Entwined modules over linear categories and Galois extensions

TL;DR

This paper explores modules over categorified fiber bundles using entwining structures involving linear categories and coalgebras, establishing Frobenius and separability conditions, and introducing Galois extensions of categories.

Contribution

It introduces the concept of C-Galois extensions of categories and characterizes entwined modules over these extensions as modules over subcategories of coinvariants.

Findings

Frobenius and separability conditions for functors on entwined modules

Definition of C-Galois extensions of categories

Equivalence of entwined modules and modules over coinvariant subcategories

Abstract

In this paper, we study modules over quotient spaces of certain categorified fiber bundles. These are understood as modules over entwining structures involving a small $K$-linear category $\mathcal D$ and a $K$-coalgebra $C$. We obtain Frobenius and separability conditions for functors on entwined modules. We also introduce the notion of a $C$-Galois extension $\mathcal E\subseteq \mathcal D$ of categories. Under suitable conditions, we show that entwined modules over a $C$-Galois extension may be described as modules over the subcategory $\mathcal E$ of $C$-coinvariants of $\mathcal D$.

Equations

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$$\psi =\{{\psi }_{XY}:C\otimes Ho{m}_{\mathcal{D}}(X,Y)⟶Ho{m}_{\mathcal{D}}(X,Y)\otimes C{\}}_{(X,Y)\in Ob(\mathcal{D}{)}^2}$$

$$\begin{array}[]{c}\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod\text{-}\mathcal{D}\qquad\mathscr{G}:Mod\text{-}\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\\ \end{array}$$

$$\begin{array}[]{ccc}\mbox{$R\longrightarrow S$ is split extension}&\qquad\Leftrightarrow&\mbox{Left adjoint $F:Mod\text{-}R\longrightarrow Mod\text{-}S$ is separable}\\ &&\\ \mbox{$R\longrightarrow S$ is separable extension}&\qquad\Leftrightarrow&\mbox{Right adjoint $G:Mod\text{-}S\longrightarrow Mod\text{-}R$ is separable}\\ &&\\ \mbox{$R\longrightarrow S$ is Frobenius extension}&\qquad\Leftrightarrow&\mbox{$(F,G)$ is Frobenius pair of functors}\\ \end{array}$$

$$(h\otimes C)(Y,X):=Hom_{\mathcal{D}}(Y,X)\otimes C\qquad\big{(}(h\otimes C)(\phi)\big{)}(f\otimes c):=\phi^{\prime\prime}f\phi^{\prime}_{\psi}\otimes c^{\psi}$$

$$\left({\theta }_X(c\otimes d)\right)\circ f={f_{\psi }}_{\psi }\circ {\theta }_Y\left(c^{\psi }\otimes d^{\psi }\right){\theta }_X(c\otimes d_1)\otimes d_2={\left({\theta }_X(c_2\otimes d)\right)}_{\psi }\otimes {c_1}^{\psi }$$

$${\theta }_X\circ {\Delta }_C={\epsilon }_C\cdot i{d}_X\forall X\in Ob(\mathcal{D})$$

$${\epsilon }_C(d)f=\sum \hat{f}\circ {\theta }_X(c_f\otimes d){\epsilon }_C(d)f=\sum {\hat{f}}_{\psi }\circ {\theta }_X(d^{\psi }\otimes c_f)$$

$$\begin{array}[]{l}(h\otimes C)(Y):=Hom_{\mathcal{D}}(-,Y)\otimes C\qquad(h\otimes C)(f)(Z)(g\otimes c):=fg\otimes c\end{array}$$

$$ca{n}_X:h{\otimes }_{\mathcal{E}}Ho{m}_{\mathcal{D}}(X,-)⟶Ho{m}_{\mathcal{D}}(X,-)\otimes C$$

$$\psi =\{{\psi }_{XY}:C\otimes Ho{m}_{\mathcal{D}}(X,Y)⟶Ho{m}_{\mathcal{D}}(X,Y)\otimes C{\}}_{(X,Y)\in Ob(\mathcal{D}{)}^2}$$

$$(gf{)}_{\psi }\otimes c^{\psi }$$

$${\epsilon }_C(c^{\psi })(f_{\psi })$$

$$f_{\psi }\otimes {\Delta }_C(c^{\psi })$$

$${\psi }_{XX}(c\otimes i{d}_X)$$

$$\rho_{\mathcal{M}(Y)}(\mathcal{M}(f)(m))=\big{(}{\mathcal{M}(f)(m)}\big{)}_{0}\otimes\big{(}{\mathcal{M}(f)(m)}\big{)}_{1}=\mathcal{M}(f_{\psi})(m_{0})\otimes{m_{1}}^{\psi}$$

$$Ho{m}_{\mathcal{M}(\psi {)}_{\mathcal{D}}^C}(\mathcal{M},\mathcal{N}):=\{\eta \in Ho{m}_{Mod\text{-}\mathcal{D}}(\mathcal{M},\mathcal{N})|\eta (X):\mathcal{M}(X)⟶\mathcal{N}(X)\text{is C-colinear}\forall X\in Ob(\mathcal{D})\}$$

$${\psi }_{XY}:C\otimes Ho{m}_{\mathcal{D}}(X,Y)⟶C\otimes Ho{m}_{\mathcal{D}}(X,Y)\otimes H\stackrel{\cong }{⟶}Ho{m}_{\mathcal{D}}(X,Y)\otimes C\otimes H⟶Ho{m}_{\mathcal{D}}(X,Y)\otimes C$$

$$\big{(}{\mathcal{M}(f)(m)}\big{)}_{0}\otimes\big{(}{\mathcal{M}(f)(m)}\big{)}_{1}=\mathcal{M}(f_{0})(m_{0})\otimes m_{1}f_{1}$$

$$\begin{array}[]{c}(\mathcal{M}\otimes C)(X):=\mathcal{M}(X)\otimes C\\ (\mathcal{M}\otimes C)(f)(m\otimes c):=\mathcal{M}(f_{\psi})(m)\otimes c^{\psi}\end{array}$$

$${\pi }_{\mathcal{M}(X)\otimes C}^r(m\otimes c):=(i{d}_{\mathcal{M}(X)}\otimes {\Delta }_C)(m\otimes c)=m\otimes c_1\otimes c_2$$

$$\begin{array}[]{ll}(\mathcal{M}(f_{\psi})(m)\otimes c^{\psi})_{0}\otimes(\mathcal{M}(f_{\psi})(m)\otimes c^{\psi})_{1}&=\mathcal{M}(f_{\psi})(m)\otimes{(c^{\psi})}_{1}\otimes{(c^{\psi})}_{2}\\ &=\mathcal{M}({f_{\psi}}_{\psi})(m)\otimes{c_{1}}^{\psi}\otimes{c_{2}}^{\psi}\quad(\text{using}~{}\eqref{eq 6.3})\\ &=(\mathcal{M}\otimes C)(f_{\psi})(m\otimes c_{1})\otimes{c_{2}}^{\psi}\end{array}$$

$$(N\otimes {\mathbf{h}}_X)(Y):=N\otimes {\mathbf{h}}_X(Y)$$

$$(N\otimes {\mathbf{h}}_X)(f)(n\otimes g):=n\otimes gf$$

$${\sigma }_{N\otimes {\mathbf{h}}_X(Y)}^r(n\otimes g):=n_0\otimes g_{\psi }\otimes {n_1}^{\psi }$$

$$\begin{array}[]{ll}(\sigma^{r}\otimes id_{C})\sigma^{r}(n\otimes g)=(\sigma^{r}\otimes id_{C})(n_{0}\otimes g_{\psi}\otimes{n_{1}}^{\psi})&=n_{0}\otimes{g_{\psi}}_{\psi}\otimes{n_{1}}^{\psi}\otimes{n_{2}}^{\psi}\\ &=n_{0}\otimes{g_{\psi}}\otimes\Delta_{C}({n_{1}}^{\psi})\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{by}~{}\eqref{eq 6.3})\\ &=(id_{N\otimes{\bf h}_{X}(Y)}\otimes\Delta_{C})\sigma^{r}(n\otimes g)\end{array}$$

$$\begin{array}[]{ll}(id_{N\otimes{\bf h}_{X}(Y)}\otimes\varepsilon_{C})\sigma^{r}(n\otimes g)&=(id_{N\otimes{\bf h}_{X}(Z)}\otimes\varepsilon_{C})(n_{0}\otimes g_{\psi}\otimes{n_{1}}^{\psi})\\ &=n_{0}\otimes\varepsilon_{C}({n_{1}}^{\psi})g_{\psi}=n_{0}\otimes\varepsilon_{C}(n_{1})g=n\otimes g\end{array}$$

$$\begin{array}[]{ll}\big{(}(N\otimes{\bf h}_{X})(f)(n\otimes g)\big{)}_{0}\otimes\big{(}(N\otimes{\bf h}_{X})(f)(n\otimes g)\big{)}_{1}&=n_{0}\otimes(gf)_{\psi}\otimes{n_{1}}^{\psi}\\ &=n_{0}\otimes g_{\psi}f_{\psi}\otimes{{n_{1}}^{\psi}}^{\psi}\quad(\text{by}~{}\eqref{eq 6.1})\\ &=(N\otimes{\bf h}_{X})(f_{\psi})(n_{0}\otimes g_{\psi})\otimes{{n_{1}}^{\psi}}^{\psi}\\ &=(N\otimes{\bf h}_{X})(f_{\psi})({(n\otimes g)}_{0})\otimes{{(n\otimes g)}_{1}}^{\psi}\end{array}$$

$$\left(({\mathbf{h}}_Y\otimes C)(f)\right){\psi }_{XY}(c\otimes g)$$

$$={\psi }_{X^{\prime }Y}(c\otimes gf)={\psi }_{X^{\prime }Y}(C\otimes {\mathbf{h}}_Y)(f)(c\otimes g)$$

$$\begin{array}[]{lll}{\big{(}\psi_{XY}(c\otimes g)\big{)}}_{0}\otimes{\big{(}\psi_{XY}(c\otimes g)\big{)}}_{1}&=g_{\psi}\otimes\Delta_{C}(c^{\psi})&\\ &={g_{\psi}}_{\psi}\otimes{c_{1}}^{\psi}\otimes{c_{2}}^{\psi}&(\text{by}~{}\eqref{eq 6.3})\\ &=(\psi_{XY}\otimes id){(c\otimes g)}_{0}\otimes{(c\otimes g)}_{1}&(\text{by}~{}\eqref{rightcomod2})\end{array}$$

$$\eta :\underset{i\in I}{⨁}{\mathbf{h}}_{|m_i|}⟶\mathcal{M}(0,...,0,i{d}_{|m_i|},0,...,0)\mapsto m_i$$

Full text

MSC(2010) Subject Classification: 16W30, 18E05

Keywords: Entwining structures, entwined modules, rings with several objects, Frobenius conditions, separability conditions, coalgebra-Galois extensions

1 Introduction

The purpose of this paper is to study a theory of modules over quotient spaces of certain categorified fiber bundles. Suppose that $X$ is an affine scheme over a field $K$ and let $G$ be an affine algebraic group scheme with a free action $\sigma:X\times G\longrightarrow X$ on $X$. Let $Y$ be the quotient given by the coequalizer

[TABLE]

If $X\longrightarrow Y$ is faithfully flat and the canonical map $can:X\times G\longrightarrow X\times_{Y}X$ is an isomorphism, then $X$ is said to be (see, for instance, [Mum], [HJS]) a principal fiber bundle over $Y$ with group $G$.

The algebraic counterpart of (1.1) consists of an algebra $A$, a Hopf algebra $H$ and a coaction $\rho:A\longrightarrow A\otimes H$ that makes $A$ into a right $H$-comodule algebra. Let B:=A^{coH}=\{\mbox{a\in A$$|$$\rho(a)=a\otimes 1_{H}}\} be the algebra of coinvariants of $A$, i.e., $B$ is given by the equalizer

[TABLE]

In this case, there is a canonical map $can:A\otimes_{B}A\longrightarrow A\otimes H$ determined by setting $can(x\otimes y)=x\cdot\rho(y)$. If the Hopf algebra $H$ has bijective antipode, $B\longrightarrow A$ is a faithfully flat extension and $can:A\otimes_{B}A\longrightarrow A\otimes H$ is an isomorphism, it was shown by Schneider [HJS] that modules over $B$ may be recovered as the category of “$(A,H)$-Hopf modules.”

We work with a small $K$-linear category $\mathcal{D}$, a $K$-coalgebra $C$ and an “entwining structure” $\psi$ consisting of a collection of morphisms

[TABLE]

satisfying conditions that we lay out in Section 2. We consider the category ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ of modules over the entwining structure $(\mathcal{D},C,\psi)$ (see Definition 2.2). These may be seen as modules over a “categorical quotient space” of $\mathcal{D}$ with respect to the coalgebra $C$ and the entwining $\psi$.

The notion of a $C$-Galois extension $\mathcal{E}\subseteq\mathcal{D}$ of categories is introduced in Section 4. Additionally, a $C$-Galois extension gives rise to a canonical entwining structure on $\mathcal{D}$. Under certain conditions, we show that modules over the category $\mathcal{E}$ of $C$-coinvariants of $\mathcal{D}$ may be described as modules over the canonical entwining structure.

Entwining structures for algebras were introduced by Brzeziński and Majid in [BrMj] and it was realized in Brzeziński [Brz00] that entwined modules provide a unifying formalism for studying diverse concepts such as relative Hopf modules, Doi-Hopf and Yetter-Drinfeld modules as well as coalgebra Galois extensions. In fact, the study of entwining structures for algebras and entwined modules over them is well developed in the literature and we refer the reader, for instance, to [Abu], [Brz00] [Br02], [BCT1], [BCT], [CG], [Jia], [Sch] for more on this subject.

Our notion of modules over an entwining structure $(\mathcal{D},C,\psi)$ builds on the analogy of Mitchell [Mit1] which says that a small $K$-linear category should be seen as a “$K$-algebra with several objects.” In particular, the category ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ also generalizes the “relative $(\mathcal{D},H)$-Hopf modules” studied in our previous work in [BBR], where $H$ is a Hopf algebra and $\mathcal{D}$ is an $H$-comodule category in the sense of Cibils and Solotar [CiSo]. In other words, $\mathcal{D}$ is a small $K$-linear category whose morphism spaces are equipped with a coaction of $H$ that is compatible with composition. When $\mathcal{D}$ has a single object, it reduces to an ordinary $H$-comodule algebra and the relative $(\mathcal{D},H)$-Hopf modules reduce to the usual notion of relative Hopf modules (see Takeuchi [Take]).

For Doi-Hopf modules, Frobenius and separability conditions were studied extensively in a series of papers [CMIZ], [CMZ1], [CMZ]. Later, Brzeziński studied Frobenius and Maschke type theorems for entwined modules in [Brz]. In this paper, we proceed in a manner analogous to the unified approach of Brzeziński, Caenepeel, Militaru and Zhu [uni] for studying Frobenius and separability conditions for entwined modules over $(\mathcal{D},C,\psi)$.

The idea is as follows: the “categorical quotient space” of $\mathcal{D}$ with respect to $C$ and $\psi$ may be thought of as a subcategory of $\mathcal{D}$ and ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ plays the role of modules over this subcategory. Although this “subcategory” of $\mathcal{D}$ need not exist in an explicit sense, we would like to study the properties of this extension of categories. In particular, we would like to know if it behaves like a separable, split or Frobenius extension of small $K$-linear categories. For this, we turn to a pair of functors

[TABLE]

Here $\mathscr{F}$ is the left adjoint and behaves like an “extension of scalars” whereas its right adjoint $\mathscr{G}$ behaves like a “restriction of scalars.” We recall here (see [uni, Theorem 1.2]) that in the classical case of an extension $R\longrightarrow S$ of rings inducing the pair of adjoint functors Mod\text{-}R\ext@arrow 3399\arrowfill@\mathrel{\raise 3.22916pt\hbox{\oalign{\scriptstyle\leftarrow$\cr\vrule width=0.0pt,height=2.15277pt$\hfil\scriptstyle\relbar\cr}}}\mathrel{\raise 3.22916pt\hbox{\oalign{\scriptstyle\relbar$\cr\vrule width=0.0pt,height=2.15277pt$\scriptstyle\relbar}}}\mathrel{\raise 3.22916pt\hbox{\oalign{\scriptstyle\relbar$\hfil\cr$\scriptstyle\vrule width=0.0pt,height=1.50694pt\smash{\rightarrow}\cr}}}{\text{\qquad F\qquad}}{\text{\qquad G\qquad}}Mod\text{-}S given by extension and restriction of scalars, we have:

[TABLE]

It is therefore natural to study criteria for the separability of the functors $\mathscr{F}$ and $\mathscr{G}$ as well as conditions for $(\mathscr{F},\mathscr{G})$ to be a Frobenius pair of functors.

