# Categorical equivalences for Hopf trusses and their modules

# Categorical equivalences for Hopf trusses and their modules

###### Abstract.

In this paper we introduce the notion of generalized invertible 1-cocycle in a
strict braided monoidal category ${\sf C}$, and we prove that the category of
Hopf trusses is equivalent to the category of generalized invertible
1-cocycles. On the other hand, we also introduce the notions of module for a
Hopf truss and for a generalized invertible 1-cocycle. We prove some
functorial results involving these categories of modules and we show that the
category of modules associated to a generalized invertible 1-cocycle is
equivalent to a category of modules associated to a suitable Hopf truss.
Finally, assuming that in ${\sf C}$ we have equalizers, we introduce the
notion of Hopf-module in the Hopf truss setting and we obtain the Fundamental
Theorem of Hopf modules associated to a Hopf truss.

Ramón González Rodríguez1 and Ana Belén Rodríguez Raposo2.

1 [https://orcid.org/0000-0003-3061-6685].

CITMAga, 15782 Santiago de Compostela, Spain.

Universidade de Vigo, Departamento de Matemática Aplicada II, E. E.
Telecomunicación, E-36310 Vigo, Spain.
email<EMAIL_ADDRESS>
2 [https://orcid.org/0000-0002-8719-5159]

Universidade de Santiago de Compostela, Departamento de Didácticas Aplicadas,
Facultade C. C. Educación, E-15782 Santiago de Compostela, Spain.
email<EMAIL_ADDRESS>
Keywords: Braided monoidal category, skew truss, Hopf truss, generalized
invertible 1-cocycle, module, Hopf-module.

2020 Mathematics Subject Classification: 18M15, 16T05.

##  1\. Introduction

The notion of skew brace was introduced recently in [6]. This algebraic object
consists of two different group structures $(T,\diamond)$ and $(T,\circ)$ on
the same set $T$ and that satisfy $\forall a,b,c\in T$ the compatibility
condition

$a\circ(b\diamond c)=(a\circ b)\diamond a^{\diamond}\diamond(a\circ c)$

where $a^{\diamond}$ denotes the inverse with respect to $\diamond$. On the
other hand, Hopf braces were introduced in [2] as the linearisation of skew
braces and then consists of two structures $(H_{1},H_{2})$ of Hopf algebras
defined on the same object that share a common coalgebra structure. As a
consequence of the compatibility condition, the Hopf algebra $H_{1}$ can be
endowed with a structure of module algebra over $H_{2}$. The relevance of this
structure comes through bijective 1-cocycles $\sigma:H_{1}\to B$, where $B$ is
a Hopf algebra that acts on $H_{1}$ (see [2]). In fact, the category of Hopf
braces with fixed $H_{1}$ is equivalent to the category of invertible
1-cocycles $\sigma:H_{1}\to B$ (see [2, Theorem 1.12]). Thus, Hopf braces are
nothing more than coalgebra isomorphisms between Hopf algebras that share the
underlying coalgebra and related by a module algebra structure.

In [3] T. Brzeziński introduces the notion of skew truss as a generalization
of the notion of skew brace in the following way: A skew truss consist of a
group structure $(T,\diamond)$ and a semigroup structure $(T,\circ)$ defined
on the same set $T$, and a map (called the cocycle of the skew truss)
$\omega:T\to T$ that satisfy $\forall a,b,c\in T$:

$a\circ(b\diamond c)=(a\circ b)\diamond\omega(a)^{\diamond}\diamond(a\circ c)$

where $\omega(a)^{\diamond}$ is the inverse in $(T,\diamond)$. This object can
be linearised to obtain the so-called Hopf truss $(H_{1},H_{2},\sigma)$ that
consist of a Hopf algebra $H_{1}$ and a non-unital bialgebra $H_{2}$ with the
same underlying coalgebra structure, and a cocycle $\sigma:H_{2}\to H_{1}$
that must satisfy the corresponding compatibility condition. Due to it,
$H_{1}$ can be endowed with a non-unital $H_{2}$-module algebra structure.
Observe that if $H_{2}$ is a Hopf algebra and $\sigma$ is the identity, the
Hopf truss is actually a Hopf brace. In the second section of this paper we
prove that there exists a relation between the categories of Hopf trusses and
skew trusses in the following way (see Theorem 2.17): The category of skew
trusses is equivalent to the full subcategory of Hopf trusses whose objects
are pointed cosemisimple Hopf trusses.

Taking into account the equivalences between the categories of Hopf braces an
invertible 1-cocycles and with the intention of extending them to the context
of Hopf trusses, in the third section of this paper we introduce the notion of
generalized invertible 1-cocycle in a strict braided monoidal category ${\sf
C}$ as a generalization of the bijective 1-cocycles proposed by I. Angiono, C.
Galindo and L. Vendramín in [2]. Thanks to this, in Theorem 3.2 we prove that
the category of Hopf trusses is equivalent to the category of generalized
invertible 1-cocycles. Moreover, in Section 4 we also introduce the notions of
module for a Hopf truss and for a generalized invertible 1-cocycle. After
doing this, in this section we prove some functorial results involving these
categories of modules and in Theorem 4.12 we show that the category of modules
associated to a generalized invertible 1-cocycle is equivalent to a category
of modules associated to a suitable Hopf truss that we can build thanks to the
generalized invertible 1-cocycle. This results are a generalization to the
Hopf truss setting of the ones proved for modules associates to Hopf braces in
[4].

On the other hand, it is well known that, if $H$ is a Hopf algebra in a
category ${\mathbb{F}}$-Vect of vector spaces over a field ${\mathbb{F}}$ and
$X$ is an object in C, the tensor product $H\otimes X$, with the action and
coaction induced by the product and the coproduct of $H$, is an object in the
category ${\sf H}$-Hopf-Mod, i.e. the category of left $H$-Hopf modules. The
objects and morphisms of this category are defined in the following way: Let
$M$ be a left $H$-module and a left $H$-comodule. In this setting, if for all
$m\in M$ and $h\in H$, we write $h.m$ for the left action and we use the
Heyenman-Sweedler notation $\rho_{M}(m)=m_{[0]}\otimes m_{[1]}$ for the
coaction, we will say that $M$ is a left $H$-Hopf module if the equality

$\rho_{M}(h.m)=h_{(1)}m_{[0]}\otimes h_{(2)}.m_{[1]}$

holds, where $\delta_{H}(h)=h_{(1)}\otimes h_{(2)}$ is the coproduct of $H$
and $h_{(1)}m_{[0]}$ is the product in $H$ of $h_{(1)}$ and $m_{[0]}$. A
morphism between two left $H$-Hopf modules is a ${\mathbb{F}}$-linear map that
is $H$-linear and $H$-colinear.

This construction introduced for $H\otimes X$ in the previous paragraph is
functorial and, as a consequence, we have a functor, called the induction
functor, $F=H\otimes-:\;$C$\;\rightarrow{\sf H}$-Hopf-Mod. Moreover, for all
$M\in H$-Hopf, the construction of subobject of coinvariants $M^{coH}=\\{m\in
M\;/\;m_{[0]}\otimes m_{[1]}=1_{H}\otimes m\\}$ also is functorial. Thus,
there exists a functor of coinvariants $G=(\;\;)^{coH}:{\sf H}$-Hopf-
Mod$\;\rightarrow$C such that $F\dashv G$. Moreover, $H\otimes M^{coH}$ and
$M$ are isomorphic in ${\sf H}$-Hopf-Mod (see [7] and [10]) and $F$ and $G$
induces an equivalence between the categories ${\sf H}$-Hopf-Mod and
${\mathbb{F}}$-Vect. The existence of the isomorphism between $H\otimes
M^{coH}$ and $M$ is the main statement of the Fundamental Theorem of Hopf
modules.

The Fundamental Theorem of Hopf modules and the categorical equivalence of the
previous paragraph remain valid for weak Hopf algebras, Hopf quasigroups, weak
Hopf quasigroups and in [5] we can find that which can also be obtained for
Hopf braces. In the last section of this paper, assuming that the braided
monoidal category ${\sf C}$ admits equalizers, we extend the previous results
to the Hopf truss setting, i.e. we introduce the notion of Hopf-module
associated to a Hopf truss, we obtain the Fundamental Theorem of Hopf modules
associated to a Hopf truss and we prove that there exists a categorical
equivalence as in the case of Hopf algebras, weak Hopf algebras, Hopf
quasigroups, weak Hopf quasigroups, Hopf braces, etc.

##  2\. Hopf trussses

Throughout this paper ${\sf C}$ denotes a strict braided monoidal category
with tensor product $\otimes$, unit object $K$ and braiding $c$. Recall that a
monoidal category is a category ${\sf C}$ together with a functor
$\otimes:{\sf C}\times{\sf C}\rightarrow{\sf C}$, called tensor product, an
object $K$ of ${\sf C}$, called the unit object, and families of natural
isomorphisms

$a_{M,N,P}:(M\otimes N)\otimes P\rightarrow M\otimes(N\otimes
P),\;\;\;r_{M}:M\otimes K\rightarrow M,\;\;\;l_{M}:K\otimes M\rightarrow M,$

in ${\sf C}$, called associativity, right unit and left unit constraints,
respectively, satisfying the Pentagon Axiom and the Triangle Axiom, i.e.,

$a_{M,N,P\otimes Q}\circ a_{M\otimes N,P,Q}=(id_{M}\otimes a_{N,P,Q})\circ
a_{M,N\otimes P,Q}\circ(a_{M,N,P}\otimes id_{Q}),$ $(id_{M}\otimes l_{N})\circ
a_{M,K,N}=r_{M}\otimes id_{N},$

where for each object $X$ in ${\sf C}$, $id_{X}$ denotes the identity morphism
of $X$ (see [9]). A monoidal category is called strict if the constraints of
the previous paragraph are identities. It is a well-known fact (see for
example [8]) that every non-strict monoidal category is monoidal equivalent to
a strict one. This lets us to treat monoidal categories as if they were strict
and, as a consequence, the results proved in a strict setting hold for every
non-strict monoidal category, for example the category ${\mathbb{F}}$-Vect of
vector spaces over a field ${\mathbb{F}}$, the category $R$-Mod of left
modules over a commutative ring $R$ , or Set the category of sets.

For simplicity of notation, given objects $M$, $N$, $P$ in ${\sf C}$ and a
morphism $f:M\rightarrow N$, we will write $P\otimes f$ for $id_{P}\otimes f$
and $f\otimes P$ for $f\otimes id_{P}$.

A braiding for a strict monoidal category ${\sf C}$ is a natural family of
isomorphisms

$c_{M,N}:M\otimes N\rightarrow N\otimes M$

subject to the conditions

$c_{M,N\otimes P}=(N\otimes c_{M,P})\circ(c_{M,N}\otimes P),\;\;c_{M\otimes
N,P}=(c_{M,P}\otimes N)\circ(M\otimes c_{N,P}).$

A strict braided monoidal category ${\sf C}$ is a strict monoidal category
with a braiding. Note that, as a consequence of the definition, the equalities
$c_{M,K}=c_{K,M}=id_{M}$ hold, for all object $M$ of ${\sf C}$. If the
braiding satisfies that $c_{N,M}\circ c_{M,N}=id_{M\otimes N},$ for all $M$,
$N$ in ${\sf C}$, we will say that ${\sf C}$ is symmetric. In this case, we
call the braiding $c$ a symmetry for the category ${\sf C}$.

######  Definition 2.1.

A monoid in ${\sf C}$ is a triple $A=(A,\eta_{A},\mu_{A})$ where $A$ is an
object in ${\sf C}$ and $\eta_{A}:K\rightarrow A$ (unit), $\mu_{A}:A\otimes
A\rightarrow A$ (product) are morphisms in ${\sf C}$ such that
$\mu_{A}\circ(A\otimes\eta_{A})=id_{A}=\mu_{A}\circ(\eta_{A}\otimes A)$ (unit
property) and $\mu_{A}\circ(A\otimes\mu_{A})=\mu_{A}\circ(\mu_{A}\otimes A)$
(associative property) hold.

Given two monoids $A=(A,\eta_{A},\mu_{A})$ and $B=(B,\eta_{B},\mu_{B})$, a
morphism $f:A\rightarrow B$ in C is an monoid morphism if
$\mu_{B}\circ(f\otimes f)=f\circ\mu_{A}$ ($f$ is multiplicative) and
$f\circ\eta_{A}=\eta_{B}$ ($f$ preserves the unit).

