# The multiplicative K-theory
of compact quantum spaces

Francesco D’Andrea  Università di Napoli “Federico II” and I.N.F.N. Sezione di
Napoli, Complesso MSA, Via Cintia, 80126 Napoli, Italy.
<EMAIL_ADDRESS>and  Tomasz Maszczyk  Instytut Matematyki,
Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we refine the classification of compact quantum spaces by
K-theory type initiated in [8]. We do it by introducing a multiplicative
K-theory functor for unital C*-algebras taking values in the category of
augmented rings, together with a natural transformation to the standard
K-theory functor, taking values in the category of abelian groups. The
multiplicative K-theory type refines the K-theory type of [8], defined in the
framework of cw-Waldhausen categories. In our refinement, we require
compatibility of morphisms with an additional structure which we call
k-topology. This is a noncommutative version of Grothendieck topology with
covering families given by compact quantum principal bundles and bases related
by suitable k-continuous maps. In the classical case of compact Hausdorff
spaces, k-topology and k-continuity reduce to the usual topology and
continuity, respectively, and our noncommutative counterpart of the product in
K-theory reduces to the standard product in topological K-theory. As an
application, we show that non-isomorphic quantizations of the standard CW-
complex structure of a complex projective space belong to the same
multiplicative K-theory type. In addition, for these complex quantum
projective spaces, we prove a noncommutative generalization of the Atiyah–Todd
calculation of the K-theory ring in terms of truncated polynomials.

###### Contents

  1. 1 Introduction
  2. 2 Categorical preliminaries
  3. 3 Cartesian morphisms of families of (quantum) group actions
    1. 3.1 The context of compact Hausdorff spaces
    2. 3.2 From spaces to algebras
    3. 3.3 From algebras to Grothendieck categories
  4. 4 k-Topology
    1. 4.1 The context of compact Hausdorff spaces
    2. 4.2 From spaces to algebras
    3. 4.3 From algebras to Grothendieck categories
  5. 5 Associated vector bundles
  6. 6 From k-topology to multiplicative K-theory
  7. 7 The Atiyah–Todd picture
    1. 7.1 The classical picture revisited
    2. 7.2 The quantum picture

##  1\. Introduction

In this paper we construct a multiplicative refinement of the notion of
K-theory type in [8]. The problem with the state of the art of noncommutative
topology, mostly based on KK-theory, is that when applied to classical spaces
it identifies very different homotopy types (e.g. disconnected and connected
ones). This means that the present noncommutative topology is not a
conservative extension of classical topology, making its name a misnomer. Our
objective here is to complete the existing approach with structures and
methods using the richness of noncommutative geometry, without changing the
content of the classical theory.

The source of the main difficulties with extending classical topology to the
noncommutative setting is lack of the diagonal of a noncommutative space,
which classically is necessary to construct a ring structure on K-theory, a
fundamental tool to obtain many important results. Our new approach to a
multiplicative K-theory is to equip compact quantum spaces with a new
additional structure, that we name k-topology, and on the side of K-theory to
replace rings by augmented rings in the sense of Cartan-Eilenberg.
Multiplicative K-theory will be then a functor from the category of compact
quantum spaces with a k-topology to the category of augmented rings. We stress
that in the classical setting our new approach is equivalent to the standard
one.

Observe that in the classical case sometimes one has to forget the ring
structure of K-theory and to work with plain abelian groups. For example, the
Mayer-Vietoris sequence relates K-groups by maps not always preserving the
ring structure. We extend this natural transformation from K-rings to K-groups
to our generalized framework of augmented K-rings. Untill the present paper,
among the two interrelated parts of the above structure, only the plain
abelian part had been extended to the noncommutative setting (one can see e.g.
[4]).

The idea of k-topology is that, although the balanced tensor product of right
modules over a noncommutative algebra doesn’t make sense, the balanced tensor
product of noncommutative vector bundles associated with a given compact
quantum principal bundle makes perfect sense since they are bimodules in a
natural way [21, Chap. 2.5]. In classical topology, it changes nothing since
every vector bundle is associated to some compact Hausdorff principal bundle
(e.g. its bundle of unitary frames) and the tensor product of the
corresponding modules of sections over the commutative algebra of functions
coincides with the tensor product of the corresponding bimodules since they
are symmetric by construction. In the noncommutative setting, however, only
the second option is viable. Since symmetric bimodules correspond to sheaves
supported on the diagonal, the tensor product of associated noncommutative
vector bundles is a replacement for a non-existent diagonal.

Classically, k-topology becomes a version of Grothendieck topology with
covering families given by compact Hausdorff principal bundles, whose bases
are related by continuous maps. In the noncommutative setting, when one tries
to use compact quantum principal bundles instead, one faces the problem that
the customary use of balanced tensor products to model fibered products, and
hence pullbacks of principal bundles, is not based on a universal property in
the category of associative unital algebras, hence it must be added as a part
of the structure. Then, we define the pullback of compact quantum principal
bundles independently, so to make it satisfy a universal property in the
commutative case. We call k-continuous those morphism along which one can
pullback compact quantum principal bundles, in the above sense. It turns out
that many aspects of the theory of compact quantum principal bundles can be
equivalently rewritten in terms of k-topology and k-continuity.

In more technical terms, our k-topology is a noncommutative version of
coverage in the sense of Johnstone [14], with compact quantum principal
bundles playing the role of covering families and k-continuous morphisms
between their bases playing the role of morphisms. This provides a solution to
the problem of extending the multiplicative structure of K-theory from the
classical to the noncommutative setting. This extension provides a presheaf on
k-topology taking values in the category of augmented rings. We stress the
fact that in the classical case, thanks to the existence of pullbacks of
(compact) principal bundles along continuous maps between (compact) spaces,
k-continuity means simply continuity and k-topology depends only on topology.
The above discussion about K-theory of compact quantum spaces can be
summarized as follows.

k-topology $\leadsto$ multiplicative topological K-theory

The formalism introduced in [8], after formal inversion of K-equivalences,
allows to group different compact quantum spaces in a single K-theory type. In
the present paper, a class of models admitting k-topologies related by
k-continuous K-equivalences defines a multiplicative K-theory type. Our main
example of a multiplicative K-theory type is based on our result that there
exists a k-continuous K-equivalence relating the weak CW-structures of a
multipullback [12, 13] and a Vaksman-Soibelman [26, 15] quantum complex
projective space. It is a multiplicative refinement of the K-equivalence
between these two (K-weak) CW-stuctures [8].

