# The structure of simply colored coalgebras

Yang Mo  Department of Mathematics, Purdue University, West Lafayette, IN
47907

###### Abstract.

A simply colored coalgebra is a coassociative counital coalgebra $C$ over an
arbitrary ring $R$, which can be decomposed into a direct sum of two
$R$-modules: one generated by set-like elements and another consisting of
conilpotent elements. Our main result is the equivalence between simply
colored coalgebras over a field $k$ and pointed coalgebras with a choice of
splitting of its coradical. Additionally, we also prove that the category of
simply colored coalgebras is both complete and cocomplete.

###### 2020 Mathematics Subject Classification:

Primary 16T15; Secondary 18M70

## Introduction

The aim of this paper is to delve into the subtleties in [KM22a] and to extend
its results. To begin, we briefly discuss the initial motivation for this
collaborative work with R. Kaufmann.

In [MP12], Peter May constructs an component coalgebra in the following sense:
Let $C_{g}=Rg\oplus\overline{C}_{g}$ and
$\overline{C}_{g}=\\{x|\Delta(x)=x\otimes g+x^{\prime}\otimes
x^{\prime\prime}+g\otimes x,\deg(x^{\prime})>0\text{ and
}\deg(x^{\prime\prime})>0\\}$ and $g$ is a set-like 111In this paper, we adopt
the unconventional term “set-like” instead of the commonly used term “group-
like” found in [MP12] [Mon93], [Rad12], [Swe69], in order to maintain
consistency. element in $C_{g}$,. In essence, $C_{g}$ is a connected graded
coalgebra over $R$ that lacks primitive elements and is indexed by the unique
set-like element in $g$. Given such a family $\\{C_{g}\\}$ of them, we define
the component coalgebra as the direct sum of coalgebras over the family, i.e.

$C=\oplus_{g}C_{g}$

This construction is motivated from the study of loop spaces. It should be
noted that each $g$ is also a set-like element in the direct sum $C$. A
component bialgebra refers to a bialgebra in which its coalgebra is a
component coalgebra. If the bialgebra is group-like, meaning every set-like
element is invertible under the multiplication, then the bialgebra becomes a
Hopf algebra.

At a certain point in our study of Hopf algebras obtained from Feynman
categories, we realized the need for a more general formulation. In a broader
sense, the components should be indexed by two potentially distinct set-like
elements, denoted as $g$ and $h$, within $C$. In other words, we have

$C=\oplus_{g,h}C_{g,h}$

The good news is that pointed coalgebras (not necessarily graded) fulfill the
required property, as shown in [Rad12]. Although each $C_{g,h}$ may not be a
coaugmented coalgebra, the existence of an antipode for a pointed coalgebra
$C$ remains the same: every set-like element is invertible. However, the bad
news is that the notion of pointed coalgebras is not well-defined when the
ground ring is not a field. This difficulty arises from the fact that the
notion of subcoalgebras may not be well-behaved.

To address this challenge, we introduce the concept of a simply colored
coalgebra. It is defined as a triple $(C,G,\delta)$, where $C$ is a coalgebra,
$G$ is a set of set-like elements in $C$ (this will be called colors), and
$\delta:C\to C[G]$ is a coalgebra retraction that maps $C$ onto the set-like
coalgebra $C[G]$. This decomposition allows us to express $C$ as the direct
sum $C=C[G]\oplus\ker{\delta}$.

The simply colored coalgebra can be defined over any commutative rings with
unity, such as $\mathbb{Z}$. Furthermore, we demonstrate that any non-
negatively graded coalgebra with its zero degree freely spanned by set-like
elements can be regarded as a simply colored coalgebra. In particular, a
connected graded coalgebra is simply colored.

This framework enables us to generalize the results presented in [GCKT20a] and
[GCKT20b]. For further details, We refer readers to [KM22a], which uses
different terminology and introduces additional variants.

In this paper, we present the following results to expand upon the topics
discussed in [KM22a]. To be more precise,

  1. (1)

The reduced comultiplication can be defined more simply in terms of a
retraction (Theorem 2.5). This allows us to extend [KM22a, Lemma 2.19] to any
comonoid in an monoidal abelian category. This demonstrates that the use of
set-like coalgebras is not necessary in the argument and that the
coassociativity of the reduced comultiplication can be proved without it.

  2. (2)

A pointed coalgebra with a splitting is a pointed coalgebra $C$ equipped with
an additional coalgebra homomorphism $\delta$ onto its coradical, such that
$C$ can be decomposed as $C=C_{0}\oplus\ker\delta$. We show that a simply
colored coalgebra over a field is equivalent to a pointed coalgebra with a
splitting. More precisely,

######  Theorem (2.16).

Let $k$ be a field,

    1. (a)

Every simply colored coalgebra $(C,G,\delta)$ over $k$ can be viewed as a
pointed coalgebra with a splitting $(C,\delta)$.

    2. (b)

Conversely, if $(C,\delta)$ is a pointed coalgebra with a splitting, it
naturally gives rise to a simply colored coalgebra structure $(C,G,\delta)$,
where $G$ is the set of all set-like elements in $C$.

In particular, it is worth noting that coaugmented conilpotent coalgebras can
be precisely characterized as pointed coalgebras with a splitting where $G$
consists of a single element. While this fact is likely known in the
literature (see, for example, [HL22]), the detailed argument supporting this
characterization is not commonly found.

  3. (3)

In the last section, we show the category of pointed coalgebras with a
splitting is both complete and cocomplete. It is worth mentioning that this
result is independently proven in [HL22]. However, we will present a different
approach that does not rely on pseudocompact algebras.

We should highlight that the characterization of reduced comultiplication (as
discussed in Theorem 2.5 and Section 2.1.2) enables us to define a ”pointed-
coalgebra-like” cooperad. It provides a generalization of the existing notion
of ”conilpotent coaugmented-like” conilpotent coaugmented cooperads. As a
consequence, it could potentially pave the way for a broader generalization of
Koszul duality of operads, similar to the approach taken in [HL22], which may
serve as a subject for future research.

### Conventions and notations

We assume that readers are familiar with the definitions and basic examples of
coalgebras, bialgebras, and Hopf algebras. Throughout this paper, we work over
a unital commutative associative ring $R$ or a field $k$. Given the
commutativity of $R$, an $R$-module naturally becomes a bimodule. The tensor
product $\otimes$ is considered to be over the ground ring $R$, unless
otherwise stated. We use $\rho_{l}$ (and $\rho_{r}$) to denote a left coaction
(and right coaction). The symbols $\omega_{l}$ and $\omega_{r}$ are reserved
for coactions induced via retraction, which we will discuss later in the
paper. It $M=N\oplus N^{\prime}$ can be decomposed, then the notation
$\pi_{N}:M\to N$ is used to represent the canonical projection onto the
submodule $N$. If the submodule is clear from the context, we omit the
subscript $B$ from $\pi_{N}$. Finally, in an unconventional manner, we will
use the term set-like elements to refer to what is commonly known as group-
like elements in most literature.

### Acknowledgement

The author would like to express gratitude to R. Kaufmann, M. Rivera, and L.
Guo for their valuable suggestions and insightful comments, which greatly
contributed to the improvement of the original ideas. I would like to extend
my special thanks to the organizers of the AMS special session on higher
structures in topology, geometry, and physics, where I had the opportunity to
present a preliminary version of this work.

##  1\. Preliminaries

Most of the material in this section can be found in textbooks such as
[Rad12], [Mon93], and [Swe69]. First, we will provide a brief overview of the
fundamental concepts

######  Definition 1.1.

  1. (1)

A cosimple coalgebra is a coalgebra whose subcoalgebras are exactly zero and
itself.

  2. (2)

A coradical of coalgebra is a sum of all its cosimple subcoalgbras.

  3. (3)

A cosemisimple coalgebra is a coalgebra which equals to its own coradical.

According to the fundamental theorem of coalgebras, any cosimple coalgebra
must be finite-dimensional. Its dual algebra corresponds precisely to a simple
algebra (see [Mon93], Chapter 1). While it is commonly referred to as a simple
coalgebra in most literature, we will retain the ”co-” prefix for consistency.
Additionally, it can be proven that the sum of cosimple coalgebras must be a
direct sum (see for example [Mon93]).

######  Example 1.2.

[Rad12] Here is an cosimple coalgebra over a field. We consider the (standard)
matrix coalgebra $M^{c}_{n}(k)$ over a field $k$. As a vector space, it is
spanned by elements $e_{ij}$ endowed with the following coalgebra structure

$\Delta(e_{ij})=\sum_{1\leq t\leq n}e_{it}\otimes e_{tj}$
$\epsilon(e_{ij})=\delta_{ij}$

$M^{c}_{n}(k)$ as a vector space is finite dimensional, so its linear dual
$(M^{c}_{n}(k))^{*}$ is a algebra over $k$. In fact, one can check it is
isomorphic to the standard matrix algebra $M_{n}(k)$ as

$e^{*}_{ij}e^{*}_{kl}=\delta_{jk}e^{*}_{il}$

Since the matrix algebra $M_{n}(k)$ is simple (the only ideal is 0 and
itself), by duality the only subcoalgebra is 0 and itself. By definition,
$M^{c}_{n}(k)$ is a simple coalgebra. $M^{c}_{n}(k)$ is pointed if and only if
$n=1$, since the dimension of simple pointed coalgebra can only have dimension
of 1 and by comparison the dimension of the matrix coalgebra $M^{c}_{n}(k)$ is
$n^{2}$ ($n^{2}>1$ for $n>1$).

######  Definition 1.3.

Let $C$ be a coalgebra over a field $k$ and $X,Y$ subspaces. Define the wedge
product $X\wedge Y$ of $X$ and $Y$ to be the kernel of the composition

${C}$${{C\otimes C}}$${{C/X\otimes
C/Y}}$$\scriptstyle{\pi\otimes\pi}$$\scriptstyle{\Delta}$

There are two important observations from the definition, which were left as
exercises in [Swe69].

######  Proposition 1.1.

[Swe69]

  1. (1)

$(X\wedge Y)\wedge Z=X\wedge(Y\wedge Z)$

  2. (2)

$X\wedge Y=\Delta^{-1}(C\otimes Y+X\otimes C)$

The first assertion follows from coassociativity $(\Delta\otimes
id)\circ\Delta=(id\otimes\Delta)\circ\Delta$ that the kernel of

${C}$${{C\otimes C\otimes C}}$${{C/X\otimes C/Y\otimes
C/Z}}$$\scriptstyle{\pi\otimes\pi\otimes\pi}$$\scriptstyle{(\Delta\otimes
id)\circ\Delta}$

is the same as the kernel of

${C}$${{C\otimes C\otimes C}}$${{C/X\otimes C/Y\otimes
C/Z}}$$\scriptstyle{\pi\otimes\pi\otimes\pi}$$\scriptstyle{(id\otimes\Delta)\circ\Delta}$

As a result, it make senses to use the notation
$\wedge^{n}X=(\wedge^{n-1}X)\wedge X$ without attention to the order. The
second assertion follows form the following linear algebra exercise.

######  Lemma 1.4.

If $f:V_{1}\to W_{1}$ and $g:V_{2}\to W_{2}$ are linear maps between vector
spaces. Then

$\ker(f\otimes g)=\ker(f)\otimes V_{2}+V_{1}\otimes\ker(g)$

###### Proof.

Every monomorphism of vector spaces splits. Thus the image of $f\otimes g$ is
$f(V_{1})\otimes f(V_{2})\subset W_{1}\otimes W_{2}$. Without loss of
generality, assume both $f$ and $g$ surjective. In this case, since the short
exact sequence splits, $f$ induces a splitting as $V_{1}=\ker f\oplus
V^{\prime}_{1}$ s.t. $V^{\prime}_{1}\cong W_{1}$ via $f$. Similarly, $g$
induces a splitting as $V_{2}=\ker g\oplus V^{\prime}_{2}$ s.t.
$V^{\prime}_{2}\cong W_{2}$ via $f$ Then

$V_{1}\otimes V_{2}=\ker f\otimes\ker g\oplus\ker f\otimes
V^{\prime}_{2}\oplus V^{\prime}_{1}\otimes\ker g\oplus V^{\prime}_{1}\otimes
V^{\prime}_{2}$

Note if $s\circ f=id$ and $t\circ g=id$, then $(s\otimes t)\circ(f\otimes
g)=(s\circ f)\otimes(t\circ g)=id\otimes id=id$. This implies given the
splitting we chose for $f$ and $g$, $im(f\otimes g)=V^{\prime}_{1}\otimes
V^{\prime}_{2}$ and

$\ker(f\otimes g)=\ker f\otimes\ker g\oplus\ker f\otimes V^{\prime}_{2}\oplus
V^{\prime}_{1}\otimes\ker g=\ker(f)\otimes V_{2}+V_{1}\otimes\ker(g)$

∎

Let $C_{0}$ be the coradical for an coalgebra $C$. It is known (pp.125
[Rad12]) the sequence of subspaces $C_{n}=\wedge^{n}C_{0}$ forms a exhaustive
coalgebra filtration; it is called coradical filtration of $C$.

