######  Proposition 3.24.

The morphisms (3.104) are natural in $V,W\in{\mathbf{P}}$, i.e. for any
morphisms $f\colon V^{\prime}\to V$ and $g\colon W\to W^{\prime}$ in
${\mathbf{P}}$ we have the commutative diagram

(3.115)

Proof. By using the formulae (3.110), (2.77) and (2.39) we obtain

$\displaystyle\theta\Big{(}\Phi_{V^{\prime},W^{\prime}}\cdot
F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(f,g)\big{)}\Big{)}=F\Big{(}\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(f,g)\big{)}\Big{)}\cdot\phi_{X,V^{\prime}}=Fg\cdot
F(\mathop{\mathrm{ev}}\nolimits)\cdot F(\mathop{\mathrm{id}}\nolimits\otimes
f)\cdot\phi_{X,V},$ $\displaystyle\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(Ff,Fg)\cdot\Phi_{V,W}\big{)}=\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(Ff,Fg)\big{)}\cdot(\Phi_{V,W}\odot\mathop{\mathrm{id}}\nolimits)=Fg\cdot\mathop{\mathrm{ev}}\nolimits\cdot(\Phi_{V,W}\odot
Ff),$

where $X=\mathop{\mathrm{\bf hom}}\nolimits(V,W)$. The right hand sides of
these equalities coincide with each other due to the naturality of $\phi$ and
the commutativity of (3.109):

$\displaystyle F(\mathop{\mathrm{id}}\nolimits\otimes
f)\cdot\phi_{X,V}=\phi_{X,V^{\prime}}\cdot(\mathop{\mathrm{id}}\nolimits\odot
Ff),$ $\displaystyle\mathop{\mathrm{ev}}\nolimits\cdot(\Phi_{V,W}\odot
Ff)=F(\mathop{\mathrm{ev}}\nolimits)\cdot\phi_{X,V^{\prime}}\cdot(\mathop{\mathrm{id}}\nolimits\odot
Ff).$

Then the bijectivity of $\theta$ implies that the diagram (3.115) commutes. ∎

######  Remark 3.25.

Even if the functor $F$ is strong monoidal, it is not guaranteed that the
natural transformation $\Phi$ is an isomorphism. A counterexample is the
monoidal embedding ${\mathbf{P}}\hookrightarrow{\mathbf{C}}$ described in
Remark 2.28: the formulae (2.92) and (2.93) imply that the components
$\Phi_{x,y}$ are not isomorphisms if $0<x<y$.

3.3.2. Translation of representations. For a monoid
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$
define morphisms $\mu_{FX}\colon FX\odot FX\to FX$ and $\eta_{FX}\colon
I_{\mathbf{D}}\to FX$ as the compositions

$\displaystyle FX\odot FX\xrightarrow{\phi_{X,X}}F(X\otimes
X)\xrightarrow{F\mu_{X}}FX$ and $\displaystyle
I_{\mathbf{D}}\xrightarrow{\varphi}FI_{\mathbf{C}}\xrightarrow{F\eta_{X}}FX$
(3.116)

respectively. Any lax monoidal functor $F\colon{\mathbf{C}}\to{\mathbf{D}}$
induces the functor

$\displaystyle\mathop{\mathbf{Mon}}\nolimits(F)\colon\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})\to\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})$
(3.117)

which translates ${\mathbb{M}}$ to the monoid
$\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}=(FX,\mu_{FX},\eta_{FX})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})$.
If $F$ is faithful, then so is $\mathop{\mathbf{Mon}}\nolimits(F)$. If $F$ is
strong monoidal and fully faithful, then $\mathop{\mathbf{Mon}}\nolimits(F)$
is fully faithful as well.

For any object $V\in{\mathbf{P}}$ the monoid $\mathop{\mathrm{\bf
end}}\nolimits(V)$ is translated to the monoid
$\mathop{\mathbf{Mon}}\nolimits(F)\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}$, which is the object $F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\in{\mathbf{D}}$ with the multiplication
$F(c_{V})\cdot\phi_{\mathop{\mathrm{\bf end}}\nolimits(V),\mathop{\mathrm{\bf
end}}\nolimits(V)}$ and unit $F(u_{V})\cdot\varphi$.

Substitution $W=V\in{\mathbf{P}}$ to (3.104) gives the morphism

$\displaystyle\Phi_{V,V}\colon F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\to\mathop{\mathrm{\bf end}}\nolimits(FV)$ (3.118)

in ${\mathbf{D}}$, where the end-object $\mathop{\mathrm{\bf
end}}\nolimits(FV)\in{\mathbf{D}}$ is defined due to $FV\in{\mathbf{Q}}$. The
first sentence of the following statement is a particular case of the points
(2) and (3) of [Bor2, Prop. 6.4.5] generalised to the relative case.

######  Proposition 3.26.

The morphism (3.118) is a morphism in
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})$ from the monoid
$\mathop{\mathbf{Mon}}\nolimits(F)\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}$ to $\mathop{\mathrm{\bf
end}}\nolimits(FV)=\big{(}\mathop{\mathrm{\bf
end}}\nolimits(FV),c_{FV},u_{FV}\big{)}$. In particular, if (3.118) is an
isomorphism in ${\mathbf{D}}$, then it gives an isomorphism of monoids
$\mathop{\mathbf{Mon}}\nolimits(F)\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\cong\mathop{\mathrm{\bf end}}\nolimits(FV)$.

Proof. The first sentence can be reformulated as the commutativity of the
diagrams

$\textstyle{F\big{(}\mathop{\mathrm{\bf end}}\nolimits(V)\big{)}\odot
F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\phi}$$\scriptstyle{\Phi_{V,V}\odot\Phi_{V,V}}$$\textstyle{F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\qquad
F(c_{V})}$$\textstyle{F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{V,V}}$$\textstyle{\mathop{\mathrm{\bf
end}}\nolimits(FV)\odot\mathop{\mathrm{\bf
end}}\nolimits(FV)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{FV}}$$\textstyle{\mathop{\mathrm{\bf
end}}\nolimits(FV)}$
$\textstyle{I_{\mathbf{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{u_{FV}}$$\textstyle{FI_{\mathbf{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(u_{V})\qquad}$$\textstyle{F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{V,V}}$$\textstyle{\mathop{\mathrm{\bf
end}}\nolimits(FV)}$

By using the formulae (2.39), (3.110),
$\phi\cdot(\varphi\odot\mathop{\mathrm{id}}\nolimits_{FV})=\mathop{\mathrm{id}}\nolimits_{FV}$
and $\theta(u_{V})=\mathop{\mathrm{id}}\nolimits_{V}$ we obtain

$\theta\big{(}\Phi_{V,V}\cdot
F(u_{V})\cdot\varphi\big{)}=\theta\big{(}\Phi_{V,V}\cdot
F(u_{V})\big{)}\cdot(\varphi\odot\mathop{\mathrm{id}}\nolimits_{FV})=F\big{(}\theta(u_{V})\big{)}\big{)}\cdot\phi\cdot(\varphi\odot\mathop{\mathrm{id}}\nolimits_{FV})=\\\
\mathop{\mathrm{id}}\nolimits_{FV}=\theta(u_{FV}).$

This implies the commutativity of the second diagram. The commutativity of the
first one is checked in the same manner:

$\theta\big{(}\Phi_{V,V}\cdot
F(c_{V})\cdot\phi\big{)}=\theta\big{(}\Phi_{V,V}\cdot
F(c_{V})\big{)}\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)=F\big{(}\theta(c_{V})\big{)}\cdot\phi\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)=\\\
F(\mathop{\mathrm{ev}}\nolimits)\cdot
F(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\phi\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi)=F(\mathop{\mathrm{ev}}\nolimits)\cdot\phi\cdot\big{(}\mathop{\mathrm{id}}\nolimits\odot
F(\mathop{\mathrm{ev}}\nolimits)\big{)}\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi)=\\\
\theta(\Phi_{V,V})\cdot\big{(}\mathop{\mathrm{id}}\nolimits\odot\theta(\Phi_{V,V})\big{)}=\mathop{\mathrm{ev}}\nolimits\cdot(\Phi_{V,V}\odot\mathop{\mathrm{id}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\odot\mathop{\mathrm{ev}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\odot\Phi_{V,V}\odot\mathop{\mathrm{id}}\nolimits)=\\\
\mathop{\mathrm{ev}}\nolimits\cdot(\mathop{\mathrm{id}}\nolimits\odot\mathop{\mathrm{ev}}\nolimits)\cdot(\Phi_{V,V}\odot\mathop{\mathrm{id}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\odot\Phi_{V,V}\odot\mathop{\mathrm{id}}\nolimits)=\theta(c_{FV})\cdot(\Phi_{V,V}\odot\Phi_{V,V}\odot\mathop{\mathrm{id}}\nolimits)=\theta\big{(}c_{FV}\cdot(\Phi_{V,V}\odot\Phi_{V,V})\big{)},$

where we also used
$\theta(c_{V})=\mathop{\mathrm{ev}}\nolimits\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)$,
$\phi\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)=\phi\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi)$,
the formula (2.38) and the naturality of $\phi$. The second sentence of Prop.
3.26 follows from the fact that the forgetful functor
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})\to{\mathbf{C}}$ reflects
isomorphisms (see e.g. [S21, Prop. 2.1]). ∎

Let us describe how the lax monoidal functor $F$ translates representations of
the monoid ${\mathbb{M}}$ on $V$ to representations of the monoid
$\widetilde{\mathbb{M}}=\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}$ on
$FV$.

######  Theorem 3.27.

Let ${\mathbf{C}}=({\mathbf{C}},\otimes)$ and
${\mathbf{D}}=({\mathbf{D}},\odot)$ be closed relative to ${\mathbf{P}}$ and
${\mathbf{Q}}$. Any lax monoidal functor
$F\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ satisfying
$F({\mathbf{P}})\subset{\mathbf{Q}}$ induces the functor

$\displaystyle
F_{{\mathbf{P}},{\mathbb{M}}}\colon\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\to\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{M}}),$
(3.119)

which maps
$(V,\rho)\in\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ to
$(FV,\widetilde{\rho})$, where
$\widetilde{\rho}\colon\widetilde{\mathbb{M}}\to\mathop{\mathrm{\bf
end}}\nolimits(FV)$ is the composition

$\displaystyle FX\xrightarrow{F\rho}F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\xrightarrow{\Phi_{V,V}}\mathop{\mathrm{\bf
end}}\nolimits(FV).$ (3.120)

The image of a morphism $f\colon(V,\rho)\to(W,\tau)$ under the functor
$F_{{\mathbf{P}},{\mathbb{M}}}$ is the morphism
$Ff\colon(FV,\widetilde{\rho})\to(FW,\widetilde{\tau})$, where
$(W,\tau)\in\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ and
$\widetilde{\tau}=\Phi_{W,W}\cdot F\tau$.

Proof. The functor $\mathop{\mathbf{Mon}}\nolimits(F)$ maps the monoid
morphism $\rho\colon{\mathbb{M}}\to\mathop{\mathrm{\bf end}}\nolimits(V)$ to
the monoid morphism
$F\rho\colon\widetilde{\mathbb{M}}\to\mathop{\mathbf{Mon}}\nolimits(F)\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}$. Hence Prop. 3.26 implies that (3.120) is a monoid
morphism $\widetilde{\mathbb{M}}\to\mathop{\mathrm{\bf end}}\nolimits(FV)$.
The commutativity of the diagram

(3.127)

follows from the commutativity of (3.14) and the naturality of $\Phi$ (Prop.
3.24). ∎

3.3.3. Translation of corepresentations. Let
${\mathbf{C}}=({\mathbf{C}},\otimes)$ and ${\mathbf{D}}=({\mathbf{D}},\odot)$
be symmetric relatively coclosed monoidal categories with the parametrising
subcategories ${\mathbf{P}}$ and ${\mathbf{Q}}$ respectively. Let
$F=(F,\varphi,\phi)\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ be a
colax monoidal functor satisfying $F({\mathbf{P}})\subset{\mathbf{Q}}$.

By considering the relatively closed monoidal categories
$({\mathbf{C}}^{\mathrm{op}},\otimes)$ and
$({\mathbf{D}}^{\mathrm{op}},\odot)$ with the parametrising subcategories
${\mathbf{P}}^{\mathrm{op}}\subset{\mathbf{C}}^{\mathrm{op}}$ and
${\mathbf{Q}}^{\mathrm{op}}\subset{\mathbf{D}}^{\mathrm{op}}$ and by using the
identification functors (3.22) and (3.23) we can apply the results of pp. 3.3
and 3.25 to the lax monoidal functor
$F^{\mathrm{op}}\colon({\mathbf{C}}^{\mathrm{op}},\otimes)\to({\mathbf{D}}^{\mathrm{op}},\odot)$.

