{X})=\chi_{F}([X_{i}]^{n+1}_{i=1})$. Now let
$x\in(P_{n}(F))(X)$. Then the above result implies

$\displaystyle(\chi_{P_{n}(F)}([X_{i}]^{n+1}_{i=1}))(x)$
$\displaystyle=(\chi_{F}([X_{i}]^{n+1}_{i=1}))(x),$
$\displaystyle=\left(F(\pi)\circ\left(\chi_{F}([X]^{n+1}_{i=1})\circ
F(\Delta^{(n+1)}_{X})\right)\right)(x),$
$\displaystyle=0~{}~{}~{}(\mathrm{since}~{}~{}~{}x\in(P_{n}(F))(X)).$

∎

######  Proposition 3.11.

The assignment of $P_{n}(F)$ to $F\in\mathcal{F}(\mathcal{C};\mathds{k})$
gives a right adjoint to the inclusion $i_{n}$. In particular, $P_{n}$ is the
$n$-th polynomial approximation.

###### Proof.

Note that Lemma 3.10 implies that $P_{n}(F)\in\mathcal{F}_{\leq
n}(\mathcal{C};\mathds{k})$, so that we obtain a functor
$P_{n}:\mathcal{F}(\mathcal{C};\mathds{k})\to\mathcal{F}_{\leq
n}(\mathcal{C};\mathds{k})$. The inclusion $\epsilon_{F}:i_{n}(P_{n}(F))\to F$
induces a natural transformation
$\mathrm{Nat}(G,P_{n}(F))\to\mathrm{Nat}(i_{n}(G),F);~{}\phi\mapsto\epsilon_{F}\circ
i_{n}(\phi)$ for $G\in\mathcal{F}_{\leq n}(\mathcal{C};\mathds{k})$ and
$F\in\mathcal{F}(\mathcal{C};\mathds{k})$. We now prove that this gives a
natural isomorphism. It suffices to show that a natural transformation
$\psi:i_{n}(G)\to F$ factors uniquely through $\epsilon_{F}$. Let $X$ be an
object of $\mathcal{C}$ and put $Y=X^{\times(n+1)}$. Then the naturality of
$\psi$ implies $(\chi_{F}([X]^{n+1}_{i=1})\circ
F(\Delta^{(n+1)}_{X}))\circ\psi_{X}=\psi_{Y}\circ(\chi_{G}([X]^{n+1}_{i=1})\circ
G(\Delta^{(n+1)}_{X}))$, with the right hand side zero due to the assumption
$\mathrm{cr}^{n+1}(G)\cong 0$. Hence, $(\chi_{F}([X]^{n+1}_{i=1})\circ
F(\Delta^{(n+1)}_{X}))\circ\psi_{X}=0$, and thus $\psi_{X}$ uniquely factors
through the inclusion $\epsilon_{F}$.

∎

######  Remark 3.12.

Dually, one can consider a small category $\mathcal{D}$ with finite coproducts
and a null object. Let $j_{n}:\mathcal{F}_{\leq
n}(\mathcal{D};\mathds{k})\to\mathcal{F}(\mathcal{D};\mathds{k})$ be the
inclusion. In a similar fashion, one can construct a cofiltration of a
$\mathcal{D}$-module $F$ by using a left adjoint $P^{n}$ to the inclusion
$j_{n}$:

${F}$${\cdots}$${j_{n+1}(P^{n+1}(F))}$${j_{n}(P^{n}(F))}$${\cdots}$

###  4\. Coradical filtration of coaugmented coalgebras

This section concerns the coradical filtration of coaugmented coalgebras.
After recalling the usual definition of the coradical filtration, we give an
equivalent definition based on idempotents by mimicking the notion of cross-
effects for functors recalled in section 3.1. The description of the $n$-th
term of the coradical filtration as a kernel in Proposition 4.4 will be useful
in section 5. To illustrate Proposition 4.4, we give in section 4.3 the
coradical filtrations of the polynomial bialgebra, the shuffle bialgebra and
finite-dimensional bialgebras.

####  4.1. Coradical filtration

Let $\mathds{k}$ be a commutative unital ring. We recall that a coaugmented
coalgebra (over $\mathds{k}$) is a quadruple $(H,\Delta,\varepsilon,\eta)$
such that $(H,\Delta,\varepsilon)$ is a coalgebra with the comultiplication
$\Delta:H\to H\otimes H$ and counit $\varepsilon:H\to\mathds{k}$ ; and
$\eta:\mathds{k}\to H$ is a coalgebra homomorphism. We abbreviate
$(H,\Delta,\varepsilon,\eta)$ to a coaugmented coalgebra $H$ if no confusion
arises, and use notations $\Delta_{H},\varepsilon_{H},\eta_{H}$ for its
structure maps. For simplicity, we introduce the notation $1_{H}$ to denote
$\eta(1_{\mathds{k}})\in H$. A morphism between coaugmented coalgebras is a
coalgebra homomorphism preserving the coaugmentations. We denote by
$\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}}$ the category of coaugmented
coalgebras and their morphisms. The tensor product of coalgebras induces a
symmetric monoidal structure on
$\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}}$.

For a coaugmented coalgebra $H$, we introduce the endomorphism
$e_{H}{:=}(\mathrm{id}_{H}-\eta_{H}\circ\varepsilon_{H})$ of the underlying
module of $H$. It is an idempotent such that
$\mathrm{Ker}(e_{H})=\mathds{k}1_{H}$ and $\mathrm{Im}(e_{H})=\overline{H}$
where $\overline{H}{:=}\mathrm{Ker}(\varepsilon_{H})$. In particular, we have
an eigenspace decomposition $H\cong\mathds{k}1_{H}\oplus\overline{H}$.

The reduced comultiplication
$\overline{\Delta}_{H}:\overline{H}\to\overline{H}\otimes\overline{H}$ is
defined by $\overline{\Delta}_{H}(a){:=}\Delta_{H}(a)-a\otimes
1_{H}-1_{H}\otimes a$. We define
$\mathrm{Prim}(H){:=}\mathrm{Ker}(\overline{\Delta}_{H})$ whose elements are
called to be primitive. It is well known that for a bialgebra $H$,
$\mathrm{Prim}(H)$ associated with the underlying coaugmented coalgebra of $H$
becomes a Lie algebra by the commutator.

The reduced comultiplication is coassociative and thus it induces a well-
defined $n$-times iterated reduced comultiplication
$\overline{\Delta}_{H}^{(n)}:\overline{H}\to\overline{H}^{\otimes n}$. This
induces the coradical filtration consisting of submodules
$P_{n}(H){:=}\mathds{k}1_{H}\oplus\mathrm{Ker}(\overline{\Delta}_{H}^{(n+1)})\subset
H$ for $n\in\mathds{N}$ (e.g. see [11, Section 1.2.4]):

(2) $\displaystyle P_{0}(H)\subset\cdots\subset P_{n}(H)\subset
P_{n+1}(H)\subset\cdots\subset H$

######  Remark 4.1.

Note that, in this paper, we use the same notation $P_{n}$ to denote the
coradical filtration of coaugmented coalgebras and also the polynomial
approximation of functors (see Definition 3.5). We investigate their relation
in section 5.

The coaugmented coalgebra $H$ has degree less than or equal to $n$ if
$P_{n}(H)=H$. We denote by
$(\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}})_{\leq n}$ the full
subcategory of $\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}}$ of
coaugmented coalgebras of degree less than or equal to $n$.

The coaugmented coalgebra $H$ is conilpotent if
$H=\bigcup_{n\in\mathds{N}}P_{n}(H)$, or, equivalently, for any
$x\in\overline{H}$ there exists $n\in\mathds{N}$ such that
$\overline{\Delta}^{(n)}_{H}(x)=0$.

The following definition should be viewed as an analogue for coaugmented
coalgebras of the explicit description of the cross-effect functor given in
(1).

######  Definition 4.2.

Let $H$ be a coaugmented coalgebra and $n\in\mathds{N}$. For $a\in H$, we
define a linear map $\delta^{n}_{H}:H\to H^{\otimes n}$ by
$\delta^{n}_{H}(a){:=}e_{H}^{\otimes n}\left(\Delta_{H}^{(n)}(a)\right)\in
H^{\otimes n}$. Here, $\Delta^{(n)}_{H}$ is the $n$-times iterated
comultiplication of $H$ and $e_{H}^{\otimes n}:H^{\otimes n}\to H^{\otimes n}$
is the $n$-fold tensor product of $e_{H}$.

######  Example 4.3.

For $n=0,1,2$, we have

$\displaystyle\delta^{0}_{H}(a)$ $\displaystyle=\varepsilon_{H}(a),$
$\displaystyle\delta^{1}_{H}(a)$
$\displaystyle=e_{H}(a)=a-\varepsilon_{H}(a)1_{H},$
$\displaystyle\delta^{2}_{H}(a)$ $\displaystyle=\Delta_{H}(a)-a\otimes
1_{H}-1_{H}\otimes a+\varepsilon_{H}(a)1_{H}^{\otimes 2}.$

######  Proposition 4.4.

Let $n\in\mathds{N}$.

  1. (1)

For $a\in\overline{H}$, we have
$\delta^{n}_{H}(a)=\overline{\Delta}_{H}^{(n)}(a)$.

  2. (2)

We have $P_{n}(H)=\mathrm{Ker}(\delta^{n+1}_{H})$.

###### Proof.

The linear map $\delta^{n}_{H}$ can be considered as an $n$-times iterated
application of $\delta^{2}_{H}:H\to H\otimes H$ as follows. Namely, we have
$(\delta^{2}_{H}\otimes\mathrm{id}_{H}^{\otimes(n-1)})\circ\delta^{n}_{H}=\delta^{n+1}_{H}$,
by the calculation below:

$\displaystyle(\delta^{2}_{H}\otimes\mathrm{id}_{H}^{\otimes(n-1)})\circ
e_{H}^{\otimes n}\circ\Delta_{H}^{(n)}=$
$\displaystyle(\delta^{2}_{H}\otimes\mathrm{id}_{H}^{\otimes(n-1)})\circ((\mathrm{id}_{H}-\eta_{H}\circ\varepsilon_{H})\otimes
e_{H}^{\otimes(n-1)})\circ\Delta_{H}^{(n)},$ $\displaystyle=$
$\displaystyle(\delta^{2}_{H}\otimes\mathrm{id}_{H}^{\otimes(n-1)})\circ(\mathrm{id}_{H}\otimes
e_{H}^{\otimes(n-1)})\circ\Delta_{H}^{(n)},$ $\displaystyle=$
$\displaystyle((e_{H}^{\otimes
2}\circ\Delta_{H})\otimes\mathrm{id}_{H}^{\otimes(n-1)})\circ(\mathrm{id}_{H}\otimes
e_{H}^{\otimes(n-1)})\circ\Delta_{H}^{(n)}$ $\displaystyle=$ $\displaystyle
e_{H}^{\otimes(n+1)}\circ\Delta_{H}^{(n+1)}.$

Here, the second equality follows from $\delta^{2}_{H}\circ\eta_{H}=0$. By
Example 4.3 and using $a\in\overline{H}$, we have:

$\delta^{2}_{H}(a)=\Delta_{H}(a)-a\otimes 1_{H}-1_{H}\otimes
a=\overline{\Delta}_{H}(a)$

for every $a\in\overline{H}$. Hence, we can inductively prove the first
assertion by:

$\delta^{n+1}_{H}(a)=((\delta^{2}_{H}\otimes\mathrm{id}_{H}^{\otimes(n-1)})\circ\delta^{n}_{H})(a)=((\overline{\Delta}_{H}\otimes\mathrm{id}_{H}^{\otimes(n-1)})\circ\overline{\Delta}_{H}^{(n)})(a)=\overline{\Delta}_{H}^{(n+1)}(a).$

We prove the second assertion. It is obvious that $\delta^{n}(1_{H})=0$, and
thus we have
$\mathrm{Ker}(\delta^{n}_{H})=\mathds{k}1_{H}\oplus\mathrm{Ker}(\overline{\Delta}_{H}^{(n)})=P_{n}(H)$,
by the above result and the definition of $P_{n}(H)$. ∎

Let $n,k\geq 1$ be natural numbers and consider a subset
$S=\\{m_{1},m_{2},\cdots,m_{k}\\}\subset\underline{n}$ with
$m_{1}<m_{2}<\cdots<m_{k}$. We define the linear map $(-)_{S}:H^{\otimes k}\to
H^{\otimes n}$ to be the linear extension of the assignment of
$1^{\otimes(m_{1}-1)}\otimes a_{1}\otimes 1^{\otimes(m_{2}-m_{1}-1)}\otimes
a_{2}\otimes\cdots\otimes 1^{\otimes(m_{k}-m_{k-1}-1)}\otimes a_{k}\otimes
1^{\otimes(n-m_{k})}$ to a tensor product $a_{1}\otimes\cdots\otimes a_{k}\in
H^{\otimes k}$. In other words, it inserts the $j$-th tensor component into
the $m_{j}$-th component.

The idempotent $e_{H}$ induces a direct sum decomposition of $H^{\otimes n}$
as follows. Let $\tau_{i}(e_{H}){:=}\mathrm{id}_{H}^{\otimes(i-1)}\otimes
e_{H}\otimes\mathrm{id}_{H}^{\otimes(n-i)}$ for $1\leq i\leq n$. For a subset
$S\subset\underline{n}$, let

$H_{n,S}{:=}\\{v\in H^{\otimes n}~{};~{}(\tau_{i}(e_{H}))(v)=v~{}(i\in
S),~{}(\tau_{i}(e_{H}))(v)=0~{}(i\not\in S)\\}.$

It is obvious that $H_{n,S}=\\{w_{S}\in H^{\otimes
n}~{};~{}w\in\overline{H}^{\otimes k}\\}\cong\overline{H}^{\otimes k}$ where
$|S|=k$. Furthermore, we have the decomposition below, since the
$\tau_{i}(e_{H})$’s are commuting idempotents.

(3) $\displaystyle H^{\otimes
n}\cong\bigoplus_{S\subset\underline{n}}H_{n,S}.$

######  Proposition 4.5.

Let $H$ be a coaugmented coalgebra. For $n\in\mathds{N}$ and $a\in H$, the
$H_{n,S}$-component of $\Delta_{H}^{(n)}(a)$ in the decomposition (3) is
$\delta^{|S|}_{H}(a)_{S}$. In particular, the following equality holds.

$\displaystyle\Delta^{(n)}_{H}(a)=\sum_{S\subset\underline{n}}\delta^{|S|}_{H}(a)_{S}.$

###### Proof.

For $S\subset\underline{n}$ such that $|S|=k$, let $\pi_{n,S}:H^{\otimes n}\to
H_{n,S}$ be the projection to the eigenspace. Note that the composition of
$\pi_{n,S}$ and the inverse of the isomorphism $(-)_{S}:\overline{H}^{\otimes
k}\to H_{n,S}$ is equal to the tensor product
$\pi^{\prime}_{n,S}=\bigotimes_{i\in
S}e_{H}\otimes\bigotimes_{i\in\underline{n}\backslash
S}\varepsilon_{H}:H^{\otimes n}\to\overline{H}^{\otimes k}$. By using the
counital property, we have
$\pi^{\prime}_{n,S}(\Delta_{H}^{(n)}(a))=\delta^{k}_{H}(a)$. Hence, we obtain
$\pi_{n,S}(\Delta_{H}^{(n)}(a))=\delta^{k}_{H}(a)_{S}$. ∎

######  Proposition 4.6.

Let $H_{1}$ and $H_{2}$ be coaugmented coalgebras. If $a\in P_{n}(H_{1})$ and
$b\in P_{m}(H_{2})$, then we have $a\otimes b\in P_{n+m}(H_{1}\otimes H_{2})$.