######  Definition 1.5.

A pointed coalgebra over a field is defined to be a coalgebra in which every
cosimple subcoalgebras is one dimensional.

Alternatively, a pointed coalgebra can be defined as a coalgebra whose
coradical is generated freely by set-like elements, since every one-
dimensional coalgebra is necessarily generated by a set-like element. It is
essential to note that the set of set-like elements is linearly independent
(refer to Lemma 2.1.12 in [Rad12] for further details).

######  Example 1.6.

  1. (1)

The simplest form of a pointed coalgebra is given by $kg$, where $g$ is a set-
like element. By duality, every one-dimensional cosimple coalgebra is
isomorphic to such a coalgebra $kg$. The element $g$ in this case is referred
to as a set-like element. Furthermore, the sum of one-dimensional cosimple
coalgebras is also pointed.

  2. (2)

According to [Swe69], a cocommutative coalgebra over an algebraically closed
field is pointed.

  3. (3)

As stated in [Rad12], a graded coalgebra with its degree zero component
spanned by group-like elements is pointed. This follows from the observation
that the coradical must be contained within the degree one component
(Proposition 4.1.2 in [Rad12]).

  4. (4)

Every path coalgebra associated with a quiver is a pointed coalgebra. This can
be verified either directly or by utilizing property (2) and noting that the
path coalgebra is graded, with its zero component being spanned by set-like
elements (vertices). The path coalgebra is coradically graded. Further
information can be found in [Chi03].

We know that the path coalgebra associated with a quiver is pointed. However,
the converse remains an open problem. Nonetheless, we can establish the
following weaker converse:

######  Proposition 1.2.

[D97] Let $C$ be a pointed coalgebra. Then there exists a quiver such that $C$
is isomorphic to an admissible subcoalgebra of the path coalgebra of the
quiver.

It has been shown in [Mon93] that every pointed coalgebra admits a splitting
of its coradical. However, the choice of splitting is not unique. To record
the information of the splitting $\delta$, we come up with the following
notion.

######  Definition 1.7.

A pointed coalgebra with a splitting is a pair $(C,\delta)$ consisting of a
pointed coalgebra $C$ and a surjective coalgebra homomorphism $\delta$ from
$C$ onto its coradical $C_{0}$.

######  Remark 1.8.

The splitting of coradical is a part of structure, and it is not property of
pointed coalgebras.

Note such a $\delta$ induces a splitting $C=C_{0}\oplus I$ where
$I=\ker\delta$. Conversely, if we are given a pointed coalgebra with
$C=C_{0}\oplus I$ where $I$ is a coideal, then the canonical projection
$\pi_{C_{0}}:C\to C_{0}$ naturally provides a splitting. Hence, we
occasionally use the notation $(C,I)$ to denote a pointed coalgebra with a
splitting.

Lastly, we demonstrate that the category of pointed coalgebras is complete and
cocomplete. We begin by presenting the following observations:

######  Lemma 1.9.

  1. (1)

Let $f:C\to D$ be a surjective coalgebra homomorphism. If $I$ is a coideal in
$C$, then $f(I)$ is also a coideal in $D$.

  2. (2)

If $f:C\to D$ is a surjective coalgebra homomorphism, where $C$ is pointed,
then $D$ is also pointed, and $f(C_{0})=D_{0}$.

  3. (3)

The category of coalgebras over a field is complete and cocomplete, meaning it
has both small limits and small colimits.

###### Proof.

  1. (1)

A direct check: $\Delta(f(I))=(f\otimes f)\Delta(I)\subset(f\otimes
f)(I\otimes C+C\otimes I)\subset f(I)\otimes D+D\otimes f(I)$ and
$\epsilon_{D}(f(I))=\epsilon_{C}(I)=0$.

  2. (2)

First, $D_{0}\subset f(C_{0})$ if $f$ is surjective (Proposition 4.1.7
[Rad12]). Second, if $g$ is set-like in $C$, then $f(g)$ is also set-like and
spans a simple subcoalgebra in $D$. Therefore, $f(C_{0})\subset D_{0}$.

  3. (3)

There are multiple approaches to prove this statement. One possible approach
is discussed in [Ago11b] and [Ago11a].

∎

######  Theorem 1.10.

  1. (1)

Every pointed coalgebra is the filtered colimit of its finite-dimensional
pointed subcoalgebra.

  2. (2)

The embedding $\iota$ from the category of pointed coalgebras to the category
of coalgebras admits a right adjoint functor $\Phi$.

  3. (3)

The category of pointed coalgebras is bicomplete.

###### Proof.

  1. (1)

Any subcoalgebra of a pointed coalgebra is also pointed, and this can be
proven using the same proof as the fundamental theorem for coalgebras.

  2. (2)

We have an explicit construction given by the formula
$\Phi(C)=\sum_{\alpha}C_{\alpha}$, where $C_{\alpha}$ runs over all pointed
subcoalgebras of $C$. Note that $\Phi(C)$ is not only a subcoalgebra of $C$
but also a pointed coalgebra. This is because, by proposition 3.4.3 in
[Rad12], every simple subcoalgebra in $\Phi(C)$ must be in one of the
$C_{\alpha}$ and thus has dimension one. Moreover, the image of a pointed
coalgebra under a coalgebra homomorphism is also a pointed coalgebra. This
follows from the fact that the image is a subcoalgebra and the map is
surjective onto its image (we can use the previous lemma in this section). A
map $f:C\to D$ induces a map $\Phi(f):\Phi(C)\to\Phi(D)$.

Now We define the unit $\eta_{X}:X\to(\Phi\iota)(X)$ to be the identity on a
pointed coalgebra $X$, and the counit $\varepsilon_{Y}:(\iota\Phi)(Y)\to Y$ as
the natural inclusion of the pointed subcoalgebra $\Phi(Y)$ in a coalgebra
$Y$. The maps $\eta$ and $\varepsilon$ are natural transformations. We need to
show that the triangular identities hold.

It is easy to check $(\Phi\iota\Phi)(X)=\Phi(X)$ and
$(\iota\Phi\iota)(Y)=\iota(Y)$. In other words, the following triangular
identities hold

$(\Phi(\varepsilon_{X})\circ\eta_{\Phi(X)})(\Phi(X))=\Phi(X)$
$(\varepsilon_{\iota(X)}\circ\iota(\eta_{X}))(\iota(X))=\iota(X)$

It follows from [Lan10] that there is an adjunction such that $\Phi$ is right
adjoint to $\iota$.

  3. (3)

(2) shows that the category of pointed coalgebras is a coreflective
subcategory of the category of coalgebras. Since the category of coalgebras is
complete and cocomplete, it follows from the corollaries in [HS71] that the
category of pointed coalgebras is also complete and cocomplete.

∎

##  2\. Algebraic perspective

In this section, we explore the algebraic structure of simply colored
coalgebras. We start by examining the general simply colored comonoid in any
monoidal abelian category. we show reduced comultiplication is coassociative.
Subsequently, we narrow our scope to simply colored coalgebras in the category
of $R$-modules. We prove the equivalence between simply colored coalgebras
over a field and pointed coalgebras with a splitting.

###  2.1. Simply colored comonoid

In this subsection, we extend the constructions and proofs presented in
Section 2.3 of [KM22a] to a monoidal category. We consider $\mathcal{C}$ to be
a small monoidal category $(\mathcal{C},\otimes,\mathds{1})$ that also
possesses an abelian structure. It is important to note that, at this stage,
we do not assume the compatibility of the monoidal structure with the abelian
structure.

######  Remark 2.1.

According to Mac Lane’s Coherence Theorem, the associators and unitors can be
assumed to be strict in the calculations. Additionally, the tensor product of
multiple objects can be specified without ambiguity. Hence, in this paper, we
assume any monoidal category is strict without loss of generality.

####  2.1.1. Reduced comultiplication

Next, let $\delta:\mathbf{C}\to\mathbf{S}$ be a comonoid retraction between
two comonoids $\mathbf{C}$ and $\mathbf{S}$ in the category. This means that
$\delta$ is the left inverse of the monomorphism of comonoids
$\iota:\mathbf{S}\to\mathbf{C}$. Since the category $\mathcal{C}$ is abelian,
we have $\mathbf{C}\cong\mathbf{S}\oplus\mathbf{I}$, where $\mathbf{I}$ is the
kernel of $\delta$. We identify $\mathbf{S}$ with the image of $\iota$, and we
consider $\delta$ as the composition $\iota\circ\delta$ (which becomes an
idempotent). Consequently, $\mathbf{S}$ is a well-defined subcomonoid.

######  Remark 2.2.

The notion of a subcomonoid is not generally well-defined. While
$\Delta(\iota(\mathbf{S}))\subset\iota(\mathbf{S})\otimes\iota(\mathbf{S})$,
it may not be a subobject of $\mathbf{C}\otimes\mathbf{C}$. However, in our
case, we require the map $\iota:\mathbf{S}\to\mathbf{C}$ to be a split
monomorphism of comonoids. The monoidal product of two split monomorphisms is
also a split monomorphism, thanks to the interchange law. Hence,
$\iota(\mathbf{S})$ is indeed a well-defined subcomonoid of $\mathbf{C}$ that
is isomorphic to $\mathbf{S}$.

Consequently, the following equations make sense.

(2.1) $\delta\circ\delta=\delta$ (2.2) $\epsilon\circ\delta=\epsilon$ (2.3)
$\Delta\circ\delta=(\delta\otimes\delta)\circ\Delta$

Now we will construct the following $\mathbf{S}$-bicomodule on $\mathbf{C}$
via $\omega_{l}=(\delta\otimes id)\circ\Delta$ and
$\omega_{r}=(id\otimes\delta)\circ\Delta$.

######  Proposition 2.1.

$\omega_{l}=(\delta\otimes id)\otimes\Delta$ and
$\omega_{r}=(id\otimes\delta)\otimes\Delta$ induces an $\mathbf{S}$-bicomodule
on $\mathbf{C}$

###### Proof.

We need to check the following equation for the right comodule structure.

(2.4) $(\omega_{r}\otimes id)\circ\omega_{r}=(id\otimes\Delta)\circ\omega_{r}$

Do the computation:

$\displaystyle(\omega_{r}\otimes id)\circ\omega_{r}$
$\displaystyle=(((id\otimes\delta)\circ\Delta)\otimes
id)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(((id\otimes\delta)\circ\Delta)\otimes\delta)\circ\Delta$
$\displaystyle=((id\otimes\delta)\otimes\delta)\circ(\Delta\otimes
id)\circ\Delta$
$\displaystyle=(id\otimes(\delta\otimes\delta))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes\Delta)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(id\otimes\Delta)\circ\omega_{r}$

Note we have used the coassociativity and the fact $\delta$ is a comonoid
homomorphism.

(2.5) $(id\otimes\epsilon)\circ\omega_{r}=id$

Do the computation again:

$\displaystyle(id\otimes\epsilon)\circ\omega_{r}$
$\displaystyle=(id\otimes\epsilon)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(id\otimes(\epsilon\circ\delta))\circ\Delta$
$\displaystyle=(id\otimes\epsilon)\circ\Delta$ $\displaystyle=id$

Here in $(\epsilon\circ\delta)=\epsilon$ we use the fact $\delta$ is a
comonoid retraction. With similar computation which we ignore, we show the
following equations for the left comodule structure.