The colax monoidal functor $F$ induces the functor

$\displaystyle\mathop{\mathbf{Comon}}\nolimits(F)\colon\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})\to\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$
(3.128)

which translates a comonoid
$(X,\Delta_{X},\varepsilon_{X})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$
to
$(FX,\Delta_{FX},\varepsilon_{FX})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$,
where

$\displaystyle\Delta_{FX}=\phi_{X,X}\cdot F\Delta_{X}\colon FX\to FX\odot FX,$
$\displaystyle\varepsilon_{FX}=\varphi\cdot F\varepsilon_{X}\colon FX\to
I_{\mathbf{D}}.$ (3.129)

The monoidal structure of $F$ defines the natural transformation $\Phi$ with
components

$\displaystyle\Phi_{V,W}=\vartheta^{-1}\big{(}\phi_{\mathop{\mathrm{\bf
cohom}}\nolimits(V,W),V}\cdot
F(\mathop{\mathrm{coev}}\nolimits_{V,W})\big{)}\colon\mathop{\mathrm{\bf
cohom}}\nolimits(FV,FW)\to F\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits(V,W)\big{)}.$ (3.130)

Some of these components give monoid morphisms:

$\displaystyle\Phi_{V,V}\colon\mathop{\mathrm{\bf
coend}}\nolimits(FV)\to\mathop{\mathbf{Comon}}\nolimits(F)\big{(}\mathop{\mathrm{\bf
coend}}\nolimits(V)\big{)}.$ (3.131)

If $\Phi_{V,V}\colon\mathop{\mathrm{\bf coend}}\nolimits(V)\to
F\big{(}\mathop{\mathrm{\bf coend}}\nolimits(V)\big{)}$ is an isomorphism in
${\mathbf{D}}$, then (3.131) is an isomorphism of comonoids
$\mathop{\mathrm{\bf
coend}}\nolimits(FV)\cong\mathop{\mathbf{Comon}}\nolimits(F)\big{(}\mathop{\mathrm{\bf
coend}}\nolimits(V)\big{)}$.

######  Theorem 3.28.

Let ${\mathbf{C}}$ and ${\mathbf{D}}$ be coclosed relative to ${\mathbf{P}}$
and ${\mathbf{Q}}$. Let $F\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$
be a colax monoidal functor such that $F({\mathbf{P}})\subset{\mathbf{Q}}$.
Let ${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$ and
$\widetilde{\mathbb{O}}=\mathop{\mathbf{Comon}}\nolimits(F){\mathbb{O}}$. Then
$F$ induces the functor

$\displaystyle
F^{\mathbb{O}}_{{\mathbf{P}}}\colon\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{O}})\to\mathop{\mathbf{Corep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{O}}),$
(3.132)

which maps
$(V,\omega)\in\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{O}})$ to
$(FV,\widetilde{\omega})$, where $\widetilde{\omega}$ is the composition

$\displaystyle\mathop{\mathrm{\bf
coend}}\nolimits(FV)\xrightarrow{\Phi_{V,V}}F\big{(}\mathop{\mathrm{\bf
coend}}\nolimits(V)\big{)}\xrightarrow{F\omega}FX.$ (3.133)

A morphism $f\colon(V,\omega)\to(W,\nu)$ is mapped to the morphism
$Ff\colon(FV,\widetilde{\omega})\to(FW,\widetilde{\nu})$, where
$W\in{\mathbf{P}}$, $\nu$ is a corepresentation of ${\mathbb{O}}$ on $W$ and
$\widetilde{\nu}=F\nu\cdot\Phi_{W,W}$.

3.3.4. Translation of actions. For any monoid
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$ a lax monoidal
functor $F=(F,\varphi,\phi)\colon{\mathbf{C}}\to{\mathbf{D}}$ induces a
functor

$\displaystyle
F_{\mathbb{M}}\colon\mathop{\mathbf{Lact}}\nolimits({\mathbb{M}})\to\mathop{\mathbf{Lact}}\nolimits(\widetilde{\mathbb{M}}),$
(3.134)

where $\widetilde{\mathbb{M}}=\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}$.
Each action $a\colon X\otimes V\to V$ of ${\mathbb{M}}$ is translated to the
action $\widetilde{a}\colon FX\odot FV\to FV$ of $\widetilde{\mathbb{M}}$
defined as the composition

$\displaystyle FX\odot FV\xrightarrow{\phi_{X,V}}F(X\otimes
V)\xrightarrow{Fa}FV,$ (3.135)

i.e. $(V,a)\mapsto(FV,\widetilde{a})$ and $f\mapsto Ff$ (see e.g. [S21, §
2.4.11] for details).

Dually, a colax monoidal functor
$F=(F,\varphi,\phi)\colon{\mathbf{C}}\to{\mathbf{D}}$ induces the functor

$\displaystyle
F^{\mathbb{O}}\colon\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})\to\mathop{\mathbf{Lcoact}}\nolimits(\widetilde{\mathbb{O}})$
(3.136)

for comonoids ${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$
and
$\widetilde{\mathbb{O}}=\mathop{\mathbf{Comon}}\nolimits(F){\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$.
It translates a coaction $\delta\colon V\to X\otimes V$ to the coaction

$\displaystyle FV\xrightarrow{F\delta}F(X\otimes
V)\xrightarrow{\phi_{X,V}}FX\odot FV.$ (3.137)

In the situation defined in p. 3.3 we have subcategories
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\subset\mathop{\mathbf{Lact}}\nolimits({\mathbb{M}})$
and
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{M}})\subset\mathop{\mathbf{Lact}}\nolimits(\widetilde{\mathbb{M}})$.
Since $F({\mathbf{P}})\subset{\mathbf{Q}}$, the restriction of the functor
(3.134) gives a functor
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\to\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{M}})$.

######  Proposition 3.29.

The restrictions of the functors (3.134) and (3.136) coincide with (3.119) and
(3.132) respectively. In other words the functor diagrams

(3.146)

commute, where the vertical arrows ‘$\hookrightarrow$’ are the embedding
functors (3.25), (3.27).

Proof. Repeats the proof of [S21, Prop. 3.11]. ∎

######  Remark 3.30.

Theorems 3.27 and 3.28 follows from Prop. 3.29 and the construction of the
functors (3.134) and (3.136) (see [S21, § 3.3.3]). By taking the
representation (3.28) as $\rho$ in Theorem 3.27 we conclude that $\Phi_{V,V}$
is a monoid morphism, so Prop. 3.26 is also a consequence the construction of
(3.134).

3.3.5. Translation under a contravariant functor. We call a contravariant
functor $F\colon{\mathbf{C}}\to{\mathbf{D}}$ lax/colax/strong monoidal from
$({\mathbf{C}},\otimes)$ to $({\mathbf{D}},\odot)$ iff the corresponding
covariant functor $\bar{F}\colon{\mathbf{C}}^{\mathrm{op}}\to{\mathbf{D}}$ is
equipped with the structure of a lax/colax/strong monoidal functor
$(\bar{F},\varphi,\phi)\colon({\mathbf{C}}^{\mathrm{op}},\otimes)\to({\mathbf{D}},\odot)$.
Equivalently, it means that the functor
$\bar{F}^{\mathrm{op}}\colon{\mathbf{C}}\to{\mathbf{D}}^{\mathrm{op}}$ is a
colax/lax/strong monoidal. Let both monoidal categories
${\mathbf{C}}=({\mathbf{C}},\otimes)$ and ${\mathbf{D}}=({\mathbf{D}},\odot)$
be symmetric. The monoidal functor $F$ is called symmetric iff
$\bar{F}\colon{\mathbf{C}}^{\mathrm{op}}\to{\mathbf{D}}$ (or, equivalently,
$\bar{F}^{\mathrm{op}}\colon{\mathbf{C}}\to{\mathbf{D}}^{\mathrm{op}}$) is
symmetric.

The results written for covariant monoidal functors can be reformulated for
contravariant monoidal functors. We consider only colax monoidal functors
applying the results of pp. 3.3 and 3.25 to the categories ${\mathbf{C}}$ and
${\mathbf{D}}^{\mathrm{op}}$ or, equivalently, the results of p. 3.27 to
${\mathbf{C}}^{\mathrm{op}}$ and ${\mathbf{D}}$. The case of contravariant lax
monoidal functor is completely analogous, while a contravariant strong
monoidal functor is a particular case of a contravariant colax monoidal
functor.

Suppose that the symmetric monoidal categories ${\mathbf{C}}$ and
${\mathbf{D}}$ are relatively closed and coclosed respectively. Let
$F=(F,\varphi,\phi)\colon{\mathbf{C}}\to{\mathbf{D}}$ be a contravariant colax
monoidal functor, where $\varphi\colon FI_{\mathbf{C}}\to I_{\mathbf{D}}$ and
$\phi_{X,Y}\colon F(X\otimes Y)\to FX\odot FY$ are structure morphisms for the
colax monoidal functor
$\bar{F}\colon{\mathbf{C}}^{\mathrm{op}}\to{\mathbf{D}}$. The contravariant
functor $F$ induces a contravariant functor

$\displaystyle\mathop{\mathbf{Comon}}\nolimits(F)\colon\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})\to\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$
(3.147)

corresponding to the composition of the identification (3.22) with the functor
(3.128) for covariant functor $\bar{F}$. A monoid
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$
is translated under (3.147) to the comonoid
$\mathop{\mathbf{Comon}}\nolimits(F){\mathbb{M}}=(FX,\Delta_{FX},\varepsilon_{FX})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$,
where $\Delta_{FX}=\phi_{X,X}\cdot F\mu_{X}$, $\varepsilon_{FX}=\varphi\cdot
F\eta_{X}$.

The natural transformation $\Phi$ for the contravariant case has the
components

$\displaystyle\Phi_{V,W}=\vartheta^{-1}\big{(}\phi_{\mathop{\mathrm{\bf
hom}}\nolimits(V,W),V}\cdot
F(\mathop{\mathrm{ev}}\nolimits_{V,W})\big{)}\colon\mathop{\mathrm{\bf
cohom}}\nolimits(FV,FW)\to F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)},$ (3.148)

$V,W,\in{\mathbf{P}}$. It gives the morphisms of comonoids
$\Phi_{V,V}\colon\mathop{\mathrm{\bf
coend}}\nolimits(FV)\to\mathop{\mathbf{Comon}}\nolimits(F)\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}$. If $\Phi_{V,V}$ is an isomorphism in
${\mathbf{D}}$, then it is an isomorphism in
$\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$.

Let ${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$ and
${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}}^{\mathrm{op}})$
be a monoid and a comonoid related via (3.22). Theorems 3.27 and 3.28 for
$\bar{F}^{\mathrm{op}}\colon{\mathbf{C}}\to{\mathbf{D}}^{\mathrm{op}}$ and
$\bar{F}\colon{\mathbf{C}}^{\mathrm{op}}\to{\mathbf{D}}$ give the covariant
functors
$(\bar{F}^{\mathrm{op}})_{{\mathbf{P}},{\mathbb{M}}}\colon\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\to\mathop{\mathbf{Rep}}\nolimits_{{\mathbf{Q}}^{\mathrm{op}}}(\widetilde{\mathbb{M}})$
and
$(\bar{F})^{\mathbb{O}}_{{\mathbf{P}}^{\mathrm{op}}}\colon\mathop{\mathbf{Corep}}\nolimits_{{\mathbf{P}}^{\mathrm{op}}}({\mathbb{O}})\to\mathop{\mathbf{Corep}}\nolimits_{{\mathbf{Q}}}(\widetilde{\mathbb{O}})$
respectively, where
$\widetilde{\mathbb{M}}=\mathop{\mathbf{Mon}}\nolimits(\bar{F}^{\mathrm{op}}){\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}}^{\mathrm{op}})$,
$\widetilde{\mathbb{O}}=\mathop{\mathbf{Comon}}\nolimits(\bar{F}){\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$.
By using (3.23) we obtain the following statement.

######  Theorem 3.31.

Let the monoidal categories ${\mathbf{C}}=({\mathbf{C}},\otimes)$ and
${\mathbf{D}}=({\mathbf{D}},\odot)$ be closed and coclosed relative to
${\mathbf{P}}$ and ${\mathbf{Q}}$ respectively. Let
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$.
Any contravariant colax monoidal functor
$F\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ such that
$F({\mathbf{P}})\subset{\mathbf{Q}}$ induces the contravariant functor

$\displaystyle
F^{\mathbb{M}}_{{\mathbf{P}}}\colon\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\to\mathop{\mathbf{Corep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{O}}),$
(3.149)

where
$\widetilde{\mathbb{O}}=\mathop{\mathbf{Comon}}\nolimits(F){\mathbb{M}}$. It
maps an object
$(V,\rho)\in\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ to
$(FV,\widetilde{\omega})\in\mathop{\mathbf{Corep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{O}})$,
where $\widetilde{\omega}\colon\mathop{\mathrm{\bf
coend}}\nolimits(FV)\to\widetilde{\mathbb{O}}$ is the corepresentation defined
as

$\displaystyle\mathop{\mathrm{\bf
coend}}\nolimits(FV)\xrightarrow{\Phi_{V,V}}F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\xrightarrow{F\rho}FX.$ (3.150)

A morphism $f\colon(V,\rho)\to(W,\tau)$ is mapped to the morphism
$Ff\colon(FV,\widetilde{\omega})\to(FW,\widetilde{\nu})$, where
$(W,\tau)\in\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ and
$\widetilde{\nu}=F\tau\cdot\Phi_{W,W}$.

3.3.6. Translation of tensor product of monoids. We denote the symmetric
structure of the both symmetric monoidal categories
${\mathbf{C}}=({\mathbf{C}},\otimes)$ and ${\mathbf{D}}=({\mathbf{D}},\odot)$
by the same letter $\sigma$.