If $A$, $B$ are monoids in ${\sf C}$, the tensor product $A\otimes B$ is also
an algebra in ${\sf C}$ where $\eta_{A\otimes B}=\eta_{A}\otimes\eta_{B}$ and
$\mu_{A\otimes B}=(\mu_{A}\otimes\mu_{B})\circ(A\otimes c_{B,A}\otimes B).$

We will say that a monoid $A$ is commutative if $\mu_{A}\circ
c_{A,A}=\mu_{A}$.

######  Definition 2.2.

A comonoid in ${\sf C}$ is a triple ${D}=(D,\varepsilon_{D},\delta_{D})$ where
$D$ is an object in ${\sf C}$ and $\varepsilon_{D}:D\rightarrow K$ (counit),
$\delta_{D}:D\rightarrow D\otimes D$ (coproduct) are morphisms in ${\sf C}$
such that $(\varepsilon_{D}\otimes
D)\circ\delta_{D}=id_{D}=(D\otimes\varepsilon_{D})\circ\delta_{D}$ (counit
property) and $(\delta_{D}\otimes
D)\circ\delta_{D}=(D\otimes\delta_{D})\circ\delta_{D}$ (coassociative
property) hold.

If ${D}=(D,\varepsilon_{D},\delta_{D})$ and
${E}=(E,\varepsilon_{E},\delta_{E})$ are comonoids, a morphism $f:D\rightarrow
E$ in C is a comonoid morphism if $(f\otimes f)\circ\delta_{D}=\delta_{E}\circ
f$ ($f$ is comultiplicative) and $\varepsilon_{E}\circ f=\varepsilon_{D}$ ($f$
preserves the counit).

Given $D$, $E$ comonoids in ${\sf C}$, the tensor product $D\otimes E$ is a
comonoid in ${\sf C}$ where $\varepsilon_{D\otimes
E}=\varepsilon_{D}\otimes\varepsilon_{E}$ and $\delta_{D\otimes E}=(D\otimes
c_{D,E}\otimes E)\circ(\delta_{D}\otimes\delta_{E}).$

We will say that a comonoid $D$ is cocommutative if
$\delta_{D}=c_{D,D}\circ\delta_{D}.$

######  Definition 2.3.

Let $D=(D,\varepsilon_{D},\delta_{D})$ be a comonoid in C. We will say that a
morphism $g:K\rightarrow D$ is a grouplike morphism if satisfy

(1) $\delta_{D}(g)=g\otimes g$

and

(2) $\varepsilon_{D}\circ g=id_{K}.$

######  Remark 2.4.

In the category of vector spaces over a field ${\mathbb{F}}$ we can find
interesting examples of comonoids. For example, if $S$ is a set, with
${\mathbb{F}}[S]$ we will denote the free ${\mathbb{F}}$-vector space on $S$,
i.e.,

${\mathbb{F}}[S]=\displaystyle\bigoplus_{s\in S}{\mathbb{F}}s.$

This vector space has a comonoid structure determined by

(3) $\delta_{{\mathbb{F}}[S]}(s)=s\otimes
s,\;\;\;\;\varepsilon_{{\mathbb{F}}[S]}(s)=1_{\mathbb{F}}.$

Let $(C,\varepsilon_{C},\delta_{C})$ be a comonoid in ${\mathbb{F}}$-Vect. A
grouplike element $c$ of $C$ is a $c\in C$ such that the linear map
$g_{c}:{\mathbb{F}}\rightarrow C$ defined by $g_{c}(1_{{\mathbb{F}}})=c$ is a
grouplike morphism in ${\mathbb{F}}$-Vect. Therefore, $c\in C$ is a grouplike
element if (3) holds for $c$, $\varepsilon_{C}$ and $\delta_{C}$. In the
following we will denote by ${\sf G}(C)$ the set of grouplike elements of $C$
and it is well known that they are linearly independent [1, Theorem 2.1.2]. If
$S$ is a set, the comonoid ${\mathbb{F}}[S]$ is called the grouplike comonoid
of $S$ and satisfies ${\sf G}({\mathbb{F}}[S])=S$. Moreover the grouplike
comonoid of ${\sf G}(C)$ is a subcomonoid of $C$ ($D$ is a subcomonoid of $C$
if $\delta_{C}(D)\subset D\otimes D$ or, in other words, $D$ is a comonoid
with the restriction of the coproduct $\delta_{C}$ and the counit
$\varepsilon_{C}$).

A pointed comonoid in ${\mathbb{F}}$-Vect is a comonoid $C$ whose simple
subcomonoids are one-dimensional. Then, $C$ is pointed if and only if its
coradical $C_{0}$ (the sum of the simple subcomonoids of $C$) is the grouplike
comonoid of ${\sf G}(C)$, i.e., $C_{0}={\mathbb{F}}[{\sf G}(C)]$. We will say
that the comonoid $C$ is cosemisimple if $C=C_{0}$. Therefore, if $C$ is
pointed cosemisimple, $C={\mathbb{F}}[{\sf G}(C)]$. On the other hand, if $G$
is a group and $C={\mathbb{F}}[G]$, we have that $C$ is pointed and
cosemisimple. Finally, if ${\mathbb{F}}$ is algebraically closed and $C$ is
cocommutative, $C$ is pointed.

######  Definition 2.5.

Let ${D}=(D,\varepsilon_{D},\delta_{D})$ be a comonoid and let
$A=(A,\eta_{A},\mu_{A})$ be an monoid. By ${\mathcal{H}}(D,A)$ we denote the
set of morphisms $f:D\rightarrow A$ in ${\sf C}$. With the convolution
operation

$f\ast g=\mu_{A}\circ(f\otimes g)\circ\delta_{D},$

${\mathcal{H}}(D,A)$ is an monoid where the unit element is
$\eta_{A}\circ\varepsilon_{D}=\varepsilon_{D}\otimes\eta_{A}$. We will say
that $f:D\rightarrow A$ is convolution invertible if there exists $f^{-1}:D\to
A$ such that $f\ast f^{-1}=f^{-1}\ast f=\varepsilon\otimes\eta$.

######  Definition 2.6.

Let $A$ be an monoid. The pair $(M,\varphi_{M})$ is a left $A$-module if $M$
is an object in ${\sf C}$ and $\varphi_{M}:A\otimes M\rightarrow M$ is a
morphism in ${\sf C}$ satisfying $\varphi_{M}\circ(\eta_{A}\otimes M)=id_{M}$
and

$\varphi_{M}\circ(A\otimes\varphi_{M})=\varphi_{M}\circ(\mu_{A}\otimes M).$

Given two left ${A}$-modules $(M,\varphi_{M})$ and $(N,\varphi_{N})$,
$f:M\rightarrow N$ is a morphism of left ${A}$-modules if
$\varphi_{N}\circ(A\otimes f)=f\circ\varphi_{M}$ (left $A$-linearity).

Then left $A$-modules with morphisms of left $A$-modules form a category that
we will denote by $\;{}_{\sf A}$Mod.

Let $B$ an object in C such that there exists an associative product
$\mu_{B}:B\otimes B\rightarrow B$. We will say that $(M,\phi_{M})$ is a non-
unital left $B$-module if
$\phi_{M}\circ(B\otimes\phi_{M})=\phi_{M}\circ(\mu_{B}\otimes M)$. A morphism
between non-unital left $B$-modules is a left $B$-linear morphism as in the
case of morphisms for modules over a monoid. Then non-unital left $B$-modules
form a category that we will denote by $\;{}_{\sf B}$mod.

######  Definition 2.7.

A non-unital bimonoid in the category $\sf C$ is a comonoid
$(B,\varepsilon_{B},\delta_{B})$ with an associative product $\mu_{B}:B\otimes
B\rightarrow B$ such that $\mu_{B}$ is a comonoid morphism. Then the following
identities hold:

(4) $\varepsilon_{B}\circ\mu_{B}=\varepsilon_{B}\otimes\varepsilon_{B},$ (5)
$\delta_{B}\circ\mu_{B}=(\mu_{B}\otimes\mu_{B})\circ\delta_{B\otimes B}.$

A bimonoid in ${\sf C}$ is a monoid $(B,\eta_{B},\mu_{B})$ and a comonoid
$(B,\varepsilon_{B},\delta_{B})$ such that $\eta_{B}$ and $\mu_{B}$ are
comonoid morphisms. Then, (4), (5),

(6) $\varepsilon_{B}\circ\eta_{B}=id_{K},$

and

(7) $\delta_{B}\circ\eta_{B}=\eta_{B}\otimes\eta_{B}$

hold.

A morphism between non-unital bimonoids $H$ and $B$ is a morphism
$f:H\rightarrow B$ in ${\sf C}$ of comonoids and multiplicative. A morphism
between bimonoids $H$ and $B$ is a morphism $f:H\rightarrow B$ in ${\sf C}$ of
monoids and comonoids.

With the composition of morphisms in C we can define a category whose objects
are non-unital bimonoids (bimonoids) and whose morphisms are morphisms of non-
unital bimonoids (bimonoids). We denote this category by ${\sf bimod}$ (${\sf
BiMod}$).

######  Definition 2.8.

Let $H$ be a bimonoid in ${\ss C}$. If there exists a morphism
$\lambda_{H}:H\rightarrow H$ in ${\sf C}$, called the antipode of $H$,
satisfying that $\lambda_{H}$ is the inverse of $id_{H}$ in
${\mathcal{H}}(H,H)$, i.e.,

(8) $id_{H}\ast\lambda_{H}=\eta_{H}\circ\varepsilon_{H}=\lambda_{H}\ast
id_{H},$

we say that $H$ is a Hopf monoid.

A morphism of Hopf monoids is an bimonoid morphism. With the composition of
morphisms in C we can define a category whose objects are Hopf monoids and
whose morphisms are morphisms of Hopf monoids. We denote this category by
${\sf Hopf}$.

######  Remark 2.9.

If $H$ is a Hopf monoid, the antipode is antimultiplicative and
anticomultiplicative

$\lambda_{H}\circ\mu_{H}=\mu_{H}\circ(\lambda_{H}\otimes\lambda_{H})\circ
c_{H,H},\;\;\;\;\delta_{H}\circ\lambda_{H}=c_{H,H}\circ(\lambda_{H}\otimes\lambda_{H})\circ\delta_{H},$

and leaves the unit and counit invariant, i.e.,

$\lambda_{H}\circ\eta_{H}=\eta_{H},\;\;\varepsilon_{H}\circ\lambda_{H}=\varepsilon_{H}.$

A Hopf monoid is commutative if it is commutative as monoid and cocommutative
if it is cocommutative as comonoid. It is easy to see that in both cases
$\lambda_{H}\circ\lambda_{H}=id_{H}$.

Note that, if $f:H\rightarrow D$ is a Hopf monoid morphism the following
equality holds:

(9) $\lambda_{D}\circ f=f\circ\lambda_{H}.$

######  Remark 2.10.

In the category Set the tensor product is the cartesian product and the unit
element is a set $\\{{\star}\\}$ with an unique element $\star$. Taking all
this into account, we have that a set $T$ is a non-unital bimonoid in Set if
and only if $T$ is a semigroup in Set with product $\diamond$. If
$(T,\diamond)$ is a semigroup the non-unital bimonoid structure is the
following: the product is the one induced by $\diamond$, the coproduct is
defined by $\delta_{T}(a)=(a,a)$ and the counit by $\varepsilon_{T}(a)=\star$.
Moreover, $T$ is a bimonoid in Set if and only if $T$ is a monoid in Set with
product $\diamond$ and unit $1_{\diamond}$. In this case we define the non-
unital bimonoid structure as in the case of semigroups and the unit is defined
by $\eta_{T}(\star)=1_{\diamond}$. Finally, $T$ is a Hopf monoid in Set if and
only if it is a group. The bimonoid structure is defined as for monoids and
the antipode by $\lambda_{T}(a)=a^{\diamond}$ where $a^{\diamond}$ is the
inverse of $a$ in $T$.

On the other hand, if ${\mathbb{F}}$ is a field and $(T,\diamond)$ is a
semigroup in Set, the direct sum

${\mathbb{F}}[T]=\displaystyle\bigoplus_{a\in T}{\mathbb{F}}a$

is a non-unital bimonoid in ${\mathbb{F}}$-Vect where $\mu_{{\mathbb{F}}[T]}$
is the unique linear map such that $\mu_{{\mathbb{F}}[T]}(a\otimes
b)=a\diamond b$ and the comonoid structure is the one defined in (3). Also, if
$(T,\diamond)$ is a monoid in Set with unit $1_{\diamond}$, ${\mathbb{F}}[T]$
is a bimonoid in ${\mathbb{F}}$-Vect, where $\eta_{{\mathbb{F}}[T]}$ is the
unique linear map such that
$\eta_{{\mathbb{F}}[T]}(1_{\mathbb{F}})=1_{\diamond}$, and the non-unital
bimonoid structure is the one introduced for semigroups. Finally, if
$(T,\diamond)$ is a group, ${\mathbb{F}}[T]$ is a Hopf monoid in
${\mathbb{F}}$-Vect with the previous bimonoid structure and antipode the
unique linear map satisfying $\lambda_{{\mathbb{F}}[T]}(a)=a^{\diamond}$,
where $a^{\diamond}$ denotes the inverse of $a$.