To prove effectiveness of our machinery, we compute the system of
multiplicative K-theories of (K-weak) hyperplane skeleta as the inverse system
of rings of truncated polynomials over the integers. This is analogous to the
Atiyah–Todd presentation of the K-theory ring of complex projective spaces as
rings of truncated polynomials over the integers, and explains the appearence
of the Atiyah–Todd-like identities in [8]. The appearence of these identities
in [8] was our original motivation to develop a theory of multiplicative
K-theory for compact quantum spaces.

Among all CW-complexes, the above-mentioned system of hyperplane embeddings of
complex projective spaces is probably the most important for mathematics and
mathematical physics. Indeed, it bridges between representation theory and
topology by providing a canonical model for the classifying space $BU(1)$,
whose K-theory ring is the adic completion of the representation ring
$R(U(1))$ with respect to the augmentation ideal. In particular, the tower of
the K-theory rings of the skeleta is a surjective inverse system of the
quotient rings of $R(U(1))$. The bridge here is built with the use of the
notion of associated vector bundle. One aim of this paper is to show that this
beautiful picture survives quantization.

In this paper we will study noncommutative spaces from three different point
of views. Motivate by the Gelfand–Naimark equivalence of categories:

{compact Hausdorff spaces} $\simeq$ {commutative unital C*-algebras}op ,

the opposite category of (not necessarily commutative) unital C*-algebras will
be called the category of compact quantum spaces, and denoted
$\mathcal{C}pct\mathcal{Q}uant$.

Motivated by the Zariski equivalence of categories:

{algebraic affine schemes over a field} $\simeq$ {commutative unital
associative algebras}op ,

the opposite category of (not necessarily commutative) unital associative
algebras will be called the category of quantum affine schemes.

The term noncommutative space will be reserved for Grothendieck categories.

For a compact Hausdorff space $X$, the relevant Grothendieck category is the
category $\mathcal{M}_{C(X)}$ of modules over $C(X)$. The importance of this
point of view in topology is justified by the fact that topological K-theory
of $X$ depends only on the subcategory of compact objects (i.e. finitely
generated and projective) in $\mathcal{M}_{C(X)}$, which makes this notion
automatically Morita-invariant.

For a scheme $X$ the relevant Grothendieck category is the category
$\mathcal{QCoh\\!}_{X}$ of quasi-coherent sheaves over $X$. The importance of
this point of view is justified by the fact that $X$ can be reconstructed from
this Grothendieck category (this is the Gabriel–Rosenberg reconstruction
theorem). This approach was very fruitful, leading to theorems by Balmer,
Garkusha, Bondal–Orlov and others about the reconstruction of a scheme $X$
from $\mathcal{QCoh\\!}_{X}$, from one of its subcategories or their
triangulated derived categories. Again, many important invariants like
K-theory, Hochschild and cyclic (co)homology, make sense in this generality
and hence they are automatically Morita-invariant.

For a compact topological group or an affine group scheme $G$, the relevant
Grothendieck category is the category $\mathcal{M}^{\mathcal{O}(G)}$ of
comodules over a commutative Hopf algebra $\mathcal{O}(G)$. For a compact
group $\mathcal{O}(G)$ is the Peter-Weyl subalgebra of $C(G)$, and for a group
scheme is the coordinate ring of $G$. A reconstruction theorem in this
framework is provided by Tannaka–Krein duality, that reconstructs $G$ from the
subcategory of finite-dimensional comodules.

Equivariant topology/algebraic geometry can be transported to the context of
Grothendieck categories as well. Any compact space/affine scheme with an
action of a compact group/affine group scheme leads to a Grothendieck category
of relative Hopf modules.

Finally, the fact that the notion of Grothendieck category is independent of
any commutativity assumption allows us to extend many aspects of equivariant
topology/algebraic geometry to the noncommutative setting.

##  2\. Categorical preliminaries

In this section we introduce some relevant notions from category theory.
Firstly, we define a Cartesian coverage restricting the notion of coverage of
Johnstone [14, Def. 2.1.9], assuming that every covering family consists of a
single morphism, that the coverage of every object contains all isomorphisms,
and that the Johnstone factorizations fit into Cartesian squares (cf. Remark
2.3). This notion make sense for a category with a distinguished class of
squares

$U^{\prime}$$U$$X^{\prime}$$X$$f$$q^{\prime}$$q$$\widetilde{f}$. (2.1)

that we name Cartesian. We assume that Cartesian squares can be composed by
horizontal pasting, and that every pair of morphisms $(f,q)$ with $q$ an
isomorphism fits into a Cartesian square (2.1).

######  Definition 2.1.

A Cartesian coverage on a category with a distinguished class of Cartesian
squares is a function $T$ assigning to every object $X$ a collection $T(X)$ of
morphisms $q:U\to X$ such that

  1. (i)

$T(X)$ contains all isomorphisms.

  2. (ii)

For every morphism $q:U\to X$ in $T(X)$ and every morphism $f:X^{\prime}\to
X$, there exists a morphism $q^{\prime}:U^{\prime}\to X^{\prime}$ in
$T(X^{\prime})$ fitting into a Cartesian square (2.1).

Next, we introduce the notion of 2-category $\mathfrak{G}$ of Grothendieck
categories. By definition, this 2-category $\mathfrak{G}$ consists of

  * •

0-cells $\mathcal{X}$ are Grothendieck categories.

  * •

1-cells $\mathcal{f}:\mathcal{X}^{\prime}\to\mathcal{X}$ are adjunctions
$\mathcal{f}=(f^{*}\dashv f_{*})$, where
$f_{*}:\mathcal{X}^{\prime}\to\mathcal{X}$ is an additive functor.

  * •

2-cells $\mathcal{f}\Longrightarrow\mathcal{g}$ are natural transformations
$f_{*}\Longrightarrow g_{*}$.