######  Lemma 3.32.

Let $F=(F,\varphi,\phi)\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ be
a lax monoidal functor. Then for any objects $X,Y,Z,W\in{\mathbf{C}}$ the
diagram

(3.155)

is commutative. If $F$ is symmetric, then we also have the following
commutative diagram:

Proof. Due to the identity
$\phi\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)=\phi\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi)$
we obtain
$\phi\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi)\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi\odot\mathop{\mathrm{id}}\nolimits)=\phi\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi)\cdot(\mathop{\mathrm{id}}\nolimits\odot\mathop{\mathrm{id}}\nolimits\odot\phi)=\phi\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\odot\mathop{\mathrm{id}}\nolimits\odot\phi)=\phi\cdot(\phi\odot\phi)$.
Then, by using (3.155) one yields
$\phi\cdot(\phi\odot\phi)\cdot\sigma^{(23)}=\phi\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)\cdot\big{(}\mathop{\mathrm{id}}\nolimits\odot(\phi\cdot\sigma)\odot\mathop{\mathrm{id}}\nolimits\big{)}=\phi\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)\cdot\big{(}\mathop{\mathrm{id}}\nolimits\odot(F\sigma\cdot\phi)\odot\mathop{\mathrm{id}}\nolimits\big{)}$.
Due to the naturality of $\phi$ this is equal to
$\phi\cdot\big{(}F(\mathop{\mathrm{id}}\nolimits\otimes\sigma)\odot\mathop{\mathrm{id}}\nolimits\big{)}\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi\odot\mathop{\mathrm{id}}\nolimits)=F(\sigma^{(23)})\cdot\phi\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi\odot\mathop{\mathrm{id}}\nolimits)=F(\sigma^{(23)})\cdot\phi\cdot(\phi\odot\phi)$.
∎

Note that $\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$ and
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})$ are both symmetric monoidal
categories (see p. 3.2). Let us describe a monoidal structure of
$\mathop{\mathbf{Mon}}\nolimits(F)\colon\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})\to\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})$
induced by the monoidal structure of $F$.

######  Proposition 3.33.

If $F$ is a symmetric lax/strong monoidal functor, then
$\mathop{\mathbf{Mon}}\nolimits(F)$ is also lax/strong monoidal and also
symmetric. The monoidal structure is given by the same morphisms
$\varphi\colon I_{\mathbf{D}}\to FI_{\mathbf{C}}$ and $\phi_{X,Y}\colon
FX\odot FY\to F(X\otimes Y)$ regarded as morphisms $\varphi\colon
I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})}\to\mathop{\mathbf{Mon}}\nolimits(F)(I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})})$
and
$\phi_{{\mathbb{M}},{\mathbb{M}}^{\prime}}=\phi_{X,Y}\colon\widetilde{\mathbb{M}}\odot\widetilde{\mathbb{M}}^{\prime}\to\mathop{\mathbf{Mon}}\nolimits(F)({\mathbb{M}}\otimes{\mathbb{M}}^{\prime})$,
where
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$,
${\mathbb{M}}^{\prime}=(Y,\mu_{Y},\eta_{Y})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$,
$\widetilde{\mathbb{M}}=\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}$ and
$\widetilde{\mathbb{M}}^{\prime}=\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}^{\prime}$.

Dually, if $F$ is a symmetric colax/strong monoidal functor, then
$\mathop{\mathbf{Comon}}\nolimits(F)$ is also colax/strong monoidal and also
symmetric. Its monoidal structure is given by the morphisms $\varphi\colon
I_{\mathbf{D}}\to FI_{\mathbf{C}}$ and $\phi_{X,Y}\colon F(X\otimes Y)\to
FX\odot FY$ regarded as morphisms between comonoids.

Proof. We need to check that $\phi_{X,Y}\colon FX\odot FY\to F(X\otimes Y)$ is
a morphism from the monoid
$\widetilde{\mathbb{M}}\odot\widetilde{\mathbb{M}}^{\prime}=\big{(}FX\odot
FY,(\mu_{FX}\odot\mu_{FY})\cdot\sigma^{(23)},\eta_{FX}\odot\eta_{FY}\big{)}$
to the monoid
$\mathop{\mathbf{Mon}}\nolimits(F)({\mathbb{M}}\otimes{\mathbb{M}}^{\prime})=\big{(}F(X\otimes
Y),F(\mu_{X}\otimes\mu_{Y})\cdot
F(\sigma)\cdot\phi,F(\eta_{X}\otimes\eta_{Y})\cdot\varphi\big{)}$, where
$\mu_{FX}=F\mu_{X}\cdot\phi$, $\eta_{FX}=F\eta_{X}\cdot\varphi$. This amounts
to the commutative diagrams

(3.160) (3.165)

The commutativity of (3.160) follows from Lemma 3.32 (for $Z=X$, $W=Y$) and
the naturality of $\phi$. The diagram (3.165) is obtained by using the formula
$\phi_{I_{\mathbf{C}},I_{\mathbf{C}}}\cdot(\varphi\odot\mathop{\mathrm{id}}\nolimits)=\mathop{\mathrm{id}}\nolimits$.
The same formula implies that $\varphi$ is a morphism from the monoid
$I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})}=(I_{{\mathbf{D}}},\mathop{\mathrm{id}}\nolimits,\mathop{\mathrm{id}}\nolimits)$
to
$\mathop{\mathbf{Mon}}\nolimits(F)I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})}=(FI_{\mathbf{C}},\phi_{I_{\mathbf{C}},I_{\mathbf{C}}},\varphi)$.
The conditions on the monoidal structure of
$\mathop{\mathbf{Mon}}\nolimits(F)$ defined in this way as well as the
agreement with the symmetric structure are valid, since they are valid for the
structure of $F$. If $F$ is strong monoidal, then $\varphi$ and $\phi_{X,Y}$
are isomorphisms in ${\mathbf{D}}$. They are isomorphisms in
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})$, since the forgetful functor
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})\to{\mathbf{D}}$ reflects
isomorphisms. ∎

3.3.7. Translation of bimonoids. Let
$F\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ be a (covariant)
symmetric strong monoidal functor. Let
${\mathbb{B}}=(X,\mu_{X},\eta_{X},\Delta_{X},\varepsilon_{X})$ be a bimonoid
in $({\mathbf{C}},\otimes)$. By definition it means that
${\mathbb{M}}=(X,\mu_{X},\eta_{X})$ and
${\mathbb{O}}=(X,\Delta_{X},\varepsilon_{X})$ are a monoid and a comonoid in
${\mathbf{C}}$ such that the comonoid structure gives the morphisms
$\Delta_{X}\colon{\mathbb{M}}\to{\mathbb{M}}\otimes{\mathbb{M}}$ and
$\varepsilon_{X}\colon{\mathbb{M}}\to
I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})}$ in
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$. The formulae (3.116) give the
monoid
$\widetilde{\mathbb{M}}=(FX,\mu_{FX},\eta_{FX})=\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}$
in ${\mathbf{D}}$. By considering $F=(F,\varphi,\phi)$ as a colax monoidal
functor with the structure morphisms $\varphi^{-1}\colon FI_{\mathbf{C}}\to
I_{\mathbf{D}}$ and $\phi^{-1}_{X,Y}\colon F(X\otimes Y)\to FX\odot FY$ we
obtain the comonoid
$\widetilde{\mathbb{O}}=(FX,\Delta_{FX},\varepsilon_{FX})=\mathop{\mathbf{Comon}}\nolimits(F){\mathbb{O}}$
in ${\mathbf{D}}$, where $\Delta_{FX}$ and $\varepsilon_{FX}$ are defined by
the formulae (3.129) but with $\varphi^{-1}$ and $\phi^{-1}$ instead of
$\varphi$ and $\phi$, i.e.

$\displaystyle\Delta_{FX}\colon FX\xrightarrow{F\Delta_{X}}F(X\otimes
X)\xrightarrow{\phi^{-1}_{X,X}}FX\odot FX,$
$\displaystyle\varepsilon_{FX}\colon
FX\xrightarrow{F\varepsilon_{X}}FI_{\mathbf{C}}\xrightarrow{\varphi^{-1}}I_{\mathbf{D}}.$
(3.166)

By applying Prop. 3.33 to the strong monoidal functor $(F,\varphi,\phi)$ we
see that $\phi_{X,X}$ and $\varphi$ are monoid isomorphisms with the inverses
$\phi^{-1}_{X,X}\colon\mathop{\mathbf{Mon}}\nolimits(F)({\mathbb{M}}\otimes{\mathbb{M}})\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\widetilde{\mathbb{M}}\odot\widetilde{\mathbb{M}}$
and
$\varphi^{-1}\colon\mathop{\mathbf{Mon}}\nolimits(F)I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})}$,
where $\widetilde{\mathbb{M}}=\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}$.
Hence the compositions (3.166) give the monoid morphisms
$\Delta_{FX}\colon\widetilde{\mathbb{M}}\to\widetilde{\mathbb{M}}\odot\widetilde{\mathbb{M}}$
and $\varepsilon_{FX}\colon\widetilde{\mathbb{M}}\to
I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})}$, so we get a bimonoid
$\widetilde{\mathbb{B}}=(FX,\mu_{FX},\eta_{FX},\Delta_{FX},\varepsilon_{FX})\in\mathop{\mathbf{Bimon}}\nolimits({\mathbf{D}})$.
We have proved the following fact.

######  Proposition 3.34.

A (covariant) symmetric strong monoidal functor
$F\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ translates bimonoids to
bimonoids. Namely, it induces the symmetric strong monoidal functor

$\displaystyle\mathop{\mathbf{Bimon}}\nolimits(F)\colon\mathop{\mathbf{Bimon}}\nolimits({\mathbf{C}},\otimes)\to\mathop{\mathbf{Bimon}}\nolimits({\mathbf{D}},\odot)$
(3.167)

agreed with the functors (3.117) and (3.128). It maps ${\mathbb{B}}$ to the
bimonoid
$\widetilde{\mathbb{B}}=\mathop{\mathbf{Bimon}}\nolimits(F){\mathbb{B}}$
defined by the formulae (3.116), (3.166).

A contravariant symmetric strong monoidal functor
$F\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ gives the contravariant
functor
$\mathop{\mathbf{Bimon}}\nolimits(F)\colon\mathop{\mathbf{Bimon}}\nolimits({\mathbf{C}},\otimes)\to\mathop{\mathbf{Bimon}}\nolimits({\mathbf{D}},\odot)$
corresponding to covariant functor
$\mathop{\mathbf{Bimon}}\nolimits(\bar{F})\colon\mathop{\mathbf{Bimon}}\nolimits({\mathbf{C}}^{\mathrm{op}},\otimes)\to\mathop{\mathbf{Bimon}}\nolimits({\mathbf{D}},\odot)$.
It translates
${\mathbb{B}}\in\mathop{\mathbf{Bimon}}\nolimits({\mathbf{C}},\otimes)$ to
$\widehat{\mathbb{B}}=\mathop{\mathbf{Bimon}}\nolimits(F){\mathbb{B}}=(FX,\mu_{FX},\eta_{FX},\Delta_{FX},\varepsilon_{FX})\in\mathop{\mathbf{Bimon}}\nolimits({\mathbf{D}},\odot)$,
where

$\displaystyle\mu_{FX}\colon FX\odot FX\xrightarrow{\phi_{X,X}}F(X\otimes
X)\xrightarrow{F\Delta_{X}}FX,$ $\displaystyle\eta_{FX}\colon
I_{\mathbf{D}}\xrightarrow{\varphi}FI_{\mathbf{C}}\xrightarrow{F\varepsilon_{X}}FX,$
(3.168) $\displaystyle\Delta_{FX}\colon FX\xrightarrow{F\mu_{X}}F(X\otimes
X)\xrightarrow{\phi_{X,X}^{-1}}FX\odot FX,$
$\displaystyle\varepsilon_{FX}\colon
FX\xrightarrow{F\eta_{X}}FI_{\mathbf{C}}\xrightarrow{\varphi^{-1}}I_{\mathbf{D}}.$
(3.169)

3.3.8. Translation of tensor product of representations. Suppose that the
parametrising subcategories ${\mathbf{P}}\subset{\mathbf{C}}$ and
${\mathbf{Q}}\subset{\mathbf{D}}$ are monoidal subcategories of the symmetric
monoidal categories $({\mathbf{C}},\otimes)$ and $({\mathbf{D}},\odot)$
respectively and that $F({\mathbf{P}})\subset{\mathbf{Q}}$ for a covariant
strong monoidal functor $F$. Let
${\mathbb{B}}=(X,\mu_{X},\eta_{X},\Delta_{X},\varepsilon_{X})\in\mathop{\mathbf{Bimon}}\nolimits({\mathbf{C}},\otimes)$
and $\widetilde{\mathbb{B}}=\mathop{\mathbf{Bimon}}\nolimits(F){\mathbb{B}}$.