Let $(T,\diamond)$, $(S,\circ)$ be semigroups in Set and let $f:T\rightarrow
S$ be a semigroup morphism in Set. Then, if ${\mathbb{F}}[f]$ denotes the
linear extension of $f$, ${\mathbb{F}}[f]$ is a non-unital bimonoid morphism
between ${\mathbb{F}}[T]$ and ${\mathbb{F}}[S]$ in ${\mathbb{F}}$-Vect. The
same property holds for a morphisms of monoids $f$ in Set, i.e.
${\mathbb{F}}[f]$ is a bimonoid morphism, and for a morphism $f$ of groups,
i.e. ${\mathbb{F}}[f]$ is a group morphism.

Therefore, if sGpr is the category of semigroups in Set, Mon is the category
of monoids in Set and Gpr denotes the category of groups, there exists three
functors

${\sf L}_{sg}:{\sf sGpr}\;\rightarrow{\sf bimon},\;\;\;\;\;{\sf L}_{m}:{\sf
Mon}\;\rightarrow{\sf BiMon},\;\;\;\;\;{\sf L}_{g}:{\sf Gpr}\;\rightarrow{\sf
Hopf},$

where bimon is the category of non-unital bimonoids in ${\mathbb{F}}$-Vect,
BiMon is the category of bimonoids in ${\mathbb{F}}$-Vect and, finally, Hopf
denotes the category of Hopf monoids in ${\mathbb{F}}$-Vect. In this setting,
the functor Lm is the restriction of Lsg to the category of monoids in ${\sf
Set}$ and Lg is the restriction of Lm to the category of groups.

In any case non-unital bimonoids, bimonoids and Hopf monoids in
${\mathbb{F}}$-Vect are comonoids. Taking this into account, if $B$ is an
object in ${\sf bimon}$, it is possible to define a semigroup structure on
${\sf G}(B)$, with product $\diamond$ induced by $\mu_{B}$ ($a\diamond
b=\mu_{B}(a\otimes b)$). Moreover, if $f:B\to B^{\prime}$ is a morphism of
non-unital bimonoids, the image of the restriction of $f$ to ${\sf G}(B)$ lies
into ${\sf G}(B^{\prime})$. Thus we have a functor ${\sf G}_{sg}$ between
${\sf bimon}$ and ${\sf sGpr}$ defined by ${\sf G}_{sg}(B)={\sf G}(B)$ on
objects and by ${\sf G}_{sg}(f)={\sf G}(B)$ on morphisms. Also, if $A$ is an
object in ${\sf BiMon}$ it is possible to define a monoid structure on ${\sf
G}(A)$, with product $\diamond$ as in the case of non-unital bimonoids and
unit $1_{\diamond}$ induced by $\eta_{A}$
($1_{\diamond}=\eta_{A}(1_{\mathbb{F}})$). Moreover, if $f:A\to A^{\prime}$ is
a morphism of bimonoids, the image of the restriction of $f$ to ${\sf G}(A)$
lies into ${\sf G}(A^{\prime})$. Then, we have a new functor ${\sf G}_{m}$
between ${\sf BiMon}$ and ${\sf Mon}$ defined by ${\sf G}_{m}(A)={\sf G}(A)$
on objects and by ${\sf G}_{m}(f)={\sf G}(f)$ on morphisms. Finally, If $H$ is
a Hopf monoid, ${\sf G}(H)$ is a group where the monoid structure is the one
defined for bimonoids and the inverse is defined by the antipode, i.e., the
inverse of $h\in{\sf G}(H)$ is $\lambda_{H}(h)$. As in the two previous cases,
this construction works well with morphisms of Hopf monoids and, as a
consequence, there exists a functor ${\sf G}_{g}$ between ${\sf Hopf}$ and
${\sf Gpr}$ defined by ${\sf G}_{g}(H)={\sf G}(H)$ on objects and by ${\sf
G}_{g}(f)={\sf G}(f)$ on morphisms.

It is true that ${\sf L}_{sg}\dashv{\sf G}_{sg}$, ${\sf L}_{m}\dashv{\sf
G}_{m}$ and ${\sf L}_{g}\dashv{\sf G}_{g}$. In the three adjunctions the unit
of every one of them is the identity. Thus, the first adjoint pair induces an
equivalence of categories between sGpr and the full subcategory of ${\sf
bimon}$ of all pointed cosemisimple non-unital bimonoids in
${\mathbb{F}}$-Vect, the second one induces an equivalence of categories
between Mon and the full subcategory of ${\sf BiMon}$ of all pointed
cosemisimple bimonoids in ${\mathbb{F}}$-Vect, and the third one induces an
equivalence of categories between Gpr and the full subcategory of ${\sf Hopf}$
of all pointed cosemisimple non-unital Hopf monoids in ${\mathbb{F}}$-Vect.
Also we have this commutative diagrams

where ${\sf I_{m}}$ and ${\sf I_{b}}$ and ${\sf I_{g}}$ denote the
corresponding inclusion functors.

######  Definition 2.11.

Let $B$ a bimonoid and let $A$ be a monoid in C. We will say that
$(A,\phi_{A})$ is a left $B$-module monoid if it is a left $B$-module with
action $\varphi_{A}:B\otimes A\rightarrow A$ such that

(10) $\varphi_{A}\circ(B\otimes\eta_{A})=\varepsilon_{B}\otimes\eta_{A}$

and

(11)
$\varphi_{A}\circ(B\otimes\mu_{A})=\mu_{A}\circ(\varphi_{A}\otimes\varphi_{A})\circ(B\otimes
c_{B,A}\otimes A)\circ(\delta_{B}\otimes A\otimes A)$

hold.

If $B$ is a non-unital bimonoid, we will say that $(A,\phi_{A})$ is a non-
unital left $B$-module monoid if $(A,\phi_{A})$ is a non-unital left
$B$-module and (10) and (11) hold.

The notion of Hopf truss was introduced by T. Brzeziński in [3] in the
category ${\mathbb{F}}$-Vect as the linearisation of the notion of skew truss.
In the monoidal setting the definition of Hopf truss is the following:

######  Definition 2.12.

Let $(H,\varepsilon_{H},\delta_{H})$ be a comonoid in C. Assume that there are
a monoid structure $(H,\eta_{H},\mu_{H}^{1})$, a product $\mu_{H}^{2}:H\otimes
H\rightarrow H$ and two endomorphism of $H$ denoted by $\lambda_{H}$ and
$\sigma_{H}$. We will say that

$(H,\eta_{H},\mu_{H}^{1},\mu_{H}^{2},\varepsilon_{H},\delta_{H},\lambda_{H},\sigma_{H})$

is a Hopf truss if:

  * (i)

$H_{1}=(H,\eta_{H},\mu_{H}^{1},\varepsilon_{H},\delta_{H},\lambda_{H})$ is a
Hopf monoid in C.

  * (ii)

$H_{2}=(H,\mu_{H}^{2},\varepsilon_{H},\delta_{H},)$ is a non-unital bimonoid
in C.

  * (iii)

The morphism $\sigma_{H}$ is a comonoid morphism and the following equality
holds:

$\mu_{H}^{2}\circ(H\otimes\mu_{H}^{1})=\mu_{H}^{1}\circ(\mu_{H}^{2}\otimes\Gamma_{H_{1}}^{\sigma_{H}})\circ(H\otimes
c_{H,H}\otimes H)\circ(\delta_{H}\otimes H\otimes H),$

where

$\Gamma_{H_{1}}^{\sigma_{H}}=\mu_{H}^{1}\circ((\lambda_{H}\circ\sigma_{H})\otimes\mu_{H}^{2})\circ(\delta_{H}\otimes
H).$

We will say that a Hopf truss is cocommutative if the comonoid
$(H,\varepsilon_{H},\delta_{H})$ is cocommutative.

Note that, a Hopf truss is a Hopf brace in the sense of I. Angiono, C. Galindo
and L. Vendramin (see [2]) if $\sigma_{H}$ is the identity and there exists a
morphism $S_{H}:H\rightarrow H$ such that
$H_{2}=(H,\eta_{H},\mu_{H}^{2},\varepsilon_{H},\delta_{H},S_{H})$ is a Hopf
monoid.

######  Notation 2.13.

Given a Hopf truss, we will denote it by
${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$. The morphism $\sigma_{H}$ is called
the cocycle of ${\mathbb{H}}$.

The proofs that we can find in Section 6 of [3] can be replicated in the
braided monoidal setting since they do not depend on the symmetry of the
category ${\mathbb{F}}$-Vect. Then we have the following properties: By [3,
Lemma 6.2 ] the cocycle $\sigma_{H}$ of a Hopf truss ${\mathbb{H}}$ in ${\sf
C}$ is fully determined by $\eta_{H}$ and the product $\mu_{H}^{2}$ in the
following way:

(12) $\sigma_{H}=\mu_{H}^{2}\circ(H\otimes\eta_{H}).$

Then, as a consequence of the associativity for the product $\mu_{H}^{2}$, we
have that

(13) $\sigma_{H}\circ\mu_{H}^{2}=\mu_{H}^{2}\circ(H\otimes\sigma_{H})$

holds. Finally, by [3, Theorem 6.5] we know that the monoid $H_{1}$ is a non-
unital left $H_{2}$-module monoid for the action
$\Gamma_{H_{1}}^{\sigma_{H}}.$

######  Definition 2.14.

Given two Hopf trusses ${\mathbb{H}}$ and ${\mathbb{B}}$, a morphism $f$
between the two underlying objects is called a morphism of Hopf trusses if
$f:H_{1}\rightarrow B_{1}$ is a Hopf monoid morphism and $f:H_{2}\rightarrow
B_{2}$ is a morphism of non-unital bimonoids. Then by [3, Proposition 6.8]

(14) $\sigma_{B}\circ f=f\circ\sigma_{H}$

holds.

Hopf trusses together with morphisms of Hopf trusses form a category which we
denote by HTr. It is obvious that Hopf braces with morphisms of Hopf braces
form a category which we denote by HBr that is a subcategory of ${\sf HTr}$.

######  Remark 2.15.

Following [3] a skew truss is a set $T$ with two binary operations
$\diamond_{1}$ and $\diamond_{2}$ and a map $\omega_{T}:T\rightarrow T$
(called the cocycle) such that the pair $T_{1}=(T,\diamond_{1})$ is a group
with unit $1_{\diamond_{1}}$, $T_{2}=(T,\diamond_{2})$ is a semigroup and the
following identity

(15)
$a\diamond_{2}(b\diamond_{1}c)=(a\diamond_{2}b)\diamond_{1}\omega(a)^{\diamond_{1}}\diamond_{1}(a\diamond_{2}c)$

holds for all $a,b,c\in T$. We will denote the previous skew truss by
${\mathbb{T}}=(T_{1},T_{2},\omega_{T})$. Then, if ${\mathbb{T}}$ is a skew
truss ${\mathbb{T}}$ is a Hopf truss in the category ${\sf Set}$. A morphism
$f$ between two skew trusses ${\mathbb{T}}=(T_{1},T_{2},\omega_{T})$ and
${\mathbb{S}}=(S_{1},S_{2},\omega_{S})$ is a map $f$ between the two
underlying sets such that $f$ is a morphism of groups between $T_{1}$ and
$S_{1}$ and of semigroups between $T_{2}$ and $S_{2}$. Then, by [3,
Proposition 2.8], the equality $\omega_{S}\circ f=f\circ\omega_{T}$ holds.
With SkTr we will denote the category of skew trusses.

Let ${\mathbb{F}}$ be a field. Let ${\mathbb{T}}$ be a skew truss. Then
${\mathbb{F}}[T]$ admits a structure of Hopf truss in ${\mathbb{F}}$-Vect
where the products, coproduct, counit and antipode are defined as in Remark
2.10 and the comonoid morphism $\sigma_{{\mathbb{F}}[T]}$ is the linear
extension of $\omega_{T}$. Also, if $f$ is a morphism between skew trusses,
its linear extension is a morphism of Hopf trusses. As a consequence, there
exists a functor

${\sf P}_{st}:{\sf SkTr}\rightarrow{\sf HTr}$

given by

${\sf P}_{st}({\mathbb{T}})=({\sf L}_{g}(T_{1}),{\sf
L}_{sg}(T_{2}),\sigma_{{\mathbb{F}}[T]})$

where ${\sf L}_{g}$, ${\sf L}_{sg}$ are the functors defined in Remark 2.10
and $\lambda_{{\mathbb{F}}[T]}=(\;)^{\diamond_{1}}$.