###### Proof.

It suffices to prove that $\delta^{n+m+1}_{H_{1}\otimes H_{2}}(a\otimes b)=0$.
The iterated comultiplication $\Delta^{(n+m+1)}(a\otimes b)$ corresponds to
$\Delta^{(n+m+1)}(a)\otimes\Delta^{(n+m+1)}(b)$ up to the switching
isomorphism

$\tau:H_{1}^{\otimes(n+m+1)}\otimes H_{2}^{\otimes(n+m+1)}\to(H_{1}\otimes
H_{2})^{\otimes(n+m+1)}.$

We apply the decomposition in Proposition 4.5 to
$\Delta^{(n+m+1)}(a),\Delta^{(n+m+1)}(b)$ respectively. By Proposition 4.4,
$\delta^{k}_{H_{1}}(a)=0$ for $k\geq n+1$ and $\delta^{l}_{H_{2}}(b)=0$ for
$l\geq m+1$. Hence, $\Delta^{(n+m+1)}(a)\otimes\Delta^{(n+m+1)}(b)$ is the sum
of
$\delta^{k}_{H_{1}}(a)_{i_{1},\cdots,i_{k}}\otimes\delta^{l}_{H_{2}}(b)_{j_{1},\cdots,j_{l}}$
for $k\leq n$, $l\leq m$, $i_{1},\cdots,i_{k}\in\underline{(n+m+1)}$ and
$j_{1},\cdots,j_{l}\in\underline{(n+m+1)}$. Note that we have

$\tau\left(\delta^{k}_{H_{1}}(a)_{i_{1},\cdots,i_{k}}\otimes\delta^{l}_{H_{2}}(b)_{j_{1},\cdots,j_{l}}\right)\in(H_{1}\otimes
H_{2})_{n+m+1,S}$

where $S=\\{i_{1},\cdots,i_{k},j_{1},\cdots,j_{l}\\}$ and $|S|<n+m+1$. Thus,
by Proposition 4.5, we see that $\delta^{n+m+1}_{H_{1}\otimes H_{2}}(a\otimes
b)=0$. ∎

######  Proposition 4.7.

The tensor product of conilpotent coaugmented coalgebras is conilpotent.

###### Proof.

Let $H_{1},H_{2}$ be conilpotent coaugmented cocommutative coalgebras. Let
$M=\mathrm{colim}_{j}P_{j}(H_{1}\otimes H_{2})$ and $u:M\to H_{1}\otimes
H_{2}$ be the canonical injection. We prove that $u$ is surjective. By
Proposition 4.6, we obtain a linear map $P_{n}(H_{1})\otimes P_{m}(H_{2})\to
P_{n+m}(H_{1}\otimes H_{2})$. This map is compatible with the filtration, so
that it induces
$v:\mathrm{colim}_{n}P_{n}(H_{1})\otimes\mathrm{colim}_{m}P_{m}(H_{2})\to M$
whose domain turns out to be $H_{1}\otimes H_{2}$. The map $u$ is surjective
since $u\circ v$ is the identity. ∎

By replacing the tensor products in Proposition 4.6 with the multiplication of
$H$, we also obtain the following statement:

######  Proposition 4.8.

Let $H$ be a bialgebra. For $a\in P_{n}(H)$ and $b\in P_{m}(H)$, we have
$a\cdot b\in P_{n+m}(H)$ .

####  4.2. Coradical filtration: the case of a field $\mathds{k}$

In this section, we assume that the ground ring $\mathds{k}$ is a field. Under
this hypothesis, we prove that the coradical filtration lifts to a coaugmented
coalgebra filtration. Furthermore, the assignment of the $n$-th coradical
filtration $P_{n}(H)$ to a coaugmented coalgebra $H$ could be described as a
right adjoint.

######  Lemma 4.9.

For linear maps $f:V_{1}\to V_{0}$ and $g:W_{1}\to W_{0}$, let
$f\tilde{\otimes}g:V_{1}\otimes W_{1}\to(V_{0}\otimes
W_{1})\oplus(V_{1}\otimes W_{0})$ be
$f\otimes\mathrm{id}_{W_{1}}\oplus\mathrm{id}_{V_{1}}\otimes g$. Then we have
$\mathrm{Ker}(f)\otimes\mathrm{Ker}(g)\cong\mathrm{Ker}(f\tilde{\otimes}g)$.

###### Proof.

Let $C_{\bullet}$ be the chain complex concentrated only on the zeroth and
first components which are given by $V_{0}$ and $V_{1}$ respectively. The only
nontrivial differential is given by $f$. Let $D_{\bullet}$ be the similar
chain complex induced by $g$. Then the claim follows from the Künneth formula
with respect to the zeroth homology. ∎

######  Proposition 4.10.

The subspace $P_{n}(H)$ is a subcoalgebra of $H$ which inherits the
coaugmentation.

###### Proof.

We prove that the composition $f:P_{n}(H)\hookrightarrow
H\stackrel{{\scriptstyle\Delta}}{{\to}}H\otimes H$ factors through
$P_{n}(H)\otimes P_{n}(H)\hookrightarrow H\otimes H$. Lemma 4.9 implies that
$P_{n}(H)\otimes
P_{n}(H)=\mathrm{Ker}(\delta^{n+1}_{H})\otimes\mathrm{Ker}(\delta^{n+1}_{H})=\mathrm{Ker}(\delta^{n+1}_{H}\tilde{\otimes}\delta^{n+1}_{H})$.
Hence, it suffices to show that
$(\delta^{n+1}_{H}\tilde{\otimes}\delta^{n+1}_{H})\circ f=0$, in particular
that $(\delta^{n+1}_{H}\otimes\mathrm{id}_{H})\circ f=0$. For $h\in P_{n}(H)$,
by the filtration (2) we have $h\in P_{n}(H)\subset P_{n+1}(H)$, so by
Proposition 4.4 we have $\delta^{n+2}_{H}(h)=0$. Note that we have

$\delta^{n+2}_{H}=\left(\delta^{n+1}_{H}\otimes(\mathrm{id}_{H}-\eta\circ\varepsilon)\right)\circ\delta_{H}.$

Hence, we have $\left((\delta^{n+1}_{H}\otimes\mathrm{id}_{H})\circ
f\right)(h)=\left((\delta^{n+1}_{H}\otimes(\eta\circ\varepsilon))\circ
f\right)(h)$. Moreover, for $h\in P_{n}(H)$, we have

$\left((\delta^{n+1}_{H}\otimes(\eta\circ\varepsilon))\circ
f\right)(h)=\delta^{n+1}_{H}(h)\otimes\eta(1)=0.$

Therefore, $f$ factors as $\Delta_{P_{n}(H)}:P_{n}(H)\to P_{n}(H)\otimes
P_{n}(H)$. All that remain is to give the counit and coaugmentation of
$P_{n}(H)$. The restriction of the counit $\varepsilon$ is the counit of
$P_{n}(H)$ ; and the coaugmentation $\eta:\mathds{k}\to H$ induces the
coaugmentation of $P_{n}(H)$. ∎

######  Corollary 4.11.

The functor
$P_{n}:\left(\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}}\right)\to\mathsf{Mod}_{\mathds{k}}$
lifts along the forgetful functor
$\left(\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}}\right)_{\leq
n}\to\mathsf{Mod}_{\mathds{k}}$. Moreover, the lift is a right adjoint of the
embedding functor $i$:

(4) ${\left(\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}}\right)_{\leq
n}}$${\left(\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}}\right)}$$\scriptstyle{i}$$\scriptstyle{P_{n}}$$\scriptstyle{\perp}$

###### Proof.

The first claim is immediate from Proposition 4.10. We sketch the proof of the
second part. Let $H$, $L$ be coaugmented coalgebras and $\varphi:L\to H$ be a
morphism. Suppose that $\mathrm{deg}(L)\leq n$, i.e. $\delta^{n+1}_{L}=0$. We
have
$\varphi^{\otimes(n+1)}\circ\delta^{n+1}_{H}=\delta^{n+1}_{L}\circ\varphi=0$
since $\varphi$ preserves the structure maps. Hence $\varphi$ uniquely factors
through the inclusion $P_{n}(H)\hookrightarrow H$. It proves
$\mathrm{Nat}(i(L),H)\cong\mathrm{Nat}(L,P_{n}(H))$. ∎

####  4.3. Examples of coradical filtrations

In this section, we give concrete examples of coradical filtrations.

#####  4.3.1. The polynomial bialgebra

Let $\mathds{k}$ be a commutative unital ring and $\mathds{k}[t]$ be the
polynomial algebra with the variable $t$ over $\mathds{k}$. It is well-known
that the algebra structure extends to a Hopf algebra structure with the
comultiplication characterized by $\Delta(t)=1\otimes t+t\otimes 1$. If
$m\cdot 1_{\mathds{k}}\in\mathds{k}$ is not a zero divisor for any nonzero
integer $m$, then the coradical filtration of $\mathds{k}[t]$ coincides with
the degree filtration:

$P_{n}(\mathds{k}[t])=\bigoplus_{i\leq n}\mathds{k}t^{i}$

For example, $\mathds{k}$ could be the integers $\mathds{Z}$ or a field of
characteristic zero.

We go further by giving an explicit calculation of $P_{n}(\mathds{k}[t])$ in
positive characteristic. The precise meaning of positive characteristic for a
general commutative ring $\mathds{k}$ is given in the statement of Theorem
4.13.

Let $p$ be a prime. For $n\in\mathds{N}$, let $(n_{i})_{i\in\mathds{N}}$ be
the representation of $n$ in base $p$. In other words,
$(n_{i})_{i\in\mathds{N}}$ is a sequence of $\mathds{N}$ such that
$n=\sum^{\infty}_{i=0}n_{i}p^{i}$ and $0\leq n_{i}<p$. We define the map
$L_{p}:\mathds{N}\to\mathds{N}$ by $L_{p}(n){:=}\sum^{\infty}_{i=0}n_{i}$.

We give a brief review of multinomial coefficients. Let $m,r\in\mathds{N}$
such that $r\geq 1$. For $k_{1},k_{2},\cdots,k_{r}\in\mathds{Z}$ such that
$\sum^{r}_{j=1}k_{j}=m$, the multinomial coefficient $\begin{pmatrix}m\\\
k_{1},k_{2},\cdots,k_{r}\end{pmatrix}\in\mathds{N}$ is characterized by the
equation,

$(z_{1}+z_{2}+\cdots+z_{r})^{m}=\sum_{\begin{subarray}{c}k_{i}\in\mathds{Z}\\\
\sum_{i}k_{i}=m\end{subarray}}\begin{pmatrix}m\\\
k_{1},k_{2},\cdots,k_{r}\end{pmatrix}z_{1}^{k_{1}}z_{2}^{k_{2}}\cdots
z_{r}^{k_{r}}.$

One can deduce the following lemma from elementary number theory, in
particular the Lucas theorem on binomial coefficients.

######  Lemma 4.12.

Let $p$ be a prime. For $m,r\in\mathds{N}$, the following conditions are
equivalent:

  * •

There exist positive integers $k_{1},k_{2},\cdots,k_{r}$ such that
$m=\sum^{r}_{j=1}k_{j}$ and the multinomial coefficient $\begin{pmatrix}m\\\
k_{1},k_{2},\cdots,k_{r}\end{pmatrix}$ is coprime to $p$.

  * •

$r\leq L_{p}(m)$.

######  Theorem 4.13.

Suppose that $p>0$ is a prime such that $p\cdot 1_{\mathds{k}}=0$ and $m\cdot
1_{\mathds{k}}\in\mathds{k}$ is not a zero divisor for $m\in\mathds{Z}$ which
is coprime to $p$. Then the coradical filtration of $\mathds{k}[t]$ is given
by,

$P_{n}(\mathds{k}[t])=\bigoplus_{L_{p}(i)\leq n}\mathds{k}t^{i}.$

In particular, the underlying coalgebra of $\mathds{k}[t]$ is conilpotent.

###### Proof.

For simplicity, write $H=\mathds{k}[t]$. By Proposition 4.4, it suffices to
compute the kernel of $\delta^{n+1}_{H}$ (see Definition 4.2). One can compute
$\delta^{n+1}_{H}(t^{m})=e^{\otimes(n+1)}_{H}(\Delta^{(n+1)}_{H}(t^{m}))$ as
follows:

$\displaystyle\delta^{n+1}_{H}(t^{m})$
$\displaystyle=\sum_{\begin{subarray}{c}0<k_{i}\\\
\sum_{i}k_{i}=m\end{subarray}}\begin{pmatrix}m\\\
k_{1},k_{2},\cdots,k_{n+1}\end{pmatrix}t^{k_{1}}\otimes
t^{k_{2}}\otimes\cdots\otimes t^{k_{n+1}}$

By Corollary 4.12 and the assumption on $p$, $\delta^{n+1}_{H}(t^{m})\neq 0$
if and only if $L_{p}(m)>n$. Furthermore, the assumption on $p$ implies that
$\\{\delta^{n+1}_{H}(t^{m})~{};~{}L_{p}(m)>n\\}$ is linearly independent since
the total degrees of $t^{k_{1}}\otimes t^{k_{2}}\otimes\cdots\otimes
t^{k_{n+1}}$, i.e. $\sum k_{i}=m$, are different from each other. ∎

######  Remark 4.14.

In section 10.1, we give a review of primitive bialgebras which are
automatically conilpotent. One can go further by using the above calculation
to prove that the polynomial bialgebra is primitive.

#####  4.3.2. The shuffle bialgebra

For a $\mathds{k}$-module $V$, we give a quick review of the shuffle-
deconcatenation bialgebra, or simply the shuffle bialgebra, $\mathrm{Sh}(V)$
whose multiplication is given by shuffles and comultiplication is given by
deconcatenation. Its underlying $\mathds{k}$-module is the tensor module
$T(V){:=}\bigoplus_{i\in\mathds{N}}V^{\otimes i}$. For
$v_{1},v_{2},\cdots,v_{n}\in V$, we introduce the notation
$\left[v_{1}|v_{2}|\cdots|v_{i}\right]{:=}v_{1}\otimes
v_{2}\otimes\cdots\otimes v_{n}\in V^{\otimes n}\subset T(V)$. For natural
numbers $p,q$ such that $p+q=n$, a permutation $\sigma\in\Sigma_{n}$ is a
$(p,q)$-shuffle if $\sigma(1)<\sigma(2)<\cdots<\sigma(p)$ and
$\sigma(p+1)<\sigma(p+2)<\cdots<\sigma(n)$. Denote by $\Sigma_{p,q}$ the
subset of $(p,q)$-shuffles. The multiplication, denoted by $\shuffle$, is
characterized by the following assignment:

$\left[v_{1}|v_{2}|\cdots|v_{i}\right]\shuffle\left[v_{i+1}|v_{i+2}|\cdots|v_{n}\right]{:=}\sum_{\sigma\in\Sigma_{i,n-i}}\left[v_{\sigma^{-1}(1)}|v_{\sigma^{-1}(2)}|\cdots|v_{\sigma^{-1}(n)}\right].$

The comultiplication $\Delta=\Delta_{\mathrm{Sh}(V)}$ is defined by
deconcatenation:

$\Delta(\left[v_{1}|v_{2}|\cdots|v_{n}\right]){:=}\left[v_{1}|\cdots|v_{n}\right]\otimes
1_{\mathds{k}}+\sum^{n-1}_{i=1}\left(\left[v_{1}|v_{2}|\cdots|v_{i}\right]\otimes\left[v_{i+1}|v_{i+2}|\cdots|v_{n}\right]\right)+1_{\mathds{k}}\otimes\left[v_{1}|\cdots|v_{n}\right].$

The above data yield a well-defined commutative bialgebra $\mathrm{Sh}(V)$.
For a free $\mathds{k}$-module $V$, $\mathrm{Sh}(V)$ is cocommutative if and
only if the rank of $V$ does not exceed $1$. The shuffle bialgebra is a Hopf
algebra since
$S(\left[v_{1}|v_{2}|\cdots|v_{n}\right])=(-1)^{n}\left[v_{n}|v_{n-1}|\cdots|v_{1}\right]$
gives the antipode.

######  Proposition 4.15.

The coradical filtration of $\mathrm{Sh}(V)$ is identified as follows:

$P_{n}(\mathrm{Sh}(V))=\bigoplus_{0\leq i\leq n}V^{\otimes i}$

In particular, the underlying coalgebra of the shuffle bialgebra
$\mathrm{Sh}(V)$ is conilpotent.

###### Proof.

Put $H=\mathrm{Sh}(V)$. One can inductively prove that the $(n+1)$-times
iterated reduced comultiplication acts on
$\left[v_{1}|v_{2}|\cdots|v_{m}\right]\in\mathrm{Sh}(V)$ as follows:

$\delta^{n+1}_{H}(\left[v_{1}|v_{2}|\cdots|v_{m}\right])=\sum_{i_{1},\cdots,i_{n}}\left(\left[v_{1}|v_{2}|\cdots|v_{i_{1}}\right]\otimes\left[v_{i_{1}+1}|v_{i_{1}+2}|\cdots|v_{i_{2}}\right]\otimes\cdots\left[v_{i_{n}+1}|v_{i_{n}+2}|\cdots|v_{m}\right]\right)$

Here, the indices $i_{1},\cdots,i_{n}$ run all over $1\leq
i_{1}<i_{2}<\cdots<i_{n}<m$. Hence, the left hand side in the statement
contains the right hand side. Moreover, if $m>n$, then the above computation
implies that $\delta^{n+1}_{H}|_{\mathrm{Sh}_{m}(V)}$ is injective where
$\mathrm{Sh}_{m}(V)$ is the subspace of $\mathrm{Sh}(V)$ generated by
$\left[v_{1}|v_{2}|\cdots|v_{m}\right]$’s. This completes the proof, since
$\delta^{n+1}_{H}(\mathrm{Sh}_{m}(V))\cap\delta^{n+1}_{H}(\mathrm{Sh}_{m^{\prime}}(V))=0$
for $m\neq m^{\prime}$. ∎

#####  4.3.3. Finite-dimensional bialgebras

In this section, we give a computation of the coradical filtration of finite-
dimensional bialgebras over a field $\mathds{k}$ of characteristic zero. In
particular, we prove that a finite-dimensional bialgebra is conilpotent if and
only if it is trivial. For this, in Lemma 4.16, we examine a compatibility of
the multiplication of a bialgebra $H$ and $\delta^{n}_{H}$ of Definition 4.2
for general characteristic.