In this paper, we will always use the following convention: for $f\in Hom_{\mathcal{D}}(Y,X)$ and $c\in C$, we write $\psi_{YX}(c\otimes f)=f_{\psi}\otimes c^{\psi}\in Hom_{\mathcal{D}}(Y,X)\otimes C$ with the summation omitted. We write $h:\mathcal{D}^{op}\otimes\mathcal{D}\longrightarrow Vect_{K}$ for the canonical $\mathcal{D}\text{-}\mathcal{D}$-bimodule $h(Y,X)=Hom_{\mathcal{D}}(Y,X)$. The entwining structure makes $h\otimes C$ into a $\mathcal{D}\text{-}\mathcal{D}$-bimodule by setting

[TABLE]

for any $(Y,X)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$, \phi:=(\phi^{\prime},\phi^{\prime\prime})\in Hom_{\mathcal{D}^{op}\otimes\mathcal{D}}\big{(}(Y,X),(Y^{\prime},X^{\prime})\big{)}, $f\in Hom_{\mathcal{D}}(Y,X)$ and $c\in C$. We consider a collection $\theta:=\{\theta_{X}:C\otimes C\longrightarrow End_{\mathcal{D}}(X)\}_{X\in Ob(\mathcal{D})}$ of $K$-linear maps satisfying the following conditions:

[TABLE]

for any $f\in Hom_{\mathcal{D}}(Y,X)$. Let $V_{1}$ be the $K$-space consisting of all such $\theta$. Our first result gives conditions for the functors $\mathscr{F}$ and $\mathscr{G}$ to be separable.

Theorem A**.**

(see 3.7, 3.8, 3.10 and 3.11) Let $\mathcal{D}$ be a small $K$-linear category, $(C,\Delta_{C},\varepsilon_{C})$ be a $K$-coalgebra and let $(\mathcal{D},C,\psi)$ be a right-right entwining structure.

• (a)

Let $V=Nat(\mathscr{G}\mathscr{F},1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}})$ be the space of natural transformations from $\mathscr{G}\mathscr{F}$ to $1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$. Then:

• (1)

There is an isomorphism $V\cong V_{1}$ of $K$-vector spaces.

• (2)

The functor $\mathscr{F}$ is separable if and only if there exists $\theta\in V_{1}$ such that

[TABLE]

• (b)

Let $W=Nat(1_{Mod\text{-}\mathcal{D}},\mathscr{F}\mathscr{G})$ be the space of natural transformations from $1_{Mod\text{-}\mathcal{D}}$ to $\mathscr{F}\mathscr{G}$. Then:

• (1)

There is an isomorphism of $K$-vector spaces from $W$ to $W_{1}=Nat(h,h\otimes C)$.

• (2)

The functor $\mathscr{G}$ is separable if and only if there exists $\eta\in W_{1}=Nat(h,h\otimes C)$ such that $(id_{h}\otimes\varepsilon_{C})\eta=id_{h}$.

The next result gives conditions for $(\mathscr{F},\mathscr{G})$ to be a Frobenius pair.

Theorem B**.**

(see 3.14) Let $\mathcal{D}$ be a small $K$-linear category, $(C,\Delta_{C},\varepsilon_{C})$ be a $K$-coalgebra and let $(\mathcal{D},C,\psi)$ be a right-right entwining structure. Then, $(\mathscr{F},\mathscr{G})$ is a Frobenius pair if and only if there exist $\theta\in V_{1}$ and $\eta\in W_{1}$ such that the following conditions hold:

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$, $d\in C$ and $\eta(X,Y)(f)=\sum\hat{f}\otimes c_{f}$.

More generally, the $\mathcal{D}\text{-}\mathcal{D}$-bimodule $h\otimes C$ may be treated as a functor $h\otimes C:\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ by setting (see Lemma 2.4)

[TABLE]

for $f\in Hom_{\mathcal{D}}(Y,X)$ and $g\otimes c\in Hom_{\mathcal{D}}(Z,Y)\otimes C$. Additionally, let $C$ be a finite dimensional coalgebra and let $C^{*}=Hom(C,K)$ be the linear dual of $C$. Then, we show that there is a functor $C^{*}\otimes h:\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$.

Theorem C**.**

(see 3.19) Let $(\mathcal{D},C,\psi)$ be an entwining structure and let $C$ be a finite dimensional coalgebra. Then, the following statements are equivalent:

(i) $(\mathscr{F},\mathscr{G})$ is a Frobenius pair.

(ii) $C^{*}\otimes h$ and $h\otimes C$ are isomorphic as functors from $\mathcal{D}$ to ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$.

In the final part of this paper, we study coalgebra Galois extensions of categories in a manner analogous to Brzeziński [Brz00], Brzeziński and Hajac [BH] and Caenepeel [Caeni]. For this, we suppose that every morphism space $Hom_{\mathcal{D}}(X,Y)$ carries the structure of a $C$-comodule $\rho_{XY}:Hom_{\mathcal{D}}(X,Y)\longrightarrow Hom_{\mathcal{D}}(X,Y)\otimes C$, $f\mapsto\sum f_{0}\otimes f_{1}$. This allows us to define a category $\mathcal{E}$ of $C$-coinvariants of $\mathcal{D}$ (see Definition 4.5). Further, we say that $\mathcal{D}$ is a $C$-Galois extension of $\mathcal{E}$ if the canonical map

[TABLE]

is an isomorphism for each $X\in Ob(\mathcal{D})$ (see Definition 4.7). We show that a $C$-Galois extension leads to a canonical entwining structure.

Theorem D**.**

(see 4.9) Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. Then, there exists a unique right-right entwining structure $(\mathcal{D},C,\psi)$ which makes $Hom_{\mathcal{D}}(-,Y)$ an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$ with its canonical $\mathcal{D}$-module structure and right $C$-coactions $\{\rho_{XY}\}_{X\in Ob(\mathcal{D})}$.

Conversely, under suitable conditions, an entwining structure $(\mathcal{D},C,\psi)$ may be used to express $\mathcal{D}$ as a $C$-Galois extension. In that case, the category ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ reduces to the category of modules over the $C$-coinvariants of $\mathcal{D}$.

Theorem E**.**

(see 4.12 and 4.21) Let $C$ be a $K$-coalgebra and $\mathcal{D}$ be a small $K$-linear category such that $Hom_{\mathcal{D}}(X,Y)$ has a right $C$-comodule structure $\rho_{XY}$ for every $X,Y\in Ob(\mathcal{D})$. Let $\mathcal{E}$ be the subcategory of $C$-coinvariants of $\mathcal{D}$. If there exists a convolution invertible collection $\Phi=\{\Phi_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)\}_{X,Y\in Ob(\mathcal{D})}$ of right $C$-comodule maps, then the following are equivalent:

(i) $\mathcal{D}$ is a $C$-Galois extension of $\mathcal{E}$.

(ii) There exists a right-right entwining structure $(\mathcal{D},C,\psi)$ such that $Hom_{\mathcal{D}}(-,Y)$ is an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$ with its canonical $\mathcal{D}$-module structure and right $C$-coactions $\{\rho_{XY}\}_{X\in Ob(\mathcal{D})}$.

(iii) For any $f\in Hom_{\mathcal{D}}(X,Y)$, the morphism $\sum f_{0}\circ\Phi^{\prime}_{ZX}(f_{1})\in Hom_{\mathcal{E}}(Z,Y)$ for every $Z\in Ob(\mathcal{D})$, where $\Phi^{\prime}$ is the convolution inverse of $\Phi$.

In this case, the categories ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ and Mod-$\mathcal{E}$ are equivalent.

Notations: Throughout the paper, $K$ is a field, $C$ is a $K$-coalgebra with comultiplication $\Delta_{C}$ and counit $\varepsilon_{C}$. We shall use Sweedler’s notation for the coproduct $\Delta_{C}(c)=c_{1}\otimes c_{2},$ and for a coaction $\rho_{M}:M\longrightarrow M\otimes C$, $\rho_{M}(m)=m_{0}\otimes m_{1}$ with the summation omitted. We denote by $C^{*}$ the linear dual of $C$. Sometimes when the coaction is clear from context, we will omit the subscript.

2 Entwining structures

In this section, we introduce a categorical generalization of entwining structures and entwined modules. We prove that the category of entwined modules is a Grothendieck category. We begin by recalling the definition of modules over a category (see, for instance, [Sten, Mit2]).

Definition 2.1**.**

A right module over a small $K$-linear category $\mathcal{D}$ is a $K$-linear functor $\mathcal{D}^{op}\longrightarrow Vect_{K}$, where $Vect_{K}$ denotes the category of $K$-vector spaces. Similarly, a left module over $\mathcal{D}$ is a $K$-linear functor $\mathcal{D}\longrightarrow Vect_{K}$. The category of all right (resp. left) modules over $\mathcal{D}$ will be denoted by $Mod$-$\mathcal{D}$ (resp. $\mathcal{D}$-$Mod$).

For each $X\in Ob(\mathcal{D})$, the representable functors ${\bf h}_{X}:=Hom_{\mathcal{D}}(-,X)$ and ${}_{X}{\bf h}:=Hom_{\mathcal{D}}(X,-)$ are examples of right and left modules over $\mathcal{D}$ respectively. Unless otherwise mentioned, by a $\mathcal{D}$-module we will always mean a right $\mathcal{D}$-module.

Let $C$ be a $K$-coalgebra and let $\mathcal{D}$ be a small $K$-linear category. Suppose that we have a collection of $K$-linear maps

[TABLE]

We use the notation $\psi_{XY}(c\otimes f)=f_{\psi}\otimes c^{\psi}$ for $c\in C$ and $f\in Hom_{\mathcal{D}}(X,Y)$. We will say that the tuple $(\mathcal{D},C,\psi)$ is a (right-right) entwining structure if the following conditions hold:

[TABLE]

for each $f\in Hom_{\mathcal{D}}(X,Y)$, $g\in Hom_{\mathcal{D}}(Y,Z)$ and $c\in C$. Throughout this paper, $(\mathcal{D},C,\psi)$ will always be an entwining structure. A morphism between entwining structures $(\mathcal{D^{\prime}},C^{\prime},\psi^{\prime})$ and $(\mathcal{D},C,\psi)$ is a pair $(\mathscr{F},\sigma)$ where $\mathscr{F}:\mathcal{D^{\prime}}\longrightarrow\mathcal{D}$ is a functor and $\sigma:C^{\prime}\longrightarrow C$ is a counital coalgebra map such that $\mathscr{F}({f^{\prime}}_{\psi^{\prime}})\otimes\sigma({c^{\prime}}^{\psi^{\prime}})=\mathscr{F}(f^{\prime})_{\psi}\otimes\sigma(c^{\prime})^{\psi}$ for any $c^{\prime}\otimes f^{\prime}\in C^{\prime}\otimes Hom_{\mathcal{D^{\prime}}}(X^{\prime},Y^{\prime})$ where $X^{\prime},Y^{\prime}\in Ob(\mathcal{D^{\prime}})$.

Definition 2.2**.**

Let $\mathcal{M}$ be a right $\mathcal{D}$-module with a given right $C$-comodule structure $\rho_{\mathcal{M}(Y)}:\mathcal{M}(Y)\longrightarrow\mathcal{M}(Y)\otimes C$ on $\mathcal{M}(Y)$ for each $Y\in Ob(\mathcal{D})$. Then, $\mathcal{M}$ is said to be an entwined module over $(\mathcal{D},C,\psi)$ if the following compatibility condition holds:

[TABLE]

for every $f\in Hom_{\mathcal{D}}(Y,X)$ and $m\in\mathcal{M}(X).$ We denote by ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ the category whose objects are entwined modules over $(\mathcal{D},C,\psi)$ and whose morphisms are given by

[TABLE]

We now give an important example of entwining structures.

Example 2.3**.**

Let $\mathcal{D}$ be a right co-$H$-category (see [CiSo] or the description in [BBR, Definition 2.4] ) and $C$ be a right $H$-module coalgebra. Then, the triple $(\mathcal{D},C,\psi)$ is an entwining structure, where $\psi$ is given by:

[TABLE]

Explicitly, we have $\psi_{XY}(c\otimes f):=f_{0}\otimes cf_{1}$ for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $c\in C$. In this case, an entwined module is precisely a right $\mathcal{D}$-module with a given right $C$-comodule structure on $\mathcal{M}(X)$ for each $X\in Ob(\mathcal{D})$ and satisfying the following compatibility condition

[TABLE]

We will refer to these modules as (right-right) Doi-Hopf modules and their category will be denoted by $\mathscr{M}_{\mathcal{D}}^{C}$. If $\mathcal{D}$ is a right co-$H$-category with a single object, i.e., an $H$-comodule algebra, then $\mathscr{M}_{\mathcal{D}}^{C}$ recovers the classical notion of Doi-Hopf modules (see [Doi]). In the particular case where $C=H$, the right-right Doi-Hopf modules have been referred to as relative Hopf modules in [BBR, $\S$ 5].

Lemma 2.4**.**

Let $(\mathcal{D},C,\psi)$ be an entwining structure and let $\mathcal{M}$ be a right $\mathcal{D}$-module. Then, we may obtain an object $\mathcal{M}\otimes C\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ by setting

[TABLE]

for $X\in\text{Ob}(\mathcal{D}),f\in Hom_{\mathcal{D}}(Y,X)$ and $m\otimes c\in\mathcal{M}(X)\otimes C$. In fact, this determines a functor from $Mod\text{-}\mathcal{D}$ to ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ .

Proof.

The fact that $\mathcal{M}\otimes C$ is a right $\mathcal{D}$-module follows from (2.1). For each $X\in Ob(\mathcal{D})$, it may be verified that $\mathcal{M}(X)\otimes C$ has a right $C$-comodule structure given by

[TABLE]

It remains to check the compatibility condition in (2.5). By definition, we have

[TABLE]

∎

Lemma 2.5**.**

Let $(\mathcal{D},C,\psi)$ be an entwining structure and $N$ be a right $C$-comodule. Then, for each $X\in Ob(\mathcal{D})$ we may obtain an object $N\otimes{\bf h}_{X}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ by setting

[TABLE]

for $Y\in Ob(\mathcal{D})$, $f\in Hom_{\mathcal{D}}(Z,Y)$, $n\otimes g\in N\otimes{\bf h}_{X}(Y)$. In fact, this determines a functor from $Comod\text{-}C$ to ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ .

Proof.

By definition, it follows that $N\otimes{\bf h}_{X}$ is a right $\mathcal{D}$-module. Further, for each $Y\in Ob(\mathcal{D})$, we define a $K$-linear map $\sigma^{r}_{N\otimes{\bf h}_{X}(Y)}:N\otimes{\bf h}_{X}(Y)\longrightarrow N\otimes{\bf h}_{X}(Y)\otimes C$ as follows

[TABLE]

We now verify that the map defined in (2.9) makes $N\otimes{\bf h}_{X}(Y)$ a right $C$-comodule. We have

[TABLE]

Moreover, using (2.2) we have

[TABLE]

It remains to verify the condition in (2.5). We have

[TABLE]

∎

It follows from Lemma 2.4 and Lemma 2.5 that both ${\bf h}_{Y}\otimes C$ and $C\otimes{\bf h}_{Y}$ are objects in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$.

Lemma 2.6**.**

Let $(\mathcal{D},C,\psi)$ be an entwining structure. Then, for each $Y\in Ob(\mathcal{D})$, we get a morphism $\Psi_{Y}:C\otimes{\bf h}_{Y}\longrightarrow{\bf h}_{Y}\otimes C$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ given by $\Psi_{Y}(X):=\psi_{XY}$.

Proof.

First we verify that $\Psi_{Y}$ is a morphism of right $\mathcal{D}$-modules. For any $f\in Hom_{\mathcal{D}}(X^{\prime},X)$, $g\in Hom_{\mathcal{D}}(X,Y)$ and $c\in C$, we have

[TABLE]

Next, we will show that $\Psi_{Y}(X)$ is $C$-colinear for every $X\in Ob(\mathcal{D})$. We have

[TABLE]

∎

We now recall from [Mit1, $\S$ 3] and [Mit2] the notion of a finitely generated module over a category. Given $\mathcal{M}\in Mod$-$\mathcal{D}$, we set $el(\mathcal{M}):=\underset{X\in Ob(\mathcal{D})}{\coprod}\mathcal{M}(X)$ to be the collection of all elements of $\mathcal{M}$. Since $\mathcal{D}$ is small, we note that $el(\mathcal{M})$ is a set. If $m\in el(\mathcal{M})$ is such that $m\in\mathcal{M}(X)$, we will write $|m|=X$.

Definition 2.7**.**

Let $\mathcal{D}$ be a small preadditive category and let $\mathcal{M}$ be a right $\mathcal{D}$-module. For each $m\in el(\mathcal{M})$, we consider the corresponding morphism $\eta_{m}:{\bf h}_{|m|}\longrightarrow\mathcal{M}$. A family of elements $\{m_{i}\in el(\mathcal{M})\}_{i\in I}$ is said to be a generating set for $\mathcal{M}$ if the induced morphism

[TABLE]

is an epimorphism in $Mod\text{-}\mathcal{D}$. In other words, every element $m\in el(\mathcal{M})$ may be expressed as a sum $m=\sum_{i\in I}\mathcal{M}(f_{i})(m_{i})$, where each $f_{i}\in Hom_{\mathcal{D}}(|m|,|m_{i}|)$ and all but finitely many $\{f_{i}\}_{i\in I}$ are zero.