Let $\mathfrak{S}$ be a sub-2-category of $\mathfrak{G}$. A weakly commutative
square in $\mathfrak{S}$ is a diagram in $\mathfrak{S}$:

$\mathcal{U}^{\prime}$$\mathcal{U}$$\mathcal{X}^{\prime}$$\mathcal{X}$$\mathcal{f}$$\mathcal{q}^{\prime}$$\mathcal{q}$$\mathcal{\widetilde{f}}$
(2.2)

with 2-cell
$\mathcal{f}\mathcal{q}^{\prime}\Longrightarrow\mathcal{q}\mathcal{\widetilde{f}}$
invertible. The latter means that a natural isomorphism of functors
$f_{*}q^{\prime}_{*}\Longrightarrow q_{*}\widetilde{f}_{*}$ is chosen.

A weakly commutative square in $\mathfrak{S}$ is weakly Cartesian if the Beck-
Chevalley condition is satisfied, i.e. the natural transformation of functors
$q^{*}f_{*}\Longrightarrow\widetilde{f}_{*}q^{\prime*}$ is an isomorphism. In
this context, $\mathcal{\widetilde{f}}$ is called a weakly Cartesian lift of
$\mathcal{f}$.

For the complicated history of the Beck-Chevalley condition, one can see [18,
Pag. 307].

######  Definition 2.2.

A weakly Cartesian coverage on $\mathfrak{S}$ is a function $T$ assigning to
every Grothendieck category $\mathcal{X}$ in $\mathfrak{S}$ a collection
$T(\mathcal{X})$ of adjunctions $\mathcal{q}:\mathcal{U}\to\mathcal{X}$
(called $T$-covering families) such that:

  1. (i)

$T(\mathcal{X})$ contains all adjoint equivalences to $\mathcal{X}$.

  2. (ii)

For every $\mathcal{q}:\mathcal{U}\to\mathcal{X}$ in $T(\mathcal{X})$ and
every adjunction $\mathcal{f}:\mathcal{X}^{\prime}\to\mathcal{X}$, there
exists an adjunction
$\mathcal{q}^{\prime}:\mathcal{U}^{\prime}\to\mathcal{X}^{\prime}$ in
$T(\mathcal{X}^{\prime})$ fitting into a weakly Cartesian square (2.2).

######  Remark 2.3.

Definition 2.1 (resp. 2.2) is inspired by the notion of Grothendieck topology
and the notion of a coverage by Johnstone [14, Def. 2.1.9]. Note, however,
that compared with Johnstone definition, here a covering family is a single
morphism (resp. adjunction), instead of a family of morphisms (resp.
adjunctions), and we assume that Johnstone’s factorization property fits into
a (weakly) Cartesian diagram. Note also that for topological spaces, any open
covering $\\{U_{i}\to X|i\in I\\}$ can be made consist of a single map
$\coprod_{i\in I}U_{i}\to X$.

Associated with the 2-category $\mathfrak{G}$ we have the 2-category of
1-cells $\mathfrak{G}(1)$. This is defined as follows:

  * •

0-cells in $\mathfrak{G}(1)$ are vertical 1-cells in $\mathfrak{G}$.

  * •

1-cells in $\mathfrak{G}(1)$ weakly commutative vertical squares between
vertical 1-cells in $\mathfrak{G}$.

  * •

2-cells in $\mathfrak{G}(1)$ are weakly commutative cubes between parallel
weakly commutative vertical squares in $\mathfrak{G}$.

##  3\. Cartesian morphisms of families of (quantum) group actions

###  3.1. The context of compact Hausdorff spaces

Let us start with a classical motivation. In the following, we tacitly assume
that every space is a compact Hausdorff topological space, every group is a
compact Hausdorff topological group, and all maps are continuous.

By a family of group actions we mean a pair $(G,U\to X)$ of a group $G$ and a
$G$-equivariant map $U\to X$ from a $G$-space $U$ to a space $X$ with trivial
$G$-action. Let us consider the category whose objects are families of group
actions and whose morphisms

$(G^{\prime},U^{\prime}\to X^{\prime})\longrightarrow(G,U\to X)$ (3.1)

are pairs consisting of a morphism $\gamma:G^{\prime}\to G$ of groups and a
commutative diagram

$U^{\prime}$$U$$X^{\prime}$$X$$f$$q^{\prime}$$q$$\widetilde{f}$ (3.2)

where $\widetilde{f}$ is $G^{\prime}$-equivariant. It is not difficult to
check that the map:

$U^{\prime}\times^{G^{\prime}}G\longrightarrow
X^{\prime}\times_{X}U\;,\qquad[(u^{\prime},g)]\mapsto\bigl{(}q^{\prime}(u^{\prime}),\widetilde{f}(u^{\prime})g\bigr{)}\;,$
(3.3)

is well-defined and it is a morphism in the category of right $G$-spaces
equipped with a continuous map to $X^{\prime}$ and a $G$-equivariant
continuous map to $U$. Notice that the diagram (3.2) decomposes now into
commutative triangles as follows

$U^{\prime}$$U$$X^{\prime}\times_{X}U$$X^{\prime}$$X$$U^{\prime}\times^{G^{\prime}}G$$\widetilde{f}$$k$$q^{\prime}$$q$$f$
(3.4)

where $k$ is our map (3.3) and the new maps are given by

$\displaystyle U^{\prime}\to U^{\prime}\times^{G^{\prime}}G,\quad
u^{\prime}\mapsto[(u^{\prime},e)],$ $\displaystyle X^{\prime}\times_{X}U\to
U,\quad(x^{\prime},u)\mapsto u,$ $\displaystyle
U^{\prime}\times^{G^{\prime}}G\to
U,\quad[(u^{\prime},g)]\mapsto\widetilde{f}(u^{\prime})g$ $\displaystyle
X^{\prime}\times_{X}U\to X^{\prime},\quad(x^{\prime},u)\mapsto x^{\prime},$
$\displaystyle U^{\prime}\times^{G^{\prime}}G\to
X^{\prime},\quad[(u^{\prime},g)]\mapsto q^{\prime}(u^{\prime})$ $\displaystyle
X^{\prime}\times_{X}U\to X,\quad(x^{\prime},u)\mapsto f(x^{\prime}).$

######  Definition 3.1.

A morphism of families of group actions (3.1) is called Cartesian if the map
(3.3) is an isomorphism.