Below, we regard $\delta_{H}^{n}(a)\in H^{\otimes n}$ as an element of the
shuffle algebra $\mathrm{Sh}(H)$ of the underlying module of $H$.

######  Lemma 4.16.

For $a\in P_{n}(H)$ and $b\in P_{m}(H)$, we have
$\delta_{H}^{n+m}(ab)=\delta_{H}^{n}(a)\shuffle\delta_{H}^{m}(b)$.

###### Proof.

Let $N=n+m$. By proposition 4.5, we have

$\Delta_{H}^{(N)}(ab)=\Delta_{H}^{(N)}(a)\Delta_{H}^{(N)}(b)=\left(\sum_{S\subset\underline{N}}\delta_{H}^{|S|}(a)_{S}\right)\left(\sum_{T\subset\underline{N}}\delta_{H}^{|T|}(b)_{T}\right).$

By the assumptions $a\in P_{n}(H)$ and $b\in P_{m}(H)$, this is equal to

$\left(\sum_{\begin{subarray}{c}S\subset\underline{N}\\\ |S|\leq
n\end{subarray}}\delta_{H}^{|S|}(a)_{S}\right)\left(\sum_{\begin{subarray}{c}T\subset\underline{N}\\\
|T|\leq
m\end{subarray}}\delta_{H}^{|T|}(b)_{T}\right)=\sum_{\begin{subarray}{c}S,T\subset\underline{N}\\\
|S|\leq n,|T|\leq
m\end{subarray}}\delta_{H}^{|S|}(a)_{S}~{}\delta_{H}^{|T|}(b)_{T}.$

Note that, by Proposition 4.5, we have $\delta_{H}^{|S|}(a)_{S}\in H_{N,S}$.
In other words, $\delta_{H}^{|S|}(a)_{S}$ is a linear combination of vectors
in $H^{\otimes N}$ whose $j$-th tensor component is $1_{H}$ for all $j\not\in
S$. $\delta_{H}^{|T|}(b)_{T}$ satisfies a similar property, so that their
product $\delta_{H}^{|S|}(a)_{S}~{}\delta^{|T|}(b)_{T}$ is a linear
combination of vectors in $H^{\otimes N}$ whose $j$-th tensor component is
$1_{H}$ for all $j\not\in S\cup T$. Hence, if $S\cup T\neq\underline{N}$, then
we have $e_{H}^{\otimes
N}\left(\delta_{H}^{|S|}(a)_{S}~{}\delta_{H}^{|T|}(b)_{T}\right)=0$. This
observation leads to

$\displaystyle\delta_{H}^{N}(ab)=(e_{H}^{\otimes
N})(\Delta_{H}^{(N)}(ab))=\sum_{\begin{subarray}{c}S,T\subset\underline{N},|S|=n\\\
S\cap
T=\emptyset\end{subarray}}\delta_{H}^{|S|}(a)_{S}~{}\delta_{H}^{|T|}(b)_{T}=\sum_{\begin{subarray}{c}S,T\subset\underline{N},|S|=n\\\
S\cap T=\emptyset\end{subarray}}\delta_{H}^{n}(a)_{S}~{}\delta_{H}^{m}(b)_{T}$

This is equal to the shuffle multiplication
$\delta_{H}^{n}(a)\shuffle\delta_{H}^{m}(b)$ by definitions. ∎

We recall a classical result on shuffle algebras. For a field $\mathds{k}$ of
characteristic zero, the shuffle algebra $\mathrm{Sh}(V)$ is isomorphic to a
polynomial algebra based on certain multi-variables [17]. Choose a basis $X$
of $V$ equipped with an order. A Lyndon word on $X$ (equipped with the order)
is a finite word on $X$ which is strictly smaller than all of its rotations
with respect to the lexicographical order. Denote by $\mathrm{Lyn}(X)$ the set
of Lyndon words on $X$. Then there exists an algebra isomorphism
$\mathrm{Sh}(V)\cong\mathds{k}[\mathrm{Lyn}(X)]$.

######  Lemma 4.17.

Let $H$ be a (not necessarily finite-dimensional) bialgebra over a field of
characteristic zero. For $a\in P_{n}(H)$ such that $a\not\in\mathds{k}1_{H}$,
the sequence $(a^{k})_{k\in\mathds{N}}$ of the powers of $a$ is linearly
independent.

###### Proof.

Without loss of generality, we may assume that $a\in P_{n}(H)\backslash
P_{n-1}(H)$. In other words, $\delta_{H}^{n+1}(a)=0$ but
$\delta_{H}^{n}(a)\neq 0$. Consider the linear map
$\delta=\sum_{m\in\mathds{N}}\delta_{H}^{m}:H\to\bigoplus_{m\in\mathds{N}}H^{\otimes
m}=\mathrm{Sh}(H)$. It suffices to show that the induced sequence
$(\delta(a^{k}))_{k\in\mathds{N}}$ is linearly independent. Note that
$\delta_{H}^{m}(a^{k})=0$ if $m>kn$, since $a^{k}\in P_{nk}(H)$ by Proposition
4.8. We claim that $\delta_{H}^{m}(a^{k})\neq 0$ if $m=kn$. Then it is obvious
that $(\delta(a^{k}))_{k\in\mathds{N}}$ is linearly independent.

We now prove the above claim. Indeed, an iterated application of Lemma 4.16
implies $\delta_{H}^{kn}(a^{k})=\delta_{H}^{n}(a)^{\shuffle k}$. The shuffle
algebra $\mathrm{Sh}(H)$ is isomorphic to a polynomial algebra since
$\mathds{k}$ is a field of characteristic zero. Hence,
$\delta_{H}^{kn}(a^{k})=\delta_{H}^{n}(a)^{\shuffle k}\neq 0$ due to
$\delta_{H}^{n}(a)\neq 0$. ∎

######  Theorem 4.18.

Let $\mathds{k}$ be a field of characteristic zero. For a finite-dimensional
bialgebra $H$, we have $P_{n}(H)\cong\mathds{k}$. In particular, $H$ is
conilpotent if and only if $H\cong\mathds{k}$.

###### Proof.

It is immediate from Lemma 4.17. ∎

######  Example 4.19.

Let $G$ be a finite group and $H=\mathds{k}G$ be the group algebra over
$\mathds{k}$ a field of characteristic zero. It is well known that $H$ is a
bialgebra whose comultiplication $\Delta$ is characterized by
$\Delta(g)=g\otimes g$ for $g\in G$. By Theorem 4.18, $H$ is conilpotent if
and only if $G$ is trivial. One can directly verify that this is true even for
arbitrary ring $\mathds{k}$.

######  Example 4.20.

For $\mathds{k}$ a field of positive characteristic $p$, there exists a
finite-dimensional conilpotent bialgebra $H$ with $\dim H>1$. Let $J$ be the
ideal of $\mathds{k}[t]$ generated by $t^{p}$. Then $J$ is a coideal of
$\mathds{k}[t]$, i.e. $\Delta(J)\subset
J\otimes\mathds{k}[t]+\mathds{k}[t]\otimes J$, so that $J$ induces the
quotient Hopf algebra $H=\mathds{k}[t]/J$. The Hopf algebra $H$ is conilpotent
since the conilpotency is inherited via a quotient map. This also follows from
a stronger result that $H$ is primitive (see section 10.1).

###  5\. Isomorphism of polynomial and coradical filtrations

Let $\mathcal{C}$ be a category satisfying the conditions given in section 2.
In this section, we prove in Theorem 5.2, that the polynomial filtration of an
exponential functor on $\mathcal{C}$ and the coradical filtration coincide.
This theorem extends a result obtained by Touzé [18, Theorem 11.10] concerning
functors on an additive category. Recall that our main example of interest,
the category $\mathsf{gr^{\mathsf{op}}}$ is not additive but satisfies the
conditions given in section 2. Our proof is based on the alternative
description of the coradical filtration of a coaugmented coalgebra given in
Proposition 4.4. Furthermore, we show that the analyticity of exponential
$\mathcal{C}$-modules could be discussed using object-wise conilpotency.

We regard the categories $\mathcal{C}$ and $\mathsf{Mod}_{\mathds{k}}$ as
symmetric monoidal categories by using the product (given by the assumption in
section 2) and the tensor product respectively. Let
$\mathcal{F}^{\mathsf{exp}}(\mathcal{C};\mathds{k})$ be the category of
exponential functors, equivalently symmetric monoidal functors, from
$\mathcal{C}$ to $\mathsf{Mod}_{\mathds{k}}$.

######  Proposition 5.1.

Let $\mathsf{Comon^{cc}}(\mathcal{C})$ be the category of cocommutative
comonoids in $\mathcal{C}$. The forgetful functor
$(\ast\downarrow\mathsf{Comon^{cc}}(\mathcal{C}))\to\mathcal{C}$ gives an
equivalence of symmetric monoidal categories.

###### Proof.

The universality of the product implies that the diagonal map is the unique
comultiplication on each object. ∎

Based on Proposition 5.1, we see that an exponential functor
$F:\mathcal{C}\to\mathsf{Mod}_{\mathds{k}}$ induces an exponential functor
$\widetilde{F}:\mathcal{C}\to(\mathds{k}\downarrow\mathsf{Coalg}^{\mathsf{cc}}_{\mathds{k}})$.
In fact, it is given by
$\widetilde{F}:\mathcal{C}\simeq(\ast\downarrow\mathsf{Comon^{cc}}(\mathcal{C}))\to(\mathds{k}\downarrow\mathsf{Comon^{cc}}(\mathsf{Mod}_{\mathds{k}}))=(\mathds{k}\downarrow\mathsf{Coalg}^{\mathsf{cc}}_{\mathds{k}})$.
In particular, for $X\in\mathcal{C}$, $\widetilde{F}(X)$ is the coaugmented
coalgebra whose underlying module is $F(X)$; and the comultiplication, counit
and coaugmentation of $\widetilde{F}(X)$ are given by

$\displaystyle F(X)$ $\displaystyle\stackrel{{\scriptstyle
F(\Delta_{X})}}{{\longrightarrow}}F(X\times X)\cong F(X)\otimes F(X),$
$\displaystyle\mathds{k}\cong F(\ast)$ $\displaystyle\stackrel{{\scriptstyle
F(\eta_{X})}}{{\longrightarrow}}F(X),$ $\displaystyle F(X)$
$\displaystyle\stackrel{{\scriptstyle
F(\varepsilon_{X})}}{{\longrightarrow}}F(\ast)\cong\mathds{k}.$

For $X\in\mathcal{C}$, we have
$\widetilde{F}(X)\in(\mathds{k}\downarrow\mathsf{Coalg}^{\mathsf{cc}}_{\mathds{k}})$,
so by section 4.1, we have the coradical filtration $P_{n}(\widetilde{F}(X))$.
In the following theorem, we compare the polynomial filtration $P_{n}(F)$ (see
Definition 3.9) applied to $X\in\mathcal{C}$ and the previous coradical
filtration.

######  Theorem 5.2.

Let $F\in\mathcal{F}^{\mathsf{exp}}(\mathcal{C};\mathds{k})$ and
$X\in\mathcal{C}$. Under the identification $F(X)=\widetilde{F}(X)$, the
polynomial filtration of $F$ evaluated at $X$ is equal to the coradical
filtration of $\widetilde{F}(X)$. In other words, we have the following
commutative diagram for $n\in\mathds{N}$:

${F(X)}$${\widetilde{F}(X)}$${(P_{n}(F))(X)}$${P_{n}(\widetilde{F}(X))}$

###### Proof.

The coaugmented coalgebra structure of $\widetilde{F}(X)$ induces the
following commutative diagram. The maps in the right two columns are induced
by the exponential isomorphism of $F$. Hence, the kernel of the composition of
maps in the first row is equal to that in the second row.

${F(X)}$${F(X)^{\otimes(n+1)}}$${F(X)^{\otimes(n+1)}}$${F(X)}$${F(X^{\times(n+1)})}$${F(X^{\times(n+1)})}$$\scriptstyle{\Delta^{(n+1)}_{\widetilde{F}(X)}}$$\scriptstyle{\cong}$$\scriptstyle{e_{\widetilde{F}(X)}^{\otimes(n+1)}}$$\scriptstyle{\cong}$$\scriptstyle{F(\Delta^{(n+1)}_{X})}$$\scriptstyle{\chi_{F}({[}X{]}^{n+1}_{i=1})}$

∎

######  Corollary 5.3.

Let $F\in\mathcal{F}^{\mathsf{exp}}(\mathcal{C};\mathds{k})$. The
$\mathcal{C}$-module $F$ is analytic if and only if the induced coaugmented
coalgebra $\widetilde{F}(X)$ is conilpotent for all $X\in\mathcal{C}$.

###### Proof.

The $\mathcal{C}$-module $F$ is analytic, i.e.
$\varinjlim_{n}(P_{n}(F))(X)=F(X)$ for any $X$, if and only if
$\widetilde{F}(X)$ is conilpotent, i.e.
$\varinjlim_{n}P_{n}(\widetilde{F}(X))=\widetilde{F}(X)$, for any $X$ by
Theorem 5.2. ∎

We give an application of the theorem to coaugmented cocommutative coalgebras.

######  Definition 5.4.

Let $H$ be a coaugmented cocommutative coalgebra. We denote by
$\mathcal{C}_{H}$ the full subcategory of
$(\mathds{k}\downarrow\mathsf{Coalg}_{\mathds{k}})$ consisting of any $n$-fold
tensor product $H^{\otimes n}$ where $n\geq 0$.

######  Proposition 5.5.

The tensor product of coaugmented cocommutative coalgebras gives the product
in the category $\mathcal{C}_{H}$.

###### Proof.

Let $X,Y$ be objects of $\mathcal{C}_{H}$. We prove that
$X\stackrel{{\scriptstyle\pi_{X}}}{{\leftarrow}}X\otimes
Y\stackrel{{\scriptstyle\pi_{Y}}}{{\to}}Y$ gives the product of $X$ and $Y$
where $\pi_{X}=\mathrm{id}_{X}\otimes\varepsilon_{Y}$ and
$\pi_{Y}=\varepsilon_{X}\otimes\mathrm{id}_{Y}$. Let $f:Z\to X$ and $g:Z\to Y$
be morphisms of $\mathcal{C}_{H}$. Then we define $h:Z\to X\otimes Y$ by
$h=(f\otimes g)\circ\Delta_{Z}$. Note that $\Delta_{Z}$ is a coalgebra
homomorphism since $Z\in\mathcal{C}_{H}$ is cocommutative. It turns out that
$h$ gives a morphism in $\mathcal{C}_{H}$ such that $\pi_{X}\circ h=f$ and
$\pi_{Y}\circ h=g$. Moreover, if $h^{\prime}:Z\to X\otimes Y$ is a morphism of
$\mathcal{C}_{H}$ such that $\pi_{X}\circ h^{\prime}=f$ and $\pi_{Y}\circ
h^{\prime}=g$, then $h^{\prime}=h$. This follows from

$\displaystyle h^{\prime}$
$\displaystyle=(\pi_{X}\otimes\pi_{Y})\circ\Delta_{X\otimes Y}\circ
h^{\prime},$ $\displaystyle=(\pi_{X}\otimes\pi_{Y})\circ(h^{\prime}\otimes
h^{\prime})\circ\Delta_{Z},$ $\displaystyle=(f\otimes g)\circ\Delta_{Z}.$

∎

######  Remark 5.6.

In fact, the same proof implies that the tensor product of (coaugmented)
coalgebras is the product in the category of cocommutative (coaugmented)
coalgebras.

By Proposition 5.5, the category $\mathcal{C}_{H}$ is a small category with
finite products and a null object as explained in section 2. The forgetful
functor from $\mathcal{C}_{H}$ to $\mathsf{Mod}_{\mathds{k}}$ gives an
exponential functor
$U_{H}\in\mathcal{F}^{\mathsf{exp}}(\mathcal{C}_{H};\mathds{k})$.