Lemma 2.8**.**

Let $(\mathcal{D},C,\psi)$ be an entwining structure and let $\mathcal{M}$ be an entwined module. We consider an element $m\in el(\mathcal{M})$. Then, there exists a finite dimensional $C$-subcomodule $V_{m}$ of $\mathcal{M}(|m|)$ containing $m$ and a morphism $\eta_{m}:V_{m}\otimes{\bf h}_{|m|}\longrightarrow\mathcal{M}$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ such that $\eta_{m}(|m|)(m\otimes id_{|m|})=m$.

Proof.

By [SCS, Theorem 2.1.7], we know that there exists a finite dimensional $C$-subcomodule $V_{m}\subseteq\mathcal{M}(|m|)$ such that $m\in V_{m}$. Now, we consider the $\mathcal{D}$-module morphism $\eta_{m}:V_{m}\otimes{\bf h}_{|m|}\longrightarrow\mathcal{M}$ defined by setting $\eta_{m}(Y)(v\otimes f):=\mathcal{M}(f)(v)$ for any $Y\in Ob(\mathcal{D}),~{}f\in Hom_{\mathcal{D}}(Y,|m|)$ and $v\in V_{m}$. We also have

[TABLE]

This shows that $\eta_{m}(Y)$ is $C$-colinear for each $Y\in Ob(\mathcal{D})$. Hence, $\eta_{m}$ is a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ such that $\eta_{m}(|m|)(m\otimes id_{|m|})=m$. ∎

Proposition 2.9**.**

Let $(\mathcal{D},C,\psi)$ be an entwining structure. Then, the category ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ of entwined modules is a Grothendieck category.

Proof.

Given a morphism $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, let $Ker(\eta)$ and $Coker(\eta)$ be respectively the kernel and cokernel in $Mod\text{-}\mathcal{D}$. Since $Comod\text{-}C$ is an abelian category, we know that $Ker(\eta)(X)$, $Coker(\eta)(X)\in Comod\text{-}C$ for each $X\in Ob(\mathcal{D})$. It is easily seen that $Ker(\eta)$ and $Coker(\eta)$ satisfy the compatibility condition in (2.5), i.e., $Ker(\eta)$, $Coker(\eta)\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Since limits and colimits in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ are obtained from those in $Mod\text{-}\mathcal{D}$ and $Comod\text{-}C$, it is clear that ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ is a cocomplete abelian category satisfying (AB5).

By Lemma 2.8, there is an epimorphism

[TABLE]

for any $\mathcal{M}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. As such, the collection $\{V\otimes{\bf h}_{X}\}$, where $X$ ranges over all objects in $\mathcal{D}$ and $V$ ranges over all (isomorphism classes of) finite dimensional $C$-comodules gives a set of generators for ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ in the sense of [Grothen, Proposition 1.9.1]. ∎

Corollary 2.10**.**

The category $\mathscr{M}_{\mathcal{D}}^{C}$ of Doi-Hopf modules is a Grothendieck category.

3 Separability and Frobenius conditions

Let $\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod$-$\mathcal{D}$ be the forgetful functor. The next result shows that the functor $\mathscr{F}$ has a right adjoint.

Lemma 3.1**.**

The forgetful functor $\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod$-$\mathcal{D}$ has a right adjoint $\mathscr{G}:Mod\text{-}\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ given by $\mathscr{G}(\mathcal{N}):=\mathcal{N}\otimes C$ for each $\mathcal{N}\in Mod\text{-}\mathcal{D}$.

Proof.

From Lemma 2.4, we know that $\mathscr{G}(\mathcal{N})=\mathcal{N}\otimes C\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for each $\mathcal{N}\in Mod\text{-}\mathcal{D}$. We define \alpha:Hom_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}\big{(}\mathcal{M},\mathscr{G}(\mathcal{N})\big{)}\longrightarrow Hom_{Mod\text{-}\mathcal{D}}(\mathscr{F}(\mathcal{M}),\mathcal{N}) by setting

[TABLE]

for each $\xi:\mathcal{M}\longrightarrow\mathcal{N}\otimes C$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, $X\in Ob(\mathcal{D})$ and $m\in\mathcal{M}(X)$.

We also define \beta:Hom_{Mod\text{-}\mathcal{D}}(\mathscr{F}(\mathcal{M}),\mathcal{N})\longrightarrow Hom_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}\big{(}\mathcal{M},\mathscr{G}(\mathcal{N})\big{)} by setting

[TABLE]

for each $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ in $Mod\text{-}\mathcal{D}$, $X\in Ob(\mathcal{D})$ and $m\in\mathcal{M}(X)$. First we check that $\alpha(\xi)$ and $\beta(\eta)$ are morphisms in $Mod\text{-}\mathcal{D}$ and ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ respectively. Using the fact that $id_{\mathcal{N}}\otimes\varepsilon_{C}:\mathcal{N}\otimes C\longrightarrow\mathcal{N}$ and $\xi$ are right $\mathcal{D}$-module morphisms, for any $f\in Hom_{\mathcal{D}}(Y,X)$, we have

[TABLE]

We also have

[TABLE]

Moreover, it is easy to see that $\beta(\eta)(X)$ is $C$-colinear for each $X\in Ob(\mathcal{D})$. We now verify that $\alpha$ and $\beta$ are inverses to each other.

[TABLE]

Further, we have \alpha\big{(}\beta(\eta)\big{)}(X)(m)=\eta(X)(m_{0})\varepsilon_{C}(m_{1})=\eta(X)(m). This proves the result. ∎

We now describe the unit $\mu:1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}\longrightarrow\mathscr{GF}$ and the counit $\nu:\mathscr{FG}\longrightarrow 1_{Mod\text{-}\mathcal{D}}$ of the adjunction in Lemma 3.1:

[TABLE]

[TABLE]

for each $\mathcal{M}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, $\mathcal{N}\in Mod\text{-}\mathcal{D}$, $X\in Ob(\mathcal{D})$.

We recall that a functor $F:\mathcal{A}\longrightarrow\mathcal{B}$ between arbitrary categories is said to be separable if the natural transformation

[TABLE]

induced by $F$ is a split monomorphism (see [NVV], [Rf, $\S$ 1]). The following result provides a characterization of separable functors.

Theorem 3.2**.**

[Rf, Theorem 1.2]* Let $F:\mathcal{A}\longrightarrow\mathcal{B}$ be a functor which has a right adjoint $G:\mathcal{B}\longrightarrow\mathcal{A}$. Let $\mu$ and $\nu$ be the unit and counit of this adjunction respectively. Then,*

• (i)

$F$* is separable if and only if there exists $\upsilon\in Nat(GF,1_{\mathcal{A}})$ such that $\upsilon\circ\mu=1_{\mathcal{A}}$, the identity natural transformation on $\mathcal{A}$.*

• (ii)

$G$* is separable if and only if there exists $\zeta\in Nat(1_{\mathcal{B}},FG)$ such that $\nu\circ\zeta=1_{\mathcal{B}}$, the identity natural transformation on $\mathcal{B}$.*

3.1 Separability conditions

Let $(\mathcal{D},C,\psi)$ be an entwining structure. We now investigate the separability of the forgetful functor $\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod$-$\mathcal{D}$. Since $\mathscr{F}$ has a right adjoint $\mathscr{G}$, it follows from Theorem 3.2 that the functor $\mathscr{F}$ is separable if and only if there exists a natural transformation $\upsilon:\mathscr{G}\mathscr{F}\longrightarrow 1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$ such that $\upsilon\circ\mu=1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$, where $\mu$ is the unit of the adjunction as explained in (3.1). Throughout Section 3, $V:=Nat(\mathscr{G}\mathscr{F},1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}})$ will denote the $K$-space of all natural transformations from $\mathscr{G}\mathscr{F}$ to $1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$. We will shortly give another useful interpretation of $V$. We start by proving few preparatory results required for this.

We recall from Lemma 2.4 and Lemma 2.5 that both ${\bf h}_{Y}\otimes C$ and $C\otimes{\bf h}_{Y}$ are objects in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$. We define a functor ${\bf h}\otimes C:\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ as

[TABLE]

for $f\in Hom_{\mathcal{D}}(Y,X),~{}g\in{\bf h}_{Y}(Z)$ and $c\in C$. Similarly, we may also obtain a functor ${\bf h}\otimes C\otimes C:\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$.

Lemma 3.3**.**

Let $f\in Hom_{\mathcal{D}}(Y,X)$. For any $\upsilon\in V$ and $c,d\in C$, we have

[TABLE]

In particular, we have

[TABLE]

Proof.

A morphism $f:Y\longrightarrow X$ in $\mathcal{D}$ induces morphisms ${\bf h}_{Y}\otimes C\longrightarrow{\bf h}_{X}\otimes C$ and ${\bf h}_{Y}\otimes C\otimes C\longrightarrow{\bf h}_{X}\otimes C\otimes C$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ as explained in (3.4). Since $\upsilon:\mathscr{G}\mathscr{F}\longrightarrow 1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$ is a natural transformation, it follows that the following diagram commutes:

[TABLE]

Thus, we have

[TABLE]

We now consider the morphism $\Psi_{Y}:C\otimes{\bf h}_{Y}\longrightarrow{\bf h}_{Y}\otimes C$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ given by $\Psi_{Y}(X):=\psi_{XY}$ as in Lemma 2.6. Then, using the naturality of $\upsilon:\mathscr{G}\mathscr{F}\longrightarrow 1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$ and (2.2) we have the following commutative diagram

[TABLE]

Using the fact that $\psi_{YY}(c\otimes id_{Y})=id_{Y}\otimes c$, we now have

[TABLE]

Combining (3.7) and (3.8), we have

[TABLE]

By putting $Y=X$ and taking $f=id_{X}$, the result of (3.6) is clear from (3.9). ∎

Lemma 3.4**.**

For any $\upsilon\in V$ and $Y\in Ob(\mathcal{D})$, we have $\upsilon(C\otimes C\otimes{\bf h}_{Y})=id_{C}\otimes\upsilon(C\otimes{\bf h}_{Y})$ as a morphism of $\mathcal{D}$-modules.

Proof.

For each $d\in C$, we define $\eta_{d}:C\otimes{\bf h}_{Y}\longrightarrow C\otimes C\otimes{\bf h}_{Y}$ by

[TABLE]

for each $X\in Ob(\mathcal{D})$, $g\in{\bf h}_{Y}(X)$ and $c\in C$. It may be easily verified that $\eta_{d}$ is a morphism of right $\mathcal{D}$-modules. We now verify that $\eta_{d}(X):C\otimes{\bf h}_{Y}(X)\longrightarrow C\otimes C\otimes{\bf h}_{Y}(X)$ is right $C$-colinear. We have

[TABLE]

Thus, $\eta_{d}:C\otimes{\bf h}_{Y}\longrightarrow C\otimes C\otimes{\bf h}_{Y}$ is a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Therefore, using the naturality of $\upsilon$, we have the following commutative diagram:

[TABLE]

Thus, for any $g\in Hom_{\mathcal{D}}(X,Y)$ and $c,c^{\prime}\in C$, we get

[TABLE]

The result follows. ∎

We now proceed to give another interpretation of $V=Nat(\mathscr{G}\mathscr{F},1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}})$. We consider a collection $\theta:=\{\theta_{X}:C\otimes C\longrightarrow End_{\mathcal{D}}(X)\}_{X\in Ob(\mathcal{D})}$ of $K$-linear maps satisfying the following conditions:

[TABLE]

for any $f\in Hom_{\mathcal{D}}(Y,X)$. Let $V_{1}$ be the $K$-space consisting of all such $\theta$.

Proposition 3.5**.**

Let $\upsilon\in V=Nat(\mathscr{G}\mathscr{F},1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}})$. For each $X\in Ob(\mathcal{D})$, we define a $K$-linear map

[TABLE]

*Then, $\theta:=\{\theta_{X}\}_{X\in Ob(\mathcal{D})}$ is an element in $V_{1}.$ *

Proof.

Since $id_{{\bf h}_{X}}\otimes\varepsilon_{C}:{\bf h}_{X}\otimes C\longrightarrow{\bf h}_{X}$ is a morphism of right $\mathcal{D}$-modules, we have

[TABLE]

for $f\in Hom_{\mathcal{D}}(Y,X)$ and $c,d\in C$. Since $\upsilon({{\bf h}_{X}\otimes C}):{\bf h}_{X}\otimes C\otimes C\longrightarrow{\bf h}_{X}\otimes C$ is a morphism of right $\mathcal{D}$-modules, we also have

[TABLE]

The morphism ${f_{\psi}}_{\psi}:Y\longrightarrow X$ in $\mathcal{D}$ induces morphisms ${\bf h}_{Y}\otimes C\longrightarrow{\bf h}_{X}\otimes C$ and ${\bf h}_{Y}\otimes C\otimes C\longrightarrow{\bf h}_{X}\otimes C\otimes C$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Therefore, we have

[TABLE]

This proves (3.11). We now verify that $\theta$ satisfies (3.12). Using Lemma 2.5, we know that $C\otimes{\bf h}_{Y}$ and $C\otimes C\otimes{\bf h}_{Y}$ belong to ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for each $Y\in Ob(\mathcal{D})$. For each $X\in Ob(\mathcal{D})$, it may be easily seen that $C\otimes{\bf h}_{Y}(X)$ is also a left $C$-comodule with coaction given by $\rho_{Y}^{l}(X):=\Delta_{C}\otimes id_{{\bf h}_{Y}(X)}$. Moreover, it may be easily verified that the following diagram commutes:

[TABLE]

This shows that $\rho_{Y}^{l}(X)$ is a morphism of right $C$-comodules. Further, for any $g\in Hom_{\mathcal{D}}(X,X^{\prime})$, we have the following commutative diagram:

[TABLE]

Thus, $\rho_{Y}^{l}:C\otimes{\bf h}_{Y}\longrightarrow C\otimes C\otimes{\bf h}_{Y}$ is a morphism of right $\mathcal{D}$-modules. This shows that $\rho_{Y}^{l}$ is a morphism in the category ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Therefore, using the naturality of $\upsilon$ and Lemma 3.4, we have the following commutative diagram:

[TABLE]

For any $c\otimes id_{X}\otimes d\in C\otimes{\bf h}_{X}(X)\otimes C$, we set a_{i}\otimes f_{i}:=\upsilon\big{(}C\otimes{\bf h}_{X}\big{)}(X)(c\otimes id_{X}\otimes d). Then, we have

[TABLE]

Now applying the map $id_{C}\otimes\varepsilon_{C}\otimes id_{{\bf h}_{X}}$ to both sides, we get

[TABLE]

Therefore, we have

[TABLE]

Since $\upsilon(C\otimes{\bf h}_{Y})(X)$ is a morphism of right $C$-comodules, we also have the following commutative diagram:

[TABLE]

Thus, we have

[TABLE]

Now, applying the map $\varepsilon_{C}\otimes id_{{\bf h}_{X}}\otimes id_{C}$ to both sides, we get

[TABLE]

Therefore,

[TABLE]

It now follows from (3.15) and (3.16) that $\theta$ satisfies (3.12). ∎

Proposition 3.6**.**

Let $\theta\in V_{1}$. Then, we have an element $\upsilon\in Nat(\mathscr{G}\mathscr{F},1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}})$ defined by

[TABLE]

for \mathcal{M}\in Ob\big{(}{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\big{)},~{}X\in Ob(\mathcal{D}),~{}m\in\mathcal{M}(X) and $c\in C$.

Proof.

We need to verify that $\upsilon(\mathcal{M}):\mathcal{M}\otimes C\longrightarrow\mathcal{M}$ is a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ and that $\upsilon$ is indeed a natural transformation. We first verify that $\upsilon(\mathcal{M})$ is a morphism of right $\mathcal{D}$-modules. Let $f\in Hom_{\mathcal{D}}(Y,X)$. Then, we have

[TABLE]

We now verify that $\upsilon(\mathcal{M})(X):\mathcal{M}(X)\otimes C\longrightarrow\mathcal{M}(X)$ is a morphism of right $C$-comodules for every $X\in Ob(\mathcal{D})$. For each $m\otimes c\in\mathcal{M}(X)\otimes C$, we have

[TABLE]

It remains to show that $\upsilon:\mathscr{G}\mathscr{F}\longrightarrow 1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$ is a natural transformation. Let $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ be a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Then, for every $X\in Ob(\mathcal{D})$ and $m\otimes c\in\mathcal{M}(X)\otimes C$, we have

[TABLE]

This proves the result. ∎

Proposition 3.7**.**

The $K$-spaces $V=Nat(\mathscr{G}\mathscr{F},1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}})$ and $V_{1}$ are isomorphic.

Proof.