######  Remark 3.2.

The decomposition (3.4) generalizes that in the Equivalence Theorem 10.3 of
[24], which concerns fibre bundles.

Several important notions of equivariant topology can be rewritten in terms of
Cartesian morphisms of families of group actions.

######  Example 3.3 (Orbit spaces).

Consider a morphism from an arbitrary family $(G,U\to X)$ of group actions to
the terminal family

$G$$\\{e\\}$$U$$\\{*\\}$$X$$\\{*\\}$$=$$q$

Then, the map (3.3)

$U\times^{G}\\{*\\}\longrightarrow
X\times_{\\{*\\}}\\{*\\}\;,\qquad[(u,e)]\mapsto\bigl{(}q(u),*\bigr{)}$

reads as

$U/G\longrightarrow X\;,\qquad[u]\mapsto q(u)\;,$

which means that it is an isomorphism if and only if $q$ is a quotient map
onto the space of orbits.

######  Example 3.4 (Slices).

Consider a morphism from an arbitrary family $(G,U\to X)$ of group actions to
the family consisting of a single action of $G$ on itself by right
translations

$G$$G$$U$$G$$X$$\\{*\\}$$=$$s$$q$

Then, the map (3.3)

$U\times^{G}G\longrightarrow
X\times_{\\{*\\}}G\;,\qquad[(u,g)]\mapsto\bigl{(}q(u),s(u)g\bigr{)}$

reads as

$U\longrightarrow X\times G\;,\qquad[u]\mapsto\bigl{(}q(u),s(u)\bigr{)}\;,$

which means that it is an isomorphism if and only if $s$ is a slice map, where
the slice is the pre-image $s^{-1}(e)\subset U$, $q$ is a quotient map onto
the space of orbits and admits a section whose the image is the slice.

######  Example 3.5 (Principal bundles).

Having a quotient map $q:U\to X$ defined as in Example 3.3, consider now a
morphism of families of group actions

$G$$G$$U\times G$$U$$U$$X$$q$$q$$\widetilde{q}$$\alpha$$=$

Here $\widetilde{q}$ is the projection onto the first Cartesian factor and
$\alpha(u,g):=ug$ the group action. The right $G$-action on $U\times G$ is on
the second factor. Then, our map (3.3)

$(U\times G)\times^{G}G\longrightarrow
U\times_{X}U\;,\qquad[((u,g_{1}),g_{2})]\mapsto\bigl{(}u,ug_{1}g_{2}\bigr{)}$

reads as the graph of the $G$-action

$U\times G\longrightarrow U\times_{X}U\;,\qquad(u,g)\mapsto(u,ug),$

which is an isomorphism if and only if $q:U\to X$ is a principal $G$-bundle.

Let us note that in this context a slice like in Example 3.4 is tantamount to
a trivialization of the principal bundle.

######  Example 3.6 (Change of the structure group).

Consider now the following morphism of principal bundles over the same base

$G^{\prime}$$G$$U^{\prime}$$U$$X$$X$$=$$q$$q^{\prime}$$\widetilde{f}$

Then, the map (3.3)

$U^{\prime}\times^{G^{\prime}}G\longrightarrow
X\times_{X}U\;,\qquad[(u^{\prime},g)]\mapsto\bigl{(}q^{\prime}(u^{\prime}),\widetilde{f}(u^{\prime})g\bigr{)}$

reads as

$U^{\prime}\times^{G^{\prime}}G\longrightarrow
U\;,\qquad[(u^{\prime},g)]\mapsto\widetilde{f}(u^{\prime})g,$

which is an isomorphism if and only if $\widetilde{f}$ is a change of
structure group. If $G^{\prime}\to G$ is an inclusion of a closed subgroup,
then $U^{\prime}\to X$ is obtained from $U\to X$ by restriction of the
structure group (and $U\to X$ is obtained from $U^{\prime}\to X$ by extension
of the structure group).

######  Example 3.7 (General morphisms of principal bundles).

Every morphism (3.1) between principal bundles is Cartesian. Indeed, notice
that the triangle

$X^{\prime}\times_{X}U$$X^{\prime}$$U^{\prime}\times^{G^{\prime}}G$$k$ (3.5)

in diagram (3.4) is a morphism of principal $G$-bundles over the same base
space $X^{\prime}$, and hence it must be an isomorphism (this is a special
case of the Equivalence Theorem 10.3 in [24], but can be also proved directly
as a simple exercise).

###  3.2. From spaces to algebras

In this section we translate equivariant topology/algebraic geometry into the
language of noncommutative algebras. Let us start with recalling the
Peter–Weyl framework. Compact quantum groups were introduced by Woronowicz in
the 80s [28], and their actions were studied first in [19]. In [11], Ellwood
gave a definition of free, proper and principal compact quantum group actions
on C*-algebras, proving that an action is principal if and only if it is free
and proper, and proving that in the case of classical compact Hausdorff groups
acting on classical locally compact Hausdorff one recovers the standard
definitions. In [2], Baum, De Commer and Hajac construct, given a compact
quantum group $G$ and the associated dense Hopf $*$-subalgebra
$\mathcal{O}(G)\subseteq C(G)$ of Woronowicz [28], their Peter-Weyl functor
from the category of compact quantum $G$-spaces to the category of
$\mathcal{O}(G)$-comodule algebras. They prove that the Ellwood freeness
condition is equivalent to the Hopf-Galois condition [2, Thm. 0.4]. (This
extends a previous result about classical compact Hausdorff principal bundles
[3].) In particular, in terms of the Peter-Weyl functor, the notion of compact
quantum principal bundles leads to Hopf-Galois extensions. This is the setting
we will use in the next section in our study of K-theory. Here, we start with
a more general algebraic framework, cf. Def. 3.8, over an arbitrary ground
field $\Bbbk$.

In the classical case, however, the theory of principal bundles depends
heavily on the existence of pullbacks of spaces. These dualize to pushouts in
the category of commutative algebras, that means balanced tensor products.
When we pass to noncommutative algebras, we face two problems: first, the
balanced tensor product of two associative algebras is in general not an
algebra; second, pushouts in such a category, i.e. amalgamated free products,
can be noncommutative even if factors are commutative and, as such, cannot
correspond to pullbacks of classical spaces. For similar reasons, Hopf
algebras and their coactions are not categorical dual to groups and their
actions. Therefore, one needs a different categorical framework to generalize
the classical equivariant topology to the context of noncommutative geometry,
which is what we provide below.