######  Corollary 5.7.

Let $H$ be a coaugmented cocommutative coalgebra. The $\mathcal{C}_{H}$-module
$U_{H}$ is analytic if and only if $H$ is conilpotent.

###### Proof.

We apply Theorem 5.2 to $\mathcal{C}=\mathcal{C}_{H}$ and $F=U_{H}$. By
Corollary 5.3, it is clear that, if the $\mathcal{C}_{H}$-module $U_{H}$ is
analytic, then $H$ is conilpotent since $\widetilde{U_{H}}(H)=H$. Conversely,
suppose that $H$ is conilpotent. By Corollary 5.3 again, it suffices to prove
that $\widetilde{U_{H}}(X)$ is analytic for any object $X\in\mathcal{C}_{H}$.
Note that any object $X$ is given by an $n$-fold tensor product $X=H^{\otimes
n}$ for some $n$. For such $X$, we have $\widetilde{U_{H}}(X)=H^{\otimes n}$
as coaugmented coalgebras by definitions. By Proposition 4.7, $H^{\otimes n}$
is conilpotent since so is $H$. ∎

##  Part II Application to $\mathsf{gr}^{\mathsf{op}}$

###  6\. The operad $\mathfrak{Ass}^{u}$ and its associated category

Every $\mathds{k}$-linear category $\mathcal{D}$ induces a bilinear functor
$\mathcal{D}^{\mathsf{op}}\times\mathcal{D}\to\mathsf{Mod}_{\mathds{k}}$ given
by the morphisms in $\mathcal{D}$. This section concerns the case where
$\mathcal{D}$ is the category $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ associated
with the operad of unital associative algebras. We recall the skeleton
$\mathsf{gr}^{\mathsf{op}}$ of the opposite category of finitely generated
free groups in section 2. The aim of this section is to recall Powell’s
assertion [14] that $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ induces a bilinear
functor
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}:\mathsf{Cat}_{\mathfrak{Lie}}^{\mathsf{op}}\times\mathds{k}\mathsf{gr}^{\mathsf{op}}\to\mathsf{Mod}_{\mathds{k}}$
where $\mathsf{Cat}_{\mathfrak{Lie}}$ is the category associated with the Lie
operad. In [14], the ground ring is assumed to be a field of characteristic
zero but the generalization to any commutative ring is straightforward as we
sketch below.

####  6.1. The operad of unital associative algebras

Let $\mathds{k}$ be a commutative ring with unit. For a small
$\mathds{k}$-linear category $\mathcal{M}$, a left (right, resp.)
$\mathcal{M}$-module is a $\mathds{k}$-linear functor from $\mathcal{M}$
($\mathcal{M}^{\mathsf{op}}$, resp.) to $\mathsf{Mod}_{\mathds{k}}$. More
generally, a $(\mathcal{M},\mathcal{N})$-bimodule is a $\mathds{k}$-bilinear
functor
$\mathcal{N}^{\mathsf{op}}\times\mathcal{M}\to\mathsf{Mod}_{\mathds{k}}$.

For a small category $\mathcal{D}$, let $\mathds{k}\mathcal{D}$ be the
$\mathds{k}$-linear category obtained by the $\mathds{k}$-linearization of
$\mathcal{D}$. Then the category of $\mathds{k}\mathcal{D}$-modules and linear
natural transformations is equivalent with
$\mathcal{F}(\mathcal{D};\mathds{k})$ the category of $\mathcal{D}$-modules in
the sense of section 3.1.

For a $\mathds{k}$-linear operad $\mathfrak{O}$, denote by
$\mathsf{Cat}_{\mathfrak{O}}$ the $\mathds{k}$-linear category associated with
$\mathfrak{O}$: its objects consist of natural numbers and the morphism sets
are given by

$\mathsf{Cat}_{\mathfrak{O}}(n,m)=\bigoplus_{f:\underline{n}\to\underline{m}}\bigotimes_{i=1}^{m}\mathfrak{O}(f^{-1}(i))$

where $f:\underline{n}\to\underline{m}$ runs over all the maps. Then the
assignment $(n,m)\mapsto\mathsf{Cat}_{\mathfrak{O}}(n,m)$ gives a
$(\mathsf{Cat}_{\mathfrak{O}},\mathsf{Cat}_{\mathfrak{O}})$-bimodule which we
also denote by $\mathsf{Cat}_{\mathfrak{O}}$.

Let $\mathfrak{Ass}^{u}$ be the $\mathds{k}$-linear operad of unital
associative algebras. The module $\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)$ is
isomorphic to the free module generated by $m$-tuples
$(f^{-1}(1),o_{1})\otimes(f^{-1}(2),o_{2})\otimes\cdots(f^{-1}(m),o_{m})$ for
maps $f:\underline{n}\to\underline{m}$ whose fibers $f^{-1}(i)$ are equipped
with a total ordering $o_{i}$. Fix a countable set of variables
$\\{x_{1},x_{2},\cdots x_{n},\cdots\\}$. For a finite set $S$ equipped with a
total ordering $o$, we introduce the notation
$x_{(S,o)}{:=}x_{a_{1}}x_{a_{2}}\cdots x_{a_{k}}$ (the formal word) if
$S=\\{a_{1},a_{2},\cdots,a_{k}\\}$ and the order $o$ is given by
$a_{i}<a_{i+1}$. If $S=\emptyset$, then $x_{(S,o)}{:=}1$. The replacement of
$(f^{-1}(i),o_{i})$ by $x_{(f^{-1}(i),o_{i})}$ induces an isomorphism:

(5)
$\displaystyle\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)\cong\bigoplus_{f:\underline{n}\to\underline{m}}\bigotimes_{i=1}^{m}\mathds{k}x_{(f^{-1}(i),o_{i})}.$

Let $A_{n}{:=}\mathds{k}\langle x_{1},\cdots,x_{n}\rangle$ be the free non-
commutative polynomial algebra generated by $n$ variables
$x_{1},\cdots,x_{n}$. We define the
$(\mathsf{Cat}_{\mathfrak{Ass}^{u}},\mathsf{Cat}_{\mathfrak{Ass}^{u}})$-bimodule
$\mathbb{A}$ by $\mathbb{A}(n,m){:=}A_{n}^{\otimes m}$. To be precise, for
each $n$, the left $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$-action on $\mathbb{A}$
is induced by the algebra structure on $A_{n}$. The right
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}$-action is induced by substitution of
variables by words. For example, $f=\mathbb{A}(x_{2}x_{3}\otimes 1\otimes
x_{4}x_{1},1):\mathbb{A}(3,1)\to\mathbb{A}(4,1)$ is an algebra homomorphism
characterized by $f(x_{1})=x_{2}x_{3}$, $f(x_{2})=1$ and
$f(x_{3})=x_{4}x_{1}$; and $\mathbb{A}(x_{2}x_{3}\otimes 1\otimes
x_{4}x_{1},m)=f^{\otimes m}$ for arbitrary $m$. The isomorphism in (5) leads
to a
$(\mathsf{Cat}_{\mathfrak{Ass}^{u}},\mathsf{Cat}_{\mathfrak{Ass}^{u}})$-bimodule
embedding $\iota:\mathsf{Cat}_{\mathfrak{Ass}^{u}}\hookrightarrow\mathbb{A}$.

####  6.2. The bimodule ${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$

In this section, we will see that using the embedding
$\iota:\mathsf{Cat}_{\mathfrak{Ass}^{u}}\hookrightarrow\mathbb{A}$ we can
define a
$(\mathds{k}\mathsf{gr}^{\mathsf{op}},\mathsf{Cat}_{\mathfrak{Lie}})$-bimodule
structure on $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$. For this, we start by
constructing a
$(\mathds{k}\mathsf{gr}^{\mathsf{op}},\mathsf{Cat}_{\mathfrak{Lie}})$-bimodule
$\mathbb{H}$ from the
$(\mathsf{Cat}_{\mathfrak{Ass}^{u}},\mathsf{Cat}_{\mathfrak{Ass}^{u}})$-bimodule
$\mathbb{A}$.

Let $H_{n}$ be the Hopf algebra with $A_{n}$ the underlying algebra structure
and the comultiplication given by the shuffle, i.e.
$\Delta(x_{i})=x_{i}\otimes 1+1\otimes x_{i}$. In particular, $H_{n}$ is
cocommutative. We define the
$(\mathds{k}\mathsf{gr}^{\mathsf{op}},\mathsf{Cat}_{\mathfrak{Lie}})$-bimodule
$\mathbb{H}$ by $\mathbb{H}(n,m){:=}H_{n}^{\otimes m}$. To be precise, for
each $n$, the left $\mathds{k}\mathsf{gr}^{\mathsf{op}}$-action on
$\mathbb{H}$ is induced by the Hopf algebra structure on $H_{n}$ (see Remark
8.1). The right $\mathsf{Cat}_{\mathfrak{Lie}}$-action is induced by
substitution of variables by Lie brackets. For example,
$g=\mathbb{H}([x_{1},x_{2}]\otimes x_{3}\otimes\cdots\otimes
x_{n},1):H_{n-1}\to H_{n}$ is a Hopf homomorphism characterized by
$g(x_{1})=x_{1}x_{2}-x_{2}x_{1}$ and $g(x_{i})=x_{i+1}$ for $i\geq 2$. The
category $\mathsf{Cat}_{\mathfrak{Lie}}$ is generated by Lie brackets and
permutations so that this example implies that $\mathbb{H}$ is well-defined.

The bimodule embedding
$\iota:\mathsf{Cat}_{\mathfrak{Ass}^{u}}\hookrightarrow\mathbb{A}$ gives
$\iota^{\prime}_{(n,m)}:{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)\hookrightarrow\mathbb{H}(n,m)$
for $n,m\in\mathds{N}$ by definitions. It turns out that the collection of
$\iota^{\prime}_{(n,m)}$ is compatible with the
$(\mathds{k}\mathsf{gr}^{\mathsf{op}},\mathsf{Cat}_{\mathfrak{Lie}})$-bimodule
structure on $\mathbb{H}$. Indeed, the assignment
$(n,m)\mapsto\iota^{\prime}_{n,m}$ preserves the left
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$-action (the right
$\mathsf{Cat}_{\mathfrak{Lie}}$-action, resp.) since it is compatible with
$\Delta,\nabla,\eta,\epsilon,s$ (the Lie brackets, resp.) and permutations
(see section 2 for the notation). We define the
$(\mathds{k}\mathsf{gr}^{\mathsf{op}},\mathsf{Cat}_{\mathfrak{Lie}})$-bimodule
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ by the bimodule structure
inherited from $\mathbb{H}$ via $\iota^{\prime}$.

######  Proposition 6.1.

For $n\in\mathds{N}$,
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
has degree $n$.

###### Proof.

We first prove that
$\mathrm{deg}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-))\leq n$, i.e.
$\mathrm{cr}^{n+1}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-))\cong 0$.
Recall the notation in section 3.1. It suffices to prove that
$\chi_{F}([m_{i}]^{n+1}_{i=1})=0$ for an arbitrary sequence
$[m_{i}]^{n+1}_{i=1}$ of objects in $\mathsf{gr}^{\mathsf{op}}$ and
$F={}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)$. If $m_{i}=0$ for some
$i$, then it is clear that $\chi_{F}([m_{i}]^{n+1}_{i=1})=0$. Suppose that
$m_{i}>0$ for all $i$. Let $N=\sum^{n+1}_{i=1}m_{i}$. By definitions, the
module ${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,N)$ is generated by
$\bigotimes^{N}_{j=1}w_{j}\in{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,N)$
such that $w_{m_{i}+1}=w_{m_{i}+2}=\cdots=w_{m_{i+1}}=1$ for some $i$ since
$(n+1)\leq N$. For such tensor products, we have
$e_{i}\left(\bigotimes^{N}_{j=1}w_{j}\right)=0$. Hence,
$\chi_{F}([m_{i}]^{n+1}_{i=1})=0$.

We now prove that
$\mathrm{deg}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-))\not\leq(n-1)$,
equivalently, $\chi_{F}([m_{i}]^{n}_{i=1})\neq 0$ for some sequence
$[m_{i}]^{n}_{i=1}$ of objects in $\mathsf{gr}^{\mathsf{op}}$. Let
$m_{i}=1\in\mathsf{gr}^{\mathsf{op}}$ for any $i$. By definition, we have

$\left(\chi_{F}([m_{i}]^{n}_{i=1})\right)(x_{1}\otimes
x_{2}\otimes\cdots\otimes x_{n})=x_{1}\otimes x_{2}\otimes\cdots\otimes
x_{n}.$

∎

####  6.3. Relation between $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ and
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$

In this section, we study how the bimodules
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ and
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ interact with each other.

We regard a formal word $w=x_{a_{1}}x_{a_{2}}\cdots x_{a_{k}}$ consisting of
the variables $x_{1},x_{2},\cdots,x_{n}$ as an element
$x_{a_{1}}x_{a_{2}}\cdots x_{a_{k}}\in\mathsf{F}_{n}$ when $k\geq 1$. If the
word $w$ is trivial, i.e. $w=1$, then $\mathcal{E}$ assigns the unit
$e\in\mathsf{F}_{n}$. By extending this observation, we introduce the
following functor:

######  Definition 6.2.

We define the linear functor

$\mathcal{E}:\mathsf{Cat}_{\mathfrak{Ass}^{u}}\hookrightarrow\mathds{k}\mathsf{gr}^{\mathsf{op}}.$

It assigns $\mathcal{E}(n){:=}n$ to an object
$n\in\mathsf{Cat}_{\mathfrak{Ass}^{u}}$. Let $w_{1}\otimes
w_{2}\otimes\cdots\otimes w_{m}\in\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)$ such
that $w_{1},w_{2},\cdots,w_{m}$ are formal words consisting of
$x_{1},x_{2},\cdots x_{n}$ (see section 6.1). By using the notation of
morphisms in $\mathsf{gr}^{\mathsf{op}}$ in section 2, the morphism
$\mathcal{E}(w_{1}\otimes w_{2}\otimes\cdots\otimes w_{m})$ from $n$ to $m$ in
$\mathsf{gr}^{\mathsf{op}}$ is defined by

$\mathcal{E}(w_{1}\otimes w_{2}\otimes\cdots\otimes
w_{m}){:=}[w_{1}|w_{2}|\cdots|w_{m}]_{n}$

where we regard the formal words $w_{1},w_{2},\cdots,w_{m}$ as elements of
$\mathsf{F}_{n}$ as above.

For example, we have $\mathcal{E}(x_{2}x_{3}\otimes 1\otimes
x_{4}x_{1})=[x_{2}x_{3}|e|x_{4}x_{1}]_{4}$.

######  Proposition 6.3.

Definition 6.2 gives a well-defined faithful linear functor.

###### Proof.

We prove that the assignment $\mathcal{E}$ preserves the identity. For an
object $n\in\mathsf{Cat}_{\mathfrak{Ass}^{u}}$, the identity of $n$ is
$x_{1}\otimes x_{2}\otimes\cdots\otimes x_{n}$ by definition. The
corresponding morphism $\mathcal{E}(x_{1}\otimes x_{2}\otimes\cdots\otimes
x_{n})=[x_{1}|x_{2}|\cdots|x_{n}]_{n}$ is the identity of $n$ in
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$.

We now prove the assignment $\mathcal{E}$ preserves the composition. Let
$f=(w_{1}\otimes w_{2}\otimes\cdots\otimes w_{m}):n\to m$ and
$f^{\prime}=(w_{1}^{\prime}\otimes w_{2}^{\prime}\otimes\cdots\otimes
w_{l}^{\prime}):m\to l$ be morphisms in the category
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}$; and put $f^{\prime}\circ
f=w_{1}^{\prime\prime}\otimes w_{2}^{\prime\prime}\otimes\cdots\otimes
w_{l}^{\prime\prime}$. By definition, if we have
$w_{i}^{\prime}=x_{a_{1}}x_{a_{2}}\cdots x_{a_{k}}$ for some $1\leq i\leq l$,
then we have $w_{a_{1}}w_{a_{2}}\cdots w_{a_{k}}=w_{i}^{\prime\prime}$. To
prove that $\mathcal{E}(f^{\prime}\circ
f)=\mathcal{E}(f^{\prime})\circ\mathcal{E}(f)$, we compute
$(\rho\circ\rho^{\prime})(x_{i})$ for $1\leq i\leq l$ where we regard
$\rho{:=}\mathcal{E}(f)$ and $\rho^{\prime}{:=}\mathcal{E}(f^{\prime})$ as
group homomorphisms: if we have $w_{i}^{\prime}=x_{a_{1}}x_{a_{2}}\cdots
x_{a_{k}}$, then we obtain

$(\rho\circ\rho^{\prime})(x_{i})=\rho(w_{i}^{\prime})=\rho(x_{a_{1}}x_{a_{2}}\cdots
x_{a_{k}})=w_{a_{1}}w_{a_{2}}\cdots w_{a_{k}}=w_{i}^{\prime\prime}.$

Hence, this proves
$\mathcal{E}(f^{\prime})\circ\mathcal{E}(f)=\mathcal{E}(f^{\prime}\circ f)$.