We define $\alpha:V\longrightarrow V_{1}$ by setting $\alpha(\upsilon)=\theta$, where $\theta$ is the collection of $K$-linear maps $\{\theta_{X}:C\otimes C\longrightarrow End_{\mathcal{D}}(X)\}_{X\in Ob(\mathcal{D})}$ defined by

[TABLE]

for $c,d\in C$. Then, $\alpha$ is a well-defined map by Proposition 3.5. We also define $\beta:V_{1}\longrightarrow V$ by setting $\beta(\theta)=\upsilon$, where $\upsilon:\mathscr{G}\mathscr{F}\longrightarrow 1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$ is defined by

[TABLE]

for \mathcal{M}\in Ob\big{(}{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\big{)},~{}X\in Ob(\mathcal{D}),~{}m\otimes c\in\mathcal{M}(X)\otimes C. By Proposition 3.6, $\beta$ is well-defined. We will now verify that $\alpha$ and $\beta$ are inverses of each other. Let $\theta\in V_{1}$. Then, for any $X,Y\in Ob(\mathcal{D})$, we have

[TABLE]

This proves that \big{(}\alpha\beta(\theta)\big{)}_{X}=\theta_{X} for all $X\in Ob(\mathcal{D})$. Therefore, $(\alpha\beta)(\theta)=\theta$. For any $\upsilon\in V$, we now verify that $(\beta\alpha)(\upsilon)=\upsilon$. We set $\theta=\alpha(\upsilon)$. Then, by definition we have

[TABLE]

For any $m^{\prime}\in\mathcal{M}(X)$, it may be easily verified that $\eta_{m^{\prime}}:{\bf h}_{X}\longrightarrow\mathcal{M}$ defined by $\eta_{m^{\prime}}(Y)(f):=\mathcal{M}(f)(m^{\prime})$ for each $f\in Hom_{\mathcal{D}}(Y,X)$ is a morphism in $Mod\text{-}\mathcal{D}$. By Lemma 2.4, this induces the morphism $\eta_{m^{\prime}}\otimes id_{C}:{\bf h}_{X}\otimes C\longrightarrow\mathcal{M}\otimes C$ in $\mathscr{M}(\psi)_{\mathcal{D}}^{C}$ defined by $(\eta_{m^{\prime}}\otimes id_{C})(Y)(f\otimes c):=\mathcal{M}(f)(m^{\prime})\otimes c$ for $f\in Hom_{\mathcal{D}}(Y,X)$ and $c\in C$. Since $\upsilon$ is a natural transformation, it follows easily that the following diagram commutes

[TABLE]

In particular, we have

[TABLE]

The comodule structure on entwined modules determines a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ as follows. We define $\tilde{\rho}:\mathcal{M}\longrightarrow\mathcal{M}\otimes C$ given by

[TABLE]

for any $\mathcal{M}\in Ob({\mathscr{M}(\psi)}_{\mathcal{D}}^{C})$ and $X\in Ob(\mathcal{D})$. We first verify that $\tilde{\rho}$ is a morphism of right $\mathcal{D}$-modules. For any $f\in Hom_{\mathcal{D}}(Y,X)$ and $m\in\mathcal{M}(X)$, we have

[TABLE]

It may be verified easily that $\tilde{\rho}(X):\mathcal{M}(X)\longrightarrow\mathcal{M}(X)\otimes C$ is right $C$-colinear. Thus, $\tilde{\rho}:\mathcal{M}\longrightarrow\mathcal{M}\otimes C$ is a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Therefore, we have the following commutative diagram

[TABLE]

Thus, we get

[TABLE]

Now applying $id_{\mathcal{M}(X)}\otimes\varepsilon_{C}$ on both sides, we obtain

[TABLE]

∎

Theorem 3.8**.**

Let $\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod$-$\mathcal{D}$ be the forgetful functor and $\mathscr{G}:Mod$-$\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, $\mathcal{N}\mapsto\mathcal{N}\otimes C$ be its right adjoint. Then, $\mathscr{F}$ is separable if and only if there exists $\theta\in V_{1}$ such that

[TABLE]

Proof.

We first recall from (3.1) that the unit of the adjunction is given by

[TABLE]

for $\mathcal{M}\in Ob({\mathscr{M}(\psi)}_{\mathcal{D}}^{C})$ and $m\in\mathcal{M}(X)$. Suppose that $\mathscr{F}$ is separable. Then, by Theorem 3.2, there exists $\upsilon\in V$ such that $\upsilon\circ\mu=1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$. Therefore, using Proposition 3.7, corresponding to $\upsilon\in V$ we can obtain an element $\theta\in V_{1}$ given by \theta_{X}(c\otimes d)=\big{(}(id_{{\bf h}_{X}}\otimes\varepsilon_{C})\upsilon({\bf h}_{X}\otimes C)\big{)}(X)(id_{X}\otimes c\otimes d) for each $c,d\in C$. Moreover, we have

[TABLE]

for any $c\in C$. Conversely, suppose that $\theta\in V_{1}$ is such that $\theta_{X}\circ\Delta_{C}=\varepsilon_{C}\cdot id_{X}$ for every $X\in Ob(\mathcal{D})$. Corresponding to $\theta\in V_{1}$ there exists $\upsilon\in V$ defined by

[TABLE]

for \mathcal{M}\in Ob\big{(}{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\big{)},~{}X\in Ob(\mathcal{D}),~{}m\in\mathcal{M}(X) and $c\in C$. Further, we have

[TABLE]

This shows that $\upsilon\circ\mu=1_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$. Hence, $\mathscr{F}$ is separable by Theorem 3.2. ∎

Next we investigate the separability of the functor $\mathscr{G}:Mod\text{-}\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ given by $\mathscr{G}(\mathcal{N})=\mathcal{N}\otimes C$ for any $\mathcal{N}\in Mod\text{-}\mathcal{D}$. Since $\mathscr{G}$ is a right adjoint of $\mathscr{F}$, it follows from Theorem 3.2 that the functor $\mathscr{G}$ is separable if and only if there exists a natural transformation $\omega:1_{Mod\text{-}\mathcal{D}}\longrightarrow\mathscr{F}\mathscr{G}$ such that $\nu\circ\omega=1_{Mod\text{-}\mathcal{D}}$, where $\nu$ is the counit of the adjunction as explained in (3.2). We set $W:=Nat(1_{Mod\text{-}\mathcal{D}},\mathscr{F}\mathscr{G})$ and proceed to give another interpretation of $W$.

We define $h:\mathcal{D}^{op}\otimes\mathcal{D}\longrightarrow Vect_{K}$ as

[TABLE]

for any $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$, \phi:=(\phi^{\prime},\phi^{\prime\prime})\in Hom_{\mathcal{D}^{op}\otimes\mathcal{D}}\big{(}(X,Y),(X^{\prime},Y^{\prime})\big{)} and $f\in Hom_{\mathcal{D}}(X,Y)$. Similarly, we define the functor $h\otimes C:\mathcal{D}^{op}\otimes\mathcal{D}\longrightarrow Vect_{K}$ as

[TABLE]

for any $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$, \phi:=(\phi^{\prime},\phi^{\prime\prime})\in Hom_{\mathcal{D}^{op}\otimes\mathcal{D}}\big{(}(X,Y),(X^{\prime},Y^{\prime})\big{)}, $f\in Hom_{\mathcal{D}}(X,Y)$ and $c\in C$. By slight abuse of notation, we will make no distinction between functors $\mathcal{D}^{op}\otimes\mathcal{D}\longrightarrow Vect_{K}$ and functors $\mathcal{D}\longrightarrow Mod\text{-}\mathcal{D}$. We observe that $h\otimes C:\mathcal{D}^{op}\otimes\mathcal{D}\longrightarrow Vect_{K}$ corresponds to $\mathscr{F}\circ({\bf h}\otimes C)$ when viewed as a functor from $\mathcal{D}\longrightarrow Mod\text{-}\mathcal{D}$.

Given a natural transformation $\eta:h\longrightarrow h\otimes C$, it is easy to see that

[TABLE]

is a morphism of right $\mathcal{D}$-modules for each $Y\in Ob(\mathcal{D})$. Similarly, for each $X\in Ob(\mathcal{D})$,

[TABLE]

is a morphism of left $\mathcal{D}$-modules.

Throughout the rest of this section, we set $W_{1}:=Nat(h,h\otimes C)$, the $K$-space consisting of all natural transformations between the functors $h$ and $h\otimes C$.

Lemma 3.9**.**

Let $\eta\in W_{1}$. We set $\eta(X,X)(id_{X})=\sum a_{X}\otimes c_{X}$ for each $X\in Ob(\mathcal{D})$ and $\eta(Y,Z)(g):=\sum\hat{g}\otimes c_{g}$ for any $g\in Hom_{\mathcal{D}}(Y,Z)$. Then,

[TABLE]

Proof.

Since $\eta(-,Z):{\bf h}_{Z}\longrightarrow{\bf h}_{Z}\otimes C$ is a morphism of right $\mathcal{D}$-modules for each $Z\in Ob(\mathcal{D})$, we have the following commutative diagram:

[TABLE]

This diagram alongwith Lemma 2.4 gives

[TABLE]

Since $\eta(Y,{-}):{{}_{Y}}{\bf h}\longrightarrow{{}_{Y}}{\bf h}\otimes C$ is a morphism of left $\mathcal{D}$-modules, we also have the following commutative diagram:

[TABLE]

This gives

[TABLE]

The result now follows from (3.22) and (3.23). ∎

Proposition 3.10**.**

The $K$-spaces $W=Nat(1_{Mod\text{-}\mathcal{D}},\mathscr{F}\mathscr{G})$ and $W_{1}=Nat(h,h\otimes C)$ are isomorphic.

Proof.

We define a $K$-linear map $\gamma:W\longrightarrow W_{1}$ by setting

[TABLE]

for any $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$. We now verify that the map is well-defined. Let \phi:=(\phi^{\prime},\phi^{\prime\prime})\in Hom_{\mathcal{D}^{op}\otimes\mathcal{D}}\big{(}(X,Y),(X^{\prime},Y^{\prime})\big{)}. Since $\omega({\bf h}_{Y}):{\bf h}_{Y}\longrightarrow{\bf h}_{Y}\otimes C$ is a morphism of right $\mathcal{D}$-modules, we have the following commutative diagram:

[TABLE]

The morphism $\phi^{\prime\prime}:Y\longrightarrow Y^{\prime}$ in $\mathcal{D}$ induces a morphism $\phi^{\prime\prime}:{\bf h}_{Y}\longrightarrow{\bf h}_{Y^{\prime}}$ of right $\mathcal{D}$-modules. Therefore, using the naturality of $\omega$, we get the following commutative diagram:

[TABLE]

We now observe that for $f\in Hom_{\mathcal{D}}(X,Y)$, we have

[TABLE]

Thus, by combining the diagrams (3.24) and (3.25), we obtain the following commutative diagram:

[TABLE]

This shows that $\eta\in W_{1}$.

Conversely, let $\eta\in W_{1}=Nat(h,h\otimes C)$. For any $Y\in Ob(\mathcal{D})$,

[TABLE]

is a morphism of right $\mathcal{D}$-modules. For any $f\in Hom_{\mathcal{D}}(X,Y)$, the naturality of $\eta$ gives us the following commutative diagram:

[TABLE]

Now, for any $\mathcal{M}$ in $Mod$-$\mathcal{D}$, we know that $\mathcal{M}=\underset{y\in el(\mathcal{M})}{colim}~{}{\bf h}_{|y|}$. Similarly, $\mathcal{M}\otimes C=\underset{y\in el(\mathcal{M})}{colim}({\bf h}_{|y|}\otimes C)$ where the colimit is taken in $Mod\text{-}\mathcal{D}$. Thus, the morphisms as in (3.26) induce a morphism $\omega(\mathcal{M}):\mathcal{M}\longrightarrow\mathcal{M}\otimes C$ of right $\mathcal{D}$-modules. Moreover, for any morphism $\mathcal{M}\overset{\zeta}{\longrightarrow}\mathcal{N}$ in $Mod$-$\mathcal{D}$, the commutative diagrams as in (3.27) induce the following equality:

[TABLE]

Therefore, for $\eta\in W_{1}$ we have obtained a natural transformation $\omega:1_{Mod\text{-}\mathcal{D}}\longrightarrow\mathscr{FG}$ in $W$. We will denote this $K$-linear map by $\delta:W_{1}\longrightarrow W$, i.e., $\delta(\eta)=\omega$ determined by $\omega({\bf h}_{Y}):=\eta({-},Y)$ for each $Y\in Ob(\mathcal{D})$. It may be easily verified that the morphisms $\gamma$ and $\delta$ are inverses of each other. ∎

Theorem 3.11**.**

Let $\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod$-$\mathcal{D}$ be the forgetful functor and $\mathscr{G}:Mod$-$\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, $\mathcal{N}\mapsto\mathcal{N}\otimes C$ be its right adjoint. Then $\mathscr{G}$ is separable if and only if there exists $\eta\in W_{1}=Nat(h,h\otimes C)$ such that

[TABLE]

Proof.

Suppose that $\mathscr{G}$ is separable. Then, by Theorem 3.2, there exists $\omega\in W=Nat(1_{Mod\text{-}\mathcal{D}},\mathscr{F}\mathscr{G})$ such that $\nu\circ\omega=1_{Mod\text{-}\mathcal{D}}$, where $\nu$ is the counit of the adjunction. Using Proposition 3.10, corresponding to $\omega\in W$, there exists an element $\eta\in W_{1}$ given by $\eta(X,Y)=\omega({\bf h}_{Y})(X)$ for every $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$. The condition (3.28) now follows from the definition of the counit in (3.2).

Conversely, let $\eta\in W_{1}$ be such that $(id_{h}\otimes\varepsilon_{C})\eta=id_{h}$. We consider $\omega:1_{Mod\text{-}\mathcal{D}}\longrightarrow\mathscr{FG}$ given by $\omega({\bf h}_{Y}):=\eta({-},Y)$ for each $Y\in Ob(\mathcal{D})$. Then, $(id_{{\bf{h}}_{Y}}\otimes\varepsilon_{C})\omega({{\bf{h}}_{Y}})=(id_{{\bf{h}}_{Y}}\otimes\varepsilon_{C})\eta({-},Y)=id_{{{\bf{h}}_{Y}}}$. Since $\mathscr{F}$ is a left adjoint and it is clear from the definition that $\mathscr{G}$ preserves colimits, we obtain that $(id_{\mathcal{N}}\otimes\varepsilon_{C})\omega(\mathcal{N})=id_{\mathcal{N}}$ for any $\mathcal{N}\in Mod$-$\mathcal{D}$, i.e, $(id\otimes\varepsilon_{C})\omega=1_{Mod{\text{-}}\mathcal{D}}$. Therefore, $\mathscr{G}$ is separable by Theorem 3.2. ∎

3.2 Frobenius conditions

Let $F:\mathcal{A}\longrightarrow\mathcal{B}$ be a functor which has a right adjoint $G:\mathcal{B}\longrightarrow\mathcal{A}$. Then, the pair $(F,G)$ is called a Frobenius pair if $G$ is both a right and a left adjoint of $F$. We recall the following characterization for Frobenius pairs (see [uni, $\S$ 1])

Theorem 3.12**.**

Let $F:\mathcal{A}\longrightarrow\mathcal{B}$ be a functor which has a right adjoint $G$. Then, $(F,G)$ is a Frobenius pair if and only if there exist $\upsilon\in Nat(GF,1_{\mathcal{A}})$ and $\omega\in Nat(1_{\mathcal{B}},FG)$ such that

[TABLE]

for all $M\in\mathcal{A}$ and $N\in\mathcal{B}$.

Lemma 3.13**.**

For any $\omega\in W=Nat(1_{Mod\text{-}\mathcal{D}},\mathscr{F}\mathscr{G})$, $N\in Comod\text{-}C$ and $Y\in Ob(\mathcal{D})$, we have $\omega(N\otimes{\bf h}_{Y})=id_{N}\otimes\omega({\bf h}_{Y})$.

Proof.

For each $n\in N$, we define $\zeta_{n}:{\bf h}_{Y}\longrightarrow N\otimes{\bf h}_{Y}$ by

[TABLE]

for any $X\in Ob(\mathcal{D})$ and $f\in Hom_{\mathcal{D}}(X,Y)$. It may be easily verified that $\zeta_{n}$ is a morphism of right $\mathcal{D}$-modules. Therefore, using the naturality of $\omega$, we have the following commutative diagram:

[TABLE]

Let $f\in Hom_{\mathcal{D}}(X,Y)$. We set $\omega({\bf h}_{Y})(X)(f)=\sum\hat{f}\otimes c_{f}$. Then, we have

[TABLE]

The result follows. ∎

Theorem 3.14**.**

Let $\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod$-$\mathcal{D}$ be the forgetful functor and $\mathscr{G}:Mod$-$\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, $\mathcal{N}\mapsto\mathcal{N}\otimes C$ be its right adjoint. Then, $(\mathscr{F},\mathscr{G})$ is a Frobenius pair if and only if there exist $\theta\in V_{1}$ and $\eta\in W_{1}$ such that the following conditions hold:

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$, $d\in C$ and $\eta(X,Y)(f)=\sum\hat{f}\otimes c_{f}$.

Proof.