######  Definition 3.8.

A Hopf-algebra-invariant algebra map is a datum $(H,A\leftarrow B)$, where $H$
is a Hopf-algebra, $A$ and $B$ are right $H$-comodule algebras, with trivial
coaction on $B$, and $B\to A$ is an $H$-comodule algebra map. A morphism
$(H^{\prime},A^{\prime}\leftarrow B^{\prime})\leftarrow(H,A\leftarrow B)$ of
Hopf-algebra-invariant algebra maps is the datum of a Hopf algebra map and a
commutative square:

$H^{\prime}$$H$$A^{\prime}$$A$$B^{\prime}$$B$$\chi$$\beta$$\varphi$$\alpha$$\varphi^{\prime}$
(3.6)

where $\alpha$ is an $H^{\prime}$-comodule algebra map, vertical arrows are
the inclusion maps of subalgebras of invariants of the coactions and $\beta$
is the $*$-homomorphism induced by $\alpha$.

Note that, thinking dually of algebras as spaces, we can think of a Hopf-
algebra-invariant algebra map $(H,A\leftarrow B)$ as a family of (quantum)
group actions with $B$ being the algebra corresponding to the parameter space.
When $B\to A$ is injective, we will call $(H,A\leftarrow B)$ an $H$-extension,
in accordance with classical Galois theory of field extensions, where a datum
$(G,L\leftarrow K)$ of a group $G$ acting by automorphisms on a field $L$ with
a $G$-fixed subfield $K$ is called a $G$-extension.

In this framework, we can replace (3.3) by a morphism in the category
${}_{B^{\prime}}\mathcal{M}^{H}_{A}$ of representations of $B^{\prime}$ on
right relative $(A,H)$ Hopf-modules, given for a diagram (3.6) by

$\kappa:B^{\prime}\otimes_{B}A\rightarrow A^{\prime}\Box^{H^{\prime}}H,\ \ \
b^{\prime}\otimes_{B}a\mapsto b^{\prime}\alpha(a_{(0)})\otimes a_{(1)},$ (3.7)

which we call the generalized canonical map.

######  Definition 3.9.

A morphism of Hopf-algebra-invariant algebra maps (3.6) is called Cartesian if
the map (3.7) is an isomorphism.

Several important notions of Hopf-Galois theory can be rewritten in terms of
Cartesian morphisms of Hopf-algebra-invariant algebra maps, as shown in the
following examples.

######  Example 3.10 (Invariants of a coaction).

Consider a morphism from the trivial $\Bbbk$-extension
$(\Bbbk,\Bbbk\leftarrow\Bbbk)$ to an arbitrary $H$-extension $(H,A\leftarrow
B)$:

$H$$\Bbbk$$A$$\Bbbk$$B$$\Bbbk$$\beta$$=$$\alpha$

Then, bijectivity of our generalized canonical map

$B=B\otimes_{\Bbbk}\Bbbk\longrightarrow A\Box^{H}\Bbbk=A^{\mathrm{co}\,H},$

means that $B\to A$ is an embedding onto $A^{\mathrm{co}\,H}\subset A$.

######  Example 3.11 (The normal basis property).

Consider a morphism from the Hopf-Galois extension $(H,H\leftarrow\Bbbk)$ to
an arbitrary Hopf-algebra-invariant algebra map $(H,A\leftarrow B)$:

$H$$H$$A$$H$$B$$\Bbbk$$=$$\alpha$

Then, the generalized canonical map

$B\otimes H\longrightarrow A\,\Box^{H}H\;,\qquad b\otimes h\mapsto
b\alpha(h_{(1)})\otimes h_{(2)}$

reads as

$B\otimes H\longrightarrow A,\qquad\quad b\otimes h\mapsto b\alpha(h).$

When $B=A^{\mathrm{co}H}\subseteq A$ is an $H$-extension, bijectivity of the
generalized canonical map is equivalent to the (right) normal basis property,
cf. [16, Def. 8.2.1].

######  Example 3.12 (Hopf-Galois condition).

Having the subalgebra $B\subset A$ of invariants for the $H$-coaction defined
as in Example 3.10, consider now a morphism of $H$-extensions:

$H$$H$$A\otimes H$$A$$A$$B$$A\otimes 1_{H}$$\alpha$$=$

Here, $\alpha:A\to A\otimes H$ is the coaction, and the coaction of $H$ on
$A\otimes H$ comes only from the second factor. Then, our generalized
canonical map

$A\otimes_{B}A\longrightarrow(A\otimes H)\Box^{H}H,$

becomes the usual canonical map of Hopf-Galois theory after composing it with
the inverse of the isomorphism $A\otimes H\xrightarrow{\,\cong\,}(A\otimes
H)\Box^{H}H$, $a\otimes h\mapsto a\otimes h_{(1)}\otimes h_{(2)}$, given by
the application of the counit to the right leg of the cotensor product.
Bijectivity of our generalized canonical map then reduces to the Hopf-Galois
condition for the $H$-extension $B\subset A$.

######  Example 3.13 (Change of the structure Hopf algebra).

Consider now the following morphism between Hopf-Galois extensions

$H^{\prime}$$H$$A^{\prime}$$A$$B$$B$$=$$\alpha$

Then, bijectivity of our generalized canonical map
$A=B\otimes_{B}A\longrightarrow A^{\prime}\Box^{H^{\prime}}H$ is equivalent to
a change of the structure Hopf algebra of a Hopf-Galois extention $B\subset
A$. If $H\to H^{\prime}$ is surjective, then $B\to A^{\prime}$ is obtained
from $B\to A$ by restriction of the structure quantum group (and $B\to A$ from
$B\to A^{\prime}$ by extension of the structure quantum group).