If ${\mathbf{C}}=({\mathbf{C}},\otimes)$ and
${\mathbf{D}}=({\mathbf{D}},\odot)$ are both relatively closed, then we have
the functors (3.119). Denote such functor for a monoid
${\mathbb{M}}=(X,\mu_{X},\eta_{X})$ by

$\displaystyle
F_{{\mathbf{P}},{\mathbb{B}}}\colon\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})\to\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{B}}).$
(3.170)

In the case, when ${\mathbf{C}}$ and ${\mathbf{D}}$ are both relatively
coclosed, the functor (3.132) for the comonoid
${\mathbb{O}}=(X,\Delta_{X},\varepsilon_{X})$ and the colax monoidal functor
$F=(F,\varphi^{-1},\phi^{-1})$ is defined and has the form

$\displaystyle
F^{\mathbb{B}}_{{\mathbf{P}}}\colon\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{B}})\to\mathop{\mathbf{Corep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{B}}).$
(3.171)

If ${\mathbf{C}}=({\mathbf{C}},\otimes)$ and
${\mathbf{D}}=({\mathbf{D}},\odot)$ are relatively closed and coclosed
respectively and $F$ is a contravariant symmetric strong monoidal functor,
then we get the contravariant functor

$\displaystyle
F^{\mathbb{B}}_{{\mathbf{P}}}\colon\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})\to\mathop{\mathbf{Corep}}\nolimits_{\mathbf{Q}}(\widehat{\mathbb{B}}),$
(3.172)

where $\widehat{\mathbb{B}}$ is defined by (3.168), $\eqref{whBBD}$.

Recall that all the categories appeared in (3.170), (3.171) and (3.172) are
monoidal categories (Theorems 3.18, 3.20).

######  Theorem 3.35.

The functors (3.170), (3.171) and (3.172) are strong monoidal functors (the
corresponding conditions are supposed: the symmetric monoidal categories
${\mathbf{C}}$ and ${\mathbf{D}}$ should be relatively closed/coclosed, the
symmetric strong monoidal $F$ should be covariant/contravariant). The monoidal
structure of the functors (3.170), (3.171) and (3.172) is inherited from $F$.

Proof. Let $\rho$ and $\rho^{\prime}$ be representations of ${\mathbb{B}}$ on
the objects $V$ and $V^{\prime}$. Their tensor product
$\rho^{\prime\prime}=\pi\cdot(\rho\otimes\rho^{\prime})\cdot\Delta_{X}$ is a
representation of ${\mathbb{B}}$ on $V^{\prime\prime}=V\otimes V^{\prime}$.
The functor $F$ translates the objects $(V,\rho)$,
$(V^{\prime},\rho^{\prime})$ and $(V^{\prime\prime},\rho^{\prime\prime})$ of
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ to the objects
$(FV,\tau)$, $(FV^{\prime},\tau^{\prime})$ and
$(FV^{\prime\prime},\tau^{\prime\prime})$ of
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{B}})$, where
$\tau=\Phi_{V,V}\cdot F\rho$, $\tau^{\prime}=\Phi_{V^{\prime},V^{\prime}}\cdot
F\rho^{\prime}$ and
$\tau^{\prime\prime}=\Phi_{V^{\prime\prime},V^{\prime\prime}}\cdot
F\rho^{\prime\prime}$. The tensor product of $\tau$ and $\tau^{\prime}$ is the
representation $\lambda=\pi\cdot(\tau\odot\tau^{\prime})\cdot\Delta_{FX}$ on
$FV\odot FV^{\prime}$. We need to show that $\phi_{V,V^{\prime}}\colon FV\odot
FV^{\prime}\to F(V\otimes V^{\prime})$ is a morphism between representations
$\lambda$ and $\tau^{\prime\prime}$. According to Definition 3.1 this amounts
to the commutative diagram

(3.177)

The representations in this diagram can be written in the form

$\displaystyle\lambda=\pi\cdot(\tau\odot\tau^{\prime})\cdot\Delta_{FX}=\pi\cdot(\Phi\odot\Phi)\cdot(F\rho\odot
F\rho^{\prime})\cdot\phi^{-1}\cdot F\Delta_{X}=$
$\displaystyle\qquad\qquad\pi\cdot(\Phi\odot\Phi)\cdot\phi^{-1}\cdot
F(\rho\otimes\rho^{\prime})\cdot F\Delta_{X},$
$\displaystyle\tau^{\prime\prime}=\Phi\cdot F\rho^{\prime\prime}=\Phi\cdot
F\pi\cdot F(\rho\otimes\rho^{\prime})\cdot F\Delta_{X},$

so we need to check the equality $\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,\phi)\cdot\pi\cdot(\Phi\odot\Phi)=\mathop{\mathrm{\bf
hom}}\nolimits(\phi,\mathop{\mathrm{id}}\nolimits)\cdot\Phi\cdot
F\pi\cdot\phi$ of two morphisms $F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(V,V)\big{)}\odot F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(V^{\prime},V^{\prime})\big{)}\to\mathop{\mathrm{\bf
hom}}\nolimits(FV\odot FV^{\prime},FV^{\prime\prime})$. By using the formulae
(2.39), (2.77) and the commutative diagrams (3.45), (3.109) we obtain their
images under $\theta$:

$\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,\phi)\big{)}\cdot(\pi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\Phi\odot\Phi\odot\mathop{\mathrm{id}}\nolimits)=\phi\cdot\mathop{\mathrm{ev}}\nolimits\cdot(\pi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\Phi\odot\Phi\odot\mathop{\mathrm{id}}\nolimits)=\\\
\phi\cdot(\mathop{\mathrm{ev}}\nolimits\odot\mathop{\mathrm{ev}}\nolimits)\cdot\sigma^{(23)}\cdot(\Phi\odot\Phi\odot\mathop{\mathrm{id}}\nolimits)=\phi\cdot(\mathop{\mathrm{ev}}\nolimits\odot\mathop{\mathrm{ev}}\nolimits)\cdot(\Phi\odot\mathop{\mathrm{id}}\nolimits\odot\Phi\odot\mathop{\mathrm{id}}\nolimits)\cdot\sigma^{(23)}=\\\
\phi\cdot\big{(}F(\mathop{\mathrm{ev}}\nolimits)\odot
F(\mathop{\mathrm{ev}}\nolimits)\big{)}\cdot(\phi\odot\phi)\cdot\sigma^{(23)}=F(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\phi\cdot(\phi\odot\phi)\cdot\sigma^{(23)},$
(3.178) $\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(\phi,\mathop{\mathrm{id}}\nolimits)\big{)}\cdot(\Phi\odot\mathop{\mathrm{id}}\nolimits)\cdot(F\pi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)=\\\
\mathop{\mathrm{ev}}\nolimits\cdot(\mathop{\mathrm{id}}\nolimits\odot\phi)\cdot(\Phi\odot\mathop{\mathrm{id}}\nolimits)\cdot(F\pi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\phi\odot\mathop{\mathrm{id}}\nolimits)=F(\mathop{\mathrm{ev}}\nolimits)\cdot\phi\cdot(F\pi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\phi\odot\phi)=\\\
F(\mathop{\mathrm{ev}}\nolimits)\cdot
F(\pi\otimes\mathop{\mathrm{id}}\nolimits)\cdot\phi\cdot(\phi\odot\phi)=F(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot
F(\sigma^{(23)})\cdot\phi\cdot(\phi\odot\phi).$ (3.179)

The right hand sides of (3.178) and (3.179) coincide by virtue of Lemma 3.32,
so the diagram (3.177) is commutative.

The unit object
$I_{\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})}=(I_{\mathbf{C}},\varepsilon_{X})$
is translated to the object $(FI_{\mathbf{C}},\widetilde{\varepsilon}_{X})$,
where

$\displaystyle\widetilde{\varepsilon}_{X}\colon
FX\xrightarrow{F\varepsilon_{X}}FI_{\mathbf{C}}=F\big{(}\mathop{\mathrm{\bf
end}}\nolimits(I_{\mathbf{C}})\big{)}\xrightarrow{\Phi_{I_{\mathbf{C}},I_{\mathbf{C}}}}\mathop{\mathrm{\bf
end}}\nolimits(FI_{\mathbf{C}})$ (3.180)

is the corresponding representation of $\widetilde{\mathbb{B}}$ on
$FI_{\mathbf{C}}$. Let us prove that $\varphi\colon I_{\mathbf{D}}\to
FI_{\mathbf{C}}$ gives the morphisms between the representations
$\varepsilon_{FX}=\varphi^{-1}\cdot F\varepsilon_{X}\colon FX\to
I_{D}=\mathop{\mathrm{\bf end}}\nolimits(I_{\mathbf{D}})$ and
$\widetilde{\varepsilon}_{X}$. Due to Prop. 2.25 the morphism
$I_{\mathbf{D}}=\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{D}},I_{\mathbf{D}})\xrightarrow{\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,\varphi)}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{D}},FI_{\mathbf{C}})=FI_{\mathbf{C}}$ coincides with
$\varphi$, so we obtain

$\displaystyle\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,\varphi)\cdot\varepsilon_{FX}=\varphi\cdot\varphi^{-1}\cdot
F\varepsilon_{X}=F\varepsilon_{X},$ $\displaystyle\mathop{\mathrm{\bf
hom}}\nolimits(\varphi,\mathop{\mathrm{id}}\nolimits)\cdot\widetilde{\varepsilon}_{X}=\mathop{\mathrm{\bf
hom}}\nolimits(\varphi,\mathop{\mathrm{id}}\nolimits)\cdot\Phi\cdot
F\varepsilon_{X}.$

Hence we need only to check that the morphism

$\displaystyle F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{C}},I_{\mathbf{C}})\big{)}\xrightarrow{\Phi}\mathop{\mathrm{\bf
hom}}\nolimits(FI_{\mathbf{C}},FI_{\mathbf{C}})\xrightarrow{\mathop{\mathrm{\bf
hom}}\nolimits(\varphi,\mathop{\mathrm{id}}\nolimits)}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{D}},FI_{\mathbf{C}})$ (3.181)

coincides with the identification $F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{C}},I_{\mathbf{C}})\big{)}=FI_{\mathbf{C}}=\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{D}},FI_{\mathbf{C}})$. Since (2.79) is the
identification, the morphism (3.181) coincides with the image of (3.181)
itself under $\theta$. Due to (2.39) and (2.77) this image equals

$\theta(\mathop{\mathrm{\bf
hom}}\nolimits(\varphi,\mathop{\mathrm{id}}\nolimits))\cdot(\Phi\odot\mathop{\mathrm{id}}\nolimits)=\mathop{\mathrm{ev}}\nolimits\cdot(\mathop{\mathrm{id}}\nolimits\odot\varphi)\cdot(\Phi\odot\mathop{\mathrm{id}}\nolimits)=\mathop{\mathrm{ev}}\nolimits\cdot(\Phi\odot\mathop{\mathrm{id}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\odot\varphi)=\\\
F(\mathop{\mathrm{ev}}\nolimits)\cdot\phi\cdot(\mathop{\mathrm{id}}\nolimits\odot\varphi)\colon
F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{C}},I_{\mathbf{C}})\big{)}\odot I_{\mathbf{D}}\to
FI_{\mathbf{C}}.$ (3.182)

Since $F(\mathop{\mathrm{ev}}\nolimits)\colon F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{C}},I_{\mathbf{C}})\otimes I_{\mathbf{C}}\big{)}\to
FI_{\mathbf{C}}$ and

$F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{C}},I_{\mathbf{C}})\big{)}\odot
I_{\mathbf{D}}\xrightarrow{\mathop{\mathrm{id}}\nolimits\odot\varphi}F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{C}},I_{\mathbf{C}})\big{)}\odot
FI_{\mathbf{C}}\xrightarrow{\phi}F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(I_{\mathbf{C}},I_{\mathbf{C}})\otimes I_{\mathbf{C}}\big{)}$

are both the identification morphisms, their composition (3.182) is the
identification as well.

By virtue of Prop. 3.2 the morphisms of representations
$\phi_{V,V^{\prime}}\colon(FV\odot
FV^{\prime},\lambda)\to(FV^{\prime\prime},\tau^{\prime\prime})$ and
$\varphi\colon(I_{\mathbf{D}},\varepsilon_{FX})\to(FX,\widetilde{\varepsilon}_{X})$
are isomorphisms in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$. ∎

######  Remark 3.36.

Recall that any symmetric strong monoidal functor $F=(F,\varphi,\phi)$ induces
strong monoidal functors

$\displaystyle
F_{\mathbb{B}}\colon\mathop{\mathbf{Lact}}\nolimits({\mathbb{B}})\to\mathop{\mathbf{Lact}}\nolimits(\widetilde{\mathbb{B}}),$
$\displaystyle
F^{\mathbb{B}}\colon\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{B}})\to\mathop{\mathbf{Lcoact}}\nolimits(\widetilde{\mathbb{B}})$
(3.183)

with the (lax) monoidal structure $(\varphi,\phi)$. The functors (3.170) and
(3.171) are restrictions of (3.183) as monoidal functors (in the sense of
Prop. 3.29 and Remark 3.22). These facts can be used for an alternative proof
of Theorem 3.35.

##  4 Examples

The general theory of representations described above was motivated by two
types of representations. First, this theory allows to unify the classical
representations on vector spaces such as representations of groups and
algebras. Second example is quantum representations on quadratic algebras
defined in [S21].