Suppose there exist $\theta\in V_{1}$ and $\eta\in W_{1}$ such that (3.31) and (3.32) hold. Then, using the isomorphisms $V\cong V_{1}$ and $W\cong W_{1}$ as in Propositions 3.7 and 3.10, there exist $\upsilon\in V$ and $\omega\in W$ corresponding to $\theta\in V_{1}$ and $\eta\in W_{1}$ respectively. We also know by Proposition 2.9 that the collection $\{N\otimes{\bf h}_{Y}\}$, where $N$ ranges over all (isomorphisms classes of) finite dimensional $C$-comodules and $Y$ ranges over all objects in $\mathcal{D}$, forms a generating set for ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Therefore, we first verify the condition (3.29) for $\mathcal{M}=N\otimes{\bf h}_{Y}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, where $N\in Comod\text{-}C$ and $Y\in Ob(\mathcal{D})$. For any $n\otimes f\in N\otimes Hom_{\mathcal{D}}(X,Y)$, we have

[TABLE]

This proves (3.29) for the generators of ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. As explained in the proof of Proposition 2.9, for any $\mathcal{M}$ in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, there is an epimorphism

[TABLE]

in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. The morphism $\eta:=\underset{m\in el(\mathcal{M})}{\bigoplus}\eta_{m}$ induces the following commutative diagram:

[TABLE]

From (3.33), it follows that $\mathscr{F}\left(\upsilon\left({V_{m}\otimes{\bf{h}}_{|m|}}\right)\right)\omega\left({\mathscr{F}\left(V_{m}\otimes{\bf{h}}_{|m|}\right)}\right)=id_{\mathscr{F}(V_{m}\otimes{\bf{h}}_{|m|})}$ for each $m\in el(\mathcal{M})$. Thus, by the commutative diagram (3.34), we have

[TABLE]

Since $\mathscr{F}$ is a left adjoint, it preserves epimorphisms. Since $\eta$ is an epimorphism, so is $\mathscr{F}(\eta)$. Therefore, (3.35) implies that $\mathscr{F}(\upsilon(\mathcal{M}))\circ\omega\left({\mathscr{F}(\mathcal{M})}\right)=id_{\mathscr{F}(\mathcal{M})}.$ This proves (3.29) for any $\mathcal{M}\in Ob({\mathscr{M}(\psi)}_{\mathcal{D}}^{C})$.

Next, we verify the condition (3.30). From the definition, it is clear that $\mathscr{G}$ preserves colimits. Since any $\mathcal{D}$-module may be expressed as the colimit of representable functors, it is enough to verify the condition (3.30) for representable functors. For any $f\otimes d\in{\bf h}_{Y}(X)\otimes C$, we have

[TABLE]

This proves (3.30). Therefore, $(\mathscr{F},\mathscr{G})$ is a Frobenius pair.

Conversely, suppose $(\mathscr{F},\mathscr{G})$ is a Frobenius pair. Then, there exist $\upsilon\in V$ and $\omega\in W$ satisfying (3.29) and (3.30). Then, using the isomorphisms $V\cong V_{1}$ and $W\cong W_{1}$ as in Propositions 3.7 and 3.10, there exist $\theta\in V_{1}$ and $\eta\in W_{1}$ corrresponding to $\upsilon\in V$ and $\omega\in W$ respectively. We will now verify the conditions (3.31) and (3.32). Taking $\mathcal{M}=C\otimes{\bf h}_{Y}$ in (3.29), for any $d\in C$ and $f\in Hom_{\mathcal{D}}(X,Y)$ we have

[TABLE]

Applying $\varepsilon_{C}\otimes id_{{\bf h}_{Y}(X)}$ on both sides, we get

[TABLE]

This proves (3.32). Now, taking $\mathcal{N}={\bf{h}}_{Y}$ in (3.30), we have

[TABLE]

Applying $id_{{\bf h}_{Y}(X)}\otimes\varepsilon_{C}$ on both sides, we get (3.31). This proves the result. ∎

3.3 Frobenius conditions in the case of a finite dimensional coalgebra

We continue with $(\mathcal{D},C,\psi)$ being an entwining structure. For each $Y\in Ob(\mathcal{D})$, we obtain an object $Hom(C,{\bf h}_{Y})$ in $Mod\text{-}\mathcal{D}$ by setting

[TABLE]

for any $X\in Ob(\mathcal{D})$, $g\in Hom_{\mathcal{D}}(X^{\prime},X)$, \phi\in Hom_{K}\big{(}C,{\bf h}_{Y}(X)\big{)} and $x\in C$. Using (3.36), we now define a functor $Hom(C,h):\mathcal{D}\longrightarrow Mod\text{-}\mathcal{D}$ as follows:

[TABLE]

for any $f\in Hom_{\mathcal{D}}(Y,X)$, \phi\in Hom_{K}\big{(}C,{\bf h}_{Y}(Z)\big{)} and $x\in C$.

For the rest of this section, we assume that $C$ is finite dimensional. Then, for each $Z\in Ob(\mathcal{D})$, we have an isomorphism

[TABLE]

Let $\{d_{i}\}_{1\leq i\leq k}$ be a basis for $C$ and $\{d_{i}^{*}\}_{1\leq i\leq k}$ be its dual basis.

Lemma 3.15**.**

Let $C$ be a finite dimensional coalgebra. Then, we have a functor

[TABLE]

Proof.

For each $Y\in Ob(\mathcal{D})$, it is clear that $C^{*}\otimes{\bf h}_{Y}\in Mod\text{-}\mathcal{D}$. We consider $f\in Hom_{\mathcal{D}}(Y,X)$ and an element $c^{*}\otimes g\in C^{*}\otimes{\bf h}_{Y}(Z)$. By the isomorphism in (3.38), $c^{*}\otimes g$ corresponds to the element \phi_{c^{*}\otimes g}\in Hom_{K}\big{(}C,{\bf h}_{Y}(Z)\big{)} given by $\phi_{c^{*}\otimes g}(x)=c^{*}(x)g$ for each $x\in C$. From the action in (3.37), the element f\cdot\phi_{c^{*}\otimes g}\in\big{(}Hom(C,{\bf h}_{X})\big{)}(Z) is given by

[TABLE]

Again, using the isomorphism in (3.38), the element in $C^{*}\otimes{\bf h}_{X}(Z)$ corresponding to $f\cdot\phi_{c^{*}\otimes g}$ is given by $\sum_{i=1}^{k}c^{*}(d_{i}^{\psi})d_{i}^{*}\otimes f_{\psi}g$. It may be easily verified that $(C^{*}\otimes h)(f):C^{*}\otimes{\bf h}_{Y}\longrightarrow C^{*}\otimes{\bf h}_{X}$ is a morphism of right $\mathcal{D}$-modules. The result now follows. ∎

Since $C$ is a coalgebra, its vector space dual $C^{*}$ is an algebra with the convolution product (c^{*}\mbox{\tiny\bullet }d^{*})(x):=\sum c^{*}(x_{1})d^{*}(x_{2}) for $c^{*},d^{*}\in C^{*}$ and $x\in C$. Let $N$ be any left $C^{*}$-module. Then, we have a $K$-linear map $\rho:N\longrightarrow Hom(C^{*},N)$ defined by $\rho(n)(c^{*}):=c^{*}n$ for $n\in N$ and $c^{*}\in C^{*}$.

In general, there is an embedding $N\otimes C\hookrightarrow Hom(C^{*},N)$ given by $(n\otimes x)(c^{*}):=c^{*}(x)n$ for $x\in C$. Since $C$ is finite dimensional, this embedding is also a surjection. This gives us a $K$-linear map $\rho:N\longrightarrow N\otimes C$ which makes $N$ a right $C$-comodule (see, for instance, [SCS, $\S$ 2.2]). Then, $\rho(n)=\sum_{i=1}^{k}d_{i}^{*}n\otimes d_{i}$. In particular, $C^{*}$ becomes a right $C$-comodule with

[TABLE]

Considering the element $\varepsilon_{C}\in C^{*}$, the coassociativity of the coaction $\rho_{C^{*}}$ may be used to verify that

[TABLE]

Proposition 3.16**.**

Let $C$ be a finite dimensional coalgebra. Then, we have a functor:

[TABLE]

Proof.

From (3.40), we know that $C^{*}$ is a right $C$-comodule. Applying Lemma 2.5, it follows that each $C^{*}\otimes{\bf h}_{Y}$ is an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Accordingly, the right $C$-comodule structure on $C^{*}\otimes{\bf h}_{Y}(Z)$ for any $Z\in Ob(\mathcal{D})$ is given by the following composition:

[TABLE]

Explicitly, we have \sigma^{r}_{C^{*}\otimes{\bf h}_{Y}(Z)}(c^{*}\otimes g)=\sum\limits_{i=1}^{k}d_{i}^{*}\mbox{\tiny\bullet }c^{*}\otimes g_{\psi}\otimes d_{i}^{\psi} for each $c^{*}\otimes g\in C^{*}\otimes{\bf h}_{Y}(Z)$. We consider $f\in Hom_{\mathcal{D}}(Y,X)$. By Lemma 3.15, this induces a morphism $C^{*}\otimes{\bf h}_{Y}\longrightarrow C^{*}\otimes{\bf h}_{X}$ in $Mod\text{-}\mathcal{D}$. In order to show that $C^{*}\otimes h:\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ is a functor, it therefore suffices to show that each morphism

[TABLE]

is right $C$-colinear. For any $c^{*}\otimes g\in C^{*}\otimes{\bf h}_{Y}(Z)$, we have

[TABLE]

∎

Since $C$ is finite dimensional, the right $C$-comodule structure on $C^{*}\otimes{\bf h}_{Y}(X)$ induces a right $C$-comodule structure on $Hom(C,{\bf h}_{Y}(X))$ for each $X,Y\in Ob(\mathcal{D})$ which we now explain. Let $\phi\in Hom(C,{\bf h}_{Y}(X))$. Then, $\phi$ corresponds to the element $\sum_{1\leq i\leq k}d_{i}^{*}\otimes\phi(d_{i})\in C^{*}\otimes{\bf h}_{Y}(X)$. We know by Proposition 3.16 that

[TABLE]

The element \sum_{i=1}^{k}d_{j}^{*}\mbox{\tiny\bullet }d_{i}^{*}\otimes(\phi(d_{i}))_{\psi}\otimes d_{j}^{\psi}\in C^{*}\otimes{\bf h}_{Y}(X)\otimes C corresponds to the element $\phi_{0}\otimes\phi_{1}\in Hom(C,{\bf h}_{Y}(X))\otimes C$ given by

[TABLE]

for $x\in C$. It now follows from (3.36), (3.37), (3.43) and Proposition 3.16 that we have a functor

[TABLE]

We also recall from (3.3) and (3.4), the functor $h\otimes C:\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, defined as follows:

[TABLE]

for $f\in Hom_{\mathcal{D}}(Y,X)$ and $g\otimes c\in{\bf h}_{Y}(Z)\otimes C$. We now set $V_{2}:=Nat(h\otimes C,C^{*}\otimes h)$.

Proposition 3.17**.**

Let $C$ be a finite dimensional coalgebra. Then,

[TABLE]

Proof.

Since $C$ is finite dimensional, we know that C^{*}\otimes{\bf h}_{Y}(X)\cong Hom_{K}\big{(}C,{\bf h}_{Y}(X)\big{)} for each $X,Y\in Ob(\mathcal{D})$. We first define a $K$-linear map $\Upsilon_{XY}:{\bf h}_{Y}(X)\otimes C\longrightarrow C^{*}\otimes{\bf h}_{Y}(X)$ given by

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $c,d\in C$. In other words, we have

[TABLE]

where $\{d_{i}\}_{1\leq i\leq k}$ is a basis for $C$ and $\{d_{i}^{*}\}_{1\leq i\leq k}$ is its dual basis.

We now define $\alpha^{\prime}:V_{1}\longrightarrow V_{2}$ by setting $\alpha^{\prime}(\theta)=\Upsilon$ with $\Upsilon:h\otimes C\longrightarrow C^{*}\otimes h$ defined as follows:

[TABLE]

for any $X,Y\in Ob(\mathcal{D})$. We now verify that $\alpha^{\prime}$ is a well-defined map. For this, we first check that $\Upsilon_{Y}:{\bf h}_{Y}\otimes C\longrightarrow C^{*}\otimes{\bf h}_{Y}$ is a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$. For any $g\in Hom_{\mathcal{D}}(X^{\prime},X)$, we need to show that the following diagram commutes:

[TABLE]

For any $f\otimes c\in{\bf h}_{Y}(X)\otimes C$, we have

[TABLE]

This shows that $\Upsilon_{Y}$ is a morphism of right $\mathcal{D}$-modules for every $Y\in Ob(\mathcal{D})$. Next we verify that $\Upsilon_{Y}(X):{\bf h}_{Y}(X)\otimes C\longrightarrow C^{*}\otimes{\bf h}_{Y}(X)$ is right $C$-colinear for every $X,Y\in Ob(\mathcal{D})$. We have

[TABLE]

Finally, we verify that $\Upsilon$ is a natural transformation from $h\otimes C$ to $C^{*}\otimes h$, i.e., the following diagram commutes for any $g\in Hom_{\mathcal{D}}(Y,Y^{\prime})$:

[TABLE]

For any $f\otimes c\in{\bf h}_{Y}(X)\otimes C$, we have

[TABLE]

This proves that $\Upsilon\in V_{2}$.

For the converse, we first observe that the functors $C^{*}\otimes h$ and $Hom(C,h)$ are isomorphic which follows from (3.38). We define $\beta^{\prime}:V_{2}\longrightarrow V_{1}$ by setting $\beta^{\prime}(\Upsilon)=\theta$ with $\theta_{X}:C\otimes C\longrightarrow End_{\mathcal{D}}(X)$ defined as follows:

[TABLE]

for any $X\in Ob(\mathcal{D})$ and $c,d\in C$. We will now verify that $\theta$ satisfies (3.11) and (3.12). For each $X\in Ob(\mathcal{D})$, we know that $\Upsilon_{X}:{\bf h}_{X}\otimes C\longrightarrow Hom(C,{\bf h}_{X})$ is a morphism of right $\mathcal{D}$-modules. Therefore, for any $f\in Hom_{\mathcal{D}}(Y,X)$, we have the following commutative diagram:

[TABLE]

Since $\Upsilon:h\otimes C\longrightarrow Hom(C,h)$ is a natural transformation, the following diagram also commutes for any $f\in Hom_{D}(Y,X)$:

[TABLE]

Therefore, we have

[TABLE]

This proves (3.11). Further, we have

[TABLE]

This proves (3.12). It remains to show that $\alpha^{\prime}$ and $\beta^{\prime}$ are inverses of each other. For every $\theta\in V_{1}$ and $c,d\in C$, it follows from (3.45) that

[TABLE]

Finally, for any $\Upsilon\in V_{2}$, $f\in Hom_{\mathcal{D}}(X,Y)$ and $c,d\in C$, we have

[TABLE]

This proves the result. ∎

Proposition 3.18**.**

Let $C$ be a finite dimensional coalgebra. Then, we have isomorphisms

[TABLE]

Proof.

Given an $\eta:h\longrightarrow h\otimes C$, we want to define $\Phi:C^{*}\otimes h\longrightarrow h\otimes C$. For each $Y\in Ob(\mathcal{D})$, we first define a $K$-linear map $\Phi_{YY}:C^{*}\otimes{\bf{h}}_{Y}(Y)\longrightarrow{\bf{h}}_{Y}(Y)\otimes C$ by the following composition:

[TABLE]

i.e., $\Phi_{YY}(c^{*}\otimes id_{Y})=\sum a_{Y}\otimes c^{*}(c_{Y_{2}})c_{Y_{1}}$, where $\sum a_{Y}\otimes c_{Y}=\eta(Y,Y)(id_{Y})$ as in the notation of Lemma 3.9. We observe that an element $c^{*}\otimes f\in C^{*}\otimes{\bf{h}}_{Y}(X)$ may be written as $c^{*}\otimes f=(C^{*}\otimes{\bf{h}}_{Y})(f)(c^{*}\otimes id_{Y})$. For each $X\in Ob(\mathcal{D})$, we now define $\Phi_{XY}:C^{*}\otimes{\bf{h}}_{Y}(X)\longrightarrow{\bf{h}}_{Y}(X)\otimes C$ as follows:

[TABLE]

for any $c^{*}\otimes f\in C^{*}\otimes{{\bf h}}_{Y}(X)$.