(2.6) $(id\otimes\omega_{l})\circ\omega_{l}=(\Delta\otimes id)\circ\omega_{l}$
(2.7) $(\epsilon\otimes id)\circ\omega_{l}=id$

Finally, for the bicomodule structure, we need to compute

(2.8) $(\omega_{l}\otimes
id)\circ\omega_{r}=(id\otimes\omega_{r})\circ\omega_{l}$
$\displaystyle(\omega_{l}\otimes id)\circ\omega_{r}$
$\displaystyle=(((\delta\otimes id)\circ\Delta)\otimes
id)\circ(id\otimes\delta)\circ\Delta$ $\displaystyle=((\delta\otimes
id)\otimes id)\circ(\Delta\otimes id)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=((\delta\otimes id)\otimes
id)\circ(\Delta\otimes\delta)\circ\Delta$ $\displaystyle=((\delta\otimes
id)\otimes id)\circ((id\otimes id)\otimes\delta)\circ(\Delta\otimes
id)\circ\Delta$ $\displaystyle=(\delta\otimes(id\otimes
id))\circ(id\otimes(id\otimes\delta))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(\delta\otimes(id\otimes
id))\circ(id\otimes((id\otimes\delta)\circ\Delta))\circ\Delta$
$\displaystyle=(\delta\otimes((id\otimes\delta)\circ\Delta))\circ\Delta$
$\displaystyle=(id\otimes((id\otimes\delta)\circ\Delta))\circ(\delta\otimes
id)\circ\Delta$ $\displaystyle=(id\otimes\omega_{r})\circ\omega_{l}$

∎

######  Remark 2.3.

In the theory of commutative rings, if $B$ is an $A$-algebra, it follows the
multiplication of $B$ is compatible with action of $A$. Analogously, the
retraction-induced bicomodule is compatible with the comonoid structure. The
compatibility means the following equations.

(2.9) $(\omega_{r}\otimes id)\circ\Delta=(id\otimes\omega_{l})\circ\Delta$

If the category $\mathcal{C}$ is abelian, this equation means the
comultiplication factors through “cotensor” over $\mathbf{S}$. We will define
what we mean by cotensor later. Now we compute:

$\displaystyle(\omega_{r}\otimes id)\circ\Delta$
$\displaystyle=((id\otimes\delta)\circ\Delta)\otimes id)\circ\Delta$
$\displaystyle=((id\otimes\delta)\otimes id)\circ(\Delta\otimes
id)\circ\Delta$ $\displaystyle=(id\otimes(\delta\otimes
id))\circ(id\otimes\Delta)\circ\Delta$ $\displaystyle=(id\otimes(\delta\otimes
id)\circ\Delta)\circ\Delta$ $\displaystyle=(id\otimes\omega_{l})\circ\Delta$
(2.10) $(id\otimes\Delta)\circ\omega_{l}=(\omega_{l}\otimes id)\circ\Delta$
$\displaystyle(id\otimes\Delta)\circ\omega_{l}$
$\displaystyle=(id\otimes\Delta)\circ(\delta\otimes id)\circ\Delta$
$\displaystyle=(\delta\otimes\Delta)\circ\Delta$
$\displaystyle=(\delta\otimes(id\otimes id))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=((\delta\otimes id)\otimes id)\circ(\Delta\otimes
id)\circ\Delta$ $\displaystyle=(((\delta\otimes id)\circ\Delta)\otimes
id)\circ\Delta$ $\displaystyle=(\omega_{l}\otimes id)\circ\Delta$

In a similar computation, it follows

(2.11) $(id\otimes\omega_{r})\circ\Delta=(\Delta\otimes id)\circ\omega_{r}$

Moreover, both $\omega_{l}$ and $\omega_{r}$ act on $\mathbf{S}$ as its own
comultiplication. Therefore, we have more equations

(2.12)
$(id\otimes\omega_{r})\circ\omega_{r}=(id\otimes\omega_{l})\circ\omega_{r}=(id\otimes\Delta)\circ\omega_{r}=(\omega_{r}\otimes
id)\circ\omega_{r}$

We will compute some of equations and the rest is left to the readers.

$\displaystyle(id\otimes\omega_{r})\circ\omega_{r}$
$\displaystyle=(id\otimes(id\otimes\delta)\circ\Delta)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(id\otimes(id\otimes\delta))\circ(id\otimes\Delta)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(id\otimes(id\otimes\delta))\circ(id\otimes(\Delta\circ\delta))\circ\Delta$
$\displaystyle=(id\otimes(id\otimes\delta))\circ(id\otimes(\delta\otimes\delta))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes(\delta\otimes(\delta\circ\delta)))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes(\delta\otimes(\delta)))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes(\delta\otimes(\delta)))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes(\delta^{2}\otimes\delta))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes(\delta\otimes
id))\circ(id\otimes(\delta\otimes\delta))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes(\delta\otimes
id))\circ(id\otimes\Delta)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(id\otimes((\delta\otimes
id))\circ\Delta)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(id\otimes\omega_{l})\circ\omega_{r}$
$\displaystyle(id\otimes\omega_{r})\circ\omega_{r}$
$\displaystyle=(id\otimes(\delta\otimes(\delta\circ\delta)))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes(\delta\otimes\delta))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=(id\otimes\Delta)\circ(id\otimes\delta)\circ\Delta$
$\displaystyle=(id\otimes\Delta)\circ\omega_{r}$

Also, in a similar computation

(2.13) $(\omega_{l}\otimes id)\circ\omega_{l}=(\omega_{r}\otimes
id)\circ\omega_{l}=(\Delta\otimes
id)\circ\omega_{l}=(id\otimes\omega_{l})\circ\omega_{l}$

######  Definition 2.4.

The reduced decomposition associated to the retraction $\delta$ is defined as
$\overline{\Delta}=\Delta-\omega_{r}-\omega_{l}$.

If the category $\mathcal{C}$ is an abelian monodial category, that is, the
following is a homomorphism of abelian groups.

$\mathcal{C}(\mathbf{X},\mathbf{Y})\otimes_{\mathbb{Z}}\mathcal{C}(\mathbf{X^{\prime}},\mathbf{Y^{\prime}})\to\mathcal{C}(\mathbf{X}\otimes\mathbf{X^{\prime}},\mathbf{Y}\otimes\mathbf{Y^{\prime}})$

In particular, it implies the tensor product is linear in both entries. We
call this condition biadditive. The reduced decomposition is coassociative in
the following sense. It is a generalization of Lemma 2.19 in [KM22a]

######  Theorem 2.5.

If $\otimes$ is biadditive, $(\overline{\Delta}\otimes
id)\overline{\Delta}=(id\otimes\overline{\Delta})\overline{\Delta}$

###### Proof.

The left hand side is
$(\overline{\Delta}\otimes id)\circ\overline{\Delta}=(\Delta\otimes
id)\circ\Delta-(\omega_{r}\otimes id)\circ\Delta-(\omega_{l}\otimes
id)\circ\Delta-(\Delta\otimes id)\circ\omega_{r}+(\omega_{r}\otimes
id)\circ\omega_{r}+(\omega_{l}\otimes id)\circ\omega_{r}-(\Delta\otimes
id)\circ\omega_{l}+(\omega_{r}\otimes id)\circ\omega_{l}+(\omega_{l}\otimes
id)\circ\omega_{l}$

The right hand side is
$(id\otimes\overline{\Delta})\circ\overline{\Delta}=(id\otimes\Delta)\circ\Delta-(id\otimes\omega_{r})\circ\Delta-(id\otimes\omega_{l})\circ\Delta-(id\otimes\Delta)\circ\omega_{r}+(id\otimes\omega_{r})\circ\omega_{r}+(id\otimes\omega_{l})\circ\omega_{r}-(\Delta\otimes
id)\circ\omega_{l}+(id\otimes\omega_{r})\circ\omega_{l}+(id\otimes\omega_{l})\circ\omega_{l}$

Note that $(\Delta\otimes id)\circ\Delta=(id\otimes\Delta)\circ\Delta$ by
coassociavity. And most of the terms are cancelled by previous computation. We
only need to check the following remaining terms.

(2.14) $(id\otimes\Delta)\circ\omega_{l}=(\omega_{l}\otimes id)\circ\Delta$
$\displaystyle(id\otimes\Delta)\circ\omega_{l}$
$\displaystyle=(id\otimes\Delta)\circ(\delta\otimes id)\circ\Delta$
$\displaystyle=(\delta\otimes\Delta)\circ\Delta$
$\displaystyle=(\delta\otimes(id\otimes id))\circ(id\otimes\Delta)\circ\Delta$
$\displaystyle=((\delta\otimes id)\otimes id)\circ(\Delta\otimes
id)\circ\Delta$ $\displaystyle=(((\delta\otimes id)\circ\Delta)\otimes
id)\circ\Delta$ $\displaystyle=(\omega_{l}\otimes id)\circ\Delta$

By similar calculation as above

(2.15) $(id\otimes\omega_{r})\circ\Delta=(\Delta\otimes id)\circ\omega_{r}$

After cancelling terms, we obtain $(\bar{\Delta}\otimes
id)\bar{\Delta}-(id\otimes\bar{\Delta})\bar{\Delta}=0$ and the proof is
completed ∎

In this case, we will refer to $\bar{\Delta}$ as the reduced comultiplication.
There are two reasons for using the term ”comultiplication.” Firstly, it is
coassociative. Secondly, when considering a coalgebra over a field, the
retraction $\delta:C\to S$ leads to a splitting $C=S\oplus I$ for a certain
coideal $I$. An elementary argument (which we will present later) shows that
$I$ is closed under the reduced comultiplication induced by $I$. Consequently,
$(I,\bar{\Delta})$ forms a coalgebra without a counit.

####  2.1.2. Short discussion on simply colored cooperads

If $\mathds{1}$ is a unit object, it naturally possesses a comonoid structure
induced by the left and right unitors
$\mathds{1}\cong\mathds{1}\otimes\mathds{1}$, and the counit map is simply an
isomorphism of comonoids. We can define the set-like comonoid $\mathbf{S}$
over a set $S$ in the category $\mathcal{C}$ as a direct sum of comonoids
$\bigoplus_{s\in S}\mathds{1}_{s}$, where $S$ is a set and
$\mathds{1}\cong\mathds{1}_{s}$ is an isomorphism of comonoids. It is expected
that simply colored comonoids can be defined in any monoidal category with an
abelian structure that may not necessarily be compatible. The challenge lies
in finding a suitable description for the conilpotent condition. The following
example of a simply colored cooperad serves as a starting point for the
concept of simply colored comonoids.

We use the notations and definitions from [LV12]. The ground ring we consider
is a field with characteristic zero. Recall that a cooperad is a comonoid in
the category of $\mathbb{S}$-modules, which is both an abelian category and a
monoidal category with $\bar{\circ}$ as the non-biadditive monoidal product.
The construction we employ is similar to that of the conilpotent cooperad in
[LV12]. We start with a cooperad $\mathbf{S}$ concentrated in arity one, where
$\mathbf{S}(0)=C[S]$ is a set-like coalgebra over a set $S$. Next, we consider
a cooperad $\mathbf{C}$ such that $\delta:\mathbf{C}\to\mathbf{S}$ is an
operadic retraction. Similar to the coalgebra case, there exists a cooperadic
coideal $\mathbf{I}$ such that $\mathbf{C}=\mathbf{S}\oplus\mathbf{I}$. We
adopt Loday’s method of defining the conilpotent condition, and we define the
following concepts in order:

$\tilde{\Delta}^{0}=id$
$\tilde{\Delta}^{1}=\Delta+(id\bar{\circ}\delta)\Delta$

Then, we can define it iteratively,

$\tilde{\Delta}^{n}=(id\bar{\circ}\tilde{\Delta})\tilde{\Delta}^{n-1}$

and

$\hat{\Delta}^{n}=\tilde{\Delta}^{n}-(id\bar{\circ}((id\bar{\circ}\delta)\Delta))\tilde{\Delta}^{n-1}$

This yields a filtration $F_{0}\mathbf{C}=\mathbf{S}$ and
$F_{n}\mathbf{C}=\ker\hat{\Delta}^{n}$. Then, we can define the simply colored
cooperad is the pair $(\mathbf{C},\delta)$ s.t. the filtration we just defined
is exhaustive (our conilpotent condition). Note that by counit axiom, it can
be shown that when $\mathbf{S}\cong\mathds{1}$ the monoidal unit , the right
coaction $\omega_{r}=(id\bar{\circ}\delta)\Delta$ amounts to “adding a level
made up of the unique set-like $|$”, which generalizes construction (see page
169 [LV12]) using $id\bar{\circ}\eta$ .