Let ${\mathbb{F}}$ be a field and ${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$ a
Hopf truss in ${\mathbb{F}}$-Vect. Let ${\sf G}(H)$ be the set of grouplike
elements of the comonoid $(H,\varepsilon_{H},\delta_{H})$. In light of Remark
2.10, on the one hand we have that ${\sf G}(H)$ is a group. On the other hand,
${\sf G}(H)$ is a semigroup. As $\sigma_{H}$ is a comonoid morphism,
$\sigma_{H}(h)\in{\sf G}(H)\;\forall\;h\in{\sf G}(H)$. If we denote by
$\omega_{\sigma_{H}}$ the restriction of $\sigma_{H}$ to ${\sf G}(H)$, we have
that $({\sf G}_{g}(H_{1}),{\sf G}_{sg}(H_{2}),\omega_{\sigma_{H}})$ is an
object in SkTr because equality 15 holds as a consequence of (iii) of
Definition 2.12. By the functoriality of ${\sf G}_{g}$ and ${\sf G}_{sg}$ any
morphism $f:{\mathbb{H}}\to{\mathbb{B}}$ of Hopf trusses induces a morphism of
skew trusses between $({\sf G}_{g}(H_{1}),{\sf
G}_{sg}(H_{2}),\omega_{\sigma_{H}})$ and $({\sf G}_{g}(B_{1}),{\sf
G}_{sg}(B_{2}),\omega_{\sigma_{B}})$ defined by ${\sf G}_{g}(f)$ or by ${\sf
G}_{sg}(f)$. Therefore

${\sf R}_{ht}:{\sf HTr}\rightarrow{\sf SkTr}$

defined by ${\sf R}_{ht}({\mathbb{H}})=({\sf G}_{g}(H_{1}),{\sf
G}_{sg}(H_{2}),\omega_{\sigma_{H}})$ on objects and by ${\sf R}_{ht}(f)={\sf
G}_{sg}(f)$ on morphisms is a functor between HTr and SkTr.

######  Definition 2.16.

Let ${\mathbb{F}}$ be a field and let ${\mathbb{H}}$ be a Hopf truss in ${\sf
Vect}_{\mathbb{F}}$. We will say that ${\mathbb{H}}$ is pointed cosemisimple
if the its subjacent coalgebra $(H,\varepsilon_{H},\delta_{H})$ is pointed and
cosemisimple.

######  Theorem 2.17.

Let ${\sf P}_{st}$ and ${\sf R}_{ht}$ be the functors defined in the previous
remark. Then, ${\sf P}_{st}\dashv{\sf R}_{ht}$ and this adjunction induces an
equivalence of categories between SkTr and the full subcategory of HTr of all
pointed cosemisimple Hopf trusses.

###### Proof.

Let ${\mathbb{T}}$ be an object in SkTr and let ${\mathbb{H}}$ be an object in
HTr. We define a map

$\Gamma_{{\mathbb{T}},{\mathbb{H}}}:{\sf Hom}_{\sf HTr}({\sf
P}_{st}({\mathbb{T}}),{\mathbb{H}})\rightarrow{\sf Hom}_{\sf
SkTr}({\mathbb{T}},{\sf R}_{ht}({\mathbb{H}}))$

in the following way: if $q:{\sf P}_{st}({\mathbb{T}})\rightarrow{\mathbb{H}}$
is a morphism in ${\sf Hom}_{\sf HTr}({\sf
P}_{st}({\mathbb{T}}),{\mathbb{H}})$,
$\Gamma_{{\mathbb{T}},{\mathbb{H}}}(q):T\rightarrow{\sf G(H)}$ is the map
defined by $\Gamma_{{\mathbb{T}},{\mathbb{H}}}(q)(a)=q(a)$. Then, using that
$q$ Hopf monoid morphism between ${\sf L}_{g}(T_{1})$ and $H_{1}$ and a non-
unital bimonoid morphism between ${\sf L}_{sg}(T_{2})$ and $H_{2}$, we obtain
that $\Gamma_{{\mathbb{T}},{\mathbb{H}}}(q)$ is a group morphism between
$T_{1}$ and ${\sf G}_{g}(H_{1})$ and a semigroup morphism between $T_{2}$ and
${\sf G}_{sg}(H_{2})$. Therefore, $\Gamma_{{\mathbb{T}},{\mathbb{H}}}(q)$ is
well defined. It is easy to show that it is natural in both components and
injective. Moreover, it is surjective because, if
$h:{\mathbb{T}}\rightarrow{\sf R}_{ht}({\mathbb{H}})$ is a morphism in SkTr,
$h=\Gamma_{{\mathbb{T}},{\mathbb{H}}}(l_{h})$, where $l_{h}$ is the unique
morphisms in HTr such that $l_{h}(a)=h(a)$ for all $a\in T$. Indeed, it is
easy to show that $l_{h}$ is a Hopf monoid morphism between ${\sf
L}_{g}(T_{1})$ and $H_{1}$ and a morphism of non-unital bimonoids between
${\sf L}_{sg}(T_{2})$ and $H_{2}$. Finally, the identity
$h=\Gamma_{{\mathbb{T}},{\mathbb{H}}}(l_{h})$ follows trivially from the
definition of $l_{h}$.

Therefore ${\sf P}_{st}\dashv{\sf R}_{ht}$ and the unit of this adjuction is
the identity while the counit is not a natural isomorphism in general. In any
case, if ${\mathbb{H}}$ is a pointed cosemisimple Hopf truss the counit is an
isomorphism, as the counits for the adjunctions given in Remark 2.10 are, and
thus we have the equivalence.

∎

##  3\. Hopf trusses and generalized invertible 1-cocycles

In [2] the authors proved that there exists a closed relation between Hopf
braces and 1-cocycles. In this section we will prove that this connection
remains valid for Hopf trusses. First we will introduce the notion of
generalized invertible 1-cocycle between a non-unital bimonoid $B$ and a Hopf
monoid $H$ in the braided monoidal category ${\sf C}$.

######  Definition 3.1.

Let $H=(H,\eta_{H},\mu_{H},\varepsilon_{H},\delta_{H},\lambda_{H})$ be a Hopf
monoid in C and let $B=(B,\mu_{B},\varepsilon_{B},\delta_{B})$ be a non-unital
bimonoid in C. Assume that $H$ is a non-unital left $B$-module monoid with
action $\phi_{H}$. Let $\pi:B\rightarrow H$ be comonoid morphism. We will say
that $\pi$ is an generalized invertible 1-cocycle if it is an isomorphism and
there exist a comonoid endomorphism $\theta_{\pi}:B\rightarrow B$ such that

(16)
$\pi\circ\mu_{B}=\mu_{H}\circ((\pi\circ\theta_{\pi})\otimes\phi_{H})\circ(\delta_{B}\otimes\pi)$

holds.

Let $\pi:B\rightarrow H$ and $\pi^{\prime}:B^{\prime}\rightarrow H^{\prime}$
be generalized invertible 1-cocycles. A morphism between them is a pair
$(f,g)$ where $f:B\rightarrow B^{\prime}$ is a morphism of non-unital
bimonoids and $g:H\rightarrow H^{\prime}$ is a morphism of Hopf monoids
satisfying the following identities:

(17) $f\circ\theta_{\pi}=\theta_{\pi^{\prime}}\circ f,$ (18)
$g\circ\pi=\pi^{\prime}\circ f,$ (19)
$g\circ\phi_{H}=\phi_{H^{\prime}}\circ(f\otimes g).$

Then, with these morphisms, generalized invertible 1-cocycles form a category
denoted by GIC. In the following lines an object in GIC will also be denoted
by the triple $(\pi:B\rightarrow H,\theta_{\pi})$.

Note that if $(\pi:B\rightarrow H,\theta_{\pi})$ is a generalized invertible
1-cocycle such that $B$ is a Hopf monoid, $(H,\phi_{H})$ is a left $B$-module
monoid and $\theta_{\pi}=id_{B}$, $(\pi:B\rightarrow H,\theta_{\pi})$ is an
invertible 1-cocycle in the sense of [2]. If we denote the category of
invertible 1-cocycles by GIC, it is obvious that it is a subcategory of GIC.

######  Theorem 3.2.

The categories GIC and HTr are equivalent.

###### Proof.

Let ${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$ be an object in HTr. Then,
$(id_{H}:H_{2}\rightarrow H_{1},\theta_{id_{H}}=\sigma_{H})$ is a generalized
invertible 1-cocycle. Indeed, trivially $id_{H}$ is a comonoid isomorphism and
$H_{1}$ is a non unital left $H_{2}$-module monoid with the action
$\Gamma_{H_{1}}^{\sigma_{H}}$ defined in (iii) of Definiton (2.12). Finally,

  * $\hskip 10.81218pt\mu_{H}^{1}\circ((id_{H}\circ\theta_{id_{H}})\otimes\Gamma_{H_{1}}^{\sigma_{H}})\circ(\delta_{H}\otimes id_{H})$

  * $=\mu_{H}^{1}\circ(\sigma_{H}\otimes(\mu_{H}^{1}\circ((\lambda_{H}\circ\sigma_{H})\otimes\mu_{H}^{2})\circ(\delta_{H}\otimes H)))\circ(\delta_{H}\otimes H)$ (by the definition of $\Gamma_{H_{1}}^{\sigma_{H}}$)

  * $=\mu_{H}^{1}\circ((\mu_{H}^{1}\circ(\sigma_{H}\otimes(\lambda_{H}\circ\sigma_{H})\circ\delta_{H}))\otimes\mu_{H}^{2})\circ\delta_{H}$ (by the associativity of $\mu_{H}^{1}$ and the coassociativity of

  * $\delta_{H}$)

  * $=\mu_{H}^{1}\circ((id_{H}\ast\lambda_{H})\otimes\mu_{H}^{2})\circ\delta_{H}$ (by the condition of comonoid morphism for $\sigma_{H}$)

  * $=id_{H}\circ\mu_{H}^{2}$ (by (8) and the properties of $\varepsilon_{H}$ and $\eta_{H}$).

On the other hand, let ${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$ and
${\mathbb{H}}^{\prime}=(H_{1}^{\prime},H_{2}^{\prime},\sigma_{H^{\prime}})$ be
objects in HTr and let $f:{\mathbb{H}}\rightarrow{\mathbb{H}}^{\prime}$ be a
morphism between them. The pair $(f,f)$ is a morphism in GIC between
$(id_{H}:H_{2}\rightarrow H_{1},\sigma_{H})$ and
$(id_{H^{\prime}}:H_{2}^{\prime}\rightarrow
H_{1}^{\prime},\sigma_{H^{\prime}})$ because $f:H_{1}\to H_{1}^{\prime}$ is a
Hopf monoid morphism, $f:H_{2}\to H_{2}^{\prime}$ is a non-unital bimonoid
morphisn, (17) holds trivially, (18) holds by (14), and, finally, (19) follows
from

  * $\hskip 10.81218ptf\circ\Gamma_{H_{1}}^{\sigma_{H}}$

  * $=\mu_{H}^{1}\circ((\lambda_{H}\circ f\circ\sigma_{H})\otimes(\mu_{H^{\prime}}^{2}\circ(f\otimes f)))\circ(\delta_{H}\otimes H)$ (by the condition of multiplicative morphism for $f$

  * and (9))

  * $=\mu_{H}^{1}\circ((\lambda_{H}\circ\sigma_{H^{\prime}}\circ f)\otimes(\mu_{H^{\prime}}^{2}\circ(f\otimes f)))\circ(\delta_{H}\otimes H)$ (by (14))

  * $=\Gamma_{H_{1}^{\prime}}^{\sigma_{H^{\prime}}}\circ(f\otimes f)$ (by the condition of comonoid morphism for $f$).