We define $\gamma^{\prime}:W_{1}\longrightarrow W_{2}$ by setting $\gamma^{\prime}(\eta)=\Phi$ with $\Phi:C^{*}\otimes h\longrightarrow h\otimes C$ given by

[TABLE]

for every $X,Y\in Ob(\mathcal{D})$. We now verify that $\gamma^{\prime}$ is a well-defined map. For this, we first check that $\Phi_{Y}:C^{*}\otimes{\bf{h}}_{Y}\longrightarrow{\bf{h}}_{Y}\otimes C$ is a morphism of right $\mathcal{D}$-modules for every $Y\in Ob(\mathcal{D})$, i.e., the following diagram commutes for any $g\in Hom_{\mathcal{D}}(X^{\prime},X)$:

[TABLE]

We have

[TABLE]

Next we verify that $\Phi_{Y}(X):C^{*}\otimes{\bf{h}}_{Y}(X)\longrightarrow{\bf{h}}_{Y}(X)\otimes C$ is right $C$-colinear for any $X,Y\in Ob(\mathcal{D})$:

[TABLE]

It follows that $\Phi_{Y}:C^{*}\otimes{\bf{h}}_{Y}\longrightarrow{\bf{h}}_{Y}\otimes C$ is a morphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. To show that $\Phi\in Nat(C^{*}\otimes h,h\otimes C)$, it remains to verify that the following diagram commutes:

[TABLE]

for any $g\in Hom_{\mathcal{D}}(Y,Z)$. For any $X\in Ob(\mathcal{D})$ and $c^{*}\otimes f\in C^{*}\otimes{\bf{h}}_{Y}(X)$, we have

[TABLE]

Conversely, we define $\delta^{\prime}:W_{2}\longrightarrow W_{1}$ by setting $\delta^{\prime}(\Phi)=\eta$ with $\eta:h\longrightarrow h\otimes C$ given by

[TABLE]

for any $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$ and $f\in Hom_{\mathcal{D}}(X,Y)$. We now verify that $\eta\in W_{1}$. Let $\phi:(X,Y)\longrightarrow(X^{\prime},Y^{\prime})$ be a morphism in $\mathcal{D}^{op}\otimes\mathcal{D}$ given by $\phi^{\prime}:X^{\prime}\longrightarrow X$ and $\phi^{\prime\prime}:Y\longrightarrow Y^{\prime}$ in $\mathcal{D}$. Then, using the fact that $\Phi_{Y}:C^{*}\otimes{\bf{h}}_{Y}\longrightarrow{\bf{h}}_{Y}\otimes C$ is a morphism of right $\mathcal{D}$-modules, we have

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$. This shows that the following diagram commutes:

[TABLE]

Now using the naturality of $\Phi:C^{*}\otimes h\longrightarrow h\otimes C$, we also have

[TABLE]

for any $g\in{\bf{h}}_{Y}(X^{\prime})$. Thus, we get the following commutative diagram:

[TABLE]

It now follows from (3.51) and (3.52) that the following diagram commutes:

[TABLE]

This shows that $\eta\in W_{2}$. It remains to check that $\gamma^{\prime}$ and $\delta^{\prime}$ are inverses of each other. First we verify that \big{(}(\delta^{\prime}\circ\gamma^{\prime})(\eta)\big{)}(X,Y)=\eta(X,Y) for all $X,Y\in Ob(\mathcal{D})$. For this, we set $\Phi=\gamma^{\prime}(\eta)$. Then, for any $f\in Hom_{\mathcal{D}}(X,Y)$, we have

[TABLE]

Next, we will show that \big{(}(\gamma^{\prime}\circ\delta^{\prime})(\Phi)\big{)}_{Y}(X)=\Phi_{Y}(X) for any $X,Y\in Ob(\mathcal{D})$. Since $C^{*}\otimes{\bf{h}}_{Y}(X)$ and ${\bf{h}}_{Y}(X)\otimes C$ are right $C$-comodules for any $X,Y\in Ob(\mathcal{D})$, they are also left $C^{*}$-modules. The left actions are respectively given by

[TABLE]

for any $d^{*},c^{*}\in C^{*}$, $f\in{\bf{h}}_{Y}(X)$ and $x\in C$. Moreover, since $\Phi_{Y}(X):C^{*}\otimes{\bf{h}}_{Y}(X)\longrightarrow{\bf{h}}_{Y}(X)\otimes C$ is right $C$-colinear, it is also left $C^{*}$-linear. We now set $\eta=\delta^{\prime}(\Phi)$. Then, for any $c^{*}\otimes f\in C^{*}\otimes{\bf{h}}_{Y}(X)$, we have

[TABLE]

This proves the result. ∎

Theorem 3.19**.**

Let $(\mathcal{D},C,\psi)$ be an entwining structure and assume that $C$ is a finite dimensional coalgebra. Let $\mathscr{F}:{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\longrightarrow Mod$-$\mathcal{D}$ be the functor forgetting the $C$-coaction and $\mathscr{G}:Mod$-$\mathcal{D}\longrightarrow{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ given by $\mathcal{N}\mapsto\mathcal{N}\otimes C$ be its right adjoint. Then, the following statements are equivalent:

(i) $(\mathscr{F},\mathscr{G})$ is a Frobenius pair.

(ii) There exist $\eta\in W_{1}$ and $\theta\in V_{1}$ such that the corresponding morphisms $\gamma^{\prime}(\eta)=\Phi:C^{*}\otimes h\longrightarrow h\otimes C\text{~{}and~{}}\alpha^{\prime}(\theta)=\Upsilon:h\otimes C\longrightarrow C^{*}\otimes h\text{~{}given by ~{}}$

[TABLE]

where $f\in{\bf{h}}_{Y}(X)$, $c^{*}\in C^{*}$ and $d\in C$, are inverses of each other.

(iii) $C^{*}\otimes h$ and $h\otimes C$ are isomorphic as objects of the category ${}_{\mathcal{D}}{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ of functors from $\mathcal{D}$ to ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$.

Proof.

(i) $\Rightarrow$ (ii) By assumption, there exist $\eta\in W_{1}$ and $\theta\in V_{1}$ satisfying (3.31) and (3.32). Then, $\alpha^{\prime}(\theta)=\Upsilon$ and $\gamma^{\prime}(\eta)=\Phi$ are morphisms in ${}_{\mathcal{D}}{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ in the notation of Proposition 3.17 and Proposition 3.18. Since $\Upsilon_{XY}:{\bf{h}}_{Y}(X)\otimes C\longrightarrow C^{*}\otimes{\bf{h}}_{Y}(X)$ and $\Phi_{XY}:C^{*}\otimes{\bf{h}}_{Y}(X)\longrightarrow{\bf{h}}_{Y}(X)\otimes C$ are right $C$-colinear, they are also left $C^{*}$-linear. Using this fact and (3.53), we have

[TABLE]

for any $c^{*}\otimes f\in C^{*}\otimes{\bf{h}}_{Y}(X)$. Thus, $\Upsilon\circ\Phi=id_{C^{*}\otimes h}$.

Using the naturality of $\Upsilon$ and $\Phi$, we have

[TABLE]

for any $f\otimes c\in{{\bf h}}_{Y}(X)\otimes C$. Thus, $\Phi\circ\Upsilon=id_{h\otimes C}$. This proves $(ii)$.

(ii) $\Rightarrow$ (iii) is obvious since both $\Phi$ and $\Upsilon$ are morphisms in ${}_{\mathcal{D}}{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$.

(iii) $\Rightarrow$ (i) Let $\Phi:C^{*}\otimes h\longrightarrow h\otimes C$ denote the isomorphism in ${}_{\mathcal{D}}{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. We consider the following morphism of $(\mathcal{D}^{op}\otimes\mathcal{D})$-modules

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$. We now set $\eta=\Phi\circ\Lambda\in W_{1}$ and $\theta=\beta^{\prime}(\Phi^{-1})\in V_{1}$ where $\beta^{\prime}$ is as in Proposition 3.17. If $\eta(X,Y)(f)=\sum\hat{f}\otimes c_{f}$, then

[TABLE]

Using the isomorphism as in (3.38) and evaluating the equality in (3.55) at $d\in C$, we get (3.32). We also have

[TABLE]

By applying the map $id_{{\bf h}_{X}(X)}\otimes\varepsilon_{C}$, we obtain

[TABLE]

Now using Lemma 3.9 and (3.56), we obtain

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $d\in C$. Applying to both sides the composition $Hom_{\mathcal{D}}(X,Y)\otimes Hom_{\mathcal{D}}(X,X)\longrightarrow Hom_{\mathcal{D}}(X,Y)$, we obtain $\sum\hat{f}\circ\left(\theta_{X}(c_{f}\otimes d)\right)=\sum fa_{X}\circ\theta_{X}(c_{X}\otimes d)=\varepsilon_{C}(d)f$. This proves (3.31). Therefore, $(\mathscr{F},\mathscr{G})$ is a Frobenius pair by Theorem 3.14. This completes the proof. ∎

4 Categorical Galois extensions and entwining structures

Let $\mathcal{D}$ be a small $K$-linear category. Let $(\mathcal{D},C,\psi)$ be a right-right entwining structure. We denote by ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$ the category of $\mathcal{D}$-$\mathcal{D}$ bimodules, i.e., the category whose objects are functors from $\mathcal{D}^{op}\otimes\mathcal{D}$ to $Vect_{K}$ and whose morphisms are natural transformations between these functors. We recall the functors $h$ and $h\otimes C$ in ${{}_{\mathcal{D}}}\mathcal{M}_{\mathcal{D}}$ from (3.20) and (3.21) respectively:

[TABLE]

for any $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$, \phi:=(\phi^{\prime},\phi^{\prime\prime})\in Hom_{\mathcal{D}^{op}\otimes\mathcal{D}}\big{(}(X,Y),(X^{\prime},Y^{\prime})\big{)} and $f\in Hom_{\mathcal{D}}(X,Y)$, $c\in C$. We refer, for instance, to [EV, $\S$ 2.2] for the tensor product which makes ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$ a monoidal category with $h\in{{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$ as the unit object.

Definition 4.1**.**

A $\mathcal{D}$-coring $\mathscr{C}$ is a coalgebra object in the monoidal category ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$. Explicitly, a $\mathcal{D}$-coring is a functor $\mathscr{C}:\mathcal{D}^{op}\otimes\mathcal{D}\longrightarrow Vect_{K}$ with two morphisms

[TABLE]

satisfying the coassociativity and counit axioms in ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$. A right $\mathscr{C}$-comodule consists of a right $\mathcal{D}$-module $\mathcal{M}$ equipped with a morphism $\rho_{\mathcal{M}}:\mathcal{M}\longrightarrow\mathcal{M}\otimes_{\mathcal{D}}\mathscr{C}$ of right $\mathcal{D}$-modules satisfying

[TABLE]

A morphism $\eta:(\mathcal{M},\rho_{\mathcal{M}})\longrightarrow(\mathcal{N},\rho_{\mathcal{N}})$ of right $\mathscr{C}$-comodules is a morphism $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ of right $\mathcal{D}$-modules satisfying

[TABLE]

The category of right $\mathscr{C}$-comodules will be denoted by $Comod$-$\mathscr{C}$.

Lemma 4.2**.**

Let $(\mathcal{D},C,\psi)$ be a right-right entwining structure. Then, the functor $h\otimes C$ is a $\mathcal{D}$-coring.

Proof.

It may be verified that $(h\otimes C)\otimes_{\mathcal{D}}(h\otimes C)\cong h\otimes C\otimes C$. This gives us morphisms

[TABLE]

in ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$. Using the coassociativity and counitality of the $K$-coalgebra $C$, it may be verified that $id_{h}\otimes\Delta_{C}$ and $id_{h}\otimes\varepsilon_{C}$ satisfy the coassociativity and counit axioms in the category ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$. Thus, $h\otimes C$ is a coalgebra object in ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$. ∎

Proposition 4.3**.**

Let $(\mathcal{D},C,\psi)$ be a right-right entwining structure. Then, the category $\mathscr{M}(\psi)_{\mathcal{D}}^{C}$ of entwined modules is identical to the category $Comod$-$(h\otimes C)$.

Proof.

Let $\mathcal{M}\in\mathscr{M}(\psi)_{\mathcal{D}}^{C}$. It may be verified that $\mathcal{M}\otimes C\cong\mathcal{M}\otimes_{\mathcal{D}}(h\otimes C)$ as right $\mathcal{D}$-modules. Then, by Lemma 2.4, $\mathcal{M}\otimes C\in\mathscr{M}(\psi)_{\mathcal{D}}^{C}$ and we have

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $m\in\mathcal{M}(Y)$. We thus obtain a morphism $\rho_{\mathcal{M}}:\mathcal{M}\longrightarrow\mathcal{M}\otimes C\cong\mathcal{M}\otimes_{\mathcal{D}}(h\otimes C)$ of right $\mathcal{D}$-modules given by $\rho_{\mathcal{M}}(X):=\rho_{\mathcal{M}(X)}$ for each $X\in Ob(\mathcal{D})$.

Applying (4.4), we have

[TABLE]

and

[TABLE]

The conditions in (4.3) now follow from the fact that $\rho_{\mathcal{M}(X)}$ is a $C$-coaction for each $X\in Ob(\mathcal{D})$. Therefore, $\mathcal{M}$ is a right $(h\otimes C)$-comodule.

Conversely, let $\mathcal{N}\in Comod$-$(h\otimes C)$. Then, $\mathcal{N}$ is a right $\mathcal{D}$-module with a given morphism $\rho_{\mathcal{N}}:\mathcal{N}\longrightarrow\mathcal{N}\otimes_{\mathcal{D}}(h\otimes C)\cong\mathcal{N}\otimes C$ of right $\mathcal{D}$-modules satisfying the conditions in (4.3). Thus, for each $Y\in Ob(\mathcal{D})$, we have a morphism $\rho_{\mathcal{N}}(Y):\mathcal{N}(Y)\longrightarrow\mathcal{N}(Y)\otimes C$ which satisfies

[TABLE]

In (4.7), we have identified $id_{\mathcal{N}}\otimes\Delta_{C}=id_{\mathcal{N}}\otimes\Delta_{h\otimes C}$ and $id_{\mathcal{N}}\otimes\varepsilon_{C}=id_{\mathcal{N}}\otimes\varepsilon_{(h\otimes C)}$ as in (4.5) and (4.6) respectively. Therefore, $\rho_{\mathcal{N}}(Y)$ defines a right $C$-comodule structure on $\mathcal{N}(Y)$ for every $Y\in Ob(\mathcal{D})$. Since $\rho_{\mathcal{N}}$ is a morphism of right $\mathcal{D}$-modules, we also have

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $n\in\mathcal{N}(Y)$. Therefore, $\mathcal{N}\in\mathscr{M}(\psi)_{\mathcal{D}}^{C}$. ∎

Lemma 4.4**.**

Let $i:\mathcal{E}\longrightarrow\mathcal{D}$ be an inclusion of small $K$-linear categories. Then, the functor $h\otimes_{\mathcal{E}}h:\mathcal{E}^{op}\otimes\mathcal{E}\longrightarrow Vect_{K}$ is a $\mathcal{D}$-coring, where $h$ is the $\mathcal{D}$-$\mathcal{D}$-bimodule as in (4.1).

Proof.

It is immediate that the functor $h\otimes_{\mathcal{E}}h$ is a $\mathcal{D}$-$\mathcal{D}$-bimodule. We need to show that $h\otimes_{\mathcal{E}}h$ is a coalgebra object in ${{}_{\mathcal{D}}}\mathscr{M}_{\mathcal{D}}$. We now define $\Delta:h\otimes_{\mathcal{E}}h\longrightarrow(h\otimes_{\mathcal{E}}h)\otimes_{\mathcal{D}}(h\otimes_{\mathcal{E}}h)\cong(h\otimes_{\mathcal{E}}h)\otimes_{\mathcal{E}}h$ as follows: for $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$, we set

[TABLE]

for any $f\otimes f^{\prime}\in{\bf h}_{Y}(Z)\otimes{{}_{X}}{\bf h}(Z)$ and $Z\in Ob(\mathcal{E})$. It is easy to check that $\Delta(X,Y)$ is well-defined. Also, it can be verified that for any morphism $(\phi^{\prime},\phi^{\prime\prime}):(X,Y)\longrightarrow(X^{\prime},Y^{\prime})$ in $\mathcal{D}^{op}\otimes\mathcal{D}$, the following diagram commutes:

[TABLE]

Thus, $\Delta$ is a morphism of $\mathcal{D}$-$\mathcal{D}$-bimodules. The map $\varepsilon:h\otimes_{\mathcal{E}}h\longrightarrow h$ is defined by composition. It may be verified that $\Delta$ and $\varepsilon$ satisfy the coassociativity and counit axioms respectively. ∎

Let $\mathcal{D}$ be a small $K$-linear category and let $C$ be a $K$-coalgebra. We consider the category ${}_{\mathcal{D}}\mathscr{M}^{C}$ of left-right Doi-Hopf modules (compare Example 2.3). Explicitly, an object in ${}_{\mathcal{D}}\mathscr{M}^{C}$ consists of a left $\mathcal{D}$-module $\mathcal{M}$ with a given right $C$-comodule structure on $\mathcal{M}(X)$ for each $X\in Ob(\mathcal{D})$ such that the following compatibility condition holds:

[TABLE]

for each $f\in Hom_{\mathcal{D}}(X,Y)$ and $m\in\mathcal{M}(X)$. A morphism $\eta:\mathcal{M}\longrightarrow\mathcal{N}$ in ${}_{\mathcal{D}}\mathscr{M}^{C}$ is a left $\mathcal{D}$-module morphism such that each $\eta(X):\mathcal{M}(X)\longrightarrow\mathcal{N}(X)$ is right $C$-colinear. By definition, $(h\otimes C)(X,-)={{}_{X}}{\bf h}\otimes C$ is a left $\mathcal{D}$-module for each $X\in Ob(\mathcal{D})$. The map $id\otimes\Delta_{C}:Hom_{\mathcal{D}}(X,Y)\otimes C\longrightarrow Hom_{\mathcal{D}}(X,Y)\otimes C\otimes C$ gives a right $C$-comodule structure on $({{}_{X}{\bf h}}\otimes C)(Y)$ for each $Y\in Ob(\mathcal{D})$. Clearly, ${}_{X}{\bf h}\otimes C\in{{}_{\mathcal{D}}}\mathscr{M}^{C}$.