As a consequence of our generalization to arbitrary monoidal categories, one
can extend the discussion for conilpotent cooperad in [LV12] to simply colored
cooperads, or even simply colored co-properads. This is due to the possibility
to define operads and properads as plethysm monoids. So their dual structures
are given by a comultiplication. The existence of such a plethysm construction
more generally has been given in [KM22b] in terms of a plus construction. The
details of this connection will be studied in future work.

###  2.2. Simply colored coalgebras

We begin by introducing the concept of set-like elements.

######  Definition 2.6.

  1. (1)

An element $g$ in a coalgebra $C$ is referred to as set-like if
$\Delta(g)=g\otimes g\in C\otimes C$, and $\epsilon(g)=1$.

  2. (2)

The set-like coalgebra over a set $G$ is a free module with the basis $G$,
where $\Delta(g)=g\otimes g$ and $\epsilon(g)=1$ for $g\in G$. In this paper,
we will denote this coalgebra as $C[G]$.

It is important to note that $C[G]=\bigoplus_{g\in G}Rg$. There is a unique
coalgebra structure on $R$ by considering $1$ as a set-like element since $R$
is unital. Recall that a retraction $\delta:C\to C[G]$ implies the existence
of a well-defined reduced comultiplication
$\overline{\Delta}=(id-\delta\otimes id-id\otimes\delta)\circ\Delta$.

######  Definition 2.7.

Let $C$ be a coalgebra, $G$ be a set of set-like elements in $C$, and
$\delta:C\to C[G]$ be a retraction from $C$ to the set-like coalgebra over
$G$. A simply colored coalgebra is a triple $(C,G,\delta:C\to C[G])$ such that
for every $x\in\ker\delta$, there exists an integer $N$ (which may depend on
$x$) such that $\bar{\Delta}^{N}(x)=0$.

######  Remark 2.8.

In the definition of a simply colored coalgebra $(C,G,\delta)$, we do not
require $G$ to contain all set-like elements of $C$. However, the fact $G$
contains all set-like elements of $C$ follows from the conilpotent condition,
as we will demonstrate in Corollary 2.5.

######  Example 2.9.

A simple example of a simply colored coalgebra is a set-like coalgebra with a
zero coideal.

In principle, one can replace $C[G]$ with any coalgebra if the representation
of that coalgebra is well-understood. Occasionally, we use the notation
$C=C[G]\oplus I$ instead of $(C,G,\delta)$ (where $I=\ker\delta$) to represent
a simply colored coalgebra.

The definition implies the following properties of set-like coalgebras and
simply colored coalgebras.

######  Proposition 2.2.

  1. (1)

Let $C=C[G]\oplus I$ be a simply colored coalgebra, then
$\overline{{\Delta}}(I)\subset I\otimes I\subset C\otimes C$. In fact, $I$ is
a $C[G]$-subbicomodule of $C$.

  2. (2)

The tensor product of two set-like coalgebras equipped the canonical coalgebra
structure 222If $C$ and $D$ are two coalgebras, the tensor product $C\otimes
D$ has a natural coalgebra structure:  $\Delta_{C\otimes
D}=(id\otimes\tau_{12}\otimes id)\circ(\Delta_{C}\otimes\Delta_{D})$ and
$\epsilon_{C\otimes D}=\epsilon_{C}\otimes\epsilon_{D}$ where
$\tau_{12}(c\otimes d)=d\otimes c$ is a linear isomorphism on $C\otimes D$ to
$D\otimes C$ is also set-like, and thus simply colored

  3. (3)

The tensor coalgebra of two simply colored coalgebras $C=C[G]\oplus I$ and
$D=C[S]\oplus J$ is also a simply colored coalgebra.

###### Proof.

  1. (1)

$I$ is a coideal, so $\Delta(I)\subset I\otimes C+C\otimes I=I\otimes
C[G]+C[G]\otimes I+I\otimes I$. It is clear that $\bar{\Delta}(I)\subset
I\otimes I$ and $(\delta\otimes id)\Delta(I)\subset C[G]\otimes I\subset
C[G]\otimes C$ (and $(id\otimes\delta)\Delta(I)\subset I\otimes C[G]\subset
C[G]\otimes C$.

  2. (2)

The tensor of two free modules is still free. So it suffices to show tensor of
set-like elements is also set-like. We compute for $g$ set-like in $C$ and $s$
set-like in $D$:

$\displaystyle\Delta_{C\otimes D}(g\otimes s)$
$\displaystyle=(id\otimes\tau_{12}\otimes
id)\circ(\Delta_{C}\otimes\Delta_{D})(g\otimes s)$ $\displaystyle=g\otimes
s\otimes g\otimes s$

  3. (3)

$C\otimes D=C[G]\otimes C[S]\oplus C[G]\otimes J\oplus I\otimes C[S]\oplus
I\otimes J$. We showed $C[G]\otimes C[S]$ is also a set-like coalgebra. Then
we need to show $C[G]\otimes J\oplus I\otimes C[S]\oplus I\otimes J=C\otimes
J+I\otimes D$ is an coideal. The counit condition is straightforward. We
finish the proof by checking the comultiplication requirement
$\Delta(C\otimes J)\subset C\otimes D\otimes C\otimes J+C\otimes J\otimes
C\otimes D$ and $\Delta(I\otimes D)\subset I\otimes D\otimes C\otimes
D+C\otimes D\otimes I\otimes D$. Therefore, $\Delta(C\otimes J+I\otimes
D)\subset(C\otimes J+I\otimes D)\otimes(C\otimes D)+(C\otimes
D)\otimes(C\otimes J+I\otimes D)$.

∎

######  Remark 2.10.

The statement in the above proposition is also true if $C[G]$ is replaced by
any coalgebra $S$.

######  Example 2.11.

Every coaugmented conilpotent coalgebra is simply colored. Recall that an
coaugmentation map is an coalgebra homomorphism $u:R\to C$ s.t. $\epsilon\circ
u=id$. If ground ring $R$ is a field $k$, the condition $\epsilon\circ u=id$
is obvious and does not need to specify. This induces a splitting
$C=Re\oplus\ker\epsilon$ in which $e=u(1)$. We claim $\ker\epsilon$ is an
coideal. It is obvious that $\epsilon(\ker\epsilon)=0$. It suffices to show
$\Delta(\ker\epsilon)\subset\ker\epsilon\otimes C+C\otimes\ker\epsilon$. Let
$x\in\ker\epsilon$. In general

$\Delta(x)\in(Re\otimes
Re)\oplus(Re\otimes\ker\epsilon)\oplus(\ker\epsilon\otimes
Re)\oplus(\ker\epsilon\otimes\ker\epsilon)$

We only need the fact $\Delta(x)$ has no terms in $Re\otimes Re$. By the left
counit identity $(\epsilon\otimes id)\circ\Delta(x)=x$, it follows there
exists $r\in R$ and $i\in\ker\epsilon$ such that $x=re+i$ and
$\Delta(x)-r(e\otimes e)-(e\otimes i)\in(\ker\epsilon\otimes
Re)\oplus(\ker\epsilon\otimes\ker\epsilon)$. However, $x\in\ker\epsilon$
implies $r=0$. Thus, we show $\ker\epsilon$ is an coideal and the projection
$\pi:C\to C/\ker\epsilon\cong Re$ is an retraction that agrees with our
notion. The reduced comultiplication then becomes
$\bar{\Delta}(x)=\Delta(x)-e\otimes x-x\otimes e$, which agrees with the usual
notion of reduced comultiplication.

The following proposition provides a major source of simply colored
coalgebras.

######  Proposition 2.3.

Any non-negatively graded coalgebra $C=\bigoplus_{i\geq 0}C(i)$ whose zeroth
homogeneous component $C(0)$ is a set-like coalgebra $C[X]$ over some set $X$
is simply colored with the choice of the coideal $\bigoplus_{i>0}C(i)$. We
will call such graded coalgebras space-like.

###### Proof.

It suffices to show the conilpotent condition. We prove by induction that if
$x\in C(i)$ for $i>0$, then $\bar{\Delta}^{n}(x)=0$ for any $n\geq i$. The
base case for $i=1$ is straightforward. Suppose it is true for $i=k$. Let
$x\in C(k+1)$, $\Delta(x)\in\sum_{i=0}^{k+1}C(i)\otimes C(n-i)$. And
$\bar{\Delta}(x)\in\sum_{i=1}^{n-1}C(i)\otimes C(n-i)$ because the reduced
comultiplication kills the parts $C(0)\otimes C(k+1)$ and $C(k+1)\otimes
C(0)$. Each $C(i)$ will be eliminated by at most $i$-iteration or reduced
comultiplication by induction hypothesis. Since reduced comultiplication is
billinear and coassociative, it follows $x$ is eliminated by at most
$k+1$-iterations of reduced comultiplication. ∎

######  Example 2.12.

  1. (1)

A connected coalgebra is simply colored. Examples of simply colored coalgebras
include Connes-Kreimer’s Hopf algebra, Goncharov’s Hopf algebra for multiple
zeta values, and Baues’ Hopf algebra for double loop spaces [GCKT20a]. These
three Hopf algebras can be unified categorically through the framework of
Feynman Categories [GCKT20a]. Specifically, if a Feynman category is
factorization finite and almost connected, the associated Hopf algebra (as
shown in Theorem 1.1 of [GCKT20b]) is connected and, therefore, simply
colored. The three mentioned Hopf algebras can be constructed by selecting
appropriate almost connected and factorization finite Feynman categories.

  2. (2)

The path coalgebra associated with a quiver is simply colored. This is because
it is known to be a (coradically) graded coalgebra whose zero homogeneous
components are spanned by set-like elements.

  3. (3)

The categorical coalgebra arising from a Feynman category [KM22a] (see Theorem
5.3) is simply colored. A similar notion of categorical coalgebra in [MRZ23]
is also simply colored.

  4. (4)

The component coalgebra [MP12] is simply colored.

######  Example 2.13.

A non-trivial example of simply colored coalgebra. Let
$C=\mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}/4\mathbf{Z}$ be a
$\mathbf{Z}$-module. We give a coalgebra structure on $C$ by setting
$g=(1,0,0)$ and $h=(0,1,0)$ two different set-like elements and $x=(0,0,1)$ a
$g$-primitive elements. $\mathbf{Z}g\oplus\mathbf{Z}h$ is a well-defined
subcoalgebra of $C$ (because the natural inclusion splits) and there is a
natural splitting of coalgebra onto the subcoalgebra just by projection where
the coideal is precisely generated by $x$. The key is that we allow the
existence of torsion elements in the coalgebra.

######  Definition 2.14.

Given $C$ a coalgebra and $A$ an algebra, the $R$-module ${\rm Hom}(C,A)$ is a
$R$-algebra with unit $\eta\circ\epsilon$ and multiplication $\star$ given by
$(f\star g)(x)=m\circ(f\otimes
g)\circ\Delta(x)=\sum_{x}\,f(x_{(1)})g(x_{(2)})$. We say $\\{{\rm
Hom}(C,A),\star,\eta\circ\epsilon\\}$ is the convolution algebra.

If for $f,g\in{\rm Hom}(C,A)$ there is $f\star g=g\star f=\eta\circ\epsilon$,
we say $f$ is $\star$-inverse (or convolution inverse) to $g$ and vice versa.
The antipode of the Hopf algebra $H$ can be equivalently formulated as the
unique convolution inverse to the identity $id$ in ${\rm Hom}(H,H)$.

The following theorem is a modification of Takeuchi’s lemma used in [KM22a]
and it allows one to simplify the problem of finding convolution inverse.

######  Theorem 2.15.

Let $F_{0}$ and $C$ be two coalgebras such that $C_{0}$ is a direct summand of
$C=F_{0}\oplus M$ as $R$-module and $C_{0}$ has a unique subcoalgebra
structure inherit from $C$. Suppose for any $x\in C$, there exists $N$ s.t.
its image under the composition of canonical projection onto $F_{0}$ with (the
kernel is $M$) and iterated comultiplications is zero.