Let us fix a basic field ${\mathbb{K}}$ of characteristic
$\mathop{\mathrm{char}}\nolimits{\mathbb{K}}\neq 2$. We suppose that any
vector space or algebra is over ${\mathbb{K}}$ unless we explicitly specify
another field.

###  4.1 Representations on vector spaces

The model example of the general representation theory is the representations
of algebras. This is the case of the closed monoidal category of vector
spaces. To describe the representations of groups we introduce a bigger
relatively closed monoidal category which contains sets as well as vector
spaces as monoidal subcategories. This gives the unification of
representations of groups and algebras. This case is generalised to groups
with a geometric structure such as Lie groups, algebraic groups etc (the
algebraic case was described in [S21, § 4.3]).

4.1.1. Representations of algebras. The category of vector spaces
$({\mathbf{Vect}},\otimes)$ with the standard tensor product $\otimes$ is a
closed symmetric monoidal category with the unit object ${\mathbb{K}}$. The
monoids in $({\mathbf{Vect}},\otimes)$ are algebras:
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{Vect}},\otimes)=({\mathbf{Alg}},\otimes)$.
For $V,W\in{\mathbf{Vect}}$ the object $\mathop{\mathrm{\bf
hom}}\nolimits(V,W)$ is a vector space of all the linear operators $V\to W$,
while $\mathop{\mathrm{\bf end}}\nolimits(V)$ is the algebra of operators on
$V$. A morphism $\rho\colon{\mathfrak{A}}\to\mathop{\mathrm{\bf
end}}\nolimits(V)$ in
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{Vect}},\otimes)={\mathbf{Alg}}$ is a
representation of an algebra ${\mathfrak{A}}\in{\mathbf{Alg}}$ on
$V\in{\mathbf{Vect}}$ in the usual sense.

4.1.2. Semi-linear sets. The category of sets ${\mathbf{Set}}$ is the category
with finite products. The usual monoids and groups are monoids and groups in
${\mathbf{Set}}$, i.e. monoids and Hopf monoids in $({\mathbf{Set}},\times)$.
Notice that the monoidal category $({\mathbf{Set}},\times)$ is closed with the
hom-objects $\mathop{\mathrm{\bf
hom}}\nolimits(X,Y)=\mathop{\mathrm{Hom}}\nolimits(X,Y)\in{\mathbf{Set}}$ and
$\mathop{\mathrm{\bf
end}}\nolimits(X)=\mathop{\mathrm{End}}\nolimits(X)\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Set}},\times)$,
where $X,Y\in{\mathbf{Set}}$.

To consider representations of monoids and groups on vector spaces in frame of
the general representation theory constructed in Section 3 we define the
category ${\mathbf{SLSet}}$. Its objects are pairs $(X,V)$, where
$X\in{\mathbf{Set}}$ and $V\in{\mathbf{Vect}}$; denote such object by $X\times
V$. Define morphisms by the formula

$\displaystyle\mathop{\mathrm{Hom}}\nolimits_{\mathbf{SLSet}}(X\times
V,Y\times
W)=\mathop{\mathrm{Hom}}\nolimits_{\mathbf{Set}}(X,Y)\times\mathop{\mathrm{Hom}}\nolimits_{\mathbf{Set}}\big{(}X,\mathop{\mathrm{Hom}}\nolimits_{\mathbf{Vect}}(V,W)\big{)}.$
(4.1)

A morphism $X\times V\to Y\times W$ is presented by a collection
$\big{(}\varphi,(f_{x})_{x\in X}\big{)}$, where $\varphi\colon X\to Y$ is an
arbitrary map and $f_{x}\colon V\to W$ are linear maps indexed by $x\in X$.

We can consider the object $X\times V\in{\mathbf{SLSet}}$ as the Cartesian
product of the sets $X$ and $V$. A morphism $F=(\varphi,f_{x})$ in
${\mathbf{SLSet}}$ can be considered as the map $F\colon X\times V\to Y\times
W$ given by the formula

$\displaystyle F(x,v)=\big{(}\varphi(x),f_{x}(v)\big{)}.$ (4.2)

Such maps are linear only in the second variable, so we call them semi-linear
maps. If $(\psi,g_{y})\colon Y\times W\to Z\times U$ is another morphism in
${\mathbf{SLSet}}$ for some $Y\in{\mathbf{Set}}$ and $U\in{\mathbf{Vect}}$,
then the composition $X\times V\xrightarrow{(\varphi,f_{x})}Y\times
W\xrightarrow{(\psi,g_{y})}Z\times U$ has the form

$\displaystyle(\psi,g_{y})\cdot(\varphi,f_{x})=(\psi\cdot\varphi,g_{\varphi(x)}\cdot
f_{x}).$ (4.3)

This formula defines the composition in ${\mathbf{SLSet}}$. Due to the
associativity of the composition of maps we indeed defined a category
${\mathbf{SLSet}}$. It is a concrete category with the forgetful functor
${\mathbf{SLSet}}\to{\mathbf{Set}}$ defined by (4.2). We call its objects
semi-linear spaces or semi-linear sets.

Let $X,X^{\prime}\in{\mathbf{Set}}$ and $V,V^{\prime}\in{\mathbf{Vect}}$. We
introduce the tensor product of the semi-linear sets $X\times V$ and
$X^{\prime}\times V^{\prime}$ by the formula

$\displaystyle(X\times V)\otimes(X^{\prime}\times V^{\prime})=(X\times
X^{\prime})\times(V\otimes V^{\prime}).$ (4.4)

The tensor product on morphisms is defined as

$\displaystyle\big{(}\varphi,(f_{x})_{x\in
X}\big{)}\otimes\big{(}\varphi^{\prime},(f^{\prime}_{x^{\prime}})_{x^{\prime}\in
X^{\prime}}\big{)}=\big{(}\varphi\times\varphi^{\prime},(f_{x}\otimes
f^{\prime}_{x^{\prime}})_{(x,x^{\prime})\in X\times X^{\prime}}\big{)},$ (4.5)

where $\varphi\colon X\to Y$, $f_{x}\colon V\to W$, $\varphi^{\prime}\colon
X^{\prime}\to Y^{\prime}$, $f^{\prime}_{x^{\prime}}\colon V^{\prime}\to
W^{\prime}$. This gives a symmetric monoidal category
$({\mathbf{SLSet}},\otimes)$ with the unit object $\\{0\\}\times{\mathbb{K}}$,
where $\\{0\\}\in{\mathbf{Set}}$ and ${\mathbb{K}}\in{\mathbf{Vect}}$ are unit
objects of $({\mathbf{Set}},\times)$ and $({\mathbf{Vect}},\otimes)$
respectively. There is one more useful ‘forgetful’ functor
${\mathbf{SLSet}}\to{\mathbf{Set}}$, $X\times V\mapsto X$. It is a strict
monoidal functor $({\mathbf{SLSet}},\otimes)\to({\mathbf{Set}},\times)$ and
hence it induces
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\otimes)\to\mathop{\mathbf{Mon}}\nolimits({\mathbf{Set}},\times)$.

An object
${\mathbb{S}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\otimes)$ is a
semi-linear set $X\times V$ with the semi-linear maps

$\displaystyle\mu_{X\times V}=\big{(}\mu_{X},(f_{x,y})_{(x,y)\in X\times
X}\big{)}\colon(X\times X)\times(V\otimes V)\to X\times V,$ (4.6)
$\displaystyle\eta_{X\times
V}=(\eta_{X},\eta_{V})\colon\\{0\\}\times{\mathbb{K}}\to X\times V$ (4.7)

satisfying the following conditions. The triple $(X,\mu_{X},\eta_{X})$ should
be a monoid in $({\mathbf{Set}},\times)$, denote $e=\eta_{X}(0)\in X$ and
$xy=\mu_{X}(x,y)$. The linear maps $f_{x,y}\colon V\otimes V\to V$ and
$\eta_{V}\colon{\mathbb{K}}\to V$ are subjected to
$f_{x,yz}\cdot(\mathop{\mathrm{id}}\nolimits\otimes
f_{y,z})=f_{xy,z}\cdot(f_{x,y}\otimes\mathop{\mathrm{id}}\nolimits)$ and
$f_{e,x}\cdot(\eta_{V}\otimes\mathop{\mathrm{id}}\nolimits)=\mathop{\mathrm{id}}\nolimits=f_{x,e}\cdot(\mathop{\mathrm{id}}\nolimits\otimes\eta_{V})$.
We call ${\mathbb{S}}=(X\times V,\mu_{X\times V},\eta_{X\times
V})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\otimes)$ a semi-linear
monoid.

The fully faithful strict monoidal functor

$\displaystyle({\mathbf{Vect}},\otimes)\hookrightarrow({\mathbf{SLSet}},\otimes),$
$\displaystyle V\mapsto\\{0\\}\times V,$ (4.8)

embeds the category ${\mathbf{Vect}}$ into ${\mathbf{SLSet}}$ as a full
monoidal subcategory of the monoidal category $({\mathbf{SLSet}},\otimes)$.

We embed ${\mathbf{Set}}$ into ${\mathbf{SLSet}}$ by the faithful strict
monoidal functor

$\displaystyle({\mathbf{Set}},\times)\hookrightarrow({\mathbf{SLSet}},\otimes),$
$\displaystyle X\mapsto X\times{\mathbb{K}}.$ (4.9)

This embedding functor is not full. It allows us to regard the category
${\mathbf{Set}}$ as a (not full) monoidal subcategory of
$({\mathbf{SLSet}},\otimes)$.

######  Remark 4.1.

The category ${\mathbf{SLSet}}$ is also a category with finite products and
$({\mathbf{Set}},\times)$ can be considered as a monoidal subcategory of
$({\mathbf{SLSet}},\times)$ via another embedding
${\mathbf{Set}}\to{\mathbf{SLSet}}$, $X\mapsto X\times 0$. The embedding
$V\mapsto\\{0\\}\times V$ does not respect the tensor product $\otimes$ of
vector spaces, it is the strict monoidal functor
$({\mathbf{Vect}},\oplus)\to({\mathbf{SLSet}},\times)$. However, these
embeddings will not give the classical representations on vector spaces.

4.1.3. Representations of usual monoids and groups. A classical notion of the
representation of a monoid
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Set}},\times)$ is a
monoid homomorphism
$\rho\colon{\mathbb{M}}\to\mathop{\mathrm{End}}\nolimits(V)$, where
$\mathop{\mathrm{End}}\nolimits(V)$ is the monoid of the linear operators on
$V$. When ${\mathbb{M}}$ is a group, the image of $\rho$ lies in the group
$GL(V)\subset\mathop{\mathrm{End}}\nolimits(V)$ consisting of the invertible
operators on $V$, so the notion of the representation of a monoid on a linear
space includes the linear representations of groups.

The $\mathop{\mathbf{Mon}}\nolimits$-functor (3.117) induced by the fully
faithful strict monoidal functor (4.8) is the category embedding
${\mathbf{Alg}}\hookrightarrow\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\times)$.
It identifies an algebra
${\mathfrak{A}}=({\mathfrak{A}},\mu_{V},\eta_{V})\in{\mathbf{Alg}}$ with the
semi-linear monoid $\\{0\\}\times{\mathfrak{A}}$; in this case $X=\\{0\\}$,
the maps $\mu_{X}$, $\eta_{X}$ are trivial and $f_{0,0,0}=\mu_{V}\colon
V\otimes V\to V$, $\eta_{V}\colon V\to{\mathbb{K}}$ are the structure
morphisms of the algebra ${\mathfrak{A}}$.

The functor (4.9) induces the embedding
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{Set}},\times)\hookrightarrow\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\otimes)$.
The monoid
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Set}},\times)$
is identified with the semi-linear monoid

$\displaystyle{\mathbb{S}}_{\mathbb{M}}=\Big{(}X\times{\mathbb{K}},\big{(}\mu_{X},(\mathop{\mathrm{id}}\nolimits_{\mathbb{K}})\big{)},(\eta_{V},\mathop{\mathrm{id}}\nolimits_{\mathbb{K}})\Big{)}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\otimes).$
(4.10)

We will see that the representations of a monoid ${\mathbb{M}}$ on $V$ are
exactly the representations of ${\mathbb{S}}_{\mathbb{M}}$ on the object $V$
identified with $\\{0\\}\times V\in{\mathbf{SLSet}}$ via the embedding (4.8).