Therefore, there exists a functor

${\sf E}:{\sf HTr}\rightarrow{\sf GIC}$

defined on objects by ${\sf E}({\mathbb{H}})=(id_{H}:H_{2}\rightarrow
H_{1},\sigma_{H})$ and on morphisms by ${\sf E}(f)=(f,f).$

The next step is to define the functor

${\sf Q}:\;{\sf IC}\rightarrow{\sf HTr}.$

Let $(\pi:B\rightarrow H,\theta_{\pi})$ be an object in GIC. Define
$\mu_{H}^{\pi}:=\pi\circ\mu_{B}\circ(\pi^{-1}\otimes\pi^{-1})$ and
$\sigma_{\pi}:=\pi\circ\theta_{\pi}\circ\pi^{-1}$. Then,
${\mathbb{H}}_{\pi}=(H,H_{\pi},\sigma_{\pi})$, where
$H_{\pi}=(H,\mu_{H}^{\pi},\varepsilon_{H},\delta_{H})$, is an object in HTr.
Indeed, trivially $H_{\pi}$ is a non-unital bimonoid in C because $\pi$ is a
comonoid isomorphism and $B$ is a non-unital bimonoid. The morphism
$\sigma_{\pi}$ is a comonoid morphism because is defined as a composition of
comonoid morphisms. Finally,

  * $\hskip 10.81218pt\mu_{H}\circ(\mu_{H}^{\pi}\otimes\Gamma_{H}^{\sigma_{\pi}})\circ(H\otimes c_{H,H}\otimes H)\circ(\delta_{H}\otimes H\otimes H)$

  * $=\mu_{H}\circ((\mu_{H}\circ((\pi\circ\theta_{\pi})\otimes\phi_{H})\circ(\delta_{B}\otimes H))\otimes(\mu_{H}\circ((((\lambda_{H}\ast id_{H})\circ\pi\circ\theta_{\pi})\otimes\phi_{H})\circ(\delta_{B}\otimes H)))$

  * $\hskip 10.81218pt\circ(B\otimes c_{B,H}\otimes H)\circ((\delta_{B}\circ\pi^{-1})\otimes H\otimes H)$ (by (16), associativity of $\mu_{H}$, coassociativity of $\delta_{H}$ and naturality

  * of $c$)

  * $=\mu_{H}\circ((\pi\circ\theta_{\pi})\otimes(\mu_{H}\circ(\phi_{H}\otimes\phi_{H})\circ(B\otimes c_{B,H}\otimes H)\circ(\delta_{B}\otimes H\otimes H)))\circ((\delta_{B}\circ\pi^{-1})\otimes H\otimes H)$

  * (by (8), the condition of comonoid morphism for $\theta_{\pi}$ and $\pi$, the properties of the unit $\eta_{H}$ and the counit $\varepsilon_{B}$ and

  * the coassociativity of $\delta_{H}$)

  * $=\mu_{H}^{\pi}\circ(H\otimes\mu_{H})$ (by the condition of non-unital left $B$-module monoid for $H$ and (16)).

Also, if $(f,g)$ is a morphism in GIC between $(\pi:B\rightarrow
H,\theta_{\pi})$ and $(\pi^{\prime}:B^{\prime}\rightarrow
H^{\prime},\theta_{\pi^{\prime}})$, $g$ is a morphism in HTr between
${\mathbb{H}}_{\pi}$ and ${\mathbb{H}}_{\pi^{\prime}}^{\prime}$ because,
$g:H\to H^{\prime}$ is a morphism of Hopf monoids and $g:H_{\pi}\to
H_{\pi^{\prime}}^{\prime}$ is a morphism in bimod because

  * $\hskip 10.81218ptg\circ\mu_{H}^{\pi}$

  * $=g\circ\pi\circ\mu_{B}\circ(\pi^{-1}\otimes\pi^{-1})$ (by definition of $\mu_{H}^{\pi}$)

  * $=\pi^{\prime}\circ f\circ\mu_{B}\circ(\pi^{-1}\otimes\pi^{-1})$ (by (18))

  * $=\pi^{\prime}\circ\mu_{B^{\prime}}\circ((f\circ\pi^{-1})\otimes(f\circ\pi^{-1}))$ (by the condition of multiplicative morphism for $f$)

  * $=\pi^{\prime}\circ\mu_{B^{\prime}}\circ((\pi^{\prime-1}\circ g)\otimes(\pi^{\prime-1}\circ g))$ (by (18))

  * $=\mu_{H^{\prime}}^{\pi^{\prime}}\circ(g\otimes g)$ (by definition of $\mu_{H^{\prime}}^{\pi^{\prime}}$).

As a consequence of these facts, we have a functor ${\sf Q}:\;{\sf
IC}\rightarrow{\sf HTr}$ defined by ${\sf Q}((\pi:B\rightarrow
H,\theta_{\pi}))={\mathbb{H}}_{\pi}$ on objects and by ${\sf Q}((f,g))=g$ on
morphisms.

These functors induce an equivalence between the two categories because,
clearly, ${\sf QE}=id_{{\sf HTr}}$. On the other hand, we have that

$(\pi,id_{H}):(\pi:B\rightarrow
H,\theta_{\pi})\rightarrow(id_{H}:H_{\pi}\rightarrow H,\sigma_{\pi})={\sf
EQ}((\pi:B\rightarrow H,\theta_{\pi}))$

is an isomorphism in GIC because $\pi:B\rightarrow H_{\pi}$ is a isomorphism
between $B$ and $H_{\pi}$ in bimod,
$\pi\circ\theta_{\pi}=\sigma_{\pi}\circ\pi$, (18) holds trivially, and

  * $\hskip 10.81218pt\Gamma_{H}^{\sigma_{\pi}}\circ(\pi\otimes id_{H})$

  * $=\mu_{H}\circ((\lambda_{H}\circ\pi\circ\theta_{\pi}\circ\pi^{-1})\otimes(\pi\circ\mu_{B}\circ(\pi^{-1}\otimes\pi^{-1})))\circ((\delta_{H}\circ\pi)\otimes H)$ (by definition of $\sigma_{\pi}$ and $\mu_{H}^{\pi}$)

  * $=\mu_{H}\circ(((\lambda_{H}\ast id_{H})\circ\pi\circ\theta_{\pi})\otimes\phi_{H})\circ(\delta_{B}\otimes H)$ (by (16), the condition of comonoid morphism for $\theta_{\pi}$ and

  * $\pi$)

  * $=\phi_{H}$ (by (8), the condition of comonoid morphism for $\theta_{\pi}$ and $\pi$ and the properties of the unit $\eta_{H}$ and the counit

  * $\varepsilon_{B}$).

Therefore, ${\sf EQ}\backsimeq{\sf Id}_{{\sf GIC}}$.

∎

######  Remark 3.3.

As is easy to see, the previous theorem is a generalization of [2, Theorem
1.12] because if we particularize this equivalence to the Hopf brace setting,
we obtain the quoted result.

##  4\. Modules for Hopf trusses and generalized invertible 1-cocycles

In this section we introduce the notion of module over a Hopf truss and over a
generalized invertible 1-cocycle. Also we will prove that, thanks to the
categorical equivalence of Theorem 3.2, there exist some interesting
categorical connections between these categories of modules.

######  Definition 4.1.

Let ${\mathbb{H}}$ be a Hopf truss. A left ${\mathbb{H}}$-module is a triple
$(M,\psi_{M}^{1},\psi_{M}^{2})$, where $(M,\psi_{M}^{1})$ is a left
$H_{1}$-module, $(M,\psi_{M}^{2})$ is a non-unital left $H_{2}$-module and the
following identity

(20)
$\psi_{M}^{2}\circ(H\otimes\psi_{M}^{1})=\psi_{M}^{1}\circ(\mu_{H}^{2}\otimes\Gamma_{M}^{\sigma_{H}})\circ(H\otimes
c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M)$

holds, where

$\Gamma_{M}^{\sigma_{H}}=\psi_{M}^{1}\circ((\lambda_{H}\circ\sigma_{H})\otimes\psi_{M}^{2})\circ(\delta_{H}\otimes
M).$

Given two left ${\mathbb{H}}$-modules $(M,\psi_{M}^{1},\psi_{M}^{2})$ and
$(N,\psi_{N}^{1},\psi_{N}^{2})$, a morphism $f:M\rightarrow N$ is called a
morphism of left ${\mathbb{H}}$-modules if $f$ is a morphism of left
$H_{1}$-modules and left non-unital $H_{2}$-modules. Left
${\mathbb{H}}$-modules with morphisms of left ${\mathbb{H}}$-modules form a
category which we denote by $\;{}_{\mathbb{H}}$Mod.

######  Example 4.2.

Let ${\mathbb{H}}$ be a Hopf truss. The triple
$(H,\psi_{H}^{1}=\mu_{H}^{1},\psi_{H}^{2}=\mu_{H}^{2})$ is an example of left
${\mathbb{H}}$-module. Also, if $K$ is the unit object of ${\sf C}$,
$(K,\psi_{K}^{1}=\varepsilon_{H},\psi_{K}^{2}=\varepsilon_{H})$ is a left
${\mathbb{H}}$-module called the trivial module.

If $X$ is an object in C, the triple

${\mathbb{H}}\otimes X=(H\otimes X,\psi_{H\otimes X}^{1}=\mu_{H}^{1}\otimes
X,\psi_{H\otimes X}^{2}=\mu_{H}^{2}\otimes X)$

is an example of left ${\mathbb{H}}$-module. Also, if $f:X\rightarrow
X^{\prime}$ is a morphism in C, $H\otimes f$ is a morphism in
$\;{}_{\mathbb{H}}$Mod between ${\mathbb{H}}\otimes X$ and
${\mathbb{H}}\otimes X^{\prime}$. Therefore, there exist a functor, called the
induction functor,

${\mathbb{H}}\otimes-:{\sf C}\rightarrow\;_{\mathbb{H}}{\sf Mod}$

defined on objects by ${\mathbb{H}}\otimes-(X)={\mathbb{H}}\otimes X$ and by
${\mathbb{H}}\otimes-(f)={\mathbb{H}}\otimes f$ on morphisms.

######  Remark 4.3.

If the a Hopf truss ${\mathbb{H}}$ is a Hopf brace and in Definition 4.1 we
assume that a $(M,\psi_{M}^{2})$ is a left $H_{2}$ module, we obtain the
definition of module over a Hopf brace introduced in [5].

######  Remark 4.4.

Using the naturality of $c$ and the coassociativity of $\delta_{H}$, it is
easy to show that (20) is equivalent to

(21)
$\psi_{M}^{2}\circ(H\otimes\psi_{M}^{1})=\psi_{M}^{1}\circ(\Lambda_{H_{1}}^{\sigma_{H}}\otimes\psi_{M}^{2})\circ(H\otimes
c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M)$

where

$\Lambda_{H_{1}}^{\sigma_{H}}=\mu_{H}^{1}\circ(\mu_{H}^{2}\otimes(\lambda_{H}\circ\sigma_{H}))\circ(H\otimes
c_{H,H})\circ(\delta_{H}\otimes H).$

Also, the following equality

(22)
$\Gamma_{M}^{\sigma_{H}}\circ(H\otimes\psi_{M}^{1})=\psi_{M}^{1}\circ(\Gamma_{H_{1}}^{\sigma_{H}}\otimes\Gamma_{M}^{\sigma_{H}})\circ(H\otimes
c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M).$

holds. Indeed:

  * $\hskip 10.81218pt\psi_{M}^{1}\circ(\Gamma_{H_{1}}^{\sigma_{H}}\otimes\Gamma_{M}^{\sigma_{H}})\circ(H\otimes c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M)$

  * $=\psi_{M}^{1}\circ((\lambda_{H}\circ\sigma_{H})\otimes(\psi_{M}^{1}\circ(\Lambda_{H_{1}}^{\sigma_{H}}\otimes\psi_{M}^{2})\circ(H\otimes c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M)))\circ(\delta_{H}\otimes H\otimes M)$

  * (by naturality of $c$, coassociativity of $\delta_{H}$, the condition of left $H_{1}$-module for $(M,\psi_{M}^{1})$ and the associativity of

  * $\mu_{H}^{1}$)

  * $=\Gamma_{M}^{\sigma_{H}}\circ(H\otimes\psi_{M}^{1})$ (by (21)).

######  Definition 4.5.

Let $(\pi:B\to H,\theta_{\pi})$ be a generalized invertible 1-cocycle. A left
module over $(\pi:B\to H,\theta_{\pi})$ is a 6-tuple
$(M,N,\phi_{M},\varphi_{M},\phi_{N},\gamma)$ where

  1. (i)

$\phi_{M}:B\otimes M\rightarrow M$ is a morphism in C.

  2. (ii)

$(M,\varphi_{M})$ is a left $H$-module.

  3. (iii)

$(N,\phi_{N})$ is a non-unitary left $B$-module.