${C}$${{C^{\otimes N}}}$${{M^{\otimes N}}}$$\scriptstyle{\pi^{\otimes
N}}$$\scriptstyle{\Delta^{N}}$

Then the following conditions are equivalent for any element $f$ of $Hom(C,A)$
with an algebra $A$:

  1. (1)

$f$ is convolution invertible

  2. (2)

$f|_{F_{0}}$ is convolution invertible in $Hom(F_{0},A)$

When $C$ is an an coalgebra over a field and $F_{0}$ is the coradical, this is
known as Takeuchi’s lemma [Tak71]. The proof for the modified version is the
same as the one in theorem 3.3 [KM22a]. The key is we translate the condition
${\rm Ext}(C/F^{QT}_{C},A)=0$ to the wedge product characterization.

######  Proposition 2.4.

Let $C=C[G]\oplus I$ be a simply colored coalgebra and $\pi:C\to I$ be the
canonical projection induced by splitting. We have $\pi^{\otimes
N}(\bar{\Delta}^{N}(x))=\pi^{\otimes N}({\Delta}^{N}(x))$ for $x\in I$.

###### Proof.

It follows because ${\Delta}(x)-\overline{\Delta}(x)\in C[G]\otimes C+C\otimes
C[G]$ and the difference will be killed when passed to the quotient.
${\Delta}^{N}(x)-\bar{\Delta}^{N}(x)$ will be also eliminated when passed to
the quotient by the same reason. ∎

It follows that

######  Corollary 2.5.

  1. (1)

If $C=C[G]\oplus I$ is simply colored, then $G$ is the set of all set-likes.

  2. (2)

Given algebra $A$, any $f\in{\rm Hom}(C,A)$ is convolution invertible if
$f|_{C[G]}\in{\rm Hom}(C|_{C[G]},A)$ is convolution invertible.

  3. (3)

We call an bialgebra is simply colored if its coalgebra structure is simply
colored. The simply colored bialgebra has an/the antipode if and only if the
set of set-likes form a group under the multiplication.

###### Proof.

We only need to show (1). Let $t$ be an set-like not in $C[G]$. It cannot be
in $I$ as well since $\epsilon(g)\neq 0$. So it has to be the form of $r+i$
for $0\neq t\in C[G]$ and $0\neq i\in I$. Since $\Delta^{n}(t)=t^{\otimes n}$,
$\pi^{\otimes n}(\bar{\Delta}^{n}(t))=\pi^{\otimes
n}({\Delta}^{n}(t))=i^{\otimes n}\neq 0$ for any $n\geq 0$. However, the
conilpotent condition implies that every $i\in I$ can be eliminated by finite
iterations of reduced comultiplication. Thus, $\pi^{\otimes
N}(\bar{\Delta}^{N}(t))=\pi^{\otimes N}(\bar{\Delta}^{N}(r))+\pi^{\otimes
N}(\bar{\Delta}^{N}(t))=\pi^{\otimes N}(\bar{\Delta}^{N}(t))=0$ for $N$
sufficiently large, which is an contradiction. ∎

####  2.2.1. Simply colored coalgebra over a field is equivalent to pointed
coalgebra with a splitting

######  Theorem 2.16.

Let $k$ be a field,

  1. (1)

Every simply colored coalgebra $(C,G,\delta)$ over $k$ can be viewed as a
pointed coalgebra with a splitting $(C,\delta)$.

  2. (2)

Conversely, if $(C,\delta)$ is a pointed coalgebra with a splitting, it
naturally gives rise to a simply colored coalgebra structure $(C,G,\delta)$,
where $G$ is the set of all set-like elements in $C$.

###### Proof.

  1. (1)

Let $C=C[G]\oplus I$ where $I=\ker\delta$. We show the coalgebra filtration
formed by $\\{\wedge^{n}(C[G])\\}_{n\geq 0}$ is exhaustive. It follows
directly from Proposition 2.4: if $\bar{\Delta}^{n}(x)=0$, then
$x\in\wedge^{n}(C[G])$. By Exercise 4.1.9 in [Rad12], the coradical
$C_{0}\subset C[G]$. Therefore, every cosimple subcoalgebra is one dimensional
and $C$ is pointed.

  2. (2)

The first statement follows again from the fact every pointed coalgebra is
isomorphic to a subcoalgebra of a path coalgebra associated to a quiver
(proposition 1.2) and the fact subcoalgebra of coradically graded coalgebra is
coradically graded (Exercise 4.4.6 [Rad12]). Now take any splitting
$C=C_{0}\oplus I$ of pointed coalgebras. The reduced comultiplication is
induced by this splitting. By discussion of proof of theorem 5.4.1 in [Mon93],
it follows $\overline{\Delta}(I_{n})\subset I_{n-1}\otimes I_{n-1}$ where
$I_{n}=C_{n}\cap I$. Since $C_{0}\cap I=\emptyset$, every element in $I$ is
eliminated through some iterations of reduced comultiplication.

∎

Therefore, we see simply colored coalgebra and pointed coalgebra with a
splitting are the same thing. Since the subcoalgebra of pointed coalgebra over
a field is well-defined and also pointed. It follows immediately that

######  Corollary 2.6.

  1. (1)

the subcoalgebra of simply colored coalgebra over a field can be realized to
be a simply colored coalgebra over a field.

  2. (2)

A simply colored coalgebra over a field is a sum of finite dimensional simply
colored subcoalgebras. Thus, every simply colored coalgebra over a field is a
filtered limit of finite dimensional simply colored coalgebras.

###### Proof.

Consider a subcoalgebra $D$ of a pointed coalgebra with a splitting
$C=C[G]\oplus I$. According to Proposition 3.4.3 [Rad12], we have $D\cap
C_{0}=D\cap C[G]=D_{0}$. Thus, $D_{0}\subseteq C[G]$ is also pointed, and in
particular, a set-like coalgebra since $C[G]$ is cosemisimple. The splitting
restricts to $D$ as $\delta|_{D}:D\to D_{0}$, resulting in a pointed
subcoalgebra with a splitting, denoted as $(D,\delta|_{D})$.

By the fundamental theorem of coalgebras, every element $x$ in the coalgebra
$C$ belongs to a finite-dimensional subcoalgebra. Moreover, any subcoalgebra
of a pointed coalgebra is also pointed. Therefore, for any element $x$, there
exists a finite-dimensional pointed subcoalgebra with a splitting where $x$
resides. Consequently, the pointed coalgebra $(C,I)$ with a splitting can be
regarded as the sum of all finite-dimensional pointed subcoalgebras with a
splitting. ∎

######  Example 2.17.

As a result, the pointed curved coalgebra in [HL22] is also a simply colored
coalgebra.

####  2.2.2. How is simply colored coalgebra ‘colored’?

The question has already been addressed in [KM22a] through the use of
orthogonal projectors. This method involves a specific choice of an orthogonal
idempotent family, which has been extensively studied in the context of
pointed coalgebras (e.g., see [Rad82] and Chapter 4 of [Rad12]). In this
section, we provide a summary of the results presented in the aforementioned
papers and books.

When $A=R$, the convolution algebra $C^{*}={\rm Hom}_{R}(C,R)$ is referred to
as the dual algebra of $C$. The multiplication in $C^{*}$ is inherited from
the multiplication in $R$, thus we omit the star notation. Any right
$C$-comodule $M$, which is also an $R$-module, induces a left $C^{*}$-module
structure on $M$ defined as $f\cdot m=(id\otimes f)\rho_{r}(m)$. Similarly,
any left $C$-comodule $M$ gives rise to a right $C^{*}$-module structure on
$M$ defined as $m\cdot f=(f\otimes id)\rho_{l}(m)$.

######  Definition 2.18.

[Rad12] Let $C$ be an coalgebra, An orthogonal family of idempotents
$\mathcal{E}$ of $C^{*}$ is a family of element $\\{e_{i}\\}_{i\in I}$ of
$C^{*}$ s.t. $e_{i}e_{i}=e_{i}$ and $e_{i}e_{j}=0$ for $i\neq j$. Furthermore,
a orthogonal family of idempotents $\mathcal{E}$ is called orthonormal if the
counit $\epsilon=\sum_{i\in I}e_{i}$ 333By the fundamental theorem of
coalgebras, a coalgebra is a direct limit of finite-dimensional
(sub)coalgebras. Thus, its dual becomes an inverse limit of algebras, which
possesses a topology that allows for certain infinite sums to make sense. This
algebra is sometimes referred to as a pseudocompact algebra in the
literature..

Denote $L_{e}(c)=e\cdot c$ and $R_{e^{g}}=c\cdot e$ for $e\in\mathcal{E}$. The
following identites are true (see pp 132 and pp 110 in [Rad12] for detail). It
still works if the ground ring is only assumed to be commutative unital.

######  Lemma 2.19.

[Rad12]

  1. (1)

$L_{e}\circ L_{e^{\prime}}=\delta_{e,e^{\prime}}L_{e}$ and $R_{e}\circ
R_{e^{\prime}}=\delta_{e,e^{\prime}}R_{e}$

  2. (2)

$\sum_{e\in\mathcal{E}}L_{e}=I_{C}=\sum_{e\in\mathcal{E}}R_{e}$ where $I_{C}$
is the identity action on $C$ of $C^{*}$.

  3. (3)

$(id\otimes L_{e})\circ\Delta=\Delta\circ L_{e}$ and $(R_{e}\otimes
id)\circ\Delta=\Delta\circ R_{e}$

  4. (4)

$\sum_{e\in\mathcal{E}}(L_{e}\otimes R_{e})\circ\Delta=\Delta$

  5. (5)

$L_{e}\circ R_{e^{\prime}}=R_{e^{\prime}}\circ L_{e}$

For a simply colored coalgebra $C=C[G]\oplus I$, we can choose the orthogonal
family to be ${\delta_{g}}_{g\in G}$, where $e_{g}\in C[G]^{*}$ represents the
dual basis to $g$. This choice is sensible due to the hypothesis that $C[G]$
is free. It can be verified that this family is also orthonormal. As a
consequence of the previous lemma, we have the following proposition:

######  Proposition 2.7.

An $C[G]$-bicomodule $M$ can be decomposed as $M=\bigoplus_{g,g^{\prime}\in
G}{{}^{g}M^{g^{\prime}}}$, where ${}^{g}M^{g^{\prime}}=e_{g^{\prime}}\cdot
M\cdot e_{g}$. Moreover, it has an explicit description
${{}^{g}M^{g^{\prime}}}={m\in{{}^{g}M^{g^{\prime}}}:\rho_{l}(m)=g\otimes
m,\rho_{r}(m)=m\otimes g^{\prime}}$.

The key observation is that every left/right/bi-comodule over $C[G]$ is
cosemisimple. In particular, a $C[G]$-bicomodule is equivalent to a
$G$-bigraded module.

######  Remark 2.20.

Alternatively, one can directly prove the result using the definition of a
bicomodule. However, it is important to note that in this case, ${{}^{g}}M\cap
M^{g^{\prime}}={{}^{g}M^{g^{\prime}}}$ holds due to the orthonormal condition.
It is not true that the equality holds in general, and one cannot deduce it
from the same argument presented in Example 1.6.7 in [Mon93]. The conditions
of freeness of $C[G]$ and being a bicomodule are essential for the result.

Move back to the main discussion. The splitting $C=C[G]\oplus I$ endows $C$
with a $C[G]$-bicomodule structure and give rise to a reduced
comultiplication.

######  Proposition 2.8.

Given a splitting $C=C[G]\oplus I$, where $I$ is an coideal, it follows

  1. (1)

$\Delta({{}^{g^{\prime}}C^{g}})\subset\sum_{s\in
G}{{}^{g^{\prime}}C^{s}}\otimes{{}^{s}C^{g}}$

  2. (2)

$I=\oplus_{g,g^{\prime}\in G}({{}^{g^{\prime}}C^{g}}\cap I)$ and
$I\cap{{}^{g^{\prime}}C^{g}}={{}^{g^{\prime}}C^{g}}$ for $g^{\prime}\neq g$

  3. (3)

As a result of (1) and (2), for $x\in{{}^{g^{\prime}}C^{g}}$, we have both
$\Delta(x)-g^{\prime}\otimes x\in I\otimes C$ and $\Delta(x)-x\otimes g\in
C\otimes I$.

###### Proof.

  1. (1)

It follows from Lemma 2.19 and Proposition 2.7.