First, we need to check that ${\mathbf{C}}=({\mathbf{SLSet}},\otimes)$ is
closed relative to ${\mathbf{P}}={\mathbf{Vect}}$. We, however, can prove that
it is closed (relative to the whole ${\mathbf{SLSet}}$). For two semi-linear
sets $X\times V,Y\times W\in{\mathbf{SLSet}}$ define an object
$\mathop{\mathrm{\bf hom}}\nolimits(X\times V,Y\times W)\in{\mathbf{SLSet}}$
as the Cartesian product (4.1), where
$\mathop{\mathrm{Hom}}\nolimits\big{(}X,\mathop{\mathrm{Hom}}\nolimits(V,W)\big{)}=\mathop{\mathrm{Hom}}\nolimits\big{(}X,\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)}$ is equipped by the structure of a linear space as
follows: the linear combination of $f,g\colon X\to\mathop{\mathrm{\bf
hom}}\nolimits(V,W)$ with the coefficients $\alpha,\beta\in{\mathbb{K}}$ is
the map $X\to\mathop{\mathrm{\bf hom}}\nolimits(V,W)$ with the values

$\displaystyle(\alpha f+\beta g)(x)=\alpha f(x)+\beta
g(x)\in\mathop{\mathrm{\bf hom}}\nolimits(V,W),$ $\displaystyle x\in X.$
(4.11)

Introduce the map $\mathop{\mathrm{ev}}\nolimits_{X\times V,Y\times
W}\colon\mathop{\mathrm{\bf hom}}\nolimits(X\times V,Y\times W)\otimes(X\times
V)\to Y\times W$, which evaluates the map
$(\varphi,f_{x})\in\mathop{\mathrm{\bf hom}}\nolimits(X\times V,Y\times W)$ on
$(x,v)\in X\times V$. It maps an element

$\displaystyle(\varphi,x,f\otimes
v)\in\mathop{\mathrm{Hom}}\nolimits(X,Y)\times
X\times\Big{(}\mathop{\mathrm{Hom}}\nolimits\big{(}X,\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)}\otimes V\Big{)}$ (4.12)

to $\big{(}\varphi(x),f(x)(v)\big{)}\in Y\times W$. One can show that
$\big{(}\mathop{\mathrm{\bf hom}}\nolimits(X\times V,Y\times
W),\mathop{\mathrm{ev}}\nolimits_{X\times V,Y\times W}\big{)}$ is a universal
arrow from the functor $-\otimes(X\times
V)\colon{\mathbf{SLSet}}\to{\mathbf{SLSet}}$ to the object $Y\times W$. Then
Theorem 2.13 implies that $({\mathbf{SLSet}},\otimes)$ is closed with the hom-
objects defined in this way.

For the case $X=Y=\\{0\\}$ we have $\\{0\\}\times\mathop{\mathrm{\bf
hom}}\nolimits(V,W)=\mathop{\mathrm{\bf hom}}\nolimits(\\{0\\}\times
V,\\{0\\}\times W)$ (this identification is the isomorphism $\Phi_{V,W}$
induced by the monoidal functor (4.8)), so the internal hom-object in
$({\mathbf{Vect}},\otimes)$ is the same as the internal hom-object in
$({\mathbf{SLSet}},\otimes)$ for the vector spaces embedded by (4.8). The
internal end is the algebra $\mathop{\mathrm{\bf end}}\nolimits(\\{0\\}\times
V)=\\{0\\}\times\mathop{\mathrm{\bf end}}\nolimits(V)$ identified with the
algebra of operators $\mathop{\mathrm{\bf end}}\nolimits(V)$.

A representation of an algebra ${\mathfrak{A}}=\\{0\\}\times{\mathfrak{A}}$ on
$V=\\{0\\}\times V$ in the sense of Definition 3.1 is exactly the algebra
morphism $\rho\colon{\mathfrak{A}}\to\mathop{\mathrm{\bf end}}\nolimits(V)$,
so it is a representation of the algebra ${\mathfrak{A}}$ in the usual sense.

Consider a monoid
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Set}},\times)$ as the
semi-linear monoid ${\mathbb{S}}_{\mathbb{M}}$ defined by (4.10). Then a
representation of ${\mathbb{M}}$ on $V=\\{0\\}\times V$ is a representation
$\widehat{\rho}\colon{\mathbb{S}}_{\mathbb{M}}\to\mathop{\mathrm{\bf
end}}\nolimits(V)$. This is a morphism
$X\times{\mathbb{K}}\to\\{0\\}\times\mathop{\mathrm{\bf end}}\nolimits(V)$ in
${\mathbf{SLSet}}$ given by the trivial map $0\colon X\to\\{0\\}$ and a
collection of linear maps
$f_{x}=\rho_{x}\colon{\mathbb{K}}\to\mathop{\mathrm{\bf end}}\nolimits(V)$.
Define the map $\rho\colon X\to\mathop{\mathrm{\bf end}}\nolimits(V)$ as
$\rho(x)=\rho_{x}(1)$. Any collection $\rho_{x}$ is given by such map $\rho$.
The morphism $\widehat{\rho}=(0,\rho_{x})$ is a representation (a morphism in
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\otimes)$) iff
$\rho(x)\rho(y)=\rho(xy)$ for any $x,y\in X$. The latter condition means
exactly that $\rho$ is a representation of the monoid ${\mathbb{M}}$ on $V$.

Thus a linear representation of a usual monoid or group ${\mathbb{M}}$ is its
representation in the monoidal category $({\mathbf{SLSet}},\otimes)$ on an
object $V$ of the subcategory ${\mathbf{Vect}}\subset{\mathbf{SLSet}}$.

4.1.4. Representations of Lie groups. Consider the category of smooth
manifolds ${\mathbf{Man}^{\infty}}$. This is a category with finite products.
A monoid
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Man}^{\infty}},\times)$
is a manifold $X$ with a smooth multiplication $X\times X\to X$ and a unity
$e\in X$. If any element $x\in X$ is invertible with respect to this
multiplication and the map $X\to X$, $x\mapsto x^{-1}$, is smooth, then
${\mathbb{M}}$ is a Lie group, so the category of Lie groups is a full
subcategory of
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{Man}^{\infty}},\times)$.

Suppose that ${\mathbb{K}}$ is an extension of the field of real numbers:
${\mathbb{R}}\subset{\mathbb{K}}$. A representation of a Lie group or monoid
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Man}^{\infty}},\times)$
on a vector space $V\in{\mathbf{FVect}}$ is a smooth monoid homomorphism
$\rho\colon{\mathbb{M}}\to\mathop{\mathrm{\bf end}}\nolimits(V)$, where
${\mathbf{FVect}}$ is the category of finite-dimensional vector spaces and the
algebra $\mathop{\mathrm{\bf end}}\nolimits(V)$ is considered as a smooth
manifold $V^{*}\otimes V$ (this is possible for finite dimensional $V$ only).
Such representations can be interpreted in terms of Section 3 by introducing a
smooth version of semi-linear spaces.

Define the category ${\mathbf{SLMan}^{\infty}}$. An object of this category is
a pair $(X,V)$ of $X\in{\mathbf{Man}^{\infty}}$, $V\in{\mathbf{FVect}}$,
denoted by $X\times V$ and regarded as a Cartesian product of smooth
manifolds. The morphisms $F\colon X\times V\to Y\times W$ are smooth maps of
the form (4.2). The objects $X\times V\in{\mathbf{SLMan}^{\infty}}$ are called
smooth semi-linear spaces or semi-linear manifolds.

We obtain the following faithful (‘forgetful’) functors. By forgetting semi-
linear structure of $X\times V$ we can consider it as an abstract manifold,
this gives the forgetful functor
${\mathbf{SLMan}^{\infty}}\to{\mathbf{Man}^{\infty}}$. By forgetting the
smoothness we obtain ${\mathbf{SLMan}^{\infty}}\to{\mathbf{SLSet}}$. Finally
we have the functor ${\mathbf{SLMan}^{\infty}}\to{\mathbf{Man}^{\infty}}$
which omits the linear part: $X\times V\mapsto X$.

The category ${\mathbf{SLMan}^{\infty}}$ has a monoidal structure with the
tensor product defined by (4.4), (4.5). Monoids in
$({\mathbf{SLMan}^{\infty}},\otimes)$ are called smooth semi-linear monoids.
The formulae $X\mapsto X\times{\mathbb{K}}$ and $V\mapsto\\{0\\}\times V$
define the embeddings (also faithful and fully faithful respectively)

$\displaystyle({\mathbf{Man}^{\infty}},\times)\hookrightarrow({\mathbf{SLMan}^{\infty}},\otimes),$
$\displaystyle({\mathbf{FVect}},\otimes)\hookrightarrow({\mathbf{SLMan}^{\infty}},\otimes).$
(4.13)

The monoidal category $({\mathbf{SLMan}^{\infty}},\otimes)$ is symmetric, but
it is not closed in the usual sense. It is closed relative to the subcategory
${\mathbf{P}}={\mathbf{FVect}}$ defined by the right embedding (4.13). The
functors (4.13) induce the embeddings

$\displaystyle\mathop{\mathbf{Mon}}\nolimits({\mathbf{Man}^{\infty}},\times)\hookrightarrow\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLMan}^{\infty}},\otimes),$
$\displaystyle{\mathbf{FAlg}}\hookrightarrow\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLMan}^{\infty}},\otimes),$
(4.14)

which identifies a monoid
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Man}^{\infty}},\times)$
(in particular, a Lie group ${\mathbb{M}}$) and a finite-dimensional algebra
${\mathfrak{A}}\in{\mathbf{FAlg}}$ with the corresponding smooth semi-linear
monoids ${\mathbb{S}}_{\mathbb{M}}$ and $\\{0\\}\times{\mathfrak{A}}$
respectively. A representation of ${\mathbb{M}}$ and of ${\mathfrak{A}}$ on a
vector space $V\in{\mathbf{FVect}}$ can be interpreted as a representation of
${\mathbb{S}}_{\mathbb{M}}$ and of $\\{0\\}\times{\mathfrak{A}}$ respectively
in the same way as for the case of p. 4.1. In particular, the linear
representations of Lie groups are representations in the monoidal category
$({\mathbf{SLMan}^{\infty}},\otimes)$.

Similarly, one can interpret the representations of a complex Lie group by
using the category of complex manifolds instead of ${\mathbf{Man}^{\infty}}$.
In this case we should suppose ${\mathbb{C}}\subset{\mathbb{K}}$.

######  Remark 4.2.

Consider the category of vector bundles over non-fixed manifold. A morphisms
of two such bundles $E$ and $E^{\prime}$ with projections $\pi\colon E\to X$
and $\pi^{\prime}\colon E^{\prime}\to X^{\prime}$ is a pair of smooth maps
$F\colon E\to E^{\prime}$ and $\varphi\colon X\to X^{\prime}$ such that
$\varphi\cdot\pi=\pi^{\prime}\cdot F$ and the induced maps
$F_{x}\colon\pi^{-1}(x)\to(\pi^{\prime})^{-1}\big{(}\varphi(x)\big{)}$ are
linear. The category ${\mathbf{SLMan}^{\infty}}$ coincides with the
subcategory of trivial vector bundles $E=X\times V$ (we use the term ‘semi-
linear space’ instead of ‘trivial vector bundle’ by the reasons explained in
[S21, § 4.3.3]). The role of $({\mathbf{C}},\otimes)$ could be played by the
category of all the locally trivial vector bundles with the tensor product
defined via local triviality, but its subcategory ${\mathbf{SLMan}^{\infty}}$
is enough to consider representations of Lie groups from the point of view of
the general representation theory.

4.1.5. Representations of affine algebraic groups. Let us fix an infinite
subfield ${\mathbb{F}}\subset{\mathbb{K}}$ and consider the category
${\mathbf{AlgSet}}_{\mathbb{F}}$ of the (affine) algebraic sets over
${\mathbb{F}}$. Since ${\mathbf{AlgSet}}_{\mathbb{F}}$ is a category with
finite products, we have the monoidal category
$({\mathbf{AlgSet}}_{\mathbb{F}},\times)$. Monoids/Hopf monoids in this
monoidal category are called (affine) algebraic monoids/groups (over
${\mathbb{F}}$). By using these category instead of ${\mathbf{Set}}$ or
${\mathbf{Man}^{\infty}}$ we can interpret representations of algebraic groups
and monoids on vector spaces in terms of the general representation theory.
The case ${\mathbb{F}}={\mathbb{K}}$ is described in details in [S21, § 4.3].

Remind the main notions. By considering the Cartesian products $X\times V$ for
$X\in{\mathbf{AlgSet}}_{\mathbb{F}}$ and $V\in{\mathbf{FVect}}$ as algebraic
sets we get the category ${\mathbf{SLAlgSet}}_{\mathbb{F}}$ (the index
${\mathbb{F}}$ means that $X$ is over ${\mathbb{F}}$, the linear part $V$ is
over ${\mathbb{K}}$). Its objects are called semi-linear (affine) algebraic
sets or (affine algebraic) semi-linear spaces. Their morphisms are regular
maps of the form (4.2); their tensor products are defined by the same formula
(4.4). The objects of
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLAlgSet}}_{\mathbb{F}},\times)$ are
called semi-linear algebraic monoids (over ${\mathbb{F}}$).

The monoidal categories $({\mathbf{AlgSet}}_{\mathbb{F}},\times)$ and
$({\mathbf{FVect}},\times)$ are identified with the monoidal subcategories of
$({\mathbf{SLAlgSet}}_{\mathbb{F}},\otimes)$ by $X\mapsto
X\times{\mathbb{K}}$, $V\mapsto\\{0\\}\times V$. Then
${\mathbf{C}}=({\mathbf{SLAlgSet}}_{\mathbb{F}},\otimes)$ is closed relative
to ${\mathbf{P}}={\mathbf{FVect}}$. A representation of an algebraic
monoid/group or finite-dimensional algebra on $V\in{\mathbf{FVect}}$ is
exactly a representation of the corresponding semi-linear algebraic monoid on
$V$.

###  4.2 Quantum representations on quadratic algebras

Now let us remind the main points of Quantum Representation Theory developed
in [S21] and apply the general representation theory to this case. Note that
the quantum representations can be considered as generalisations of the
representations of algebras, algebraic groups and semi-linear algebraic
monoids.