######  Example 3.14 (General morphism of Hopf-Galois extensions).

Contrary to the case of principal bundles, given a morphism of noncommutative
Hopf-Galois extensions over the same base and with the same structure Hopf
algebra, there is no reason to believe that it should be an isomorphism. The
main obstacle to mimicking the classical proof is the lack of points and the
lack of the universal property of the balanced tensor product of
noncommutative algebras. To partially overcome this difficulty, instead of
general morphisms of principal bundles we may consider Cartesian morphisms of
Hopf-Galois extensions, where the Cartesian property is a replacement of the
fact that in the classical case every morphism of principal $G$-bundles must
be a pullback. However, for extending to the noncommutative setting the fact
that a morphism of principal bundles over the same base must be necessarily an
isomorphism (cf. Example 3.7), this algebraic language is still insufficient.
A proper solution to this problem will be given in categorical language in
Example 3.34.

######  Remark 3.15.

For a Hopf-Galois extension the normal basis property, cf. Example 3.11, is
tantamount to cleftness [16, Thm. 8.2.4]. In the context of affine group
scheme actions, principal bundles are equivalent to faithfully flat Hopf-
Galois extension [17, Def. 0.10].

###  3.3. From algebras to Grothendieck categories

To overcome the lack of functorial pullback of quantum spaces in the approach
based on a balanced tensor product of algebras, we introduce a categorical
framework allowing us to speak about Cartesian morphisms between Hopf-algebra-
invariant algebra maps. In the previous section, we passed from diagrams (3.1)
of topological spaces to diagrams (3.6) of algebras. The aim of this section
is to think of noncommutative spaces as Grothendieck categories, and pass from
diagrams (3.6) of algebras to diagrams (2.2) of Grothendieck categories which
resemble the diagrams (3.1) of spaces.

One of the benefits of such a reformulation is that if will produce an algebra
structure on the domain of the generalized canonical map, in such a way that
it becomes a morphism of algebras. It should be stressed that in defining it
we do not rely on the algebra structure of the codomain, but we rather rely on
category theory only, what makes this result quite remarkable. To this end, we
will need some preliminary results about this passage from Algebra to Category
Theory.

Our Grothendieck categories will be the categories of relative Hopf modules
[9]. Let us recall that, given a Hopf algebra $H$ and a right $H$-comodule
algebra $A$, a relative $(A,H)$-Hopf module is a right $A$-module right
$H$-comodule $M$ satisfying the condition

$(ma)_{(0)}\otimes(ma)_{(1)}=m_{(0)}a_{(0)}\otimes m_{(1)}a_{(1)},$

for all $m\in M$ and $a\in A$ (we adopt the usual Sweedler notation for
coactions). We denote by $\mathcal{M}^{H}_{A}$ the category of all relative
$(A,H)$-Hopf modules, with morphisms given by $A$-module-$H$-comodule maps.

The next lemma explains how to produce new relative Hopf modules from old ones
by changing the Hopf algebra $H$ and the $H$-comodule algebra $A$ in a
compatible way.

######  Lemma 3.16.

If $M\in\mathcal{M}^{H}_{A}$, we let $M\otimes_{A}A^{\prime}$ be the right
$A^{\prime}$-module and left $H^{\prime}$-comodule with $A^{\prime}$-action by
right multiplication on the second factor and $H^{\prime}$-coaction given by:

$(m\otimes_{A}a^{\prime})_{(0)}\otimes(m\otimes_{A}a^{\prime})_{(1)}:=(m_{(0)}\otimes_{A}a^{\prime}_{(0)})\otimes\chi(m_{(1)})a^{\prime}_{(1)}.$
(3.8)

For $M^{\prime}\in\mathcal{M}^{H^{\prime}}_{A^{\prime}}$, we let
$M^{\prime}\Box^{H^{\prime}}H$ be the right $A$-module and left $H$-comodule
with $H$-coaction on second factor of the cotensor product and right
$A$-action given by:

$\Big{(}\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}\Big{)}a:=\sum\nolimits_{i}m^{\prime}_{i}\alpha(a_{(0)})\otimes
h_{i}a_{(1)}.$ (3.9)

Then,

  * (i)

the coaction (3.8) is well-defined;

  * (ii)

the action (3.9) is well-defined (its image is in
$M^{\prime}\Box^{H^{\prime}}H$);

  * (iii)

$M\otimes_{A}A^{\prime}$ is an object of
$\mathcal{M}_{A^{\prime}}^{H^{\prime}}$;

  * (iv)

$M^{\prime}\Box^{H^{\prime}}H$ is an object of $\mathcal{M}_{A}^{H}$.

###### Proof.

(i) We have to verify that the relations defining the balanced tensor product
are mapped to zero by (3.8). This is a simple check:

$\displaystyle(m\otimes_{A}\alpha(a)a^{\prime}-ma\otimes_{A}a^{\prime})_{(0)}\otimes(m\otimes_{A}\alpha(a)a^{\prime}-ma\otimes_{A}a^{\prime})_{(1)}$
$\displaystyle=\;$
$\displaystyle(m\otimes_{A}\alpha(a)a^{\prime})_{(0)}\otimes(m\otimes_{A}\alpha(a)a^{\prime})_{(1)}-(ma\otimes_{A}a^{\prime})_{(0)}\otimes(ma\otimes_{A}a^{\prime})_{(1)}$
$\displaystyle=\;$
$\displaystyle\big{(}m_{(0)}\otimes_{A}(\alpha(a)a^{\prime})_{(0)}\big{)}\otimes\chi(m_{(1)})(\alpha(a)a^{\prime})_{(1)}-\big{(}(ma)_{(0)}\otimes_{A}a^{\prime}_{(0)}\big{)}\otimes\chi((ma)_{(1)})a^{\prime}_{(1)}$
$\displaystyle=\;$
$\displaystyle\big{(}m_{(0)}\otimes_{A}\alpha(a)_{(0)}a^{\prime}_{(0)}\big{)}\otimes\chi(m_{(1)})\alpha(a)_{(1)}a^{\prime}_{(1)}-\big{(}m_{(0)}a_{(0)}\otimes_{A}a^{\prime}_{(0)}\big{)}\otimes\chi(m_{(1)})\chi(a_{(1)})a^{\prime}_{(1)}$
$\displaystyle=\;$
$\displaystyle\big{(}m_{(0)}\otimes_{A}\alpha(a_{(0)})a^{\prime}_{(0)}\big{)}\otimes\chi(m_{(1)})\chi(a_{(1)})a^{\prime}_{(1)}-\big{(}m_{(0)}a_{(0)}\otimes_{A}a^{\prime}_{(0)}\big{)}\otimes\chi(m_{(1)})\chi(a_{(1)})a^{\prime}_{(1)}$
$\displaystyle=\;$
$\displaystyle\Big{(}m_{(0)}\otimes_{A}\alpha(a_{(0)})a^{\prime}_{(0)}-m_{(0)}a_{(0)}\otimes_{A}a^{\prime}_{(0)}\Big{)}\otimes\chi(m_{(1)}a_{(1)})a^{\prime}_{(1)}=0.$