From this point onwards, we suppose additionally that each $Hom_{\mathcal{D}}(X,Y)$ has a given right $C$-comodule structure denoted by

[TABLE]

Definition 4.5**.**

Let $\mathcal{E}\subseteq\mathcal{D}$ be the subcategory with $Ob(\mathcal{E})=Ob(\mathcal{D})$ and

[TABLE]

We will say that $\mathcal{E}$ is the subcategory of $C$-coinvariants of $\mathcal{D}$.

Example 4.6**.**

Let $H$ be a Hopf algebra over $K$ and let $\mathcal{D}$ be a right co-$H$-category. In this case, the subcategory $\mathcal{E}$ of $H$-coinvariants of $\mathcal{D}$ is given by setting $Ob(\mathcal{E})=Ob(\mathcal{D})$ and $Hom_{\mathcal{E}}(X,Y)=Hom_{\mathcal{D}}(X,Y)^{coH}$.

It follows that the right $C$-comodule structures $\rho_{XY}:Hom_{\mathcal{D}}(X,Y)\longrightarrow Hom_{\mathcal{D}}(X,Y)\otimes C$ induce a morphism ${{}_{X}{\bf h}}\longrightarrow{{}_{X}{\bf h}}\otimes C$ of left $\mathcal{E}$-modules for each $X\in Ob(\mathcal{D})$. Further, for every $Y\in Ob(\mathcal{D})$, this induces a morphism

[TABLE]

where $f\in Hom_{\mathcal{D}}(Z,Y),~{}f^{\prime}\in Hom_{\mathcal{D}}(X,Z)$ and $Z\in Ob(\mathcal{E})$. It may be easily verified that the coaction in (4.10) makes $h\otimes_{\mathcal{E}}{{}_{X}{\bf h}}$ an object of ${{}_{\mathcal{D}}}\mathscr{M}^{C}$.

We obtain therefore canonical morphisms of $K$-vector spaces given by the following composition

[TABLE]

For each $X\in Ob(\mathcal{D})$, this induces a morphism in ${{}_{\mathcal{D}}}\mathscr{M}^{C}$ as follows

[TABLE]

Definition 4.7**.**

Let $C$ be a $K$-coalgebra and $\mathcal{D}$ be a small $K$-linear category such that $Hom_{\mathcal{D}}(X,Y)$ has a right $C$-comodule structure for every $X,Y\in Ob(\mathcal{D})$. Let $\mathcal{E}$ be a $K$-linear subcategory of $\mathcal{D}$. Then, $\mathcal{D}$ is called a $C$-Galois extension of $\mathcal{E}$ if

• (i)

$Ob(\mathcal{E})=Ob(\mathcal{D})$* and $Hom_{\mathcal{E}}(X,Y)=Hom^{C}_{Mod\text{-}\mathcal{D}}({{\bf h}_{X}},{{\bf h}_{Y}})$.*

• (ii)

The induced canonical morphism $can_{X}:h\otimes_{\mathcal{E}}{{}_{X}{\bf h}}\longrightarrow{{}_{X}{\bf h}}\otimes C$ is an isomorphism in ${}_{\mathcal{D}}\mathscr{M}^{C}$ for each $X\in Ob(\mathcal{D})$.

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. For each $X\in Ob(\mathcal{D})$, we define

[TABLE]

We refer to these as the translation maps of the Galois extension.

Lemma 4.8**.**

*Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. Let $\{\tau_{X}:C\longrightarrow{{\bf h}_{X}}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}\}_{X\in Ob(\mathcal{D})}$ be the associated translation maps. We use the notation $\tau_{X}(c)=c^{(1)}\otimes c^{(2)}$ (summation omitted). Then,

(i) $\tau_{X}$ is right $C$-colinear i.e., $c^{(1)}\otimes{c^{(2)}}_{0}\otimes{c^{(2)}}_{1}=(c_{1})^{(1)}\otimes(c_{1})^{(2)}\otimes c_{2}$.

(ii) For any $f\in Hom_{\mathcal{D}}(X,Y)$, we have $f_{0}(f_{1})^{(1)}\otimes(f_{1})^{(2)}=id_{Y}\otimes f\in{{\bf h}_{Y}}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}$.

(iii) $c^{(1)}c^{(2)}=\varepsilon_{C}(c)\cdot id_{X}$.*

Proof.

The $C$-colinearity of $\tau_{X}$ follows from the $C$-colinearity of $can_{XX}^{-1}$. Explicitly, for any $c\in C$, we have

[TABLE]

This proves (i). Since $can^{-1}_{X}:{{}_{X}{\bf h}}\otimes C\longrightarrow h\otimes_{\mathcal{E}}{{}_{X}{\bf h}}$ is a morphism of left $\mathcal{D}$-modules for each $X\in Ob(\mathcal{D})$, we also have

[TABLE]

This proves (ii). Again using the definition of $can_{XX}$ and $\tau_{X}$, we have $(can_{XX}\circ\tau_{X})(c)=id_{X}\otimes c$. Thus,

[TABLE]

Now, by applying the map $id\otimes\varepsilon_{C}$ to both sides, we get (iii). ∎

Theorem 4.9**.**

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. We denote by $\rho_{XY}:Hom_{\mathcal{D}}(X,Y)\longrightarrow Hom_{\mathcal{D}}(X,Y)\otimes C$ the right $C$-comodule structure maps. Then, there exists a unique right-right entwining structure $(\mathcal{D},C,\psi)$ which makes ${\bf h}_{Y}$ an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$ with its canonical $\mathcal{D}$-module structure and right $C$-coactions $\{\rho_{XY}\}_{X\in Ob(\mathcal{D})}$.

This entwining structure $(\mathcal{D},C,\psi)$ is given by

[TABLE]

Proof.

Using Lemma 4.8, the proof will follow essentially in the same way as that of [BH, Theorem 2.7]. ∎

Lemma 4.10**.**

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. Then, $h\otimes_{\mathcal{E}}h\cong h\otimes C$ as $\mathcal{D}$-corings.

Proof.

We define $can:h\otimes_{\mathcal{E}}h\longrightarrow h\otimes C$ by setting $can(X,Y):=can_{XY}$ for each $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$. We first verify that $can$ is a morphism of $\mathcal{D}$-$\mathcal{D}$-bimodules. Clearly, $can(X,-)=can_{X}$ which, by definition, is a morphism of left $\mathcal{D}$-modules. Therefore, it suffices to show that $can(-,Y)$ is a morphism of right $\mathcal{D}$-modules, i.e., the following diagram commutes for any $g\in Hom_{\mathcal{D}}(Z,Z^{\prime})$:

[TABLE]

By Theorem 4.9, we know that ${\bf h}_{W}$ is an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for each $W\in Ob(\mathcal{D})$. Thus, for any $f\in{\bf h}_{W}(Z^{\prime})$, we have

[TABLE]

Therefore, for any $f^{\prime}\otimes f\in{\bf h}_{Y}(W)\otimes_{\mathcal{E}}{{}_{Z^{\prime}}}{\bf h}(W)$, we obtain

[TABLE]

It remains to verify that $can$ is also a coalgebra morphism. First, we show that the following diagram commutes:

[TABLE]

For any $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$ and $w\otimes w^{\prime}\in{\bf h}_{Y}(W)\otimes{{}_{X}}{\bf h}(W)$, we have

[TABLE]

It may be verified easily that $can$ is compatible with counits. Since $can$ is a morphism in the category of $\mathcal{D}$-$\mathcal{D}$-bimodules and $can(X,Y)=can_{XY}$ is an isomorphism for each $(X,Y)\in Ob(\mathcal{D}^{op}\otimes\mathcal{D})$, it follows that $can$ is an isomorphism with inverse given by $can^{-1}(X,Y):=can^{-1}_{XY}$. This proves the result. ∎

Definition 4.11**.**

Let $\mathcal{D}$ be a small $K$-linear category such that $Hom_{\mathcal{D}}(X,Y)$ is a right $C$-comodule for every $X,Y\in Ob(\mathcal{D})$. Let $\Phi_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)$ and $\Phi_{YZ}:C\longrightarrow Hom_{\mathcal{D}}(Y,Z)$ be two $C$-comodule maps. Then, their convolution product is given by

[TABLE]

A collection of right $C$-comodule maps $\Phi=\{\Phi_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)\}_{X,Y\in Ob(\mathcal{D})}$ is said to be convolution invertible if there exists a collection $\Phi^{\prime}=\{\Phi^{\prime}_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)\}_{X,Y\in Ob(\mathcal{D})}$ of $C$-comodule maps such that

[TABLE]

for every $c\in C$.

Theorem 4.12**.**

Let $C$ be a $K$-coalgebra and $\mathcal{D}$ be a small $K$-linear category such that $Hom_{\mathcal{D}}(X,Y)$ has a right $C$-comodule structure $\rho_{XY}$ for every $X,Y\in Ob(\mathcal{D})$. Let $\mathcal{E}$ be the subcategory of $C$-coinvariants of $\mathcal{D}$. If there exists a convolution invertible collection $\Phi=\{\Phi_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)\}_{X,Y\in Ob(\mathcal{D})}$ of right $C$-comodule maps, then the following are equivalent:

(i) $\mathcal{D}$ is a $C$-Galois extension of $\mathcal{E}$.

(ii) There exists a right-right entwining structure $(\mathcal{D},C,\psi)$ such that ${\bf h}_{Y}$ is an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$ with its canonical $\mathcal{D}$-module structure and right $C$-coactions $\{\rho_{XY}\}_{X\in Ob(\mathcal{D})}$.

(iii) For any $f\in Hom_{\mathcal{D}}(X,Y)$, the morphism $f_{0}\circ\Phi^{\prime}_{ZX}(f_{1})\in Hom_{\mathcal{E}}(Z,Y)$ for every $Z\in Ob(\mathcal{D})$, where $\Phi^{\prime}$ is the convolution inverse of $\Phi$.

Proof.

By Theorem 4.9, we have $(i)\Rightarrow(ii)$. To prove $(ii)\Rightarrow(iii)$, we will use the equality

[TABLE]

for any $c\in C$. We first give a proof of this. Since ${\bf h}_{Y}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$, we have

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$. Also, for any $c\in C$, we have

[TABLE]

Now, using (4.14), we have

[TABLE]

This proves the equality (4.12).

For any $f\in Hom_{\mathcal{D}}(X,Y)$, consider the morphism $f_{0}\circ\Phi^{\prime}_{ZX}(f_{1}):Z\longrightarrow Y$ in $\mathcal{D}$. Then $f_{0}\circ\Phi^{\prime}_{ZX}(f_{1})$ induces a morphism of right $\mathcal{D}$-modules ${\bf{h}}_{Z}\longrightarrow{\bf{h}}_{Y}$ which we denote by $\tilde{f}$. We now verify that the map $\tilde{f}(X^{\prime}):{\bf{h}}_{Z}(X^{\prime})\longrightarrow{\bf{h}}_{Y}(X^{\prime})$ is right $C$-colinear for each $X^{\prime}\in Ob(\mathcal{D})$. Since ${\bf{h}}_{Y}$ is an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Y\in Ob(\mathcal{D})$, the following diagram commutes for any $g\in Hom_{\mathcal{D}}(X^{\prime},Z)$:

[TABLE]

Thus, we have

[TABLE]

Therefore, $\tilde{f}\in Hom^{C}_{Mod\text{-}\mathcal{D}}({{\bf h}_{Z}},{{\bf h}_{Y}})=Hom_{\mathcal{E}}(Z,Y)$.

For $(iii)\Rightarrow(i)$, we start by showing that $can_{XY}:{\bf h}_{Y}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}\longrightarrow Hom_{\mathcal{D}}(X,Y)\otimes C$ is an isomorphism for each $X,Y\in Ob(\mathcal{D})$. We define $can^{-1}_{XY}:Hom_{\mathcal{D}}(X,Y)\otimes C\longrightarrow{\bf h}_{Y}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}$ by

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$ and $c\in C$. Then, using the $C$-colinearity of $\Phi_{XY}$, we have

[TABLE]

On the other hand, by assumption, we obtain

[TABLE]

for any $g\otimes_{\mathcal{E}}g^{\prime}\in{\bf h}_{Y}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}$. From the definition in (4.16), it is clear that setting $can_{X}^{-1}(Y):=can_{XY}^{-1}$ for each $Y\in Ob(\mathcal{D})$ determines a morphism in ${}_{\mathcal{D}}\mathscr{M}^{C}$ which is inverse to $can_{X}$. This completes the proof. ∎

Example 4.13**.**

Let $H$ be a Hopf algebra over $K$. If $\mathcal{C}$ is a left $H$-module category, then the smash product category $\mathcal{C}\#H$ (see [CiSo]) is a right co-$H$-category with the right $H$-coaction determined by $f\#h\mapsto f\#h_{1}\otimes h_{2}$ on each $Hom_{\mathcal{C}\#H}(X,Y)=Hom_{\mathcal{C}}(X,Y)\otimes H$. By definition, we know that $Ob(\mathcal{C})=Ob(\mathcal{C}\#H)$. It is easy to see that $Hom_{\mathcal{C}}(X,Y)=Hom_{\mathcal{C}\#H}(X,Y)^{coH}$.

We claim that $\mathcal{C}\#H$ is an $H$-Galois extension of $\mathcal{C}$. We first observe that for any $f\#h\in Hom_{\mathcal{C}\#H}(Z,Y)$ and $f^{\prime}\#h^{\prime}\in Hom_{\mathcal{C}\#H}(X,Z)$, we have

[TABLE]

Thus, $can_{XY}:{\bf h}_{Y}\otimes_{\mathcal{C}}{{}_{X}{\bf h}}\longrightarrow Hom_{\mathcal{C}\#H}(X,Y)\otimes H$ has the following form

[TABLE]

for each $X,Y\in Ob(\mathcal{C}\#H)$, Then, it may be verified that for each $X,Y\in Ob(\mathcal{C}\#H)$, $can_{XY}$ is an isomorphism with inverse $can_{XY}^{-1}:Hom_{\mathcal{C}\#H}(X,Y)\otimes H\longrightarrow{\bf h}_{Y}\otimes_{\mathcal{C}}{{}_{X}{\bf h}}$ determined by

[TABLE]

Proposition 4.14**.**

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. If there exists a convolution invertible collection $\Phi=\{\Phi_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)\}_{X,Y\in Ob(\mathcal{D})}$ of right $C$-comodule maps, then

[TABLE]

for each $X\in Ob(\mathcal{E})=Ob(\mathcal{D})$.

Proof.

Let $\Phi^{\prime}$ be the convolution inverse of $\Phi$. Given $f\in Hom_{\mathcal{D}}(X,Y)$, it follows from Theorem 4.12 that $f_{0}\circ\Phi^{\prime}_{ZX}(f_{1})\in Hom_{\mathcal{E}}(Z,Y)$ for every $Z\in Ob(\mathcal{D})$. We define

[TABLE]

Using Definition 4.5, we see that $\rho_{XY^{\prime}}(gf)=gf_{0}\otimes f_{1}$ for any $g\in Hom_{\mathcal{E}}(Y,Y^{\prime})$. Hence, we have

[TABLE]

Using (4.17), it may be easily seen that $\eta$ is a morphism of left $\mathcal{E}$-modules. Using the coassociativity of the $C$-coactions $\{\rho_{XY}\}_{X,Y\in Ob(\mathcal{D})}$, it is also clear that $\eta$ is objectwise $C$-colinear. Therefore, $\eta$ is a morphism in ${{}_{\mathcal{E}}}\mathscr{M}^{C}$.

Conversely, we define $\zeta:Hom_{\mathcal{E}}(X,-)\otimes C\longrightarrow Hom_{\mathcal{D}}(X,-)$ given by $\zeta(Y)(f^{\prime}\otimes c):=f^{\prime}\circ\Phi_{XX}(c)$ for $Y\in Ob(\mathcal{E})$. It is immediate that $\zeta$ is a morphism of left $\mathcal{E}$-modules. Moreover,

[TABLE]

where the last equality follows from the fact that $\Phi_{XX}$ is $C$-colinear. It follows that $\zeta(Y)$ is $C$-colinear for each $Y\in Ob(\mathcal{E})$ and hence $\zeta$ is a morphism in ${{}_{\mathcal{E}}}\mathscr{M}^{C}$. It may be verified that $\zeta$ is the inverse of $\eta$. ∎

Definition 4.15**.**

Let $\mathcal{D}$ be a small $K$-linear category and $\mathcal{E}$ be a $K$-subcategory. Let $(\mathscr{C},\Delta_{\mathscr{C}},\varepsilon_{\mathscr{C}})$ be a $\mathcal{D}$-coring. Then, a collection

[TABLE]

is said to be group-like for $\mathscr{C}$ with respect to $\mathcal{E}$ if

• (i)

$\Delta_{\mathscr{C}}(X,X)(s_{X})=s_{X}\otimes s_{X}$* and $\varepsilon_{\mathscr{C}}(s_{X})=id_{X}$ for any $X\in Ob(\mathcal{E})$,*

• (ii)

For any $f\in Hom_{\mathcal{E}}(X,Y)$, we have

[TABLE]

Example 4.16**.**

(i) If $\mathcal{E}$ is a subcategory of $\mathcal{D}$, then the collection $\{id_{X}\otimes id_{X}\in{\bf h}_{X}\otimes_{\mathcal{E}}{{}_{X}}{\bf h}\}_{X\in Ob(\mathcal{E})}$ is group-like for $h\otimes_{\mathcal{E}}h$ with respect to $\mathcal{E}$.