  2. (2)

For $i\in I$, we know $i=\epsilon\cdot i\cdot\epsilon$. Let
$i_{gg^{\prime}}=e_{g}^{\prime}\cdot i\cdot e_{g}$ so $i=\sum_{g,g^{\prime}\in
G}i_{gg^{\prime}}$ a direct sum where
$i_{gg^{\prime}}=e_{g}(i_{1})e_{g^{\prime}}(i_{2})i_{3}$. Note if
$g^{\prime}\neq g$, $i_{gg^{\prime}}\in g\cdot(C[G])\cdot g^{\prime}\oplus
g\cdot I\cdot g^{\prime}=g\cdot I\cdot g^{\prime}$. If $g=g^{\prime}$, then
$i_{gg^{\prime}}=c_{g}g\oplus i^{\prime}_{gg^{\prime}}$ where $g\in G$ and
$i^{\prime}_{gg^{\prime}}\in g\cdot I\cdot g^{\prime}\subset I$. $C[G]\oplus
I=C$ implies $c_{g}=0$ for all $g$. We conclude then
$I=\oplus_{g,g^{\prime}\in G}(I\cap{{}^{g^{\prime}}C^{g}})$. To show
${{}^{g^{\prime}}C^{g}}\subset I\cap{{}^{g^{\prime}}C^{g}}$, just apply
$\epsilon=\sum_{s\in G}e_{s}$ to $e_{g^{\prime}}\cdot x\cdot e_{g}$ for $x\in
C$ and check $\sum_{s\in G}e_{g^{\prime}}(x_{1})e_{s}(x_{2})e_{g}(x_{3})=0$ by
orthogonality.

  3. (3)

Observe
$\Delta({{}^{g^{\prime}}x^{g}})-g^{\prime}\otimes{{}^{g^{\prime}}x^{g}}\in\sum_{s\in
G;s\neq g^{\prime}}{{}^{g^{\prime}}C^{s}}\otimes{{}^{s}C^{g}}$ and
$\epsilon({{}^{g^{\prime}}C^{s}})=0$. Thus $\Delta(x)-g^{\prime}\otimes x\in
I\otimes C$. The other is similar.

∎

We show a simply colored coalgebra has the following structure:

  * •

$C$ is a $G$-bigraded module: $C=\bigoplus_{g,h\in G}{{}^{g}C^{h}}$ and $I$ is
a $G$-bigraded submodule of $C$ that $I=\oplus_{g,h\in G}({{}^{g}C^{h}}\cap
I)$. Both are compatible with both comultiplication and the
retraction:${{}^{g}C^{g}}=Rg\oplus{{}^{g}I^{g}}$, if $g\neq h$ then
${{}^{g}C^{h}}={{}^{g}I^{h}}$ and $\Delta({{}^{g}C^{h}})\subset\sum_{t\in
G}{{}^{g}C^{t}}\otimes{{}^{t}C^{h}}$.

  * •

There is a well-defined coassociative reduced comultiplication $\bar{\Delta}$
induced by $\delta$ s.t. For $x\in{{}^{g}C^{h}}$,
$\bar{\Delta}(x)=\Delta(x)-x\otimes h-g\otimes x$. Furthermore, for any $i\in
I$, there exists a non-negative integer $n$ (that depends on x) s.t.
$\bar{\Delta}^{n}(x)=0$.

The algebraic structure also determines a simply colored coalgebra. We will
take a categorical viewpoint on the algebraic structure of simply colored
coalgebras in the next section.

##  3\. Categorical perspective

In this section, we explore the categorical perspective of simply colored
coalgebras. The key result we establish is that the category of simply colored
coalgebras over a field (equivalently, pointed coalgebras with a splitting)
admits all small limits and colimits. The main challenge lies in demonstrating
that the category has both products and coequalizers. To overcome this
challenge, we introduce the notion of reduced colored coalgebras, which bears
resemblance to the approach used in studying conilpotent coaugmented
coalgebras by disregarding the coaugmentation.

###  3.1. Three categories related to simply colored coalgebras

Note for this part we do not assume the ground ring is a field.

######  Definition 3.1.

The category of simply colored coalgebras is defined as follows:

  * •

The objects are precisely simply colored coalgebras $(C,G,\delta)$.

  * •

a morphism between $(C,G,\delta)$ and $(D,H,\mu)$ is precisely a coalgebra
homomorphism $f:C\to D$ such that $\mu\circ f=f|_{C[G]}\circ\delta$.

######  Remark 3.2.

We have shown that the image of a set-like element under a morphism between
simply colored coalgebras is also a set-like element. Therefore, the map
$f|_{C[G]}$ has the codomain $C[H]$. Additionally, according to the
fundamental theorems of coalgebra homomorphism [Swe69], a morphism between
$(C,G,\delta)$ and $(D,H,\mu)$ in the category of simply colored coalgebras
can be equivalently described as a coalgebra homomorphism $f$ between $C$ and
$D$ such that $f(\ker\delta)\subseteq\ker\mu$.

Recall that we have shown how the reduced comultiplication makes a simply
colored coalgebra $C=C[G]\oplus I$ to a coalgebra without counit
$(I,\overline{\Delta})$. This coalgebra without counit possesses the property
that every element in $I$ is eliminated by a finite number of iterations of
the reduced comultiplication. Specifically, it is reduced in the following
sense:

######  Definition 3.3.

A coalgebra without counit $\overline{C}$ is called reduced simply colored
coalgebra if

  * •

It is conilpotent: for every $x\in\overline{C}$, there exists a finite natural
number $n$ s.t. $\overline{\Delta}^{n}(x)=0$. 444Since counit axiom does not
hold, the comultiplication is no longer guaranteed to be injective.

  * •

There exists a set $G$ and $\overline{C}$ is a $C[G]$-bicomodule. In
particular, $\overline{C}$ is a naturally $G$-bigraded comodule
$\overline{C}=\bigoplus_{g,h\in G}{{}^{g}\overline{C}{}^{h}}$.

  * •

Furthermore, $\overline{C}$ is $G$-bigraded coalgebra over $R$:
$\overline{C}=\bigoplus_{g,h\in G}{{}^{g}\overline{C}{}^{h}}$ and
$\Delta({{}^{g}\overline{C}{}^{h}})\subset\oplus_{s\in
S}{{}^{g}\overline{C}{}^{s}}\otimes{{}^{s}\overline{C}{}^{h}}$.

The reduced colored coalgebra is then characterized as a pair
$(\overline{C},G)$.

######  Definition 3.4.

The category of reduced (simply) colored coalgebras is defined as follows:

  * •

The objects are precisely reduced colored coalgebras $(\overline{C},G)$.

  * •

The morphism are precisely pairs $(\overline{f},i)$ where
$\overline{f}:\overline{C}\to\overline{D}$ is a coalgebra homomorphism that
only preserves comultiplication and $i:G\to S$ is a morphism of sets.

There is a third category $(V,G)$ formed by forgetting the coalgebra structure
of the reduced colored coalgebras.

######  Definition 3.5.

The category of colored modules (over a ring $R$) has the following:

  * •

The objects are precisely pairs $(V,G)$ where $V$ is a $G$ bigraded $R$-module
with $V=\bigoplus_{g,h\in G}{{}^{g}V^{h}}$

  * •

The morphisms are precisely pairs $(f,i):(V,G)\to(W,S)$ where $i:G\to S$ is a
morphism of sets and $f({{}^{g}V^{h}})\subset{{}^{i(g)}W^{i(h)}}$ 555In other
word, $(f,i)=\bigoplus_{g,h\in G}f|_{{}^{g}V^{h}}$ and ${\rm
Im}\,(f|_{{}^{g}V^{h}})\subset{{}^{i(g)}W^{i(h)}}$. On the other hands, if we
have a sequence of maps $\phi_{g,h}$ on ${{}^{g}V{}^{h}}$ s.t. ${\rm
Im}\,(\phi_{g,h})\subset{{}^{i(g)}W^{i(h)}}$ given a function of sets $i:G\to
S$. Then we can construct a morphism of colored modules via gluing
$(\phi,i)=\bigoplus_{g,h\in G}\phi_{g,h}$..

What are the relationships between these categories? In the upcoming
subsection, we will see in the next part that there exists a cofree functor
from the category of colored vector spaces (colored modules over a field) to
the category of pointed coalgebras with a splitting (simply colored coalgebras
over a field). Unfortunately, this adjunction may not hold for over arbitrary
ring $R$. For the rest of the subsection, we will establish an equivalence
between the category of simply colored coalgebras and the category of reduced
colored coalgebras. As discussed earlier, we already know how to obtain a
reduced colored coalgebra from a simply colored coalgebra. Next, we will show
the reverse process by extending the reduced comultiplication
$\overline{\Delta}$. The procedure is as follows:

  * •

Join the set-like coalgebra $C[G]$ over the set $S$ as a direct summand to
$\overline{C}$, so $C=C[S]\oplus\overline{C}$.

  * •

Extend the counit map $\epsilon$ from $C[G]$ by setting
$\epsilon(\overline{C})=0$

  * •

The comultiplication $\Delta$ on $C[G]\oplus\overline{C}$ is defined as
follows: its restriction on $C[S]$ is the same as comultiplication on set-like
coalgebra. Its restriction on $\overline{C}=\bigoplus_{g,h\in
G}{{}^{g}\overline{C}{}^{h}}$ is constructed as
$\Delta(x)=\rho_{l}(x)+\overline{\Delta}(x)+\rho_{r}(x)$ for
$x\in\overline{C}$. It is helpful to read component-wise that
$\Delta(x)=g\otimes x+\overline{\Delta}(x)+x\otimes h$ for
$x\in{{}^{g}\overline{C}^{h}}$.

We need to show the resulting construction is indeed a coalgebra, and
furthermore, a simply colored coalgebra.

######  Proposition 3.1.

The construction $C[G]\oplus\overline{C}$ described above produces a simply
colored coalgebra $C=C[G]\oplus I$ where $I=\overline{C}$ is an coideal of
$C$.

###### Proof.

First, we need to verify both counit and coassociativity axioms are satisfied.
The counit axiom is straightforward once the constructed comultiplication is
well-defined, whereas the coassociativity requires computations. It is enough
to check component-wise. For simplicity, we use the notation
${{}^{g}x^{h}}\in{{}^{g}\overline{C}^{h}}$ and
$\overline{\Delta}(^{g}x^{h})=\sum_{s\in G}{{}^{g}x^{s}}\otimes{{}^{s}x^{h}}$.

$(\Delta\otimes id)\Delta({{}^{g}x^{h}})=(\Delta\otimes
id)(g\otimes{{}^{g}x^{h}}+\sum_{s\in
G}{{}^{g}x^{s}}\otimes{{}^{s}x^{h}}+{{}^{g}x^{h}}\otimes h)$ $=g\otimes
g\otimes{{}^{g}x^{h}}+\sum_{s\in G}(g\otimes{{}^{g}x^{s}}+\sum_{t\in
G}{{}^{g}x^{t}}\otimes{{}^{t}x^{s}}+{{}^{g}x^{s}}\otimes
s)\otimes{{}^{s}x^{h}}+$ $(g\otimes{{}^{g}x^{h}}+\sum_{t\in
G}{{}^{g}x^{t}}\otimes{{}^{t}x^{s}}+{{}^{g}x^{h}}\otimes h)\otimes h$

and

$(id\otimes\Delta)\Delta({{}^{g}x^{h}})=(id\otimes\Delta)(g\otimes{{}^{g}x^{h}}+\sum_{s\in
G}{{}^{g}x^{s}}\otimes{{}^{s}x^{h}}+{{}^{g}x^{h}}\otimes h)$
$=g\otimes(g\otimes{{}^{g}x^{h}})+\sum_{s\in
G}{{}^{g}x^{s}}\otimes{{}^{s}x^{h}}+{{}^{g}x^{h}}\otimes h)+$ $\sum_{s\in
G}{{}^{g}x^{s}}\otimes(s\otimes{{}^{s}x^{h}}+\sum_{{t^{\prime}}\in
G}{{}^{s}x^{t^{\prime}}}\otimes{{}^{t^{\prime}}x^{h}}+{{}^{s}x^{h}}\otimes
h)+{{}^{g}x^{h}}\otimes h\otimes h$

Observe that due to coaasociativity of $\overline{\Delta}$666The Sweddler’s
notation is omitted, but it is easy to see this equation corresponds to
$(\overline{\Delta}\otimes
id)\overline{\Delta}({{}^{g}x^{h}})=(id\otimes\overline{\Delta})\overline{\Delta}({{}^{g}x^{h}})$,

$\sum_{s\in G}\sum_{t\in
G}{{}^{g}x^{t}}\otimes{{}^{t}x^{s}}\otimes{{}^{s}x^{h}}=\sum_{s\in
G}\sum_{{t^{\prime}}\in
G}{{}^{g}x^{s}}\otimes{{}^{s}x^{t^{\prime}}}\otimes{{}^{t^{\prime}}x^{h}}$

Then it follows that $(\Delta\otimes
id)\circ\Delta=(id\otimes\Delta)\circ\Delta$. Finally, $I=\overline{C}$ is a
coideal can be checked easily. ∎

To extend the construction of Proposition 3.1 to a functor, we now define how
it acts on each morphism.