The functor $\mathcal{E}$ is faithful since it assigns the formal word basis
of $\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)$ to the basis
$\mathsf{gr}^{\mathsf{op}}(n,m)$ of
$\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)$. ∎

On the one hand, the operad homomorphism $\mathfrak{Lie}\to\mathfrak{Ass}^{u}$
induces the functor:

$\mathcal{B}:\mathsf{Cat}_{\mathfrak{Lie}}\to\mathsf{Cat}_{\mathfrak{Ass}^{u}}$

By definitions, the diagram below commutes up to a natural isomorphism.

${\mathsf{Cat}_{\mathfrak{Ass}^{u}}^{\mathsf{op}}\times\mathsf{Cat}_{\mathfrak{Ass}^{u}}}$${\mathsf{Mod}_{\mathds{k}}}$${\mathsf{Cat}_{\mathfrak{Lie}}^{\mathsf{op}}\times\mathsf{Cat}_{\mathfrak{Ass}^{u}}}$${\mathsf{Cat}_{\mathfrak{Lie}}^{\mathsf{op}}\times\mathds{k}\mathsf{gr}^{\mathsf{op}}}$$\scriptstyle{\mathsf{Cat}_{\mathfrak{Ass}^{u}}}$$\scriptstyle{\mathcal{B}\times\mathrm{id}}$$\scriptstyle{\mathrm{id}\times\mathcal{E}}$$\scriptstyle{{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}}$

###  7\. On a left ideal of $\mathds{kr}\mathsf{gr}^{\mathsf{op}}$

The functor
$\mathcal{E}:\mathsf{Cat}_{\mathfrak{Ass}^{u}}\hookrightarrow\mathds{k}\mathsf{gr}^{\mathsf{op}}$
introduced in Definition 6.2 is not full. For example, the morphism
$\Delta:1\to 2$ is not in the image of $\mathcal{E}$. The aim of this section
is to define a certain left ideal $\mathcal{I}$ of
$\mathds{kr}\mathsf{gr}^{\mathsf{op}}$ for which the composition
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)\stackrel{{\scriptstyle\mathcal{E}}}{{\hookrightarrow}}\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)\stackrel{{\scriptstyle\pi}}{{\to}}\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)/\mathcal{I}(n,m)$
is surjective (see Theorem 7.7). In other words, all the morphisms in
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$ arises from $\mathcal{E}$ modulo
$\mathcal{I}$. We will use this to define the
$\mathsf{Cat}_{\mathfrak{Lie}}$-module $\mathcal{P}(F)$ later (see Definition
9.11).

In this section, we freely use the notations of morphisms in
$\mathsf{gr}^{\mathsf{op}}$ following section 2. Note that we use the notation
$[\rho(x_{1})|\cdots|\rho(x_{m})]_{n}$ to denote morphisms in the opposite
category $\mathsf{gr}^{\mathsf{op}}$, not the category $\mathsf{gr}$.

In the following, we recall the notion of semicategory. It is similar to that
of category, so it has objects and morphisms with composition, but we do not
require the identity morphisms.

######  Definition 7.1.

Let $\theta:1\to 2$ be the morphism in $\mathds{k}\mathsf{gr}^{\mathsf{op}}$
defined by

$\theta{:=}[x_{1}|x_{1}]_{1}-[e|x_{1}]_{1}-[x_{1}|e]_{1}.$

We define $\mathcal{I}(n,m)$ to be the submodule of
$\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)$ generated by
$f\circ(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})$ for all $1\leq
i\leq n$ and $f\in\mathds{k}\mathsf{gr}^{\mathsf{op}}(n+1,m)$. We denote by
$\mathcal{I}$ the subsemicategory of $\mathds{k}\mathsf{gr}^{\mathsf{op}}$
which consists of objects $\mathds{N}$ and morphisms $\mathcal{I}(n,m)$ for
$n,m\in\mathds{N}$.

######  Proposition 7.2.

The subsemicategory $\mathcal{I}$ is a left ideal of
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$. In other words, if
$h\in\mathds{k}\mathsf{gr}^{\mathsf{op}}(m,l)$ and $g\in\mathcal{I}(n,m)$,
then $h\circ g\in\mathcal{I}(n,l)$.

###### Proof.

The morphism $g$ is a linear combination of
$f\circ(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})$ for some $f$ and
$i$. The result follows from $(h\circ
f)\circ(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})\in\mathcal{I}(n,l)$.
∎

We investigate the quotient
$\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)/\mathcal{I}(n,m)$ so as to define
the primitive filtration in section 9.1. For
$\rho,\rho^{\prime}\in\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)$, we write
$\rho\equiv\rho^{\prime}$ if $\rho-\rho^{\prime}\in\mathcal{I}(n,m)$. In
particular, for $\rho\in\mathds{k}\mathsf{gr}^{\mathsf{op}}(n+1,m)$, we have

$\displaystyle\rho\circ[x_{1}|\cdots|x_{i-1}|x_{i}|x_{i}|x_{i+1}|\cdots|x_{n}]_{n}$
$\displaystyle\equiv$
$\displaystyle\rho\circ[x_{1}|\cdots|x_{i-1}|x_{i}|e|x_{i+1}|\cdots$
$\displaystyle|x_{n}]_{n}+\rho\circ[x_{1}|\cdots|x_{i-1}|e|x_{i}|x_{i+1}|\cdots|x_{n}]_{n}.$

######  Example 7.3.

For $[x_{1}^{2}]_{1}\in\mathds{k}\mathsf{gr}^{\mathsf{op}}(1,1)$, we have
$[x_{1}^{2}]_{1}=[x_{1}x_{2}]_{2}\circ[x_{1}|x_{1}]_{1}\equiv[x_{1}x_{2}]_{2}\circ[e|x_{1}]_{1}+[x_{1}x_{2}]_{2}\circ[x_{1}|e]_{1}=2[x_{1}]_{1}$.
Similarly, one may obtain $[x_{1}^{k}]_{1}\equiv k[x_{1}]_{1}$.

For $1\leq i\leq n$, let $\mathsf{F}_{n}^{(i)}$ be the subgroup of
$\mathsf{F}_{n}$ generated by $x_{1},\cdots,x_{i-1},x_{i+1},\cdots,x_{n}$.

######  Lemma 7.4.

Let $n,m\in\mathds{N}$.

  1. (1)

For $1\leq i\leq n$, we have
$[x_{1}|x_{2}|\cdots|x_{i-1}|x_{i+1}|\cdots|x_{n}]_{n}\equiv 0$.

  2. (2)

Let $w_{1},w_{2},\cdots,w_{m}\in\mathsf{F}_{n}$. If there exists $1\leq i\leq
n$ such that $w_{1},w_{2},\cdots,w_{m}\in\mathsf{F}_{n}^{(i)}$, then
$[w_{1}|\cdots|w_{m}]_{n}\equiv 0$.

###### Proof.

Without loss of generality, we prove the first claim for $i=n$, i.e.
$[x_{1}|x_{2}|\cdots|x_{n-1}]_{n}\equiv 0$. This follows from

$\displaystyle[x_{1}|x_{2}|\cdots|x_{n-1}]_{n},$ $\displaystyle=$
$\displaystyle[x_{1}|x_{2}|\cdots|x_{n-1}]_{n+1}\circ[x_{1}|x_{2}|\cdots|x_{n-1}|x_{n}|x_{n}]_{n},$
$\displaystyle\equiv$
$\displaystyle[x_{1}|x_{2}|\cdots|x_{n-1}]_{n+1}\circ[x_{1}|x_{2}|\cdots|x_{n-1}|x_{n}|e]_{n}+[x_{1}|x_{2}|\cdots|x_{n-1}]_{n+1}\circ[x_{1}|x_{2}|\cdots|x_{n-1}|e|x_{n}]_{n},$
$\displaystyle=$
$\displaystyle[x_{1}|x_{2}|\cdots|x_{n-1}]_{n}+[x_{1}|x_{2}|\cdots|x_{n-1}]_{n}.$

Hence, we have $[x_{1}|x_{2}|\cdots|x_{n-1}]_{n}\equiv 0$.

We now prove the second claim. Let $\rho:\mathsf{F}_{m}\to\mathsf{F}_{n}$ be
the homomorphism such that $\rho(x_{k})=w_{k}$ for $1\leq k\leq m$; and
$e_{i}:\mathsf{F}_{n-1}\to\mathsf{F}_{n}$ be the homomorphism such that

$\displaystyle e_{i}(x_{k})=\begin{cases}x_{k}&(k<i),\\\ x_{k+1}&(k\geq
i).\end{cases}$

By the assumption $w_{1},w_{2},\cdots,w_{m}\in\mathsf{F}_{n}^{(i)}$, the
homomorphism $\rho$ factors as follows.

${\mathsf{F}_{m}}$${\mathsf{F}_{n}}$${\mathsf{F}_{n-1}}$$\scriptstyle{\rho}$$\scriptstyle{\exists\rho^{\prime}}$$\scriptstyle{e_{i}}$

By definitions, this diagram is rephrased as

$[w_{1}|\cdots|w_{m}]_{n}=[\rho^{\prime}(x_{1})|\cdots|\rho^{\prime}(x_{m})]_{n-1}\circ[x_{1}|x_{2}|\cdots|x_{i-1}|x_{i+1}|\cdots|x_{n}]_{n}$

in the category $\mathsf{gr}^{\mathsf{op}}$. Hence, the second claim is
immediate from the first claim and Proposition 7.2. ∎

######  Lemma 7.5.

Let $1\leq i\leq n$. For $w_{1},\cdots,w_{k}\in\mathsf{F}_{n}^{(i)}$ and
$a_{1},\cdots,a_{k}\in\mathds{Z}$, we have

$[w_{1}x_{i}^{a_{1}}w_{2}x_{i}^{a_{2}}\cdots
w_{k}x_{i}^{a_{k}}]_{n}\equiv\sum^{k}_{j=1}[w_{1}w_{2}\cdots
w_{j-1}(w_{j}x_{i}^{a_{j}})w_{j+1}w_{j+2}\cdots w_{k}]_{n}.$

###### Proof.

We prove the claim for $i=n$ by induction with respect to $k$. Let
$\rho=[w_{1}x_{n}^{a_{1}}w_{2}x_{n+1}^{a_{2}}\cdots
w_{k}x_{n+1}^{a_{k}}]_{n+1}$.

$\displaystyle[w_{1}x_{n}^{a_{1}}w_{2}x_{n}^{a_{2}}\cdots
w_{k}x_{n}^{a_{k}}]_{n}$
$\displaystyle=\rho\circ[x_{1}|x_{2}|\cdots|x_{n-1}|x_{n}|x_{n}]_{n},$
$\displaystyle\equiv\rho\circ[x_{1}|x_{2}|\cdots|x_{n-1}|x_{n}|e]_{n}+\rho\circ[x_{1}|x_{2}|\cdots|x_{n-1}|e|x_{n}]_{n},$
$\displaystyle=[w_{1}x_{n}^{a_{1}}w_{2}w_{3}\cdots
w_{k}]_{n}+[w_{1}w_{2}x_{n}^{a_{2}}w_{3}x_{n}^{a_{3}}\cdots
w_{k}x_{n}^{a_{k}}]_{n}$

We now focus on the right summand
$[w_{1}w_{2}x_{n}^{a_{2}}w_{3}x_{n}^{a_{3}}\cdots w_{k}x_{n}^{a_{k}}]_{n}$. We
put $w^{\prime}_{1}=w_{1}w_{2}$ and $w^{\prime}_{s}=w_{s+1}$ for $2\leq
s\leq(k-1)$; and apply the assumption of induction so as to complete the
proof. The proof for $i$ other than $n$ is straightforward by applying the
permutation of variables. ∎

######  Lemma 7.6.

For $\sigma\in\Sigma_{n}$, we have
$[x_{\sigma(1)}^{a_{1}}x_{\sigma(2)}^{a_{2}}\cdots
x_{\sigma(n)}^{a_{n}}]_{n}\equiv(a_{1}a_{2}\cdots
a_{n})[x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}]_{n}$.

###### Proof.

We consider the case that $\sigma$ is the identity, since the proof for the
general case is similar. Lemma 7.4 implies that, if $a_{j}=0$ for some $j$,
then $[x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}]_{n}\equiv 0$. Let
$1\leq j\leq n$ such that $a_{j}\neq 0$. For simplicity, let $j=n$. Suppose
$a_{n}>0$. For $w=x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n-1}^{a_{n-1}}$, we have

$\displaystyle[wx_{n}^{a_{n}}]_{n}$
$\displaystyle=[wx_{n}^{a_{n}-1}x_{n+1}]_{n+1}\circ[x_{1}|\cdots|x_{n-1}|x_{n}|x_{n}]_{n},$
$\displaystyle\equiv[wx_{n}^{a_{n}-1}x_{n+1}]_{n+1}\circ[x_{1}|\cdots|x_{n-1}|x_{n}|e]_{n}+[wx_{n}^{a_{n}-1}x_{n+1}]_{n+1}\circ[x_{1}|\cdots|x_{n-1}|e|x_{n}]_{n},$
$\displaystyle=[wx_{n}^{a_{n}-1}]_{n}+[wx_{n}]_{n}$

Inductively, we obtain $[wx_{n}^{a_{n}}]_{n}=a_{n}[wx_{n}]_{n}$. If $a_{n}<0$,
then we have
$[wx_{n}^{a_{n}}x_{n}^{-a_{n}}]_{n}\equiv[wx_{n}^{a_{n}}]_{n}+[wx_{n}^{-a_{n}}]_{n}\equiv[wx_{n}^{a_{n}}]_{n}-a_{n}[wx_{n}]_{n}$.
By the second part of Lemma 7.4, we have $[wx_{n}^{a_{n}}]_{n}=[w]_{n}\equiv
0$ so that the result follows. ∎

######  Theorem 7.7.

For $n,m\in\mathds{N}$, the composition of the maps below is surjective:

$\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)\stackrel{{\scriptstyle\mathcal{E}}}{{\to}}\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)\stackrel{{\scriptstyle\pi}}{{\to}}\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,m)/\mathcal{I}(n,m).$

###### Proof.

We first deal with the case $m=1$. Let
$\rho\in\mathds{k}\mathsf{gr}^{\mathsf{op}}(n,1)$. Note that by the second
part of Lemma 7.4 we have $\rho\equiv\sum^{r}_{j=1}a_{j}[w_{j}]_{1}$ where
$w_{j}\in\mathsf{F}_{n}$ and $w_{j}\not\in\mathsf{F}_{n}^{(i)}$ for any $i$.
We iteratively apply Lemma 7.5 to the linear combination. Then $\rho$ is a
linear combination of $[x_{\sigma(1)}^{a_{1}}x_{\sigma(2)}^{a_{2}}\cdots
x_{\sigma(n)}^{a_{n}}]_{n}$ up to $\mathcal{I}(n,1)$ where $\sigma$ is a
permutation. By Lemma 7.6, such summand up to $\mathcal{I}(n,1)$ is induced by
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,1)$ via $\mathcal{E}$.

We sketch the proof for general $m$. Note that one can prove the formulae
analogous to Lemma 7.5, 7.6 even if there are some bars inserted between the
words in the parenthesis $[~{}]$. For example, by considering $m=2$, we have

$\displaystyle[w_{1}x_{i}^{a_{1}}\cdots
w_{k}x_{i}^{a_{k}}|w_{k+1}x_{i}^{a_{k+1}}\cdots w_{l}x_{i}^{a_{l}}]_{n}$
$\displaystyle\equiv$ $\displaystyle\sum^{k}_{j=1}[w_{1}\cdots
w_{j-1}(w_{j}x_{i}^{a_{j}})w_{j+1}\cdots|\cdots
w_{l}]_{n}+\sum^{l}_{j=k+1}[w_{1}\cdots|\cdots
w_{j-1}(w_{j}x_{i}^{a_{j}})w_{j+1}\cdots w_{k}]_{n}.$

This holds even if there are more than one bar; the proof is parallel.
Similarly, for instance, we have

$[x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{k}^{a_{k}}|x_{k+1}^{a_{k+1}}\cdots
x_{n}^{a_{n}}]_{n}\equiv(a_{1}a_{2}\cdots a_{n})[x_{1}x_{2}\cdots
x_{k}|x_{k+1}\cdots x_{n}]_{n}.$

By using these extended formulae, one can prove our assertion as above. ∎

###  8\. Application to exponential $\mathsf{gr}^{\mathsf{op}}$-modules

The opposite category of finitely generated free groups
$\mathsf{gr}^{\mathsf{op}}$ is a small category which satisfies the
assumptions of $\mathcal{C}$. Let $\mathds{k}$ be a commutative unital ring
unless otherwise specified. By [13] (see also [7, Remark 2.1, Theorem 2.2] and
[4]), the evaluation functor
$\mathrm{ev}_{1}:\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\to\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}$
induces an equivalence of categories (see Remark 8.1). In this section, we
give an explicit description of some subcategories of exponential
$\mathsf{gr}^{\mathsf{op}}$-modules under this equivalence.