(ii) We now check that the right hand side of (3.9) belongs to the cotensor
product $M^{\prime}\Box^{H^{\prime}}H$. Indeed,

$\displaystyle\sum\nolimits_{i}\big{(}m^{\prime}_{i}\alpha(a_{(0)})\big{)}_{(0)}\otimes\big{(}m^{\prime}_{i}\alpha(a_{(0)})\big{)}_{(1)}\otimes
h_{i}a_{(1)}$ $\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i(0)}\alpha(a_{(0)})_{(0)}\otimes
m^{\prime}_{i(1)}\alpha(a_{(0)})_{(1)}\otimes h_{i}a_{(1)}$ $\displaystyle=\;$
$\displaystyle\big{(}\sum\nolimits_{i}m^{\prime}_{i(0)}\otimes
m^{\prime}_{i(1)}\otimes
h_{i}\big{)}\big{(}\alpha(a_{(0)})_{(0)}\otimes\alpha(a_{(0)})_{(1)}\otimes
a_{(1)}\big{)}$ $\displaystyle=\;$
$\displaystyle\big{(}\sum\nolimits_{i}m^{\prime}_{i}\otimes\chi(h_{i(1)})\otimes
h_{i(2)}\big{)}\big{(}\alpha(a_{(0)})\otimes\chi(a_{(1)})\otimes
a_{(2)}\big{)}$ $\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}\alpha(a_{(0)})\otimes\chi(h_{i(1)}a_{(1)})\otimes
h_{i(2)}a_{(2)}$ $\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}\alpha(a_{(0)})\otimes\chi\big{(}(h_{i}a_{(1)})_{(1)}\big{)}\otimes(h_{i}a_{(1)})_{(2)}$

for all $\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i}\in
M^{\prime}\Box^{H^{\prime}}H$ and $a\in A$.

(iii) Next, we prove that $M\otimes_{A}A^{\prime}$ is an
$(A^{\prime},H^{\prime})$-relative Hopf module. Indeed, denoting
$\mu^{\prime}:=m\otimes_{A}a^{\prime}_{1}$, one has

$\displaystyle(\mu^{\prime}a^{\prime})_{(0)}\otimes(\mu^{\prime}a^{\prime})_{(1)}$
$\displaystyle=\;$
$\displaystyle\big{(}m_{(0)}\otimes_{A}(a^{\prime}_{1}a^{\prime})_{(0)}\big{)}\otimes\chi(m_{(1)})(a^{\prime}_{1}a^{\prime})_{(1)}$
$\displaystyle=\;$
$\displaystyle\big{(}m_{(0)}\otimes_{A}a^{\prime}_{1(0)}a^{\prime}_{(0)}\big{)}\otimes\chi(m_{(1)})a^{\prime}_{1(1)}a^{\prime}_{(1)}$
$\displaystyle=\;$
$\displaystyle\big{(}(m_{(0)}\otimes_{A}a^{\prime}_{1(0)})\otimes\chi(m_{(1)})a^{\prime}_{1(1)}\big{)}(1\otimes
a^{\prime}_{(0)}\otimes a^{\prime}_{(1)})$ $\displaystyle=\;$
$\displaystyle\mu^{\prime}_{(0)}a^{\prime}_{(0)}\otimes\mu^{\prime}_{(1)}a^{\prime}_{(1)}.$

(iv) Finally, we prove that $M^{\prime}\Box^{H^{\prime}}H$ is an
$(A,H)$-relative Hopf module. Indeed, denoting
$\mu:=\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i}$, then

$\displaystyle(\mu a)_{(0)}\otimes(\mu a)_{(1)}$ $\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}\alpha(a_{(0)})\otimes(h_{i}a_{(1)})_{(1)}\otimes(h_{i}a_{(1)})_{(2)}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}\alpha(a_{(0)})\otimes
h_{i(1)}a_{(1)}\otimes h_{i(2)}a_{(2)}$ $\displaystyle=\;$
$\displaystyle\Big{(}\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i(1)}\otimes
h_{i(2)}\Big{)}(\alpha(a_{(0)})\otimes a_{(1)}\otimes a_{(2)})$
$\displaystyle=\;$ $\displaystyle\mu_{(0)}a_{(0)}\otimes\mu_{(1)}a_{(1)}.$

∎

Now we are going to construct adjunctions between our Grothendieck categories.
First, using the module and comodule structure of the previous lemma we define
a pair of functors:

$\mathcal{M}^{H^{\prime}}_{A^{\prime}}$$\mathcal{M}^{H}_{A}$$\widetilde{f}_{*}$$\widetilde{f}^{*}$
(3.10)

given on objects by

$\widetilde{f}^{*}M:=M\otimes_{A}A^{\prime}\qquad\text{and}\qquad\widetilde{f}_{*}M^{\prime}:=M^{\prime}\;\Box^{H^{\prime}}H,$

defined using the maps $\alpha$ and $\chi$ in (3.6), respectively, and acting
on morphisms in the obvious way.

Next, we prove that $\widetilde{f}^{*}$ and $\widetilde{f}_{*}$ form an
adjoint pair of functors. For this, we need the following preliminary lemma.