  4. (iv)

The equality

(23)
$\phi_{M}\circ(B\otimes\varphi_{M})=\varphi_{M}\circ(\phi_{H}\otimes\phi_{M})\circ(B\otimes
c_{A,H}\otimes M)\circ(\delta_{B}\otimes H\otimes M).$

holds.

  5. (v)

$\gamma:N\to M$ is an isomorphism in $\sf C$ such that

(24)
$\gamma\circ\phi_{N}=\varphi_{M}\circ((\pi\circ\theta_{\pi})\otimes\phi_{M})\circ(\delta_{B}\otimes\gamma).$

######  Definition 4.6.

Let $(M,N,\phi_{M},\varphi_{M},\phi_{N},\gamma)$ and
$(M^{\prime},N^{\prime},\phi_{M^{\prime}},\varphi_{M^{\prime}},\phi_{N^{\prime}},\gamma^{\prime})$
be left modules over a generalized invertible 1-cocycle $(\pi:B\to
H,\theta_{\pi})$. A morphism between them is a pair $(h,l)$ of morphisms in C
such that:

  * (i)

The morphism $h:M\rightarrow M^{\prime}$ satisfies
$f\circ\phi_{M}=\phi_{M^{\prime}}\circ(B\otimes f)$ and is left $H$-linear.

  * (ii)

The morphism $l:N\rightarrow N^{\prime}$ is left $B$-linear.

  * (iii)

The following identity holds:

(25) $h\circ\gamma=\gamma^{\prime}\circ l.$

Note that, by (25), the morphism $l$ is determined by $h$ because
$l=(\gamma^{\prime})^{-1}\circ h\circ\gamma.$

With the obvious composition of morphisms, left modules over a generalized
invertible 1-cocycle $(\pi:B\to H,\theta_{\pi})$ with action $\phi_{H}$ form a
category that we will denote by $\;{}_{(\pi,\phi_{H},\theta_{\pi})}$Mod.

######  Remark 4.7.

If $(M,N,\phi_{M},\varphi_{M},\phi_{N},\gamma)$ is a left module over the
generalized invertible 1-cocycle $(\pi:B\rightarrow H,\theta_{\pi})$, by (24),
we obtain that $\phi_{N}$ is determined by $\phi_{M}$ and $\varphi_{M}$ in the
following way:

(26)
$\phi_{N}=\gamma^{-1}\circ\varphi_{M}\circ((\pi\circ\theta_{\pi})\otimes\phi_{M})\circ(\delta_{B}\otimes\gamma).$

Also, composing in both sides of the equality (24) with
$(((\lambda_{H}\circ\pi)\otimes B)\circ\delta_{B})\otimes\gamma^{-1}$ on the
right and with $\varphi_{M}$ on the left we obtain the identity

(27)
$\phi_{M}=\varphi_{M}\circ((\lambda_{H}\circ\pi\circ\theta_{\pi})\otimes(\gamma\circ\phi_{N}))\circ(\delta_{B}\otimes\gamma^{-1})$

because:

  * $\hskip 10.81218pt\varphi_{M}\circ((\lambda_{H}\circ\pi\circ\theta_{\pi})\otimes(\gamma\circ\phi_{N}))\otimes(\delta_{B}\otimes\gamma^{-1})$

  * $=\varphi_{M}\circ((\lambda_{H}\circ\pi\circ\theta_{\pi})\otimes(\varphi_{M}\circ((\pi\circ\theta_{\pi})\otimes\phi_{M})\circ(\delta_{B}\otimes\gamma))\otimes(\delta_{B}\otimes\gamma^{-1})$ (by (24))

  * $=\varphi_{M}\circ(((\lambda_{H}\ast id_{H})\circ\pi\circ\theta_{\pi})\otimes\phi_{M})\otimes(\delta_{B}\otimes M)$ (by the condition of comonoid morphism for $\theta_{\pi}$ and $\pi$, the coassociativity of $\delta_{B}$)

  * $=\phi_{M}$ (by (8) and the unit counit properties)).

######  Example 4.8.

It is easy to see that if $(\pi:B\rightarrow H,\theta_{\pi})$ is a generalized
invertible 1-cocycle, the 6-tuple $(H,B,\phi_{H},\mu_{H},\mu_{B},\pi)$ is an
example of left module over $(\pi:B\rightarrow H,\theta_{\pi})$.

######  Theorem 4.9.

Assume that $(f,g)$ is a morphism between the generalized invertible
1-cocycles $(\pi:B\rightarrow H,\theta_{\pi})$ and $(\pi:B^{\prime}\rightarrow
H^{\prime},\theta_{\pi^{\prime}})$. Then, there exists a functor

${\sf
M}_{(f,g)}:\;_{(\pi^{\prime},\phi_{H^{\prime}},\theta_{\pi^{\prime}})}{\sf
Mod}\;\rightarrow\;_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}$

defined on objects by

${\sf
M}_{(f,g)}((P,Q,\phi_{P},\varphi_{P},\phi_{Q},\tau))=(P,Q,\phi_{P}^{\pi}=\phi_{P}\circ(f\otimes
P),\varphi_{P}^{\pi}=\varphi_{P}\circ(g\otimes
P),\phi_{Q}^{\pi}=\phi_{Q}\circ(f\otimes Q),\tau)$

and on morphisms by the identity.

###### Proof.

The existence of the functor ${\sf M}_{(f,g)}$ is a consequence of the
following facts: Trivially $(P,\varphi_{P}^{\pi})$ is a left $H$-module and
$(Q,\phi_{Q}^{\pi})$ is a non-unitary left $B$-modules and . Also,

  * $\hskip 10.81218pt\varphi_{P}^{\pi}\circ(\phi_{H}\otimes\phi_{P}^{\pi})\circ(B\otimes c_{B,H}\otimes P)\circ(\delta_{B}\otimes H\otimes P)$

  * $=\varphi_{P}\circ((g\circ\phi_{H})\otimes(\phi_{P}\circ(f\otimes P)))\circ(B\otimes c_{B,H}\otimes P)\circ(\delta_{B}\otimes H\otimes P)$ (by definition of $\phi_{P}^{\pi}$ and $\varphi_{P}^{\pi}$)

  * $=\varphi_{P}\circ(\phi_{H^{\prime}}\otimes\phi_{P})\circ(B^{\prime}\otimes c_{B^{\prime},H^{\prime}}\otimes P)\circ(((f\otimes f)\circ\delta_{B})\otimes g\otimes P)$ (by (19) and naturality of $c$)

  * $=\varphi_{P}\circ(\phi_{H^{\prime}}\otimes\phi_{P})\circ(B^{\prime}\otimes c_{B^{\prime},H^{\prime}}\otimes P)\circ((\delta_{B^{\prime}}\circ f)\otimes g\otimes P)$ (by the comonoid morphism condition for

  * $f$)

  * $=\phi_{P}^{\pi}\circ(B\otimes\varphi_{P}^{\pi})$(by (23)),

and

  * $\hskip 10.81218pt\varphi_{P}^{\pi}\circ((\pi\circ\theta_{\pi})\otimes\phi_{P}^{\pi})\circ(\delta_{B}\otimes\tau)$

  * $=\varphi_{P}\circ((g\circ\pi\circ\theta_{\pi})\otimes(\phi_{P}\circ(f\otimes P)))\circ(\delta_{B}\otimes\tau)$ (by definition of $\phi_{P}^{\pi}$ and $\varphi_{P}^{\pi}$)

  * $=\varphi_{P}\circ((\pi^{\prime}\circ\theta_{\pi^{\prime}}\circ f)\otimes(\phi_{P}\circ(f\otimes P)))\circ(\delta_{B}\otimes\tau)$ (by (17) and (18))

  * $=\varphi_{P}\circ((\pi^{\prime}\circ\theta_{\pi^{\prime}})\otimes\phi_{P})\circ((\delta_{B^{\prime}}\circ f)\otimes\tau)$ (by the comonoid morphism condition for $f$)

  * $=\tau\circ\phi_{Q}^{\pi}$ (by (24))

Then $(P,Q,\phi_{P}^{\pi},\varphi_{P}^{\pi},\phi_{Q}^{\pi},\tau)$ is an object
in $\;{}_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}$. Finally, it is obvious that
if $(h,l)$ is a morphism in
$\;{}_{(\pi^{\prime},\phi_{H^{\prime}},\theta_{\pi^{\prime}})}{\sf Mod}$,
$(h,l)$ is a morphism in $\;{}_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}$. ∎

######  Remark 4.10.

If $(f,g)$ is an isomorphism defined between the generalized invertible
1-cocycles $(\pi:B\rightarrow H,\theta_{\pi})$ and $(\pi:B^{\prime}\rightarrow
H^{\prime},\theta_{\pi^{\prime}})$ with inverse $(f^{-1},g^{-1})$, the functor
${\sf M}_{(f,g)}$ is an isomorphism of categories with inverse ${\sf
M}_{(f^{-1},g^{-1})}$. For example, in the proof of Theorem 3.2 we proved
that, for all generalized invertible 1-cocycle $(\pi:B\rightarrow
H,\theta_{\pi})$ , $(\pi,id_{H})$ is an isomorphism between the generalized
invertible 1-cocycles $(\pi:B\rightarrow H,\theta_{\pi})$ , $(\pi,id_{H})$ and
$(id_{H}:H_{\pi}\rightarrow H,\sigma_{\pi})$ where
$\sigma_{\pi}=\pi\circ\theta_{\pi}\circ\pi^{-1}$ and the action is
$\Gamma_{H}^{\sigma_{\pi}}$, i.e. the action is $\phi_{H}\circ(\pi^{-1}\otimes
H)$ where $\phi_{H}$ is the action associated to $(\pi:B\rightarrow
H,\theta_{\pi})$. Therefore, the functor

${\sf
M}_{(\pi,id_{H})}:\;_{(id_{H},\Gamma_{H}^{\sigma_{\pi}},\sigma_{\pi})}{\sf
Mod}\;\rightarrow\;_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}$

is an isomorphism of categories with inverse

${\sf M}_{(\pi^{-1},id_{H})}:\;_{(\pi,\phi_{H},\theta_{\pi})}{\sf
Mod}\;\rightarrow\;_{(id_{H},\Gamma_{H}^{\sigma_{\pi}},\sigma_{\pi})}{\sf
Mod}.$

######  Theorem 4.11.

Let ${\mathbb{H}}$ be a Hopf truss and let ${\sf E}({\mathbb{H}})$ be the
invertible 1-cocycle induced by the functor E introduced in the proof of
Theorem 3.2. There exists a functor

${\sf G}_{{\mathbb{H}}}:\;_{{\mathbb{H}}}{\sf
Mod}\;\rightarrow\;_{(id_{H},\Gamma_{H_{1}}^{\sigma_{H}},\sigma_{H})}{\sf
Mod}$

defined on objects by

${\sf
G}_{{\mathbb{H}}}((M,\psi_{M}^{1},\psi_{M}^{2}))=(M,M,\widehat{\phi}_{M}=\Gamma_{M}^{\sigma_{H}},\widehat{\varphi}_{M}=\psi_{M}^{1},\overline{\phi}_{M}=\psi_{M}^{2},id_{M})$

and on morphisms by ${\sf G}_{{\mathbb{H}}}(f)=(f,f)$.

###### Proof.

By assumption, $(M,\widehat{\varphi}_{M}=\psi_{M}^{1})$ is a left
$H_{1}$-module and $(M,\overline{\phi}_{M}=\psi_{M}^{2})$ is a non-unital left
$H_{2}$-module. On the other hand, by (22) we have that

$\widehat{\phi}_{M}\circ(H\otimes\widehat{\varphi}_{M})=\widehat{\varphi}_{M}\circ(\Gamma_{H_{1}}\otimes\widehat{\phi}_{M})\circ(H\otimes
c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M)$

and then, (23) holds. Also,

  * $\hskip 10.81218pt\widehat{\varphi}_{M}\circ(\sigma_{H}\otimes\widehat{\phi}_{M})\circ(\delta_{H}\otimes M)$

  * $=\psi_{M}^{1}\circ(\sigma_{H}\otimes\Gamma_{M}^{\sigma_{H}})\circ(\delta_{H}\otimes M)$ (by definition of $\widehat{\varphi}_{M}$ and $\widehat{\phi}_{M}$)

  * $=\psi_{M}^{1}\circ(\sigma_{H}\otimes(\psi_{M}^{1}\circ((\lambda_{H}\circ\sigma_{H})\otimes\psi_{M}^{2})\circ(\delta_{H}\otimes M)))\circ(\delta_{H}\otimes M)$ (by definition of $\Gamma_{M}^{\sigma_{H}}$)

  * $=\psi_{M}^{1}\circ(((id_{H}\ast\lambda_{H}^{1})\circ\sigma_{H})\otimes\psi_{M}^{2})\circ(\delta_{H}\otimes M)$ (by the condition of left $H_{1}$-module of $(M,\psi_{M}^{1})$, the

  * coassociativity of $\delta_{H}$ and the condition of comonoid morphism for $\sigma_{H}$)

  * $=\overline{\phi}_{M}$ (by (8), the counit properties, the condition of left $H_{1}$-module of $(M,\psi_{M}^{1})$ and the definition of $\overline{\phi}_{M}$).