######  Remark 8.1.

We sketch a construction of $\mathrm{ev}_{1}$ here. Recall the notation in
section 2. Let
$F\in\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$. Then
the underlying module of the Hopf algebra $\mathrm{ev}_{1}(F)$ is $F(1)$. The
multiplication, unit, comultiplication, counit and antipode are given as
follows respectively:

$\displaystyle\nabla_{\mathrm{ev}_{1}(F)}:$ $\displaystyle F(1)\otimes
F(1)\cong F(2)\stackrel{{\scriptstyle F(\nabla)}}{{\to}}F(1),$
$\displaystyle\eta_{\mathrm{ev}_{1}(F)}:$ $\displaystyle\mathds{k}\cong
F(0)\stackrel{{\scriptstyle F(\eta)}}{{\to}}F(1),$
$\displaystyle\Delta_{\mathrm{ev}_{1}(F)}:$ $\displaystyle
F(1)\stackrel{{\scriptstyle F(\Delta)}}{{\to}}F(2)\cong F(1)\otimes F(1),$
$\displaystyle\varepsilon_{\mathrm{ev}_{1}(F)}:$ $\displaystyle
F(1)\stackrel{{\scriptstyle F(\epsilon)}}{{\to}}F(0)\cong\mathds{k},$
$\displaystyle S_{\mathrm{ev}_{1}(F)}:$ $\displaystyle
F(1)\stackrel{{\scriptstyle F(\gamma)}}{{\to}}F(1).$

The quasi-inverse
$\alpha:\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}\to\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
is given as follows. For an object
$H\in\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}$, we set
$\left(\alpha(H)\right)(n)=U(H)^{\otimes n}$ where $U(H)$ is the underlying
module of $H$. The cocommutative Hopf algebra structure on $H$ induces a
contravariant assignment of morphisms.

####  8.1. Analytic exponential $\mathsf{gr^{\mathsf{op}}}$-modules

In this section, we give an equivalence between conilpotent cocommutative Hopf
algebras, and analytic exponential $\mathsf{gr}^{\mathsf{op}}$-modules.

Let $\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}$ be the full subcategory
of $\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}$ consisting of conilpotent
cocommutative Hopf algebras over $\mathds{k}$.

Let
$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$ be
the full subcategory of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$ consisting
of analytic $\mathsf{gr^{\mathsf{op}}}$-modules.

######  Theorem 8.2.

The functor $\mathrm{ev}_{1}$ induces an equivalence of categories:

$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}.$

###### Proof.

We apply Corollary 5.3 to prove our claim. By Corollary 5.3, the restriction
of $\mathrm{ev}_{1}$ to
$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
factors through
$\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}\hookrightarrow\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}$.
Denote the factorized functor by
$\mathrm{ev}_{1}^{\prime}:\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\to\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}$.
Consider $\alpha$ introduced in Remark 8.1. On the one hand, the composition
$\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}\hookrightarrow\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}\stackrel{{\scriptstyle\alpha}}{{\to}}\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
factors through the embedding
$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\hookrightarrow\mathcal{F}^{\mathsf{exp}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$.
In fact, the functor $\alpha(H)$ assigned to a conilpotent cocommutative Hopf
algebra $H$ is analytic since $(\alpha(H))(n)=H^{\otimes n}$ is conilpotent
for any $n$, by Proposition 4.7. Denote the factorized functor by
$\alpha^{\prime}:\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}\to\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$.
Then $\mathrm{ev}_{1}^{\prime}$ and $\alpha^{\prime}$ induce an equivalence of
categories since so do the functors $\mathrm{ev}_{1}$ and $\alpha$. ∎

####  8.2. Outer exponential $\mathsf{gr}^{\mathsf{op}}$-modules

In this section, we prove an equivalence between bicommutative Hopf algebras
and outer exponential $\mathsf{gr}^{\mathsf{op}}$-modules. An outer
$\mathsf{gr}^{\mathsf{op}}$-module is a functor
$F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ such that the induced
action of inner automorphisms on $F(n)$ is trivial for any $n$. The notion of
outer $\mathsf{gr}^{\mathsf{op}}$-module is introduced in [16], motivated by
the study of representations of $\mathrm{Out}(\mathsf{F}_{n})$ coming from
Hochschild homology associated to a wedge of circles.

Let $\mathsf{ab}$ be the category consisting of the free abelian groups
$\mathds{Z}^{n}$ for $n\in\mathds{N}$ and group homomorphisms. The evaluation
$\mathrm{ev}_{\mathds{Z}}:\mathcal{F}^{\mathsf{exp}}(\mathsf{ab};\mathds{k})\to\mathsf{Hopf}^{\mathsf{bc}}_{\mathds{k}}$
at $\mathds{Z}\in\mathsf{ab}$ induces an equivalence of categories [18]. The
assignment of the dual $X^{\vee}{:=}\mathrm{Hom}(X,\mathds{Z})$ to
$X\in\mathsf{ab}$ induces an equivalence of categories
$(-)^{\vee}:\mathsf{ab}^{\mathsf{op}}\to\mathsf{ab}$. We denote by
$\mathrm{ev}_{\mathds{Z}}:\mathcal{F}^{\mathsf{exp}}(\mathsf{ab}^{\mathsf{op}};\mathds{k})\to\mathsf{Hopf}^{\mathsf{bc}}_{\mathds{k}}$
their composition with a slight abuse of notation. Especially,
$\mathrm{ev}_{\mathds{Z}}$ gives an equivalence of categories with a
commutative diagram below where $\mathfrak{a}:\mathsf{gr}\to\mathsf{ab}$ is
the abelianization functor:

(6)
${\mathcal{F}^{\mathsf{exp}}(\mathsf{ab}^{\mathsf{op}};\mathds{k})}$${\mathsf{Hopf}^{\mathsf{bc}}_{\mathds{k}}}$${\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})}$${\mathsf{Hopf}^{\mathsf{cc}}_{\mathds{k}}}$$\scriptstyle{\mathfrak{a}^{\ast}}$$\scriptstyle{\mathrm{ev}_{\mathbb{Z}}}$$\scriptstyle{\mathrm{ev}_{1}}$

Let
$\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
be the full subcategory of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ consisting
of outer $\mathsf{gr}^{\mathsf{op}}$-modules.

######  Theorem 8.3.

The functor $\mathfrak{a}^{\ast}$ induces an equivalence of categories:

$\mathcal{F}^{\mathsf{exp}}(\mathsf{ab}^{\mathsf{op}};\mathds{k})\simeq\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$

In particular, the functor $\mathrm{ev}_{1}$ induces an equivalence of
categories:

$\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\simeq\mathsf{Hopf}^{\mathsf{bc}}_{\mathds{k}}.$

###### Proof.

By (6), $\mathfrak{a}^{\ast}$ is a faithfully full embedding. It is clear that
the induced automorphism $\mathfrak{a}(g)$ is trivial for any inner
automorphism $g$. Hence, the essential image of $\mathfrak{a}^{\ast}$ is
contained in
$\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$.
It suffices to prove that every
$F\in\mathcal{F}^{\mathsf{exp}}_{\mathsf{out}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
arises from a precomposition with $\mathfrak{a}$. Equivalently, we prove that
the cocommutative Hopf algebra $H=\mathrm{ev}_{1}(F)$ is commutative. For the
group presentation $\mathsf{F}_{2}=\langle x_{1},x_{2}\rangle$, consider the
inner automorphism $\rho:\mathsf{F}_{2}\to\mathsf{F}_{2}$ induced by
$x_{1}\in\mathsf{F}_{2}$, i.e. $\rho(x_{i})=x_{1}x_{i}x_{1}^{-1}$. We denote
by $S$ the antipode on $H$, and freely use the Sweedler notation of
comultiplication below. For $x,y\in H$, one can calculate as follows by
definition:

$\displaystyle(F(\rho))(x\otimes y)=x_{(1)}\otimes x_{(2)}yS(x_{(3)}).$

By the hypothesis that $F$ is outer, we have $(F(\rho))(x\otimes y)=x\otimes
y$. This gives $x\otimes y=x_{(1)}\otimes x_{(2)}yS(x_{(3)})$. Hence, for any
$x,y\in H$, we have
$\varepsilon(x)y=\varepsilon(x_{(1)})x_{(2)}yS(x_{(3)})=x_{(1)}yS(x_{(2)})$.
This result leads to the following calculation for $x,y\in H$:

$\displaystyle yx$ $\displaystyle=\varepsilon(x_{(1)})yx_{(2)},$
$\displaystyle=x_{(1)}yS(x_{(2)})x_{(3)},$
$\displaystyle=x_{(1)}y\varepsilon(x_{(2)})=xy.$

∎

####  8.3. Analytic $\mathsf{gr}^{\mathsf{op}}$-modules and Lie algebras

In this section, we apply the previous results to prove an equivalence between
some exponential $\mathsf{gr}^{\mathsf{op}}$-modules and Lie algebras. The
proof depends heavily on classical structure theorems for bialgebras (e.g. see
[11, Theorem 1.3.4]).

In this section, we assume that the ground ring $\mathds{k}$ is a field of
characteristic zero. Let $\mathsf{Lie}_{\mathds{k}}$ be the category of Lie
algebras over $\mathds{k}$ and Lie homomorphisms. Then the assignment of
primitive elements to Hopf algebras induce an equivalence of categories [12,
Section 5]:

(7)
$P:\mathsf{Hopf}^{\mathsf{cc,conil}}_{\mathds{k}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathsf{Lie}_{\mathds{k}}.$

The quasi-inverse is given by the universal enveloping algebra construction
$U$. This equivalence leads to the following corollary.

Let
$\mathcal{F}^{\mathsf{exp}}_{\omega,\mathsf{out}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})$
be the full subcategory of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ consisting
of analytic and outer $\mathsf{gr}^{\mathsf{op}}$-modules.

Let $\mathsf{Lie}^{\mathsf{ab}}_{\mathds{k}}$ be the full subcategory of
$\mathsf{Lie}_{\mathds{k}}$ consisting of abelian Lie algebras.

######  Corollary 8.4.

If the characteristic of the ground field $\mathds{k}$ is zero, then the
composition of the evaluation on $1\in\mathsf{gr}^{\mathsf{op}}$ with the
functor $P$ induces equivalences of categories:

  1. (1)

$\mathcal{F}^{\mathsf{exp}}_{\omega}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{Lie}_{\mathds{k}}$.

  2. (2)

$\mathcal{F}^{\mathsf{exp}}_{\omega,\mathsf{out}}(\mathsf{gr^{\mathsf{op}}};\mathds{k})\simeq\mathsf{Lie}^{\mathsf{ab}}_{\mathds{k}}$.

###### Proof.

The first claim is immediate from Theorem 8.2. The second one follows from the
second part of Theorem 8.3. ∎

######  Remark 8.5.

Let $\mathfrak{Lie}$ be the Lie operad. In [14], Powell constructs an
equivalence between analytic $\mathsf{gr}^{\mathsf{op}}$-modules and
$\mathfrak{Lie}$-modules of characteristic of zero. In particular, it yields
an embedding
$\mathsf{Lie}_{\mathds{k}}\hookrightarrow\mathcal{F}_{\mathfrak{Lie}}\simeq\mathcal{F}_{\omega}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$.
The equivalences in Corollary 8.4 are compatible with the embedding.

Let
$\mathcal{F}^{\mathsf{exp}}_{<\infty}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
be the full subcategory of
$\mathcal{F}^{\mathsf{exp}}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ consisting
of $\mathsf{gr}^{\mathsf{op}}$-modules with finite degrees.

######  Corollary 8.6.

If the characteristic of the ground field $\mathds{k}$ is zero, then the
category
$\mathcal{F}^{\mathsf{exp}}_{<\infty}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
is equivalent with the one point category.

###### Proof.

Let $F$ be an exponential $\mathsf{gr}^{\mathsf{op}}$-module with finite
degree $n\geq 0$. In particular, the functor $F$ is analytic and thus, $F$ is
induced by a Lie algebra $\mathfrak{g}$ by the first part of Corollary 8.4. By
applying Theorem 5.2 to $X=1$, we obtain $\left(P_{n}(F)\right)(X)\cong
P_{n}(U(\mathfrak{g}))$. We have $U(\mathfrak{g})=P_{n}(U(\mathfrak{g}))$ due
to the degree assumption on $F$. If $\mathfrak{g}\not\cong 0$, then we choose
a nonzero $x\in\mathfrak{g}$. By Poincaré-Birkhoff-Witt, we have $0\neq
x^{n+1}\in U(\mathfrak{g})$. On the other hand,
$\delta^{n+1}(x^{n+1})=(n+1)!x^{\otimes(n+1)}\neq 0$ since the characteristic
is zero. Hence, $x\not\in P_{n}(U(\mathfrak{g}))=U(\mathfrak{g})$ which leads
to a contradiction so that we have $\mathfrak{g}\cong 0$. Thus, the
equivalence in the first part of Corollary 8.4 reduces to
$\mathcal{F}^{\mathsf{exp}}_{<\infty}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\simeq\ast$.
∎

###  9\. Primitive filtration of $\mathsf{gr}^{\mathsf{op}}$-modules

In this section, we introduce the primitive filtration of
$\mathsf{gr}^{\mathsf{op}}$-modules. For exponential
$\mathsf{gr}^{\mathsf{op}}$-modules, it reproduces the primitive filtration of
cocommutative Hopf algebras. The primitive filtration is induced by the
primitive part (see Definition 9.2) which turns out to have an action from the
Lie operad $\mathfrak{Lie}$. It generalizes the Lie algebra structure on the
primitive part of Hopf algebras.

####  9.1. Primitive filtration

In this section, we define the primitive filtration of
$\mathsf{gr}^{\mathsf{op}}$-modules. Let $\mathds{k}$ be a commutative unital
ring unless otherwise specified. The primitive filtration of
$F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ is constructed from a
$\Sigma$-module $\mathcal{P}(F)\in\mathcal{F}(\Sigma;\mathds{k})$ which is
induced by the first step of the polynomial filtration related with $F$.
Moreover, we compare the polynomial filtration in section 3.2 and the
primitive filtration.

Recall the cross effect functor in section 3.1. By using the explicit
construction of polynomial approximation in Definition 3.9, we define

$\widetilde{P}_{1}(F){:=}P_{1}(\mathrm{cr}^{1}(F)),~{}F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k}).$

In the following, recall that we are identifying natural numbers with the
objects of $\mathsf{gr}^{\mathsf{op}}$ so that the product in the category
$\mathsf{gr}^{\mathsf{op}}$ is realized as the sum of natural numbers.

######  Definition 9.1.

Let $F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$.

  * •

Let $[X_{i}]^{n-1}_{i=1}$ be a sequence of objects in
$\mathsf{gr}^{\mathsf{op}}$. For $1\leq m\leq n$, we define
$\hat{F}_{m}([X_{i}]^{n-1}_{i=1})\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$
by
$\left(\hat{F}_{m}([X_{i}]^{n-1}_{i=1})\right)(Y){:=}F(\sum^{m-1}_{i=1}X_{i}+Y+\sum^{n-1}_{i=m}X_{i})$
for an object $Y$, and
$\left(\hat{F}_{m}([X_{i}]^{n-1}_{i=1})\right)(f){:=}F(\Asterisk^{m-1}_{i=1}\mathrm{id}_{X_{i}}\ast
f\ast\Asterisk^{n-1}_{i=m}\mathrm{id}_{X_{i}})$ for a morphism $f$.