(ii) Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. Then $h\otimes C$ is a $\mathcal{D}$-coring (by Theorem 4.9 and Lemma 4.2) and the collection $\{{id_{X}}_{0}\otimes{id_{X}}_{1}\in Hom_{\mathcal{D}}(X,X)\otimes C\}_{X\in Ob(\mathcal{E})}$ is group-like for $h\otimes C$ with respect to $\mathcal{E}$. Since ${\bf h}_{Y}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for each $Y\in Ob(\mathcal{D})$, we have

[TABLE]

for any $f\in Hom_{\mathcal{D}}(X,Y)$. But, if $f\in Hom_{\mathcal{E}}(X,Y)$, then we also have

[TABLE]

Proposition 4.17**.**

Let $\mathcal{E}\subseteq\mathcal{D}$ be a subcategory and $\mathscr{C}$ be a $\mathcal{D}$-coring. Let $\{s_{X}\}_{X\in Ob(\mathcal{E})}$ be a group-like collection for $\mathscr{C}$ with respect to $\mathcal{E}$. For a right $\mathscr{C}$-comodule $(\mathcal{N},\rho_{\mathcal{N}})$, the $\mathcal{E}$-submodule $\mathcal{N}^{co\mathscr{C}}:\mathcal{E}^{op}\longrightarrow Vect_{K}$ of coinvariants of $\mathcal{N}$ is given by:

[TABLE]

for any $X\in Ob(\mathcal{E})$, $f\in Hom_{\mathcal{E}}(X,Y)$ and $n^{\prime}\in\mathcal{N}^{co\mathscr{C}}(Y)$.

Proof.

We will show that for any $f\in Hom_{\mathcal{E}}(X,Y)$, the morphism $\mathcal{N}^{co\mathscr{C}}(f):\mathcal{N}^{co\mathscr{C}}(Y)\longrightarrow\mathcal{N}^{co\mathscr{C}}(X)$ is well-defined. Since $\rho_{\mathcal{N}}:\mathcal{N}\longrightarrow\mathcal{N}\otimes_{\mathcal{D}}\mathscr{C}$ is a morphism of right $\mathcal{D}$-modules, we have the following commutative diagram:

[TABLE]

Let $n^{\prime}\in\mathcal{N}^{co\mathscr{C}}(Y)$ so that $\rho_{\mathcal{N}}(Y)(n^{\prime})=n^{\prime}\otimes s_{Y}$. Since $f\in Hom_{\mathcal{E}}(X,Y)$, using (4.18) we have

[TABLE]

This shows that $\mathcal{N}(f)(n^{\prime})=\mathcal{N}^{co\mathscr{C}}(f)(n^{\prime})\in\mathcal{N}^{co\mathscr{C}}(X)$. The result follows. ∎

The next result shows that in the case of a $C$-Galois extension $\mathcal{E}\subseteq\mathcal{D}$, we recover the notion of coinvariants as in Definition (4.5).

Lemma 4.18**.**

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. Consider the collection $\{{id_{X}}_{0}\otimes{id_{X}}_{1}\in Hom_{\mathcal{D}}(X,X)\otimes C\}_{X\in Ob(\mathcal{D})}$ which is group-like for $h\otimes C$ with respect to $\mathcal{E}$. Then, $\left(Hom_{\mathcal{D}}(-,Y)\right)^{co(h\otimes C)}(X)=Hom_{\mathcal{E}}(X,Y)$ for any $X$, $Y\in Ob(\mathcal{D})=Ob(\mathcal{E})$.

Proof.

Since $\mathcal{D}$ is a $C$-Galois extension of $\mathcal{E}$, we know that there is a canonical entwining $(\mathcal{D},C,\psi)$ such that ${\bf h}_{Y}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. Using Proposition 4.3, ${\bf h}_{Y}$ may be treated as an object of $Comod\text{-}(h\otimes C)$. Let $g\in\left(Hom_{\mathcal{D}}(-,Y)\right)^{co(h\otimes C)}(X)$. Then, $\rho_{XY}(g)=g\circ{id_{X}}_{0}\otimes{id_{X}}_{1}$. Using the fact that ${\bf h}_{Y}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ we have

[TABLE]

for any $f\in Hom_{\mathcal{D}}(Z,X)$. Therefore, $g\in Hom_{\mathcal{E}}(X,Y)$. The converse follows directly using the Definition (4.5). ∎

Lemma 4.19**.**

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$ and let $(\mathcal{D},C,\psi)$ be the canonical entwining structure associated to it. We denote by $\rho_{XY}:Hom_{\mathcal{D}}(X,Y)\longrightarrow Hom_{\mathcal{D}}(X,Y)\otimes C$ the right $C$-comodule structure maps. Then, for any $\mathcal{M}\in$ $Mod\text{-}\mathcal{E}$, we may obtain an object $\mathcal{M}\otimes_{\mathcal{E}}h\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ by setting

[TABLE]

for $f\in Hom_{\mathcal{D}}(X,Y)$ and $m\otimes g\in\mathcal{M}(Z)\otimes{{}_{Y}{\bf h}}(Z)$. In fact, this determines a functor from $Mod\text{-}\mathcal{E}$ to ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$.

Proof.

Clearly, $\mathcal{M}\otimes_{\mathcal{E}}h\in$ $Mod\text{-}\mathcal{D}$. For each $Y\in Ob(\mathcal{D})$, it may be verified that $\mathcal{M}\otimes_{\mathcal{E}}{{}_{Y}{\bf h}}$ has a right $C$-comodule structure given by

[TABLE]

for any $g\in Hom_{\mathcal{D}}(Y,Z)$ and $m\in\mathcal{M}(Z)$. By Theorem 4.9, ${\bf h}_{Z}$ is an object in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ for every $Z\in Ob(\mathcal{D})$ with its canonical $\mathcal{D}$-module structure and right $C$-coactions $\{\rho_{XZ}\}_{X\in Ob(\mathcal{D})}$. Therefore, we have $\rho_{XZ}({\bf h}_{Z}(f)(g)))=(gf)_{0}\otimes(gf)_{1}=g_{0}f_{\psi}\otimes g_{1}^{\psi}$ for any $f\in Hom_{\mathcal{D}}(X,Y)$. Consequently, we have

[TABLE]

This shows that $\mathcal{M}\otimes_{\mathcal{E}}h\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. ∎

Lemma 4.20**.**

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$. If there exists a convolution invertible collection $\Phi=\{\Phi_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)\}_{X,Y\in Ob(\mathcal{D})}$ of right $C$-comodule maps, then

(i) $Hom_{\mathcal{D}}(X,-)$ is flat as a left $\mathcal{E}$-module.

(ii) $\underset{X\in Ob(\mathcal{D})}{\bigoplus}Hom_{\mathcal{D}}(X,-)$ is faithfully flat as a left $\mathcal{E}$-module.

(iii) For any $\mathcal{M}\in$ $Mod\text{-}\mathcal{E}$, there is a monomorphism $\mathcal{M}\hookrightarrow\mathcal{M}\otimes_{\mathcal{E}}h$ in $Mod\text{-}\mathcal{E}$ given by $m\mapsto m\otimes id_{X}$ for any $m\in\mathcal{M}(X)$.

Proof.

(i) Let $i:\mathcal{M}_{1}\hookrightarrow\mathcal{M}_{2}$ be a monomorphism of right $\mathcal{E}$-modules. By Proposition 4.14, it follows that the induced map $\mathcal{M}_{1}\otimes_{\mathcal{E}}Hom_{\mathcal{D}}(X,-)\longrightarrow\mathcal{M}_{2}\otimes_{\mathcal{E}}Hom_{\mathcal{D}}(X,-)$ coincides with the map $\mathcal{M}_{1}(X)\otimes C\xrightarrow{i(X)\otimes id_{C}}\mathcal{M}_{2}(X)\otimes C$ for each $X\in Ob(\mathcal{E})=Ob(\mathcal{D})$. Since $i(X)\otimes id_{C}$ is clearly a monomorphism, it follows that $Hom_{\mathcal{D}}(X,-)$ is flat as a left $\mathcal{E}$-module.

(ii) This is clear from the fact that $\mathcal{M}(X)\otimes C=\mathcal{M}\otimes_{\mathcal{E}}Hom_{\mathcal{D}}(X,-)=0$ $\Rightarrow$ $\mathcal{M}(X)=0$.

(iii) Since $\underset{Y\in Ob(\mathcal{D})}{\bigoplus}Hom_{\mathcal{D}}(Y,-)$ is faithfully flat as a left $\mathcal{E}$-module, it is enough to prove that for each $Y\in Ob(\mathcal{D})$, we have a monomorphism

[TABLE]

This is true because the morphism in (4.20) has a section

[TABLE]

for any $m^{\prime}\in\mathcal{M}(Z)$ and $g^{\prime}\otimes f^{\prime}\in{{}_{X}{\bf h}}(Z)\otimes{{}_{Y}{\bf h}}(X)$. ∎

Theorem 4.21**.**

Let $\mathcal{D}$ be a $C$-Galois extension of $\mathcal{E}$ and let $(\mathcal{D},C,\psi)$ be the canonical entwining structure associated to it. Suppose there exists a convolution invertible collection $\Phi=\{\Phi_{XY}:C\longrightarrow Hom_{\mathcal{D}}(X,Y)\}_{X,Y\in Ob(\mathcal{D})}$ of right $C$-comodule maps. Then, the categories ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$ and Mod-$\mathcal{E}$ are equivalent.

Proof.

We consider the collection $\{{id_{X}}_{0}\otimes{id_{X}}_{1}\in Hom_{\mathcal{D}}(X,X)\otimes C\}_{X\in Ob(\mathcal{E})}$ which is group-like for the coring $h\otimes C$ with respect to $\mathcal{E}$. We define

[TABLE]

Using Lemma 4.19 and Proposition 4.17, we see that the functors $\mathscr{F}$ and $\mathscr{G}$ are well-defined. We now verify that $\mathscr{G}\circ\mathscr{F}\cong id_{{Mod\text{-}}\mathcal{E}}$ i.e., $(\mathcal{M}\otimes_{\mathcal{E}}h)^{co(h\otimes C)}\cong\mathcal{M}$ for any $\mathcal{M}\in Mod\text{-}\mathcal{E}$.

From Lemma 4.10, we know that $h\otimes C\cong h\otimes_{\mathcal{E}}h$ as $\mathcal{D}$-corings. Under this isomorphism, the collection $\{{id_{X}}_{0}\otimes{id_{X}}_{1}\in Hom_{\mathcal{D}}(X,X)\otimes C\}_{X\in Ob(\mathcal{E})}$ maps to the collection $\{{id_{X}}\otimes{id_{X}}\in{\bf h}_{X}\otimes{{}_{X}}{\bf h}\}_{X\in Ob(\mathcal{E})}$ which is group-like for $h\otimes_{\mathcal{E}}h$ with respect to $\mathcal{E}$. Therefore, it suffices to show that $\mathcal{M}\cong(\mathcal{M}\otimes_{\mathcal{E}}h)^{co(h\otimes_{\mathcal{E}}h)}$.

By Lemma (4.20)(iii), we have an inclusion $i:\mathcal{M}\longrightarrow\mathcal{M}\otimes_{\mathcal{E}}h$ of right $\mathcal{E}$-modules. It is clear that $i(M)\subseteq(\mathcal{M}\otimes_{\mathcal{E}}h)^{co(h\otimes_{\mathcal{E}}h)}$. By definition, $\tilde{\rho}=\rho_{\mathcal{M}\otimes_{\mathcal{E}}h}:\mathcal{M}\otimes_{\mathcal{E}}h\longrightarrow(\mathcal{M}\otimes_{\mathcal{E}}h)\otimes_{\mathcal{D}}(h\otimes_{\mathcal{E}}h)$ is determined by

[TABLE]

for each $X\in Ob(\mathcal{D})$. The coinvariants $(\mathcal{M}\otimes_{\mathcal{E}}h)^{co(h\otimes_{\mathcal{E}}h)}:\mathcal{E}^{op}\longrightarrow Vect_{K}$ are given by

[TABLE]

For ${\sum}m_{Y}\otimes f_{Y}\in(\mathcal{M}\otimes_{\mathcal{E}}h)^{co(h\otimes_{\mathcal{E}}h)}(X)$, we now have

[TABLE]

We set $\mathcal{P}:=(\mathcal{M}\otimes_{\mathcal{E}}h)/\mathcal{M}\in Mod\text{-}\mathcal{E}$ and consider the following short exact sequence:

[TABLE]

Then $\eta$ induces the morphism $\eta\otimes id_{h}:(\mathcal{M}\otimes_{\mathcal{E}}h)\otimes_{\mathcal{E}}h\longrightarrow\mathcal{P}\otimes_{\mathcal{E}}h$ of right $\mathcal{E}$-modules which for each $X\in Ob(\mathcal{D})$ is given by

[TABLE]

where $m^{\prime}\in\mathcal{M}(Z)$, $f^{\prime}\in Hom_{\mathcal{D}}(Y,Z)$, $g^{\prime}\in Hom_{\mathcal{D}}(X,Y)$ and $Y,Z\in Ob(\mathcal{E})$. Applying $(\eta\otimes id_{h})(X)$ to (4.22), we obtain

[TABLE]

Applying Lemma (4.20)(iii) to the inclusion $\mathcal{P}\hookrightarrow\mathcal{P}\otimes_{\mathcal{E}}h$, it follows from (4.23) that $\sum\eta(X)(m_{Y}\otimes_{\mathcal{E}}f_{Y})=0$ for every $X\in Ob(\mathcal{E})$. Therefore, ${\sum}m_{Y}\otimes f_{Y}\in i(\mathcal{M})(X)$. This proves that $\mathcal{M}\cong(\mathcal{M}\otimes_{\mathcal{E}}h)^{co(h\otimes_{\mathcal{E}}h)}$.

It remains to show that $\mathscr{F}\circ\mathscr{G}\cong id_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$. Let $\mathcal{N}\in{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}\cong$ $Comod\text{-}(h\otimes C)$. Then, $\mathcal{N}$ is a right $\mathcal{D}$-module with a given morphism

[TABLE]

in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. By definition, $\mathcal{N}^{co(h\otimes C)}$ is the equalizer of the following morphisms

[TABLE]

where $j$ is given by

[TABLE]

for every $X\in Ob(\mathcal{D})$. By Lemma 4.20(i), it follows that $\mathcal{N}^{co(h\otimes C)}\otimes_{\mathcal{E}}{{}_{X}}{\bf h}$ is the equalizer of the following morphisms

[TABLE]

Comparing with (4.9), we observe that $j\otimes id=id_{\mathcal{N}}\otimes_{\mathcal{E}}\Delta_{h\otimes_{\mathcal{E}}h}(X,-)$. Using the coassociativity of $\rho_{\mathcal{N}}:\mathcal{N}\longrightarrow\mathcal{N}\otimes_{\mathcal{E}}{h}$, it follows from (4.25) that $\rho_{\mathcal{N}}(X)$ factorises through $\mathcal{N}^{co(h\otimes C)}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}$, which is denoted by $\rho^{\prime}_{\mathcal{N}}(X):\mathcal{N}(X)\longrightarrow\mathcal{N}^{co(h\otimes C)}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}\subseteq\mathcal{N}\otimes_{\mathcal{E}}{{}_{X}{\bf h}}$.

We claim that $\rho^{\prime}_{\mathcal{N}}:\mathcal{N}\longrightarrow\mathcal{N}^{co(h\otimes C)}\otimes_{\mathcal{E}}{h}$ is an isomorphism in ${\mathscr{M}(\psi)}_{\mathcal{D}}^{C}$. From the counit property, we know that $(id_{\mathcal{N}}\otimes_{\mathcal{D}}\varepsilon_{h\otimes_{\mathcal{E}}h})\circ\rho_{\mathcal{N}}=id_{\mathcal{N}}$. Hence, $\rho_{\mathcal{N}}$ is a monomorphism and so is $\rho^{\prime}_{\mathcal{N}}$. It remains to show that $\rho^{\prime}_{\mathcal{N}}(X)$ is an epimorphism for each $X\in Ob(\mathcal{D})$. For each $X\in Ob(\mathcal{D})$, we define

[TABLE]

Since $\rho^{\prime}_{\mathcal{N}}$ is a morphism of right $\mathcal{D}$-modules, we now have

[TABLE]

This shows that $\mathscr{F}\circ\mathscr{G}\cong id_{{\mathscr{M}(\psi)}_{\mathcal{D}}^{C}}$.

∎