Finally, it is easy to show that if $f$ is a morphism in
$\;{}_{{\mathbb{H}}}{\sf Mod}$ between the objects
$(M,\psi_{M}^{1},\psi_{M}^{2})$ and
$(M^{\prime},\psi_{M^{\prime}}^{1},\psi_{M^{\prime}}^{2})$, the pair $(f,f)$
is a morphism in $\;{}_{(id_{H},\Gamma_{H_{1}}^{\sigma_{H}},\sigma_{H})}{\sf
Mod}$ between ${\sf G}_{{\mathbb{H}}}((M,\psi_{M}^{1},\psi_{M}^{2}))$ and
${\sf
G}_{{\mathbb{H}}}((M^{\prime},\psi_{M^{\prime}}^{1},\psi_{M^{\prime}}^{2}))$.
∎

######  Theorem 4.12.

Let $(\pi:B\rightarrow H,\theta_{\pi})$ be a generalized invertible 1-cocycle.
Then the categories ${}_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}$ and
$\;{}_{{\mathbb{H}}_{\pi}}$Mod are equivalent.

###### Proof.

First of all we will prove that there exists a functor

${\sf H}_{{\sf tr}}^{\pi}:\;_{(\pi,\phi_{H},\theta_{\pi})}{\sf
Mod}\;\rightarrow\;_{{\mathbb{H}}_{\pi}}{\sf Mod}$

defined on objects by

${\sf H}_{{\sf
tr}}^{\pi}((M,N,\phi_{M},\varphi_{M},\phi_{N},\gamma))=(M,\overline{\psi}_{M}^{1}=\varphi_{M},\overline{\psi}_{M}^{2}=\gamma\circ\phi_{N}\circ(\pi^{-1}\otimes\gamma^{-1}))$

and on morphisms by ${\sf H}_{{\sf tr}}^{\pi}((h,l))=h$. Indeed: By
assumption, $(M,\overline{\psi}_{M}^{1}=\varphi_{M})$ is a left $H$-module
and, using the condition of non-unital left $B$-module of $N$, we obtain that

$(M,\overline{\psi}_{M}^{2}=\gamma\circ\phi_{N}\circ(\pi^{-1}\otimes\gamma^{-1}))$

is a non-unital left $H_{\pi}$-module. Also, by (27), we have that the
identity

(28) $\Gamma_{M}^{\sigma_{\pi}}=\phi_{M}\circ(\pi^{-1}\otimes M)$

holds, where
$\Gamma_{M}^{\sigma_{\pi}}=\overline{\psi}_{M}^{1}\circ((\lambda_{H}\circ\sigma_{\pi})\otimes\overline{\psi}_{M}^{2})\circ(\delta_{H}\otimes
M)$. Then, (20) holds because:

  * $\hskip 10.81218pt\overline{\psi}_{M}^{2}\circ(H\otimes\overline{\psi}_{M}^{1})$

  * $=\varphi_{M}\circ((\pi\circ\theta_{\pi})\otimes\phi_{M})\circ((\delta_{B}\circ\pi^{-1})\otimes\varphi_{M})$ (by (26))

  * $=\varphi_{M}\circ((\pi\circ\theta_{\pi})\otimes(\varphi_{M}\circ(\phi_{H}\otimes\phi_{M})\circ(B\otimes c_{B,H}\otimes M)\circ(\delta_{B}\otimes H\otimes M)))\circ((\delta_{B}\circ\pi^{-1})\otimes H\otimes M)$

  * (by (23))

  * $=\varphi_{M}\circ((\mu_{H}\circ((\pi\circ\theta_{\pi})\otimes\phi_{H})\circ(\delta_{B}\otimes\pi))\otimes\phi_{M})\circ(B\otimes c_{B,B}\otimes M)\circ((\delta_{B}\circ\pi^{-1})\otimes\pi^{-1}\otimes M)$

  * (by the condition of left $H$-module for $M$, the coassociativity of $\delta_{B}$, the naturality of $c$ and the condition of

  * isomorphism for $\pi$)

  * $=\varphi_{M}\circ((\pi\circ\mu_{B})\otimes\phi_{M})\circ(B\otimes c_{B,B}\otimes M)\circ((\delta_{B}\circ\pi^{-1})\otimes\pi^{-1}\otimes M)$ (by (16))

  * $=\varphi_{M}\circ(\mu_{H_{\pi}}\otimes(\phi_{M}\circ(\pi^{-1}\otimes M)))\circ(H\otimes c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M)$ (by the condition of coalgebra

  * isomorphism for $\pi$ and the naturality of $c$)

  * $=\overline{\psi}_{M}^{1}\circ(\mu_{H_{\pi}}\otimes\Gamma_{M}^{\sigma_{\pi}})\circ(H\otimes c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes M)$ (by (28))

On the other hand, if $(h,l)$ is a morphisms in
$\;{}_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}$ between
$(M,N,\phi_{M},\varphi_{M},\phi_{N},\gamma)$ and
$(M^{\prime},N^{\prime},\phi_{M^{\prime}},\varphi_{M^{\prime}},\phi_{N^{\prime}},\gamma^{\prime})$,
we have that $h$ is a morphism in $\;{}_{{\mathbb{H}}_{\pi}}{\sf Mod}$ between
$(M,\overline{\psi}_{M}^{1},\overline{\psi}_{M}^{2})$ and
$(M^{\prime},\overline{\psi}_{M^{\prime}}^{1},\overline{\psi}_{M^{\prime}}^{2})$
because, using that $h$ is a morphism of left $H$-modules, we have
$h\circ\overline{\psi}_{M}^{1}=\overline{\psi}_{M^{\prime}}^{1}\circ(H\otimes
h)$ and, by (25) and the condition of morphism of non-unital left $B$-modules
for $l$, we have that
$h\circ\overline{\psi}_{M}^{2}=\overline{\psi}_{M^{\prime}}^{2}\circ(H\otimes
h)$.

Taking into account the functors ${\sf H}_{{\sf tr}}^{\pi}$, ${\sf
G}_{{\mathbb{H}}_{\pi}}$ and ${\sf M}_{(\pi,id_{H})}$, it is easy to show that

${\sf H}_{{\sf br}}^{\pi}\circ({\sf M}_{(\pi,id_{H})}\circ{\sf
G}_{{\mathbb{H}}_{\pi}})={\sf id}_{\;{}_{{\mathbb{H}}_{\pi}}{\sf Mod}}$

and

$(({\sf M}_{(\pi,id_{H})}\circ{\sf G}_{{\mathbb{H}}_{\pi}})\circ{\sf H}_{{\sf
br}}^{\pi})((M,N,\phi_{M},\varphi_{M},\phi_{N},\gamma))=(M,M,\phi_{M},\varphi_{M},\overline{\phi}_{M}^{\pi}=\gamma\circ\phi_{N}\circ(A\otimes\gamma^{-1}),id_{M})$

holds. Then,

$({\sf M}_{(\pi,id_{H})}\circ{\sf G}_{{\mathbb{H}}_{\pi}})\circ{\sf H}_{{\sf
br}}^{\pi}\backsimeq{\sf id}_{{}_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}}$

because $(id_{M},\gamma)$ is an isomorphism in
${}_{(\pi,\phi_{H},\theta_{\pi})}{\sf Mod}$ between the objects
$(M,N,\phi_{M},\varphi_{M},\phi_{N},\gamma)$ and
$(M,M,\phi_{M},\varphi_{M},\overline{\phi}_{M}^{\pi},id_{M}).$

∎

######  Remark 4.13.

When we particularize the previous results to Hopf braces we obtain the
categorical equivalences obtained in [4]. More concretely, [4, Theorem 2.26]
is the Hopf brace version of Theorem 28.

##  5\. The fundamental theorem of Hopf modules for Hopf trusses

In this section we will assume that ${\sf C}$ admits equalizers. As a
consequence every idempotent morphism in ${\sf C}$ splits, i.e., if
$q:M\rightarrow M$ is a morphism in ${\sf C}$ such that $q=q\circ q$, there
exists an object $I(q)$, called the image of $q$, and morphisms
$i:I(q)\rightarrow M$ and $p:M\rightarrow I(q)$ such that $q=i\circ p$ and
$p\circ i=id_{I(q)}$. The morphisms $p$ and $i$ will be called a factorization
of $q$. Note that $I(q)$, $p$ and $i$ are unique up to isomorphism.

Under the previous condition, we will introduce the category
${\mathbb{H}}$-Hopf of left Hopf modules over a Hopf truss ${\mathbb{H}}$ and
we will to obtain the Fundamental Theorem of Hopf modules for Hopf trusses
that is a generalization of the one proved in [5] for Hopf brace.

######  Definition 5.1.

Let $D$ be a comonoid in C. The pair $(M,\rho_{M})$ is a left $D$-comodule if
$M$ is an object in C and $\rho_{M}:M\rightarrow D\otimes M$ is a morphism in
C satisfying $(\varepsilon\otimes M)\circ\rho_{M}=id_{M}$,
$(D\otimes\rho_{M})\circ\rho_{M}=(\delta\otimes M)\circ\rho_{M}$. Given two
left ${D}$-comodules $(M,\rho_{M})$ and $(N,\rho_{N})$, a morphism
$f:M\rightarrow N$ in C is a morphism of left $D$-comodules if $(D\otimes
f)\circ\rho_{M}=\rho_{N}\circ f$. Left $D$-comodules with morphisms of left
$D$-comodules form a category which we denote by $\;{}^{{\sf D}}$Comod.

######  Definition 5.2.

Let $D$ be a comonoid such that there exits a comonoid morphism
$e:K\rightarrow D$. Let $(M,\rho_{M})$ be a left $D$-comodule. We define the
subobject of coinvariants of $M$, denoted by $M^{coD}_{e}$, as the equalizer
object of $\rho_{M}$ and $e\otimes M$. Then, we have an equalizer diagram

where $j_{M}^{e}$ denotes the equalizer (inclusion) morphism.

If $H$ is a Hopf monoid, the unit $\eta_{H}$ is a comonoid morphism. Then, Let
$(M,\rho_{M})$ be a left $D$-comodule, we will denote the equalizer object of
$\rho_{M}$ and $\eta_{H}\otimes M$ by $M^{coD}$ and the equalizer morphism by
$j_{M}$.

######  Definition 5.3.

Let $B$ be a non-unital bimonoid. A non-unital left $H$-Hopf module over $B$
is a triple $(M,\varphi_{M},\rho_{M})$ where $(M,\varphi_{M})$ is a non-unital
left $B$-module, $(M,\rho_{M})$ is a left $B$-comodule and

(29) $\varphi_{M}\circ\rho_{M}=(\mu_{B}\otimes\varphi_{M})\circ(B\otimes
c_{B,B}\otimes M)\circ(\delta_{B}\otimes\rho_{M})$

holds. Non-unital left $B$-Hopf modules over $B$ with left linear and colinear
morphisms form a category which we denote by ${\sf B}$-Hopf-mod.

The definition for left $H$-Hopf modules over a Hopf monoid $H$ is similar
changing non-unital left $H$-modules by left $H$-modules. Then, in this case
we will denote the category of $H$-Hopf modules over $H$ by ${\sf H}$-Hopf-Mod

######  Remark 5.4.

Let $H$ be a Hopf monoid, it easy to show that, if $(M,\varphi_{M},\rho_{M})$
is a left $H$-Hopf module over $H$, the endomorphism $q_{M}:M\rightarrow M$,
defined by

$q_{M}=\varphi_{M}\circ(\lambda_{H}\otimes M)\circ\rho_{M}$

is idempotent and satisfies $\rho_{M}\circ q_{M}=\eta_{H}\otimes q_{M}$.
Therefore, there exists a unique morphism

$t_{M}:M\rightarrow M^{coH}$

such that $t_{M}\circ j_{M}=q_{M}.$ Let $I(q_{M})$ be the image of the
idempotent morphism $q_{M}$ and let $i_{M}:I(q_{M})\rightarrow M$ and
$p_{M}:M\rightarrow I(q_{M})$ be the morphisms such that that
$q_{M}=i_{M}\circ p_{M}$ and $p_{M}\circ i_{M}=id_{I(q_{M})}$. The morphism

$\omega_{M}=t_{M}\circ i_{M}:I(q_{M})\rightarrow M^{coH}$

is an isomorphism with inverse $\omega_{M}^{-1}=p_{M}\circ j_{M}$. Moreover,
$t_{M}\circ\varphi_{M}=\varepsilon_{H}\otimes t_{M}$ and, as a consequence,
$(M^{coH},t_{M})$ is the coequalizer of $\varphi_{M}$ and
$\varepsilon_{H}\otimes M$.