  * •

For a sequence $[X_{i}]^{n}_{i=1}$, we define
$R_{n}(F)\in\mathcal{F}(\left(\mathsf{gr}^{\mathsf{op}}\right)^{\times
n};\mathds{k})$ by

$\left(R_{n}(F)\right)\left([X_{i}]^{n}_{i=1}\right){:=}\bigcap^{n}_{i=1}\widetilde{P}_{1}\left(\hat{F}_{i}(X_{1},\cdots,X_{i-1},X_{i+1},\cdots,X_{n})\right)(X_{i}).$

For $[X_{i}]^{n}_{i=1}$ a sequence of objects in $\mathsf{gr}^{\mathsf{op}}$,
the symmetry on $\mathsf{gr}^{\mathsf{op}}$ induces a $\Sigma_{n}$-action:
$\sigma\in\Sigma_{n}$ induces an isomorphism

$F(X_{1}+X_{2}+\cdots+X_{n})\to
F(X_{\sigma^{-1}(1)}+X_{\sigma^{-1}(2)}+\cdots+X_{\sigma^{-1}(n)}).$

The restriction to $\left(R_{n}(F)\right)\left([X_{i}]^{n}_{i=1}\right)\subset
F(X_{1}+X_{2}+\cdots+X_{n})$ induces an isomorphism

$\left(R_{n}(F)\right)\left([X_{i}]^{n}_{i=1}\right)\to\left(R_{n}(F)\right)\left([X_{\sigma^{-1}(i)}]^{n}_{i=1}\right).$

By substitution of $X_{i}=1$ for any $i$, we define the following.

######  Definition 9.2.

We define $\mathcal{P}(F)\in\mathcal{F}(\Sigma;\mathds{k})$ the primitive part
of $F$ by

$\left(\mathcal{P}(F)\right)(n):=\left(R_{n}(F)\right)(\overbrace{1,1,\cdots,1}^{n}).$

######  Remark 9.3.

It turns out that $\mathcal{P}(F)$ can be enhanced to a
$\mathsf{Cat}_{\mathfrak{Lie}}$-module (see section 9.2).

######  Example 9.4.

Let $\mathds{k}$ be a field. If $F$ is exponential, then there exists a
cocommutative Hopf algebra $H$ such that $F(n)=H^{\otimes n}$ (explicitly,
$F=\alpha(H)$ in Remark 8.1). In this case, we have a natural isomorphism
$\left(\mathcal{P}(F)\right)(n)\cong\mathrm{Prim}(H)^{\otimes n}$. This will
be proved in Lemma 10.14.

Recall the notation in Definition 7.1.

######  Proposition 9.5.

For $n\in\mathds{N}$, we have

$\left(\mathcal{P}(F)\right)(n)=\bigcap^{n}_{i=1}\mathrm{Ker}\left(F(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})\right).$

###### Proof.

For $X_{1}=X_{2}=\cdots=X_{n}=1\in\mathsf{gr}^{\mathsf{op}}$, we have

$\widetilde{P}_{1}\left(\hat{F}_{i}(X_{1},\cdots,X_{i-1},X_{i+1},\cdots,X_{n})\right)(X_{i})=\mathrm{Ker}\left(F(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})\right)$

by definition. ∎

The category $\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ is equivalent
with the category of $\mathds{k}$-linear functors from
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$ to $\mathsf{Mod}_{\mathds{k}}$ via the
linear extension. Based on this observation, we regard
$F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ as the corresponding
linear functor
$F:\mathds{k}\mathsf{gr}^{\mathsf{op}}\to\mathsf{Mod}_{\mathds{k}}$. Recall
the functor
$\mathcal{E}:\mathsf{Cat}_{\mathfrak{Ass}^{u}}\hookrightarrow\mathds{k}\mathsf{gr}^{\mathsf{op}}$
in Definition 6.2. We define

$\mathcal{E}^{\ast}F{:=}F\circ\mathcal{E}.$

For $n\in\mathds{N}$, the structure of $\mathcal{E}^{\ast}F$ yields a
$\mathds{k}$-linear natural transformation,

$J_{n}:\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\otimes
F(n)\to\mathcal{E}^{\ast}F;~{}f\otimes
v\mapsto\left((\mathcal{E}^{\ast}F)(f)\right)(v).$

By applying the tensor-hom adjunction, we obtain a homomorphism

$\check{J}_{n}:F(n)\to\mathrm{Nat}\left(\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),\mathcal{E}^{\ast}F\right).$

In the following statements, we recall the notation
$(x_{1}\otimes\cdots\otimes x_{n})\in\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,n)$
of the identity and the generators $\Delta,\nabla,\eta,\epsilon,\gamma$ of the
category $\mathsf{gr}^{\mathsf{op}}$ (see section 6).

######  Theorem 9.6.

Let $n\in\mathds{N}$. For
$F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$, we consider
$\check{J}_{n}$ defined as above. The restriction of $\check{J}_{n}$ along the
inclusion $\left(\mathcal{P}(F)\right)(n)\subset F(n)$ factors through
$\mathrm{Nat}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F\right)$.
In other words, there exists a homomorphism
$\check{I}_{n}:(\mathcal{P}(F))(n)\to\mathrm{Nat}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F\right)$
making the following diagram commutative:

${F(n)}$${\mathrm{Nat}\left(\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),\mathcal{E}^{\ast}F\right)}$${(\mathcal{P}(F))(n)}$${\mathrm{Nat}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F\right).}$$\scriptstyle{\check{J}_{n}}$$\scriptstyle{\check{I}_{n}}$

###### Proof.

Let $v\in\left(\mathcal{P}(F)\right)(n)$. Consider $\Phi$ the
$\mathds{k}$-linear natural transformation $\check{J}_{n}(v)$ from
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)$ to $\mathcal{E}^{\ast}F$. In
particular, we have

$\Phi_{n}(x_{1}\otimes x_{2}\cdots\otimes x_{n})=v.$

We prove that $\Phi$ gives a natural transformation from $K_{n}$ to $F$. Put
$K_{n}={}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$.
Note that $\mathcal{E}^{\ast}K_{n}=\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)$.
Let $f:m\to l$ be a morphism in $\mathds{k}\mathsf{gr}^{\mathsf{op}}$. It
suffices to prove that the diagram below commutes:

${K_{n}(m)}$${F(m)}$${K_{n}(l)}$${F(l)}$$\scriptstyle{\Phi_{m}}$$\scriptstyle{K_{n}(f)}$$\scriptstyle{F(f)}$$\scriptstyle{\Phi_{l}}$

Let $a\in K_{n}(m)$. Below, we prove that $(\Phi_{l}\circ
K_{n}(f))(a)=\left(F(f)\circ\Phi_{m}\right)(a)$. Note that we have

$a=\left(\mathcal{E}^{\ast}K_{n}(a)\right)(x_{1}\otimes x_{2}\cdots\otimes
x_{n}).$

where we regard $a$ on the right hand side as an element of
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)$. By Theorem 7.7, there exists
$g\in\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,l)$ such that
$\mathcal{E}(g)-f\circ\mathcal{E}(a)\in\mathcal{I}(n,l)$.

$\displaystyle(\Phi_{l}\circ K_{n}(f))(a)$ $\displaystyle=(\Phi_{l}\circ
K_{n}(f)\circ\mathcal{E}^{\ast}K_{n}(a))(x_{1}\otimes x_{2}\cdots\otimes
x_{n}),$ $\displaystyle=\left(\Phi_{l}\circ
K_{n}(f\circ\mathcal{E}(a))\right)(x_{1}\otimes x_{2}\cdots\otimes x_{n}),$
$\displaystyle=(\Phi_{l}\circ K_{n}(\mathcal{E}(g)))(x_{1}\otimes
x_{2}\cdots\otimes x_{n}).$

where the final equality follows from the fact
$\left(K_{n}(\mathcal{I}(n,l))\right)(x_{1}\otimes x_{2}\cdots\otimes
x_{n})=0$, which follows from
$\left(K_{n}(\mathrm{id}_{i-1}\ast\theta\ast\mathrm{id}_{n-i})\right)(x_{1}\otimes
x_{2}\cdots\otimes x_{n})=0$. The assumption
$\Phi\in\mathrm{Nat}(\mathcal{E}^{\ast}K_{n},\mathcal{E}^{\ast}F)$ implies
that

$\displaystyle(\Phi_{l}\circ K_{n}(\mathcal{E}(g)))(x_{1}\otimes
x_{2}\cdots\otimes x_{n})$
$\displaystyle=(F(\mathcal{E}(g))\circ\Phi_{n})(x_{1}\otimes
x_{2}\cdots\otimes x_{n}),$ $\displaystyle=\left(F(\mathcal{E}(g))\right)(v).$

Note that we have $\left(F(\mathcal{I}(l,n))\right)(v)=0$, since
$v\in\left(\mathcal{P}(F)\right)(n)$ by the hypothesis and Proposition 9.5.
Thus, we obtain
$\left(F(\mathcal{E}(g))\right)(v)=\left(F(f\circ\mathcal{E}(a))\right)(v)$.
By the assumption
$\Phi\in\mathrm{Nat}(\mathcal{E}^{\ast}K_{n},\mathcal{E}^{\ast}F)$ again, we
obtain

$\displaystyle\left(F(f\circ\mathcal{E}(a))\right)(v)$
$\displaystyle=\left(F(f)\circ\mathcal{E}^{\ast}F(a)\circ\Phi_{n}\right)(x_{1}\otimes
x_{2}\cdots\otimes x_{n}),$
$\displaystyle=\left(F(f)\circ\Phi_{m}\circ\mathcal{E}^{\ast}K_{n}(a)\right)(x_{1}\otimes
x_{2}\cdots\otimes x_{n}),$ $\displaystyle=\left(F(f)\circ\Phi_{m}\right)(a).$

∎

######  Definition 9.7.

We denote by $I_{n}$ the induced natural transformation by applying the
tensor-hom adjunction to $\check{I}_{n}$:

$I_{n}:~{}{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\otimes\left(\mathcal{P}(F)\right)(n)\to
F.$

Then the following is well-defined since $I_{k}$’s are
$\mathsf{gr}^{\mathsf{op}}$-homomorphisms by Theorem 9.6.

######  Definition 9.8.

For $F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$, we define the
primitive filtration of $F$:

${F}$${\cdots}$${Q_{n}(F)}$${Q_{n+1}(F)}$${\cdots}$

where $Q_{n}(F)\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ is the
image of $\sum_{k\leq n}I_{k}:\bigoplus_{k\leq
n}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,-)\otimes\left(\mathcal{P}(F)\right)(k)\right)\to
F$. The $\mathsf{gr}^{\mathsf{op}}$-module $F$ is primitive if
$F\cong\mathrm{colim}_{n}~{}Q_{n}(F)$.

######  Proposition 9.9.

Let $F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$. We have
$Q_{n}(F)\subset P_{n}(F)$. In particular, every primitive
$\mathsf{gr}^{\mathsf{op}}$-module is analytic.

###### Proof.

Recall $I_{n}$ in Theorem 9.6. We apply the polynomial approximation functor
$P_{n}:\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\to\mathcal{F}_{\leq
n}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ to the natural transformation
$\sum_{k\leq n}I_{k}$. By Proposition 6.1, the application yields a natural
transformation,

$\bigoplus_{k\leq
n}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(k,-)\otimes\left(\mathcal{P}(F)\right)(k)\right)\to
P_{n}(F).$

The definition of $Q_{n}(F)$ implies that $Q_{n}(F)\subset P_{n}(F)$. ∎

####  9.2. Representability of the induced
$\mathsf{Cat}_{\mathfrak{Lie}}$-module

In this section, we show that the $\Sigma$-module structure on
$\mathcal{P}(F)$ extends to a left $\mathsf{Cat}_{\mathfrak{Lie}}$-module
structure. Especially, we have a functor
$\mathcal{P}:\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\to\mathcal{F}_{\mathfrak{Lie}}$.
Furthermore, we go further by proving that the
$(\mathds{k}\mathsf{gr}^{\mathsf{op}},\mathsf{Cat}_{\mathfrak{Lie}})$-bimodule
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ represents the functor
$\mathcal{P}:\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\to\mathcal{F}_{\mathfrak{Lie}}$.

Let
$\mathcal{B}:\mathsf{Cat}_{\mathfrak{Lie}}\to\mathsf{Cat}_{\mathfrak{Ass}^{u}}$
be the functor induced by the operad homomorphism
$\mathfrak{Lie}\to\mathfrak{Ass}^{u}$.

######  Lemma 9.10.

For $n,m\in\mathds{N}$, the functor $\mathcal{B}$ induces a linear map

$\widetilde{\mathcal{B}}_{n,m}:\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\to\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\right)(m).$

###### Proof.

Recall the notation in Definition 7.1. Let
$f_{r}=\mathrm{id}_{r-1}\ast\theta\ast\mathrm{id}_{m-r}:m\to(m+1)$ and put
$L_{r}=\mathrm{Ker}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,f_{r})\right)$
for $1\leq r\leq m$. We have
$\bigcap^{m}_{r=1}L_{r}=\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\right)(m)$
by Proposition 9.5, so that it suffices to give a factorization of
$\mathcal{B}$ as $\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\to L_{r}$ for arbitrary
$r$. Recall that in section 6.2 we introduce
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)$ as the submodule of
$\mathbb{H}(n,m)$ the tensor product of noncommutative polynomial bialgebras.
Let $M_{r}$ be the kernel of
$\mathbb{H}(n,f_{r}):\mathbb{H}(n,m)\to\mathbb{H}(n,m+1)$. By definition of
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ (in section 6.2), we obtain the
following commutative diagram whose rows and columns are all exact:

${0}$${0}$${0}$${0}$${L_{r}}$${{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)}$${{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m+1)}$${0}$${M_{r}}$${\mathbb{H}(n,m)}$${\mathbb{H}(n,m+1).}$$\scriptstyle{{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n{,}f_{r})}$$\scriptstyle{\mathbb{H}(n{,}f_{r})}$

One may apply this diagram to prove that $L_{r}$ is the pullback of the
inclusions
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)\to\mathbb{H}(n,m)$ and
$M_{r}\to\mathbb{H}(n,m)$. Hence, it suffices to construct
$h:\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\to M_{r}$ such that the diagram below
commutes:

${\mathsf{Cat}_{\mathfrak{Lie}}(n,m)}$${L_{r}}$${{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,m)}$${M_{r}}$${\mathbb{H}(n,m).}$$\scriptstyle{\mathcal{B}}$

The iterated Lie brackets of variables $x_{1},x_{2},\cdots,x_{n}$ lie in
$\mathrm{Prim}(H_{n})$. Hence, we obtain a linear map
$h^{\prime}:\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\to\mathrm{Prim}(H_{n})^{\otimes
m}$. On the one hand, the inclusion $i:\mathrm{Prim}(H_{n})\to H_{n}$ induces
$\mathrm{Prim}(H_{n})^{\otimes m}\to M_{r}$ due to the following commutative
diagram and the exactness of the row:

${0}$${M_{r}}$${\mathbb{H}(n,m)}$${\mathbb{H}(n,m+1)}$${\mathrm{Prim}(H_{n})^{\otimes
m}}$$\scriptstyle{\mathbb{H}(n{,}f_{r})}$$\scriptstyle{0}$$\scriptstyle{h^{\prime\prime}}$

Therefore, $h=h^{\prime\prime}\circ
h^{\prime}:\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\to M_{r}$ is the linear map
which satisfies our requirement. ∎

The inclusion
$\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\right)\to{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)$
induces a natural transformation
$\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\right)\otimes
M\to{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\otimes M$ for a
$\mathds{k}$-module $M$. By definition, this factors through the inclusion
$\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\otimes
M\right)\to{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\otimes M$, and we
obtain the factorization:

$\mathcal{U}^{M}_{n}:\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\right)\otimes
M\to\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\otimes
M\right).$

By using this observation with $\widetilde{\mathcal{B}}_{n,m}$ in Lemma 9.10
and $I_{n}$ in Definition 9.7, we introduce the following.

######  Definition 9.11.

For $n,m\in\mathds{N}$, we define
$\mathcal{L}_{n,m}:\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\otimes(\mathcal{P}(F))(n)\to\left(\mathcal{P}(F)\right)(m)$
by the composition below.

${\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\otimes(\mathcal{P}(F))(n)}$${\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\right)(m)\otimes(\mathcal{P}(F))(n)}$${\left(\mathcal{P}(F)\right)(m)}$${\mathcal{P}\left({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)\otimes(\mathcal{P}(F))(n)\right)(m).}$$\scriptstyle{\widetilde{\mathcal{B}}_{n{,}m}\otimes\mathrm{id}}$$\scriptstyle{\mathcal{U}^{(\mathcal{P}(F))(n)}_{n}}$$\scriptstyle{\mathcal{P}(I_{n})}$

The definition of $\mathcal{L}_{n,m}$ is unpacked as follows. Let
$a\in\mathsf{Cat}_{\mathfrak{Lie}}(n,m)$ and
$x\in\left(\mathcal{P}(F)\right)(n)$. By definitions, we have

(8) $\displaystyle\mathcal{L}_{n,m}(a\otimes x)=J_{n}(\mathcal{B}(a)\otimes
x).$

######  Proposition 9.12.

Let $F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$. The family
$\mathcal{L}_{n,m}$ for $n,m\in\mathds{N}$ gives a
$\mathsf{Cat}_{\mathfrak{Lie}}$-module structure on $\mathcal{P}(F)$, which is
a sub $\mathsf{Cat}_{\mathfrak{Lie}}$-module of
$\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$.