4.2.1. Quantum linear spaces. For two ${\mathbb{N}}_{0}$-graded algebras
${\mathcal{A}}=\bigoplus\limits_{k\geqslant 0}{\mathcal{A}}_{k}$ and
${\mathcal{B}}=\bigoplus\limits_{k\geqslant 0}{\mathcal{B}}_{k}$ we denote
${\mathcal{A}}\circ{\mathcal{B}}=\bigoplus\limits_{k\geqslant
0}({\mathcal{A}}_{k}\otimes{\mathcal{B}}_{k})$. This operation was introduced
by Manin in [Man87, Man88] for quadratic algebras. We call it (white) Manin
product, it defines a symmetric monoidal structure on the category of all the
${\mathbb{N}}_{0}$-graded algebras ${\mathbf{GrAlg}}$. The unit object is the
polynomial algebra ${\mathbb{K}}[u]=\bigoplus\limits_{k\geqslant
0}{\mathbb{K}}u^{k}$.

The finitely generated quadratic algebras over ${\mathbb{K}}$ (i.e. connected)
form a monoidal subcategory
$({\mathbf{FQA}},\circ)\subset({\mathbf{GrAlg}},\circ)$. This subcategory is a
coclosed monoidal category with the internal cohom-objects
$\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{A}},{\mathcal{B}})={\mathcal{A}}^{!}\bullet{\mathcal{B}}\in{\mathbf{FQA}}$,
${\mathcal{A}},{\mathcal{B}}\in{\mathbf{FQA}}$ (see [Man88, § 4], [S21, §
4.4]). Manin interpreted the opposite category ${\mathbf{FQA}}^{\mathrm{op}}$
as a quantum (non-commutative) version of ${\mathbf{FVect}}$ and called its
objects the quantum linear spaces. The classical finite-dimensional vector
spaces are lifted to the quantum level by the contravariant colax monoidal
embedding
$S^{*}\colon({\mathbf{FVect}},\otimes)\hookrightarrow({\mathbf{FQA}},\circ)$,
$V\mapsto SV^{*}$. Alternatively one can use the contravariant strong monoidal
embedding
$T^{*}\colon({\mathbf{FVect}},\otimes)\hookrightarrow({\mathbf{FQA}},\circ)$,
$V\mapsto TV^{*}$ (see [S21] for details).

The corepresentations of a comonoid
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{FQA}},\circ)$
on a quadratic algebra ${\mathcal{B}}\in{\mathbf{FQA}}$ or, equivalently, the
representations of a monoid in $({\mathbf{FQA}}^{\mathrm{op}},\circ)$ on a
quantum linear space ${\mathcal{B}}\in{\mathbf{FQA}}^{\mathrm{op}}$ are
quantum analogues for the finite-dimensional representations of finite-
dimensional algebras.

The category ${\mathbf{QA}}$ consisting of all the quadratic algebras over
${\mathbb{K}}$ (including infinitely generated) is a monoidal subcategory of
$({\mathbf{GrAlg}},\circ)$. We proved in [S21, § 4.4.5] that for any
${\mathcal{B}}\in{\mathbf{FQA}}$ the functor
$-\circ{\mathcal{B}}\colon{\mathbf{QA}}\to{\mathbf{QA}}$ has a left adjoint
(relative to the whole ${\mathbf{QA}}$), hence the symmetric monoidal category
$({\mathbf{QA}},\circ)$ is coclosed relative to ${\mathbf{P}}={\mathbf{FQA}}$
due to the point (3) of Prop. 2.19 applied to the monoidal category
${\mathbf{C}}=({\mathbf{QA}},\circ)$. The corepresentations of comonoids
${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{QA}},\circ)$ on
${\mathcal{B}}\in{\mathbf{FQA}}$ generalise the finite-dimensional
representations of infinite-dimensional algebras at least partially.

4.2.2. Quantum semi-linear spaces. The algebraic set can be considered as a
particular case the affine scheme. Categorically this fact is interpreted via
a fully faithful functor
$A\colon{\mathbf{AlgSet}}_{\mathbb{F}}\hookrightarrow{\mathbf{CommAlg}}_{\mathbb{F}}^{\mathrm{op}}$,
where ${\mathbf{CommAlg}}_{\mathbb{F}}$ is the category of the commutative
algebras over ${\mathbb{F}}$ and $A(X)$ is the algebra of regular
${\mathbb{F}}$-valued functions on the algebraic set
$X\in{\mathbf{AlgSet}}_{\mathbb{F}}$. Hence ${\mathbf{AlgSet}}_{\mathbb{F}}$
is contravariantly embedded into ${\mathbf{Alg}}_{\mathbb{F}}$, the category
of algebras over ${\mathbb{F}}$. This is a standard way to interpret the
algebras as non-commutative affine algebraic sets/varieties/schemes.

The contravariant functor $S^{*}\colon{\mathbf{FVect}}\to{\mathbf{FQA}}$ is
extended to the contravariant faithful functor
$S^{*}\colon{\mathbf{SLAlgSet}}_{\mathbb{F}}\to{\mathbf{GrAlg}}$ as
$S^{*}(X\times V)=A(X)\otimes_{\mathbb{F}}SV^{*}=A(X)^{e}\otimes SV^{*}$,
where ${\mathfrak{R}}^{e}={\mathfrak{R}}\otimes_{\mathbb{F}}{\mathbb{K}}$ is
the extension of scalars. In particular, an object $X\times{\mathbb{K}}$ (that
is an algebraic set $X\in{\mathbf{AlgSet}}_{\mathbb{F}}$ embedded into
${\mathbf{SLAlgSet}}_{\mathbb{F}}$) is lifted to the quantum level as
$A(X)\otimes_{\mathbb{F}}{\mathbb{K}}[u]$, the latter is a graded algebra
${\mathcal{A}}\in{\mathbf{GrAlg}}$ with components ${\mathcal{A}}_{k}\cong
A(X)^{e}$. The contravariant functor $S^{*}$ has a colax monoidal structure
$({\mathbf{SLAlgSet}}_{\mathbb{F}},\otimes)\to({\mathbf{GrAlg}},\circ)$.
Similarly, the functor $T^{*}$ is extended to a contravariant faithful strong
monoidal functor
$T^{*}\colon({\mathbf{SLAlgSet}}_{\mathbb{F}},\otimes)\to({\mathbf{GrAlg}},\circ)$,
$T^{*}(X\times V)=A(X)^{e}\otimes TV^{*}$ (if ${\mathbb{F}}={\mathbb{K}}$,
then $S^{*}$ and $T^{*}$ are fully faithful).

The images of ${\mathbf{SLAlgSet}}_{\mathbb{F}}$ under the functors $S^{*}$
and $T^{*}$ lie in the category ${\mathbf{QA}}_{\mathrm{s}c}$ of semi-
connected quadratic algebras. This is a full subcategory
${\mathbf{QA}}_{\mathrm{s}c}\subset{\mathbf{GrAlg}}$ consisting of the
quadratic algebras ${\mathfrak{R}}\otimes{\mathcal{B}}$, where
${\mathfrak{R}}\in{\mathbf{Alg}}$, ${\mathcal{B}}\in{\mathbf{QA}}$. Moreover,
these images lie in ${\mathbf{FQA}}_{\mathrm{s}c}$, the full subcategory of
the quadratic algebras ${\mathfrak{R}}\otimes{\mathcal{B}}$, where
${\mathfrak{R}}\in{\mathbf{Alg}}$, ${\mathcal{B}}\in{\mathbf{FQA}}$. The
objects of ${\mathbf{QA}}_{\mathrm{s}c}^{\mathrm{op}}$ or at least of
${\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}}$ can be considered as quantum
analogues of the semi-linear algebraic sets over ${\mathbb{K}}$ (see details
in [S21]).

It was shown in [S21, § 4.4.6] that for any ${\mathcal{B}}\in{\mathbf{FQA}}$
the functor
$-\circ{\mathcal{B}}\colon{\mathbf{QA}}_{\mathrm{s}c}\to{\mathbf{QA}}_{\mathrm{s}c}$
has a left adjoint. Due to the point (3) of Prop. 2.19 the monoidal category
${\mathbf{C}}=({\mathbf{QA}}_{\mathrm{s}c},\circ)$ is coclosed relative to
${\mathbf{P}}={\mathbf{FQA}}$. Since $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{A}},{\mathcal{B}})\in{\mathbf{FQA}}\subset{\mathbf{FQA}}_{\mathrm{s}c}$
for any ${\mathcal{A}},{\mathcal{B}}\in{\mathbf{FQA}}$, Prop. 2.20 implies
that the monoidal subcategory
${\mathbf{C}}^{\prime}=({\mathbf{FQA}}_{\mathrm{s}c},\circ)$ is also coclosed
relative to ${\mathbf{P}}={\mathbf{FQA}}$.

Corepresentations of comonoids
${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{QA}}_{\mathrm{s}c},\circ)$
on ${\mathcal{B}}\in{\mathbf{FQA}}$ are quantum analogues of representations
of semi-linear algebraic monoids. In particular, they give a quantum version
of the representations of the affine algebraic groups. A classical
representation is considered as a corepresentation on a quadratic algebra via
the functor (3.149), where
$F\colon({\mathbf{SLAlgSet}}_{\mathbb{F}},\otimes)\to({\mathbf{QA}}_{\mathrm{s}c},\circ)$
is $S^{*}$ or $T^{*}$. For the case of an algebraic group or monoid one need
to compose these functors with the embedding
${\mathbf{AlgSet}}_{\mathbb{F}}\hookrightarrow{\mathbf{SLAlgSet}}_{\mathbb{F}}$.
In the case of the strong monoidal functor $F=T^{*}$ we obtain a contravariant
monoidal embedding of representations (3.172).

One can lift the representations of groups, monoids
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Set}},\times)$ and,
more generally, semi-linear monoids
${\mathbb{S}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLSet}},\otimes)$ to
the quantum level by the same formulae with $A(X)$ replaced by the algebra of
all the functions $f\colon X\to{\mathbb{K}}$ on $X\in{\mathbf{Set}}$.

4.2.3. The case of all graded algebras. The category
${\mathbf{QA}}_{\mathrm{s}c}$ is enough to lift the classical representations
of algebraic monoids and (finite-dimensional) algebras to the quantum level,
however, sometimes we need to consider more general ${\mathbb{N}}_{0}$-graded
and ${\mathbb{Z}}$-graded algebras (see e.g. [S21, § 6.2.2]).

Let us extend the monoidal category $({\mathbf{GrAlg}},\circ)$ to the case of
all the ${\mathbb{Z}}$-graded algebras. Denote it by
${\mathbb{Z}}$-$\mathbf{GrAlg}$. The Manin product of the algebras
${\mathcal{A}},{\mathcal{B}}\in\textbf{${\mathbb{Z}}$-GrAlg}$ is the
${\mathbb{Z}}$-graded vector space with the components
$({\mathcal{A}}\circ{\mathcal{B}})_{k}={\mathcal{A}}_{k}\otimes{\mathcal{B}}_{k}$,
$k\in{\mathbb{Z}}$, and multiplication $(a_{k}\otimes
b_{k})(a^{\prime}_{l}\otimes
b^{\prime}_{l})=(a_{k}a^{\prime}_{l})\otimes(b_{k}b^{\prime}_{l})\in{\mathcal{A}}_{k+l}\otimes{\mathcal{B}}_{k+l}$,
where $a_{k}\in{\mathcal{A}}_{k}$, $a^{\prime}_{l}\in{\mathcal{A}}_{l}$,
$b_{k}\in{\mathcal{B}}_{k}$, $b^{\prime}_{l}\in{\mathcal{B}}_{l}$.

In [S21] we defined a quantum representation of a quantum algebra
${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{GrAlg}},\circ)$ on
${\mathcal{B}}\in{\mathbf{FQA}}$ as a morphism
$\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ in
$\mathop{\mathbf{Comon}}\nolimits({\mathbf{GrAlg}},\circ)$. More generally, we
can define it as a morphism in
$\mathop{\mathbf{Comon}}\nolimits({\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}},\circ)$
from the same $\mathop{\mathrm{\bf end}}\nolimits({\mathcal{B}})$ to a
comonoid
${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}},\circ)$.
This notion can be interpreted in terms of Section 3 as corepresentation (or
as a representation in the opposite category) due to the following statement.

######  Theorem 4.3.

The monoidal category
${\mathbf{C}}=({\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}},\circ)$ is coclosed
relative to ${\mathbf{P}}={\mathbf{FQA}}$.