######  Proposition 3.2.

A coalgebra homomorphism between simply colored coalgebras induces an
homomorphism of coalgebras without counit. On the other hand, A coalgebra
homomorphism without counit between reduced colored coalgebras has an
extenstion to a coalgebra homomorphism to the simply colored coalgebras.

###### Proof.

The first statement is clear from the structure of simply colored coalgebras
because coalgebra homomorphism maps set-like elements to set-like elements ,
$i(g)=f(g)$ for $f:C\to D$ and
$\overline{f}=f|_{\overline{C}}:\overline{C}\to\overline{D}$.
For the second statement, it is due to the way we construct the reduced
comultiplication. If $(\overline{f},i):(\overline{C},G)\to(\overline{D},S)$
between two reduced colored coalgebras, then there is an extension to the
morphism between simply colored coalgebras as $f(g)=i(g)$ for $g\in G$ and
$f(x)=\overline{f}(x)$ for $x\in\overline{C}$. We check it is well-defined:

  * •

$\Delta(f(x))=i(g)\otimes\overline{f}(x)+\overline{f}(x^{\prime})\otimes\overline{f}(x^{\prime\prime})+\overline{f}(x)\otimes
i(h)$ for $x\in{{}^{g}\overline{C}{}^{h}}$ where
$\overline{f}(\overline{\Delta}(x))=(\overline{\Delta}\otimes\overline{\Delta})\overline{f}(x)=\overline{f}(x^{\prime})\otimes\overline{f}(x^{\prime\prime})$.
In this case, the $x\in{{}^{g}\overline{C}{}^{h}}$ is sent to
$\overline{f}(x)\in{{}^{i(g)}\overline{C}{}^{i(h)}}$. Moreover,
$\Delta(f(g))=\Delta(i(g))=i(g)\otimes i(g)$.

  * •

$\epsilon(f(x))=0$ if $x\in\overline{C}$ and $\epsilon(f(g))=\epsilon(i(g))=1$

∎

######  Corollary 3.3.

There is a functor $F$ from the category of simply colored coalgebras to the
category of reduced colored coalgebras and a functor $G$ that goes in the
other direction such that $F$ (and $G$) gives an isomorphism of categories.

###### Proof.

Define $F((C,I))=(I,G)$ ($C=C[G]\oplus I$) and $F(f)=(f|_{I},f|_{C[G]})$.
Also, define $G$ based on Proposition 3.1 and Proposition 3.2. It is
straightforward to check both $F$ and $G$ are functors and $F\circ G={\rm
Id}\,$ and $G\circ F={\rm Id}\,$. ∎

######  Remark 3.6.

From now on, we will write a simply colored coalgebra as
$C=C[G]\oplus\overline{C}$ where $\overline{C}$ is the associated reduced
colored coalgebra.

###  3.2. Interlude: cofree construction

From this point forward, we assume that the ground ring is a field. As
established by previous theorems, a simply colored coalgebra is equivalent to
a pointed coalgebra with a splitting. In his work [Rad82], Radford
demonstrates that the forgetful functor from the category of pointed
coalgebras with a splitting to the category of colored vector spaces (colored
bicomodules over $k$) admits a left adjoint. This left adjoint is known as the
cofree pointed coalgebra (with a splitting) over a pair $(V,G)$, and it is
constructed using a cotensor coalgebra. Let us recall the notion of the
cotensor product.

Given a right $C^{\prime}$-comodule $M$ with a right coaction $\rho_{R}$ and a
left $C^{\prime}$-comodule $N$ with a left coaction $\rho_{L}$, we define the
cotensor product $M\square_{C^{\prime}}N$ as the equalizer of
$id_{M}\otimes\rho_{L}$ and $\rho_{R}\otimes id_{N}$:

${M\square_{C^{\prime}}N}$${M\otimes N}$${M\otimes C^{\prime}\otimes
N}$$\scriptstyle{\rho_{r}\otimes id_{N}}$$\scriptstyle{id_{M}\otimes\rho_{l}}$

Since $M\otimes N$ is a vector space, $M\square_{C^{\prime}}N$ is the kernel
of $(\rho_{R}\otimes id_{N}-id\otimes\rho_{L})$ and subvector space of
$M\otimes N$. When the underlying space is a vector space, there is an
associativity $(M\square_{C^{\prime}}N)\square_{C^{\prime}}Q\cong
M\square_{C^{\prime}}(N\square_{C^{\prime}}Q)$. Next, we can construct the
cotensor coalgebra ([Nic78]) over a pair $(S,M)$ of a coalgebra $S$ and
$S$-bicomodule $M$ as follows (It is clear in this case the underlying coring
of a comodule, so we omit the subscript $S$ in $\square_{S}$ as $\square$):

$CoT_{S}(M)=\bigoplus_{n\geq 0}M^{\Box n}=S\oplus M\oplus(M\square
M)\oplus(M\square M\square M)\oplus...$

Since the ground ring is a field, $M^{\Box n}$ is a vector subspace in
$M^{\otimes n}$. Every element in $M^{\Box n}$ could be written as the normal
bar notation $[x_{1}|x_{2}|...x_{n}]$ without confusion. This cotensor
construction can be endowed with a natural coalgebra structure by
deconcatenation:

$\Delta([x_{1}|...|x_{n}])=[\rho_{l}(x_{1})|...|x_{i}]+\sum_{i=1}^{n-1}[x_{1}|...|x_{i}]\square[x_{i+1}|...|x_{n}]+[x_{1}|x_{2}|...|\rho_{r}(x_{n})]$

and counit

$\epsilon|_{S}=\epsilon_{S}\text{~{}and~{}}\epsilon|_{M^{\Box
n}}=0\text{~{}for~{}all~{}}n$

Note the cotensor coalgebra has a clear grading by number of tensor factors of
$M$. Radford shows that $C(V,G)=CoT_{C[G]}(V)$ exhibiting the following
universal property:

######  Theorem 3.7 (Radford, [Rad82]).

Let $C=k[G(C)]\oplus\overline{C}$ be a pointed coalgebra with a splitting, and
$C(S,V)$ constructed from pair $(S,V)$. Let $f$ be a linear map from
$\overline{C}$ to $V$. If $f$ is compatible with both bigrading in
$\overline{C}$ and $V$, i.e. there is an set function $\phi:G(C)\to S$ s.t.
$f({{}^{g}\overline{C}{}^{h}})\subset^{\phi(g)}V^{\phi(h)}$. Then there exists
a unique coalgebra map $F:C\to C(G,V)$ such that the following diagram of
$k$-linear maps commutes

${\overline{C}}$${V}$${{C(G,V)}}$$\scriptstyle{f}$$\scriptstyle{F|_{\overline{C}}}$$\scriptstyle{\pi_{V}}$

where $\pi_{V}$ is the projection from $C(S,V)$ to $V$ and
$F({{}^{g}C^{h}})\subset{{}^{\phi(g)}C(G,V)^{\phi(h)}}$ for $x,y\in G(C)$.

$CoT_{C[G]}(V)$ is pointed with the splitting given by $\delta|_{C[G]}=id$ and
$\delta|_{{}^{\Box n}}=0$. Now we set
$\overline{C(S,V)}=\bigoplus_{n>0}M^{\Box n}$ as the reduced colored coalgebra
associated to $C(S,V)$. By the equivalence of reduced colored coalgebras and
simply colored colagbras, the universal property becomes

${\overline{C}}$${V}$${{\overline{C(G,V)}}}$$\scriptstyle{f}$$\scriptstyle{\bar{F}}$$\scriptstyle{\pi_{V}}$

Then, it becomes the cofree construction in the category of reduced colored
coalgebras, and we will called it reduced cofree colored coalgebra over a
field.

######  Example 3.8.

This theorem generalizes the known fact that the tensor coalgebra is cofree in
the category of (coaugmented) conilpotent coalgebras by letting $G$ to be a
Singleton.

###  3.3. Limits and colimits

Finally, we conclude this section by demonstrating that the category of
pointed coalgebras with a splitting (equivalently, the category of reduced
colored coalgebras over a field) is both complete and cocomplete. Our approach
will differ from that of [HL22].

######  Theorem 3.9.

The category of pointed coalgebras with a splitting has all small coproducts
and equalizers

###### Proof.

Recall that the coproduct of coalgebras corresponds to the direct sum of
coalgebras. Given that the sum of pointed coalgebras is also pointed, and the
sum of coideals is also a coideal, the coproduct of pointed coalgebras with a
splitting inherits the structure from the coproduct in the category of
coalgebras. It is precisely the direct sum of pointed coalgebras with a
splitting induced by the direct sum of coideals. The argument for the
equalizer follows the same reasoning as presented in Theorem 1.1 of [Ago11b].
∎

Now we come to the more challenging aspect. Instead of dealing with pointed
coalgebras with a splitting, we will shift our focus to reduced colored
coalgebras as they offer a less complex framework.

######  Lemma 3.10.

The category of colored vector spaces has all small products.

###### Proof.

Given a family $\Lambda$ of colored vector spaces
$(\overline{C_{\alpha}},G_{\alpha})$ for $\alpha\in\lambda$, we define the
colored product $\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}})$ as
the vector subspace of $\prod_{\alpha\in\Lambda}\overline{C_{\alpha}}$
generated by the sequence of homogeneous elements
$({{}^{g^{\prime}_{\alpha}}x_{\alpha}}{}^{g_{\alpha}})_{\alpha}$ where each
${{}^{g^{\prime}_{\alpha}}x_{\alpha}}{}^{g_{\alpha}}\in{{}^{g^{\prime}_{\alpha}}\overline{C_{\alpha}}}{}^{g_{\alpha}}$
for some $g^{\prime}_{\alpha}$ and $g_{\alpha}$ in $G_{\alpha}$.
$\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}})$ is
$(\prod_{\alpha\in\Lambda}G_{\alpha})$ bigraded by construction. The natural
projection from product of vector spaces $p_{\alpha}$ from
$\prod_{\alpha\in\Lambda}\overline{C_{\alpha}}\to\overline{C_{\beta}}$ for a
$\lambda\in\Lambda$ restricts to the subspace
$\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}})$. The restriction
$\tilde{p}_{\alpha}$ of each projection $p_{\alpha}$ from the universal
property on the subspace
$\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}})$ is also
surjective. The universal property of product of sets induces a map
$\pi_{\lambda}:\prod_{\alpha^{\prime}\in\Lambda}G_{\alpha^{\prime}}\to
G_{\lambda}$.
Now we would like to show
$((\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}}),\prod_{\alpha\in\Lambda}G_{\alpha}),\\{(\tilde{p}_{\alpha},\pi_{\alpha})\\}_{\alpha\in\lambda})$
satisfies the universal property of product.
Let $(\overline{D^{\prime}},S)$ be a colored vector space and a family of map
$(\psi_{\alpha},\xi_{\alpha}):(\overline{D^{\prime}},S)\to(\overline{C_{\alpha}},G_{\alpha})$
for each $\alpha$ in a family $\Lambda$ of a colored vector spaces
$(\overline{C_{\alpha}},G_{\alpha})$. We have
$\psi_{\alpha}({{}^{g}\overline{D^{\prime}}{}^{h}})\subset{{}^{\xi_{\alpha}(g)}\overline{C_{\alpha}}{}^{\xi_{\alpha}(h)}}$
and $\xi_{\alpha}(S)\subset G_{\alpha}$. There are three observations:

  * •

The construction of product of sets implies that $\xi_{\alpha}(S)\subset
G_{\alpha}$ for $\alpha\in\lambda$ induces a morphism of sets
$\xi(S)\subset\prod_{\alpha\in\Lambda}G_{\alpha}$. The map $\xi$ sends $s\in
S$ to a sequence of elements $(\xi_{\alpha}(s))_{{\alpha\in\Lambda}}$.