On the other hand, the object $H\otimes M^{coH}$ is a left $H$-Hopf module
with action $\varphi_{H\otimes M^{coH}}=\mu_{H}\otimes M^{coH}$ and coaction
$\rho_{H\otimes M^{coH}}=\delta_{H}\otimes M^{coH}$. The Fundamental Theorem
of Hopf modules asserts that $H\otimes M^{coH}$ is isomorphic to $M$ in the
category ${\sf H}$-Hopf-Mod. The isomorphism is defined by

$\theta_{M}=\varphi_{M}\circ(H\otimes j_{M}):H\otimes M^{coH}\rightarrow M$

where $\theta_{M}^{-1}=(H\otimes t_{M})\circ\rho_{M}$. In the same way as in
the case of $M^{coH}$, if $X$ is an object in C, the tensor product $H\otimes
X$, with the action and coaction induced by the product and the coproduct of
$H$, is a left $H$-Hopf module. Then, there exists a functor ${\sf
F}=H\otimes-:{\sf C}\;\rightarrow{\sf H}$-Hopf-Mod called the induction
functor. Also, for all $M\in{\sf H}$-Hopf-Mod, the construction of $M^{coH}$
is functorial. Thus, there exists a new functor ${\sf G}=(\;\;)^{coH}:{\sf
H}$-Hopf-Mod$\;\rightarrow{\sf C}$, called the functor of coinvariants, such
that ${\sf F}\dashv{\sf G}$. Moreover, ${\sf F}$ and ${\sf G}$ induce an
equivalence between the categories ${\sf H}$-Hopf-Mod and ${\sf C}$.

Let us see now the definition of left Hopf module over a Hopf truss
${\mathbb{H}}$.

######  Definition 5.5.

Let ${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$ be a Hopf truss. A left Hopf
module over ${\mathbb{H}}$ (left ${\mathbb{H}}$-Hopf module) is a $4$-tuple
$(M,\psi_{M}^{1},\psi_{M}^{2},\rho_{M})$ such that:

  * (i)

The triple $(M,\psi_{M}^{1},\psi_{M}^{2})$ is a left ${\mathbb{H}}$-module.

  * (ii)

The triple $(M,\psi_{M}^{1},\rho_{M})$ is a left $H_{1}$-Hopf module.

  * (iii)

The triple $(M,\psi_{M}^{2},\rho_{M})$ is a non-unital left $H_{2}$-Hopf
module.

  * (iv)

If $j_{M}$ is the equalizer morphism of $\rho_{M}$ and $\eta_{H}\otimes M$,
the identity

$\psi_{M}^{1}\circ(\sigma_{H}\otimes j_{M})=\psi_{M}^{2}\circ(H\otimes j_{M})$

holds.

A morphism of left Hopf modules over ${\mathbb{H}}$ is a morphism of left
${\mathbb{H}}$-modules and left $H$-comodules.

Left Hopf modules over ${\mathbb{H}}$ with morphisms of left Hopf modules form
a category which we denote by ${\mathbb{H}}$-Hopf-Mod. Note that, this
definition is a generalization to the Hopf truss setting of the notion of Hopf
module over a Hopf brace introduced in [5].

######  Example 5.6.

Let $X$ be an object in C and let ${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$ be a
Hopf truss. Then, the $4$-tuple

$(H\otimes X,\psi_{H\otimes X}^{1}=\mu_{H}^{1}\otimes X,\psi_{H\otimes
X}^{2}=\mu_{H}^{2}\otimes X,\rho_{H\otimes X}=\delta_{H}\otimes X)$

is a left ${\mathbb{H}}$-Hopf module. Indeed, by Example 4.2, the triple
$(H\otimes X,\psi_{H\otimes X}^{1},\psi_{H\otimes X}^{2})$ is a left
${\mathbb{H}}$-module. Moreover, $(H\otimes X,\psi_{H\otimes
X}^{1},\rho_{H\otimes X})$ is a left $H_{1}$-Hopf module and $(H\otimes
X,\psi_{H\otimes X}^{2},\rho_{H\otimes X})$ is a non-unital left $H_{2}$-Hopf
module. Finally,

is an equalizer diagram and this implies that $(H\otimes X)^{coH_{1}}=X$ and
$j_{H\otimes X}=\eta_{H}\otimes X$. Then, (iv) of Definition 5.5 holds,
because by (12), we have that $\varphi_{H\otimes X}\circ(\sigma_{H}\otimes
j_{H\otimes X})=\sigma_{H}\otimes X=\psi_{H\otimes X}\circ(H\otimes
j_{H\otimes X}).$ Note that, for $X=K$, we obtain that

$(H,\mu_{H}^{1},\mu_{H}^{2},\delta_{H})$

is an object in ${\mathbb{H}}$-Hopf-Mod where $H^{coH_{1}}=K$.

On the other hand, $q_{H\otimes X}=\varepsilon_{H}\otimes\eta_{H}\otimes X$
and, as a consequence, $I(q_{H\otimes X})=X$, $p_{H\otimes
X}=\varepsilon_{H}\otimes X$ and $i_{H\otimes X}=\eta_{H}\otimes X.$

Finally, it is obvious that, if $f:X\rightarrow Y$ is a morphism in C,
$H\otimes f$ is a morphism in ${\mathbb{H}}$-Hopf-Mod.

Then, as a consequence of the facts presented in the previous example, we have
the following results:

######  Theorem 5.7.

Let ${\mathbb{H}}$ be a Hopf truss. There exists a functor ${\sf
F}=H\otimes-:{\sf C}\;\rightarrow{\mathbb{H}}$-Hopf, called the induction
functor, defined on objects by ${\sf F}(X)=(H\otimes X,\psi_{H\otimes
X}^{1},\psi_{H\otimes X}^{2},\rho_{H\otimes X})$ and on morphisms by ${\sf
F}(f)=H\otimes f.$

######  Theorem 5.8.

(Fundamental Theorem of Hopf modules) Let
${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$ be a Hopf truss and let
$(M,\varphi_{M},\psi_{M},\rho_{M})$ be an object in ${\mathbb{H}}$-Hopf-Mod.
Then $(M,\psi_{M}^{1},\psi_{M}^{2},\rho_{M})$ and $F(M^{coH_{1}})$ are
isomorphic in ${\mathbb{H}}$-Hopf-Mod.

###### Proof.

Let $(M,\psi_{M}^{1},\psi_{M}^{2},\rho_{M})$ be an object in
${\mathbb{H}}$-Hopf. By Theorem 5.7, the $4$-tuple

$(H\otimes M^{coH_{1}},\psi_{H\otimes M^{coH_{1}}}^{1},\psi_{H\otimes
M^{coH_{1}}}^{2},\rho_{H\otimes M^{coH_{1}}})$

is a left ${\mathbb{H}}$-Hopf module. Let $\theta_{M}:H\otimes
M^{coH_{1}}\rightarrow M$ be the morphism defined by
$\theta_{M}=\psi_{M}^{1}\circ(H\otimes j_{M})$. Then, by the general theory of
Hopf modules exposed at the beginning of this section, $\theta_{M}$ is an
isomorphism of $H_{1}$-Hopf modules with inverse $\theta_{M}^{-1}=(H\otimes
t_{M})\circ\rho_{M}$. Also, $\theta_{M}$ is an isomorphism of non-unital
$H_{2}$-Hopf modules because

  * $\hskip 10.81218pt\psi_{M}^{2}\circ(H\otimes\theta_{M})$

  * $=\psi_{M}^{1}\circ(\mu_{H}^{2}\otimes\Gamma_{M}^{\sigma_{H}})\circ(H\otimes c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes j_{M})$ (by 20)

  * $=\psi_{M}^{1}\circ(\mu_{H}^{2}\otimes(\psi_{M}^{1}\circ(H\otimes\psi_{M}^{1})\circ((((\lambda_{H}\circ\sigma_{H})\otimes\sigma_{H})\circ\delta_{H})\otimes M)))\circ(H\otimes c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes j_{M})$

  * (by (v) of definition 5.5)

  * $=\psi_{M}^{1}\circ(\mu_{H}^{2}\otimes(\psi_{M}^{1}\circ((\lambda_{H}\ast id_{H})\circ\sigma_{H})\otimes M))\circ(H\otimes c_{H,H}\otimes M)\circ(\delta_{H}\otimes H\otimes j_{M})$ (by the

  * condition of comonoid morphism for $\sigma_{H}$ and the condition of left $H_{1}$-module for $M$)

  * $=\psi_{M}^{1}\circ(\mu_{H}^{2}\otimes j_{M})$ (by (8), the naturality of $c$, the counit properties and the condition of left $H_{1}$-module for $M$)

  * $=\theta_{M}\circ\psi_{H\otimes M^{coH_{1}}}^{2}$ (by definition.)

Therefore, $\theta_{M}$ is an isomorphism in ${\mathbb{H}}$-Hopf-Mod because,
by (ii) of Definition 5.5, the properties of $j_{M}$, the naturality of $c$
and the properties of $\eta_{H}$, we have that $\theta_{M}$ is a morphism of
left $H$-comodules.

∎

In Theorem 5.7 we construct the induction functor ${\sf F}=H\otimes-:{\sf
C}\rightarrow{\mathbb{H}}$-Hopf-Mod for a Hopf truss
${\mathbb{H}}=(H_{1},H_{2},\sigma_{H})$. As in the Hopf case, for all
$M\in{\mathbb{H}}$-Hopf-Mod, the construction of the subobject of coinvariants
$M^{co{\mathbb{H}}}:=M^{coH_{1}}$ is functorial. Thus, there exists a functor
of coinvariants ${\sf W}=(\;\;)^{co{\mathbb{H}}}:{\mathbb{H}}$-Hopf-
Mod$\rightarrow{\sf C}$ defined on objects by $={\sf
W}((M,\psi_{M}^{1},\psi_{M}^{2},\rho_{M}))=M^{co{\mathbb{H}}}$ where
$M^{co{\mathbb{H}}}=M^{coH_{1}}$ and on morphisms $f:M\rightarrow N$ by ${\sf
W}(f)=f^{co{\mathbb{H}}}$, where $f^{co{\mathbb{H}}}:=f^{coH_{1}}$ is the
unique morphism such that $j_{N}\circ f^{coH_{1}}=f\circ j_{M}$.

In the end of this section we prove that there exists a categorical
equivalence between C and the category ${\mathbb{H}}$-Hopf for a Hopf truss
${\mathbb{H}}$.

######  Theorem 5.9.

Let ${\mathbb{H}}$ a Hopf truss. The induction functor ${\sf F}=H\otimes-:{\sf
C}\rightarrow{\mathbb{H}}$-Hopf-Mod is left adjoint of the functor of
coinvariants ${\sf W}=(\;\;)^{co{\mathbb{H}}}:{\mathbb{H}}$-Hopf-
Mod$\rightarrow{\sf C}$ and they induce a categorical equivalence between
${\mathbb{H}}$-Hopf and C.

###### Proof.

Let $X$ be an object in C. Then the unit of the adjunction is
$\alpha_{X}=id_{X}$ because as we proved in Example 5.6, ${\sf W}({\sf
F}(X))=X$. For all $(M,\psi_{M}^{1},\psi_{M}^{2},\rho_{M})$ in
${\mathbb{H}}$-Hopf-Mod, the counit is defined by $\beta_{M}=\theta_{M}$ where
$\theta_{M}$ is the isomorphism introduced in the proof of Theorem 5.8. The
triangular identities hold, because they hold for the adjunction between the
categories C and $H_{1}$-Hopf-Mod. ∎

If we particularize the previous theorems to the case of Hopf braces we have
the Fundamental Theorem of Hopf Modules and the associated categorical
equivalence obtained in [5].

## Funding

The authors were supported by Ministerio de Ciencia e Innovación of Spain.
Agencia Estatal de Investigación. Unión Europea - Fondo Europeo de Desarrollo
Regional (FEDER). Grant PID2020-115155GB-I00: Homología, homotopía e
invariantes categóricos en grupos y álgebras no asociativas.

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