###### Proof.

It suffices to prove that the diagram below commutes:

${\mathsf{Cat}_{\mathfrak{Lie}}(m,l)\otimes\mathsf{Cat}_{\mathfrak{Lie}}(n,m)\otimes\left(\mathcal{P}(F)\right)(n)}$${\mathsf{Cat}_{\mathfrak{Lie}}(n,l)\otimes\left(\mathcal{P}(F)\right)(n)}$${\mathsf{Cat}_{\mathfrak{Lie}}(m,l)\otimes\left(\mathcal{P}(F)\right)(m)}$${\left(\mathcal{P}(F)\right)(l).}$$\scriptstyle{\mathrm{id}\otimes\mathcal{L}_{n{,}m}}$$\scriptstyle{\circ\otimes\mathrm{id}}$$\scriptstyle{\mathcal{L}_{n{,}l}}$$\scriptstyle{\mathcal{L}_{m{,}l}}$

It follows from

$\displaystyle\mathcal{L}_{m,l}(b\otimes\mathcal{L}_{n,m}(a\otimes x))$
$\displaystyle=J_{m}(\mathcal{B}(b)\otimes J_{n}(\mathcal{B}(a)\otimes x)),$
$\displaystyle=J_{n}\left((\mathcal{B}(b)\circ\mathcal{B}(a))\otimes
x\right),$ $\displaystyle=J_{n}\left(\mathcal{B}(b\circ a)\otimes x\right),$
$\displaystyle=\mathcal{L}_{n,l}((b\circ a)\otimes x),$

where
$a\in\mathsf{Cat}_{\mathfrak{Lie}}(n,m),b\in\mathsf{Cat}_{\mathfrak{Lie}}(m,l),x\in\left(\mathcal{P}(F)\right)(n)$.
Furthermore, $\mathcal{P}(F)$ is a sub $\mathsf{Cat}_{\mathfrak{Lie}}$-module
of $\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$ since the
$\mathsf{Cat}_{\mathfrak{Lie}}$-action is inherited from that of
$\mathcal{B}^{\ast}\mathcal{E}^{\ast}(F)$ as we see in (8). ∎

For the category $\mathsf{Cat}_{\mathfrak{Ass}^{u}}$, the Yoneda embedding is
given by
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}^{\mathsf{op}}\to\mathcal{F}_{\mathfrak{Ass}^{u}};n\mapsto\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)$
where $\mathcal{F}_{\mathfrak{Ass}^{u}}$ is the category of
$\mathsf{Cat}_{\mathfrak{Ass}^{u}}$-modules and linear natural
transformations. For $G\in\mathcal{F}_{\mathfrak{Ass}^{u}}$, the Yoneda lemma
implies that the evaluation on the identity $(x_{1}\otimes\cdots\otimes
x_{n})\in\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,n)$ (see section 6 for the
notation) induces an isomorphism:

(9) $\displaystyle\mathrm{ev}_{x_{1}\otimes\cdots\otimes
x_{n}}:\mathrm{Nat}(\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),G)\to G(n).$

This isomorphism is natural with respect to
$n\in\mathsf{Cat}_{\mathfrak{Ass}^{u}}$.

Let $F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$. For
$G=\mathcal{E}^{\ast}F$ (see Definition 6.2 for $\mathcal{E}$), note that
$\mathrm{Nat}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F)$ is
naturally embedded into
$\mathrm{Nat}(\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),G)$. We interpret the
subspace via the Yoneda isomorphism in Theorem 9.14.

In the following statements, we use the notation of the generators
$\Delta,\nabla,\eta,\epsilon,s$ of the category $\mathsf{gr}^{\mathsf{op}}$
(see section 6.2).

######  Lemma 9.13.

Let $F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ and
$n\in\mathds{N}$. For
$\xi\in\mathrm{Nat}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F)$, we
have,

$\xi(x_{1}\otimes\cdots\otimes x_{n})\in\left(\mathcal{P}(F)\right)(n).$

###### Proof.

It suffices to prove that $\xi(x_{1}\otimes\cdots\otimes
x_{n})\in\mathrm{Ker}\left(F(\mathrm{id}_{m-1}\ast\theta\ast\mathrm{id}_{n-m})\right)$
for any $1\leq m\leq n$ by Proposition 9.5. For simplicity, let
$K_{n}{:=}~{}{}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-)$ be the
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$-module. Since $\xi$ preserves
$\mathds{k}\mathsf{gr}^{\mathsf{op}}$-actions, we have,

$(F(\Delta\ast\mathrm{id}_{n-1}))(\xi(x_{1}\otimes\cdots\otimes
x_{n}))=\xi((K_{n}(\Delta\ast\mathrm{id}_{n-1}))(x_{1}\otimes\cdots\otimes
x_{n})).$

By definition, we have

$(K_{n}(\Delta\ast\mathrm{id}_{n-1}))(x_{1}\otimes\cdots\otimes
x_{n})=x_{1}\otimes 1\otimes x_{2}\otimes\cdots x_{n}+1\otimes x_{1}\otimes
x_{2}\otimes\cdots\otimes x_{n},$

which leads to the above claim for $m=1$. One can proceed similarly for any
$m$. ∎

We now prove that the
$(\mathds{k}\mathsf{gr}^{\mathsf{op}},\mathsf{Cat}_{\mathfrak{Lie}})$-bimodule
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$ represents the functor
$\mathcal{P}:\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})\to\mathcal{F}_{\mathfrak{Lie}}$.
In the following statement, recall that $\mathcal{P}(F)$ is a left
$\mathsf{Cat}_{\mathfrak{Lie}}$-module by Proposition 9.12. The assignment
$n\mapsto\mathrm{Nat}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F)$
gives a left $\mathsf{Cat}_{\mathfrak{Lie}}$-module arising from the right
$\mathsf{Cat}_{\mathfrak{Lie}}$-module structure of
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}$.

######  Theorem 9.14.

Let $F\in\mathcal{F}(\mathsf{gr}^{\mathsf{op}};\mathds{k})$ and
$n\in\mathds{N}$. The evaluation $\mathrm{ev}_{x_{1}\otimes\cdots\otimes
x_{n}}$ in (9) factors through an isomorphism which is natural with respect to
$n\in\mathsf{Cat}_{\mathfrak{Lie}}$:

$\mathrm{Nat}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F)\cong\mathcal{P}(F)(n).$

###### Proof.

The map $\mathrm{ev}_{x_{1}\otimes\cdots\otimes x_{n}}$ factors into
$h_{n}:\mathrm{Nat}(\mathsf{Cat}_{\mathfrak{Ass}^{u}}(n,-),F)\to\mathcal{P}(F)(n)$
due to Lemma 9.13. Moreover, $h_{n}$ is injective since
$\mathrm{ev}_{x_{1}\otimes\cdots\otimes x_{n}}$ is an isomorphism. The
naturality of $h_{n}$ with respect to $n\in\mathsf{Cat}_{\mathfrak{Lie}}$
follows from that of (9). Recall the map $I_{n}$ in Theorem 9.6. By
definitions, we have $h_{n}\circ I_{n}$ is the identity. Hence, $h_{n}$ is
surjective. ∎

######  Corollary 9.15.

For $\mathds{k}$ a field of characteristic zero, a
$\mathsf{gr}^{\mathsf{op}}$-module is primitive if and only if it is analytic.

###### Proof.

Let $F$ be a $\mathsf{gr}^{\mathsf{op}}$-module. By Proposition 9.9, if $F$ is
primitive, then it is analytic. Conversely, if $F$ is analytic, then the
evaluation gives
${}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}}\otimes_{\mathsf{Cat}_{\mathfrak{Lie}}}\mathrm{Nat}({}_{\Delta}\mathsf{Cat}_{\mathfrak{Ass}^{u}},F)\cong
F$ by [14] since the characteristic is zero. We take a composition with the
isomorphism in Theorem 9.14 to deduce that $\sum_{k\in\mathds{N}}I_{k}$ is
surjective. Therefore, $F$ is primitive. ∎

######  Remark 9.16.

For the ground field $\mathds{k}$ of positive characteristic, there exist
analytic and non-primitive $\mathsf{gr}^{\mathsf{op}}$-modules. Some examples
arise by combining Theorem 8.2 and Remark 10.16.

###  10\. Application to primitive exponential
$\mathsf{gr}^{\mathsf{op}}$-modules

In this section, we prove an equivalence between primitive cocommutative Hopf
algebras and primitive exponential $\mathsf{gr}^{\mathsf{op}}$-modules (see
Definition 9.8 for the definition of primitive
$\mathsf{gr}^{\mathsf{op}}$-modules).

####  10.1. Primitive filtration and coradical filtration

In this section, we give a comparison of the coradical filtration and the
primitive filtration of bialgebras which is reviewed below.

Let $\mathds{k}$ be a commutative unital ring. For a bialgebra $H$ over
$\mathds{k}$, its underlying coaugmented coalgebra induces the coradical
filtration $P_{\bullet}(H)$. The algebra structure gives a submodule
$P_{1}(H)^{n}\subset H$ generated by multiples of $n$ elements in $P_{1}(H)$.
It yields the primitive filtration of $H$:

$\mathds{k}=P_{1}(H)^{0}\subset\cdots\subset P_{1}(H)^{n}\subset
P_{1}(H)^{n+1}\subset\cdots\subset H.$

The bialgebra $H$ is primitive (or primitively generated) if
$H=\bigcup_{n\in\mathds{N}}P_{1}(H)^{n}$.

We first prove that the coradical filtration includes the primitive
filtration, properly in general. We also discuss some conditions implying that
the two filtrations coincide. In particular, for $\mathds{k}$ a field of
characteristic zero, if the bialgebra is either cocommutative, primitive or
finite-dimensional, then the filtrations are equal.

######  Proposition 10.1.

Let $H$ be a bialgebra over $\mathds{k}$. The coradical filtration contains
the primitive filtration of $H$:

$P_{1}(H)^{n}\subset P_{n}(H).$

###### Proof.

The result follows from an iterated application of Proposition 4.8. ∎

We say that a bialgebra is good if its coradical filtration is equal to the
primitive filtration.

######  Example 10.2.

We suppose that $m\cdot 1_{\mathds{k}}\in\mathds{k}$ is not a zero divisor for
any nonzero integer $m$. An application of the results in section 4.3.1
implies that polynomial bialgebras are good.

######  Example 10.3.

A bialgebra is not good in general. Consider the shuffle bialgebra
$H=\mathrm{Sh}(V)$ (see section 4.3.2). Note that for
$v_{1},v_{2},\cdots,v_{k}\in V\subset\mathrm{Sh}(V)$, the multiplication
$v_{1}\shuffle v_{2}\shuffle\cdots\shuffle v_{k}$ lies in the $k$-th symmetric
tensor module contained in $T(V)=\mathrm{Sh}(V)$, i.e. the invariant subspace
under the symmetric group $\Sigma_{k}$ action. Hence, if $V$ is a free
$\mathds{k}$-module with $\dim V\geq 2$, then $P_{1}(H)^{n}\neq P_{n}(H)$ for
$n\geq 2$ by Proposition 4.15. Furthermore, one can prove directly that, for
$\mathds{k}$ a field of positive characteristic, $\mathrm{Sh}(V)$ is not good
even if $\dim V=1$.

######  Example 10.4.

If $\mathds{k}$ is a field of characteristic zero, then finite-dimensional
bialgebras are good by Theorem 4.18.

We now suppose that the ground ring $\mathds{k}$ is a field. Below, we give
general properties satisfied by good bialgebras (see Theorem 10.8).

######  Proposition 10.5.

If $\mathds{k}$ is a field, then we have
$\delta_{H}^{n}(P_{n}(H))\subset\mathrm{Prim}(H)^{\otimes n}$.

###### Proof.

Recall the notations in section 4.1. By definitions, we have
$(\mathrm{id}_{H}^{\otimes(i-1)}\otimes\varepsilon_{H}\otimes\mathrm{id}_{H}^{\otimes(n-i)})\circ\delta_{H}^{n}=0$
for $1\leq i\leq n$. This leads to the following by applying Lemma 4.9:

$\delta_{H}^{n}(P_{n}(H))\subset\bigcap_{i=1}^{n}\mathrm{Ker}(\mathrm{id}_{H}^{\otimes(i-1)}\otimes\varepsilon_{H}\otimes\mathrm{id}_{H}^{\otimes(n-i)})=\overline{H}^{\otimes
n}$

Hence, to prove our claim, it suffices to show that
$(\mathrm{id}_{\overline{H}}^{\otimes(i-1)}\otimes\overline{\Delta}_{H}\otimes\mathrm{id}_{\overline{H}}^{\otimes(n-i)})\circ\delta_{H}^{n}(x)=0$
for $x\in P_{n}(H)$ by Lemma 4.9. Note that Proposition 4.4 implies that

$(\mathrm{id}_{\overline{H}}^{\otimes(i-1)}\otimes\overline{\Delta}_{H}\otimes\mathrm{id}_{\overline{H}}^{\otimes(n-i)})\circ\delta_{H}^{n}=\delta_{H}^{n+1}.$

We now use the assumption $x\in P_{n}(H)$, equivalently
$\delta_{H}^{n+1}(x)=0$. ∎

By Proposition 10.5, the map $\delta_{H}^{n}:H\to H^{\otimes n}$ factors
through $f_{n}:P_{n}(H)\to\mathrm{Prim}(H)^{\otimes n}$. By Proposition 10.1,
the iterated multiplication $\nabla^{(n)}:H^{\otimes n}\to H$ induces a linear
map $g_{n}:P_{1}(H)^{\otimes n}\to P_{n}(H)$. Let
$h_{n}:\mathrm{Prim}(H)^{\otimes n}\to P_{n}(H)$ be the restriction of
$g_{n}$. Let $s_{n}:\mathrm{Prim}(H)^{\otimes n}\to\mathrm{Prim}(H)^{\otimes
n}$ be the linear extension of $s_{n}(a_{1}\otimes a_{2}\otimes\cdots\otimes
a_{n}){:=}\sum_{\sigma\in\Sigma_{n}}a_{\sigma(1)}\otimes
a_{\sigma(2)}\otimes\cdots\otimes a_{\sigma(n)}$.

######  Lemma 10.6.

For $n\in\mathds{N}$, we have $f_{n}\circ h_{n}=s_{n}$.

###### Proof.

Let $a_{1},a_{2},\cdots,a_{n}\in\mathrm{Prim}(H)$.

$\displaystyle(f_{n}\circ h_{n})(a_{1}\otimes a_{2}\otimes\cdots\otimes
a_{n})$ $\displaystyle=\delta^{n}(a_{1}a_{2}\cdots a_{n}),$
$\displaystyle=\delta_{H}^{1}(a_{1})\shuffle\delta_{H}^{1}(a_{2})\shuffle\cdots\shuffle\delta_{H}^{1}(a_{n})~{}(\mathrm{by}~{}\mathrm{Lemma}~{}\ref{202209010909}),$
$\displaystyle=a_{1}\shuffle a_{2}\shuffle\cdots\shuffle
a_{n}~{}(\mathrm{by}~{}\varepsilon(a_{j})=0),$

By definitions, we have $a_{1}\shuffle a_{2}\shuffle\cdots\shuffle
a_{n}=\sum_{\sigma\in\Sigma_{n}}a_{\sigma(1)}\otimes
a_{\sigma(2)}\otimes\cdots\otimes a_{\sigma(n)}$. ∎

In the following, consider the idempotent $e_{H}$ on $P_{1}(H)$ defined by
$e_{H}(x)=x-\varepsilon_{H}(x)\cdot 1_{H}$.

######  Lemma 10.7.

Let $W$ be the kernel of $e_{H}^{\otimes n}$. Then we have $g_{n}(W)\subset
P_{n-1}(H)$.

###### Proof.

Note that $W$ is the subspace generated by $a_{1}\otimes
a_{2}\otimes\cdots\otimes a_{n}\in P_{1}(H)^{\otimes n}$ such that
$a_{j}\in\mathds{k}1_{H}$ for some $1\leq j\leq n$. Hence, by Proposition
10.1, we obtain $g_{n}(W)\subset P_{n-1}(H)$ . ∎

######  Theorem 10.8.

Let $H$ be a bialgebra over a field $\mathds{k}$. Then the first part below
implies the second part.

  * •

The bialgebra $H$ is good.

  * •

For $n\in\mathds{N}$, we have