  * •

The construction of product of vector spaces implies that
$\psi_{\alpha}(\overline{D^{\prime}})\subset\overline{C_{\alpha}}$ induces a
morphism of vector spaces
$\phi:\overline{D^{\prime}}\to\prod_{\alpha\in\lambda}\overline{C}_{\alpha}$.
The map sends $d\in\overline{D^{\prime}}$ to a sequence of vectors
$(\phi_{\alpha}(d))_{{\alpha\in\Lambda}}$

  * •

It is an important observation that the “homogeneous” map
$\phi:\overline{D^{\prime}}\to\prod_{\alpha\in\Lambda}\overline{C_{\alpha}}$
sends $d\in{{}^{g}\overline{D^{\prime}}{}^{h}}$ to
$(\phi_{\alpha}(d))_{{\alpha\in\Lambda}}\in\prod_{\alpha\in\Lambda}{{}^{\xi_{\alpha}(g)}\overline{C_{\alpha}}{}^{\xi_{\alpha}(h)}}$
because
$\psi_{\alpha}(d)\in{{}^{\xi_{\alpha}(g)}\overline{C_{\alpha}}{}^{\xi_{\alpha}(h)}}$.
Moreover, ${{}^{\xi_{\alpha}(g)}\overline{C_{\alpha}}{}^{\xi_{\alpha}(h)}}$ is
in the image of $\tilde{p}_{\alpha}$.

Therefore, the image of $\phi$ is in the colored vector spaces
$\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}})$ and preserves the
degrees based on the degree map $\xi$. It follows the family of maps
$\\{\phi_{\alpha},\xi_{\alpha}\\}_{\alpha\in\lambda}$ induces a map
$(\phi|_{\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}})},\xi)$ from
$(\overline{D^{\prime}},S)$ to
$(\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}}),\prod_{\alpha\in\Lambda}G_{\alpha})$.
The uniqueness of
$(\phi|_{\mathbb{G}(\prod_{\alpha\in\Lambda}\overline{C_{\alpha}})},\xi)$
follows from the fact that $\xi$ and $\phi$ are unique up to unique
isomorphism because of universal properties of products of sets and vector
spaces respectively. ∎

######  Remark 3.11.

The infinite product of vector spaces presents several subtleties. One notable
instance is the existence of a non-trivial topology: while many vectors in the
product can be expressed as an infinite sum of a set of vectors, not all
infinite sums of vectors may belong to this product. This issue is reflected
in the fact that $\oplus_{i}\prod_{j}M_{i,j}\ncong\prod_{j}\oplus_{i}M_{i,j}$,
and the usual grading cannot be applied. To address this, we are motivated by
the concept of the graded product of $\mathbb{N}$-graded modules. By
leveraging the property of homogeneous maps, $\oplus_{i}\prod_{j}M_{i,j}$
becomes the appropriate choice.

######  Theorem 3.12.

The category of reduced colored coalgebras over a field has all small
coequalizers and small products

###### Proof.

We start with the coeqaulizer first. We would like to coequalizer the
morphisms $(f,a),(g,b):(\overline{C},G)\to(\overline{D},S)$. It begins with
the three observations:

  * •

In the category of sets, there exists the coequalizer (up to unique
isomorphism) of morphisms of sets $a,b:G\to S$, which is a quotient set
$\frac{S}{\sim}$ and the equivalence relation $\sim$ is generated by $a(x)\sim
b(x)$ for all $x\in G$. Let the natural map from $S$ to $\frac{S}{\sim}$ be
$p$.

  * •

$\overline{D}$ can be endowed with a new bigrading based on $\frac{S}{\sim}$.
We set ${}^{[g]}\overline{D}{}^{[h]}$ equal $\bigoplus\limits_{s,t\in
S;p(s)=p(g),p(t)=p(h)}{{}^{g}\overline{D}{}^{h}}$ for $[g]=p(g),g\in S$. The
comultiplication is compatible with the new bigrading: if
$\Delta(^{g}{\overline{D}}{}^{h})\subset\sum_{t\in
S}^{g}{\overline{D}}{}^{t}\otimes^{t}{\overline{D}}{}^{h}$, then
$\Delta(^{[g]}{\overline{D}}{}^{[h]})\subset\sum_{t\in
S}{{}^{[g]}{\overline{D}}{}^{[t]}}\otimes{{}^{[t]}{\overline{D}}{}^{[h]}}=\sum_{[t]\in\frac{S}{\sim}}{{}^{[g]}{\overline{D}}{}^{[t]}}\otimes{{}^{[t]}{\overline{D}}{}^{[h]}}$
because ${}^{s}{\overline{D}}{}^{t}\subset{{}^{[s]}{\overline{D}}{}^{[t]}}$
for any $s,t\in S$ by construction.

  * •

$f(x)-g(x)\in{}^{[a(g)]}\overline{D}{}^{[a(h)]}$ for
$x\in{}^{g}\overline{C}{}^{h}$ and $p(a(g))=p(b(g))=[a(g)]$
($p(a(h))=p(b(h))=[a(h)]$). It implies that

${\rm
Im}\,(f-g)=\bigoplus_{[s],[t]\in\frac{S}{\sim}}({}^{[s]}\overline{D}{}^{[t]}\cap{\rm
Im}\,(f-g))$

i.e. it is generated by homogeneous elements. $f$ and $g$ that restrict to
${}^{g}\overline{C}{}^{h}$ is coequalized in
$\frac{{}^{[a(g)]}\overline{D}{}^{[a(h)]}}{{}^{[a(g)]}\overline{D}{}^{[a(h)]}\cap{\rm
Im}\,(f-g)}$. In addition, ${\rm Im}\,(f-g)$ is a graded coideal
$\bigoplus_{\text{cond}_{2}}{{}^{s^{\prime}}\overline{C}{}^{t^{\prime}}}$. The
check is the same as in the case of normal coalgebra, but we do not need to
check counit condition.

We claim
$(\bigoplus_{[s],[t]\in\frac{S}{\sim}}\frac{{}^{[s]}\overline{D}{}^{[t]}}{{}^{[s]}\overline{D}{}^{[t]}\cap{\rm
Im}\,(f-g)},(\pi,p))$ is the coequalizer (up to unique isomorphism) of $(f,a)$
and $(g,b)$ where $\pi$ is the natural projection which is homogeneous
according to $p$. Suppose there is another $((\overline{B},Q),(h,c))$ that
coequalizes the two morphisms. On the level of set, there exists a unique
morphism $\tilde{c}$ from $\frac{S}{\sim}\to Q$ s.t. $\tilde{c}\circ p=c$.

${G}$${S}$${\frac{S}{\sim}}$${Q}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{p}$$\scriptstyle{c}$$\scriptstyle{\exists!\tilde{c}}$

This induces the following diagram of homogeneous maps on the level of vector
spaces. Fixing two element $[s],[t]$ in $\frac{S}{\sim}$. Name condition 1 to
be $p(a(m))=p(b(m))=[s]$, $p(a(n))=p(b(n))=[t]$ and $m,n\in G$. Likewise, name
the condition 2 to be $p(s^{\prime})=[s]$, $p(t^{\prime})=[t]$ and
$s^{\prime},t^{\prime}\in S$. The goal is to simplify the writing under the
plus sign. There exists a coequalizer on the level of vector spaces.

${\bigoplus_{\text{cond}_{1}}{{}^{m}\overline{C}{}^{n}}}$${\bigoplus_{\text{cond}_{2}}{{}^{s^{\prime}}\overline{D}{}^{t^{\prime}}}}$${{\frac{{}^{[s]}\overline{D}{}^{[t]}}{{\rm
Im}\,(f-g)_{\text{res}}}}}$${{{}^{c(s)}\overline{B}{}^{c(t)}}}$$\scriptstyle{(f,a)_{\text{res}}}$$\scriptstyle{(g,b)_{\text{res}}}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{(h,c)}$$\scriptstyle{\tilde{h}_{[s],[t]}}$

Note
$\bigoplus_{\text{cond}_{2}}{{}^{s^{\prime}}\overline{D}{}^{t^{\prime}}}={{}^{[s]}\overline{D}{}^{[t]}}$
as vector spaces (but not colored ones). Thus the natural projection
$\tilde{\pi}$ is in fact the restriction of $(\pi,p)$. The dash arrow is
unique due to isomorphism theorems of vector spaces; it is defined as
$\tilde{h}_{[s],[t]}(\tilde{\pi}(s))=h(s)$ (remember $\tilde{\pi}$ is
surjective). ${\rm Im}\,(f-g)_{\text{res}}$ is the image of $f-g$ that
restricts on $\bigoplus_{\text{cond}_{1}}{{}^{m}\overline{C}{}^{n}}$. Because
of previous observations, ${\rm Im}\,(f-g)_{\text{res}}$ is Furthermore,

$\frac{{}^{[s]}\overline{D}{}^{[t]}}{{}^{[s]}\overline{D}{}^{[t]}\cap{\rm
Im}\,(f-g)}=\frac{{}^{[s]}\overline{D}{}^{[t]}}{{\rm Im}\,(f-g)_{\text{res}}}$

Then glue $\tilde{h}_{[s],[t]}$ for each element $[s],[t]\in\frac{S}{\sim}$;
it yields
$(\tilde{h},\tilde{c})=\bigoplus_{[s],[t]\in\frac{S}{\sim}}\tilde{h}_{[s],[t]}$
as a morphism between colored vector spaces. Since ${\rm Im}\,(f-g)$ is an
coideal, this morphism is furthermore a morphism between reduced colored
coalgebras, which is unique because $\tilde{c}$ is unique on the level of sets
and $\tilde{c}$ is unqieu on the level of vector spaces (and respect the
bigrading). It shows
$(\bigoplus_{[s],[t]\in\frac{S}{\sim}}\frac{{}^{[s]}\overline{D}{}^{[t]}}{{}^{[s]}\overline{D}{}^{[t]}\cap{\rm
Im}\,(f-g)},(\pi,p))$ is the coequalizer between $(f,a)$ and $(g,b)$.

Next, the idea of constructing the product is the same as the one in theorem
1.1 in [Ago11b], but there are some modification to the argument: Let
$(\overline{C_{i}},G_{i})_{i\in I}$ be a a family of reduced colored
coalgebras and (Lemma 3.10) the associated reduced cofree colored coalgebra
$\overline{C(\mathbb{G}(\prod_{i\in I}\overline{C_{i}}),\prod_{i\in I}G_{i})}$
over the colored vector spaces $\prod_{i\in I}\overline{C_{i}}$ where we
forget the coalgebra structure of $\overline{C_{i}}$. The rest follows
[Ago11b]. Let $(D,\prod_{i\in I}G_{i})$ be the sum of all subcoalgebras
$(E,\prod_{i\in I}G_{i})$ of $\overline{C(\mathbb{G}(\prod_{i\in
I}\overline{C_{i}}),\prod_{i\in I}G_{i})}$ such that
$(\tilde{p_{i}},\pi_{i})\circ(\pi_{\mathbb{G}(\prod_{i\in
I}\overline{C_{i}})},id)\circ(j_{E},id)$ where $(j_{E},id)$ is the canonical
inclusion from $E$ to $\mathbb{G}(\prod_{i\in I}\overline{C_{i}})$. Then one
can complete the proof by following the same method in [Ago11b] to check

$((D,\prod_{i\in
I}G_{i}),((\tilde{p_{i}},\pi_{i})\circ(\pi_{\mathbb{G}(\prod_{i\in
I}\overline{C_{i}})},id)\circ(j_{E},id))_{i\in I})$

satisfies the universal property and thus the product of the family $I$ in the
category of reduced colored coalgebras.

∎

Lastly,

######  Corollary 3.4.

The category of pointed coalgebras with a splitting (or equivalently, the
category of reduced colored coalgebras) has all small limits and colimits

######  Remarks 3.5.

Our argument also works if the underlying $R$-module is either
$\mathbb{N}$-graded or differential $\mathbb{N}$-graded. First, we need to
endow $C[G]$ with a trivial graded structure that it lies only in zero degree
and trivial differentials. Second, every morphisms considered in the paper
should be also homogeneous and commutes with differentials.

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