Proof. Note that the category ${\mathbf{QA}}$ is a monoidal subcategory of
$({\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}},\circ)$ and that
$({\mathbf{QA}},\circ)$ is coclosed relative to ${\mathbf{P}}={\mathbf{FQA}}$.
In order to apply Theorem 2.26 we need to check that
${\mathbf{C}}^{\prime}={\mathbf{QA}}$ is a coreflective subcategory of
${\mathbf{C}}={\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}}$. For a
${\mathbb{Z}}$-graded algebra ${\mathcal{A}}$ denote by
$G^{\prime}{\mathcal{A}}$ the subalgebra of ${\mathcal{A}}$ generated by the
subspace ${\mathcal{A}}_{1}$. This is an ${\mathbb{N}}_{0}$-graded algebra
$G^{\prime}{\mathcal{A}}$ generated by its first order component
$V={\mathcal{A}}_{1}=(G^{\prime}{\mathcal{A}})_{1}$, so we have the graded
epimorphism $a\colon TV\twoheadrightarrow G^{\prime}{\mathcal{A}}$, where
$TV\in{\mathbf{GrAlg}}$ is the tensor algebra. The kernel of $a$ is a graded
ideal $I=\bigoplus\limits_{k\geqslant 2}I_{k}$ in $TV$, where $I_{k}=I\cap
V^{\otimes k}$, and we obtain the isomorphism $G^{\prime}{\mathcal{A}}\cong
TV/I$ in ${\mathbf{GrAlg}}$. Note that $TV/I$ is a quotient algebra of the
quadratic algebra $TV/(I_{2})\in{\mathbf{QA}}$, where $(I_{2})$ is the ideal
generated by $I_{2}\subset V^{\otimes 2}$. Let $G{\mathcal{A}}=TV/(I_{2})$ and
$\varepsilon_{\mathcal{A}}\colon G{\mathcal{A}}\to{\mathcal{A}}$ be the
composition

$\displaystyle\varepsilon_{\mathcal{A}}\colon
G{\mathcal{A}}=TV/(I_{2})\twoheadrightarrow TV/I\cong
G^{\prime}{\mathcal{A}}\hookrightarrow{\mathcal{A}}.$ (4.15)

Its first graded component is the identity morphism
$(\varepsilon_{\mathcal{A}})_{1}=\mathop{\mathrm{id}}\nolimits_{V}\colon
V\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}{\mathcal{A}}_{1}$.
Consider a quadratic algebra ${\mathcal{B}}=TW/(S)\in{\mathbf{QA}}$ and a
graded homomorphism $f\colon{\mathcal{B}}\to{\mathcal{A}}$, where
$W\in{\mathbf{Vect}}$ and $S$ is a subspace of $W^{\otimes 2}$. Since
${\mathcal{B}}$ is generated by ${\mathcal{B}}_{1}=W$, the image of $f$ lies
in the subalgebra $G^{\prime}{\mathcal{A}}\subset{\mathcal{A}}$, so it factors
as in the commutative diagram

(4.20)

Such morphism $f^{\prime}\colon{\mathcal{B}}\to G^{\prime}{\mathcal{A}}$ is
unique, since the embedding
$G^{\prime}{\mathcal{A}}\hookrightarrow{\mathcal{A}}$ is a monomorphism.
Moreover, one can show that any $f^{\prime}\colon{\mathcal{B}}\to TV/I$
factors as

(4.25)

through a unique $h\colon{\mathcal{B}}\to TV/(I_{2})$. Indeed, the first
component of $TV/(I_{2})\twoheadrightarrow TV/I$ is
$\mathop{\mathrm{id}}\nolimits_{V}$, so the commutativity of (4.25) is
possible only if the first components of $f^{\prime}$ and $h$ coincide. Since
$(f^{\prime}_{1}\otimes f^{\prime}_{1})S\subset I_{2}$, the formula
$h_{1}=f^{\prime}_{1}$ correctly defines the graded homomorphism
$h\colon{\mathcal{B}}\to TV/(I_{2})$ and the diagram (4.25) is commutative.
Thus for any $f\colon{\mathcal{B}}\to{\mathcal{A}}$ there is a unique
$h\colon{\mathcal{B}}\to G{\mathcal{A}}$ such that
$\varepsilon_{\mathcal{A}}\cdot h=f$. By virtue of Theorem 2.13 we get a right
adjoint $G\colon{\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}}\to{\mathbf{QA}}$ for
the embedding
${\mathbf{QA}}\hookrightarrow{\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}}$ such
that $h=Gf$. By taking the first graded components in the commutative diagrams
(4.20), (4.25) we see that the first component
$(Gf)_{1}\colon{\mathcal{B}}_{1}\to V$ coincides with
$f_{1}\colon{\mathcal{B}}_{1}\to{\mathcal{A}}_{1}=V$. Since ${\mathcal{B}}$ is
generated by ${\mathcal{B}}_{1}$, this formula uniquely defines the whole
graded homomorphism $Gf\colon{\mathcal{B}}\to G{\mathcal{A}}=TV/(I_{2})$.

Now let us prove that
$\phi_{{\mathcal{A}},{\mathcal{A}}^{\prime}}=G(\varepsilon_{\mathcal{A}}\circ\varepsilon_{{\mathcal{A}}^{\prime}})\colon
G{\mathcal{A}}\circ G{\mathcal{A}}^{\prime}\to
G({\mathcal{A}}\circ{\mathcal{A}}^{\prime})$ is an isomorphism for arbitrary
${\mathbb{Z}}$-graded algebras
${\mathcal{A}},{\mathcal{A}}^{\prime}\in{\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}}$.
Let $V={\mathcal{A}}_{1}$, $V^{\prime}={\mathcal{A}}^{\prime}_{1}$, the
morphism $\phi_{{\mathcal{A}},{\mathcal{A}}^{\prime}}$ is uniquely defined by
its first order component
$(\phi_{{\mathcal{A}},{\mathcal{A}}^{\prime}})_{1}=\big{(}G(\varepsilon_{\mathcal{A}}\circ\varepsilon_{{\mathcal{A}}^{\prime}})\big{)}_{1}=(\varepsilon_{\mathcal{A}}\circ\varepsilon_{{\mathcal{A}}^{\prime}})_{1}=(\varepsilon_{\mathcal{A}})_{1}\otimes(\varepsilon_{{\mathcal{A}}^{\prime}})_{1}=\mathop{\mathrm{id}}\nolimits_{V\otimes
V^{\prime}}$. Denote by $I=\bigoplus\limits_{k\geqslant 2}I_{k}$,
$I^{\prime}=\bigoplus\limits_{k\geqslant 2}I^{\prime}_{k}$ the kernels of
$a\colon TV\twoheadrightarrow G^{\prime}{\mathcal{A}}$ and $a^{\prime}\colon
TV^{\prime}\twoheadrightarrow G^{\prime}{\mathcal{A}}^{\prime}$ respectively,
where $I_{k}=I\cap V^{\otimes k}$,
$I^{\prime}_{k}=I^{\prime}\cap(V^{\prime})^{\otimes k}$. The kernel of the
composition $T(V\otimes V^{\prime})\cong TV\circ
TV^{\prime}\xrightarrow{a\circ a^{\prime}}G^{\prime}{\mathcal{A}}\circ
G^{\prime}{\mathcal{A}}^{\prime}\subset{\mathcal{A}}\circ{\mathcal{A}}^{\prime}$
is a graded ideal $J=\bigoplus\limits_{k\geqslant 2}J_{k}$ with components
$J_{k}=J\cap(V\otimes V^{\prime})^{\otimes k}$. The subalgebra
$G^{\prime}{\mathcal{A}}\circ
G^{\prime}{\mathcal{A}}^{\prime}\subset{\mathcal{A}}\circ{\mathcal{A}}^{\prime}$
coincides with $G^{\prime}({\mathcal{A}}\circ{\mathcal{A}}^{\prime})$, hence
$G({\mathcal{A}}\circ{\mathcal{A}}^{\prime})=T(V\otimes V^{\prime})/(J_{2})$.
By taking into account the isomorphisms $G^{\prime}{\mathcal{A}}\cong TV/I$
and $G^{\prime}{\mathcal{A}}^{\prime}\cong TV^{\prime}/I^{\prime}$ we see that
the ideal $J$ is the kernel of the composition

$\displaystyle T(V\otimes V^{\prime})\cong TV\circ
TV^{\prime}\twoheadrightarrow TV/(I_{2})\circ
TV^{\prime}/(I^{\prime}_{2})\twoheadrightarrow TV/I\circ
TV^{\prime}/I^{\prime},$ (4.26)

where the epimorphisms $\twoheadrightarrow$ are the Manin products of the
corresponding projections. The second order component of the right epimorphism
in (4.26) is the identity map, so the kernel of the second order component of

$\displaystyle T(V\otimes V^{\prime})\cong TV\circ
TV^{\prime}\twoheadrightarrow TV/(I_{2})\circ TV^{\prime}/(I^{\prime}_{2})$
(4.27)

equals $J_{2}$. Since (4.27) is a morphism of quadratic algebras, its kernel
is the ideal $(J_{2})$. Hence we obtain the isomorphism $TV/(I_{2})\circ
TV^{\prime}/(I^{\prime}_{2})\cong T(V\otimes V^{\prime})/(J_{2})$, whose first
order component is the identity map $\mathop{\mathrm{id}}\nolimits_{V\otimes
V^{\prime}}$. This isomorphism is exactly the morphism
$\phi_{{\mathcal{A}},{\mathcal{A}}^{\prime}}\colon G{\mathcal{A}}\circ
G{\mathcal{A}}^{\prime}\to G({\mathcal{A}}\circ{\mathcal{A}}^{\prime})$. In
particular, $\phi_{{\mathcal{A}},{\mathcal{A}}^{\prime}}$ is an isomorphism
for any ${\mathcal{A}}\in{\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}}$,
${\mathcal{A}}^{\prime}\in{\mathbf{FQA}}$, so the conditions of Theorem 2.26
are fulfilled. ∎

Since $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{A}},{\mathcal{B}})\in{\mathbf{FQA}}\subset{\mathbf{GrAlg}}$
for any ${\mathcal{A}},{\mathcal{B}}\in{\mathbf{FQA}}$, Prop. 2.20 implies
that the category $({\mathbf{GrAlg}},\circ)$ is also coclosed relative to
${\mathbf{FQA}}$. The internal cohom-functor in
${\mathbf{C}}=({\mathbf{GrAlg}},\circ)$ or
$({\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}},\circ)$ with parametrising
subcategory ${\mathbf{P}}={\mathbf{FQA}}$ equals to the composition of the
functor $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{FQA}}^{\mathrm{op}}\times{\mathbf{FQA}}\to{\mathbf{FQA}}$
and the embedding ${\mathbf{FQA}}\hookrightarrow{\mathbf{GrAlg}}$ or
${\mathbf{FQA}}\hookrightarrow{\text{${\mathbb{Z}}$-$\mathbf{GrAlg}$}}$.

## Conclusion

General representation theory. Relative adjunctions introduced by Ulmer in
[Ulm] can be used to generalise the notion of closed monoidal category in such
way that the approach of [S21, § 3] works literally for this more general
case. Thus we obtain a general representation theory for relatively closed
monoidal categories and for monoidal functors between such categories. In
particular, we described the tensor product of representations and how it is
translated by a monoidal functor. Any representation can be considered as a
left action, however we proved all the statements about representations
independently from the theory of actions, we used the natural transformations
$\pi$ and $\Phi$.

Classical representations and semi-linear spaces. This general approach covers
the classical types of representations on vector spaces: representations of
algebras, groups, monoids, Lie groups and algebraic groups. To interpret
representations of groups and monoids as representations in the general sense
we introduced semi-linear spaces with some monoidal product. The monoids in
the monoidal category of semi-linear spaces (the semi-linear monoids)
generalise algebras, monoids and groups, so the representation theory of such
monoids includes the classical representations. The same can be done for the
groups and monoids with geometric structures.

Quantum Representation Theory. We proved that the categories of the
${\mathbb{N}}_{0}$-graded and, more generally, of the ${\mathbb{Z}}$-graded
algebras with the Manin product ‘$\circ$’ are relatively closed, where the
parametrising subcategory consists of the connected finitely generated
quadratic algebras: ${\mathbf{P}}={\mathbf{FQA}}$. This means that the general
representation theory described here is applicable for quantum representations
introduced in [S21, § 5.1]. We expect that it will be also applied to
generalise quantum representations for other cases such as super-case,
topological case, corepresentations on the connected algebras, a quantum
version of the projective representations.

## References

  * [Bor2] Borceux, F.: Handbook of Categorical Algebra: Volume 2, Categories and Structures, Encyclopedia of Mathematics and its Applications, Cambridge University Press, (1995).
  * [MacLane] MacLane, S.: Categories for the Working Mathematician (second edition), Graduate Texts in Mathematics 5, Springer, (1998).
  * [Man87] Manin, Yu. I.: Some remarks on Koszul algebras and quantum groups, Ann. de l’Inst. Fourier 37, no. 4, (1987), pp. 191–205.
  * [Man88] Manin, Yu. I.: Quantum Groups and Non-Commutative Geometry, University of Montreal, Centre de Recherches Mathématiques, Montreal, QC, (1988), 91 pp.
  * [Man91] Manin, Yu. I.: Topics in Non Commutative Geometry, M. B. Porter Lectures, Princeton University Press, (1991), 164 pp.
  * [Par] Pareigis, B.: A non-commutative non-cocommutative Hopf algebra in “nature”, J. Algebra 70, (1981), 356–374.
  * [S21] Silantyev, A.: Quantum Representation Theory and Manin matrices I: finite-dimensional case, arXiv:2108.00269 [math.QA].
  * [Ulm] Ulmer, F.: Properties of dense and relative adjoint functors, J. Algebra 8, 1, (1968), 77--95.