######  Lemma 3.17.

Let $M^{\prime}$ be an object of $\mathcal{M}^{H^{\prime}}_{A^{\prime}}$.
Then, there is an isomorphism of right $B$-modules given by

$M^{\prime\textrm{co}H^{\prime}}\to(M^{\prime}\Box^{H^{\prime}}H)^{\textrm{co}H},\qquad
m^{\prime}\mapsto m^{\prime}\otimes 1_{H},$ (3.11)

and with inverse

$M^{\prime\textrm{co}H^{\prime}}\leftarrow(M^{\prime}\Box^{H^{\prime}}H)^{\textrm{co}H},\qquad\sum\nolimits_{i}m^{\prime}_{i}\epsilon(h_{i})\;\reflectbox{$\mapsto$}\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}$ (3.12)

where $\epsilon$ is the counit of $H$.

###### Proof.

First, we check that the two maps are well defined. If $m^{\prime}$ is
invariant under the $H^{\prime}$-coaction, then $m^{\prime}\otimes 1_{H}$
belongs to the cotensor product and is invariant under the $H$ coaction (on
the second factor). Concerning the map (3.12),
$\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i}$ belongs to the cotensor product
if and only if

$\sum\nolimits_{i}m^{\prime}_{i(0)}\otimes m_{i(1)}\otimes
h_{i}=\sum\nolimits_{i}m^{\prime}_{i}\otimes\chi(h_{i(1)})\otimes h_{i(2)}$
(3.13)

and is $H$-invariant if and only if

$\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i(1)}\otimes
h_{i(2)}=\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i}\otimes 1_{H}.$ (3.14)

Applying the counit of $H$ to the third leg of (3.13) and multiplying we get

$\sum\nolimits_{i}m^{\prime}_{i(0)}\otimes
m_{i(1)}\epsilon(h_{i})=\sum\nolimits_{i}m^{\prime}_{i}\otimes\chi(h_{i(1)})\epsilon(h_{i(2)})=\sum\nolimits_{i}m^{\prime}_{i}\otimes\chi(h_{i})$

Applying the counit of $H^{\prime}$, which we also denote by $\epsilon$, to
the second leg of (3.14) and multiplying we get

$\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}=\sum\nolimits_{i}m^{\prime}_{i}\otimes\epsilon(h_{i(1)})h_{i(2)}=\sum\nolimits_{i}m^{\prime}_{i}\epsilon(h_{i})\otimes
1_{H}.$ (3.15)

From the latter two equations we get

$\sum\nolimits_{i}m^{\prime}_{i(0)}\epsilon(h_{i})\otimes
m^{\prime}_{i(1)}=\sum\nolimits_{i}m^{\prime}_{i}\epsilon(h_{i})\otimes
1_{H^{\prime}},$

which means that (3.12) is well defined (its image is in the set of invariants
of the $H^{\prime}$-coaction).

Composing (3.11) with (3.12) we get the identity

$m^{\prime}\mapsto m^{\prime}\otimes 1_{H}\mapsto
m^{\prime}\epsilon(1_{H})=m^{\prime}.$

Composing the two maps in the opposite order we get

$\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}\mapsto\sum\nolimits_{i}m^{\prime}_{i}\epsilon(h_{i})\mapsto\sum\nolimits_{i}m^{\prime}_{i}\epsilon(h_{i})\otimes
1_{H},$

which is the identity thanks to (3.15). ∎

######  Lemma 3.18.

$\widetilde{f}^{*}$ is left adjoint to $\widetilde{f}_{*}$.

###### Proof.

First, we construct a unit $\eta:1\to\widetilde{f}_{*}\widetilde{f}^{*}$.
Given an object $M$ of $\mathcal{M}_{A}^{H}$, the $M$-component of the unit is
defined as:

$\eta_{M}:M\to\widetilde{f}_{*}\widetilde{f}^{*}M=(M\otimes_{A}A^{\prime})\Box^{H^{\prime}}H,\qquad
m\mapsto(m_{(0)}\otimes_{A}1)\otimes m_{(1)}.$

Next, we prove that $\eta_{M}$ is well-defined, i.e. that each image is
contained in the appropriate cotensor product as above. One has

$\displaystyle(m_{(0)}\otimes_{A}1)_{(0)}\otimes(m_{(0)}\otimes_{A}1)_{(1)}\otimes
m_{(1)}$ $\displaystyle=\;$
$\displaystyle\big{(}m_{(0)}\otimes_{A}1\big{)}\otimes\chi(m_{(1)})\otimes
m_{(2)}$ $\displaystyle=\;$
$\displaystyle\big{(}m_{(0)}\otimes_{A}1\big{)}\otimes\chi(m_{(1)(1)})\otimes
m_{(1)(2)}.$

Next, we check that $\eta_{M}$ is an $A$-module map. For all $m\in M$ and
$a\in A$, one has

$\displaystyle\eta_{M}(ma)=\;$
$\displaystyle((ma)_{(0)}\otimes_{A}1)\otimes(ma)_{(1)}$
$\displaystyle=(m_{(0)}a_{(0)}\otimes_{A}1)\otimes m_{(1)}a_{(1)}$
$\displaystyle=\big{(}m_{(0)}\otimes_{A}\alpha(a_{(0)})\big{)}\otimes
m_{(1)}a_{(1)}$ $\displaystyle=\big{(}(m_{(0)}\otimes_{A}1)\otimes
m_{(1)}\big{)}\cdot\big{(}(1\otimes_{A}\alpha(a_{(0)}))\otimes a_{(1)}\big{)}$
$\displaystyle=\eta_{M}(m)a.$

Then, we prove that $\eta_{M}$ is an $H$-comodule map. For $m\in M$, one has

$\eta_{M}(m)_{(0)}\otimes\eta_{M}(m)_{(1)}=(m_{(0)}\otimes_{A}1)\otimes
m_{(1)}\otimes m_{(2)}=\eta_{M}(m_{(0)})\otimes m_{(1)}.$

Next, we construct a counit $\epsilon:\widetilde{f}^{*}\widetilde{f}_{*}\to
1$. Given $M^{\prime}\in\mathcal{M}_{A^{\prime}}^{H^{\prime}}$, its
$M^{\prime}$-component is defined as:

$\epsilon_{M^{\prime}}:(M^{\prime}\Box^{H^{\prime}}H)\otimes_{A}A^{\prime}=\widetilde{f}^{*}\widetilde{f}_{*}M^{\prime}\to
M^{\prime},\qquad\Big{(}\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}\Big{)}\otimes_{A}a^{\prime}\mapsto\sum\nolimits_{i}m^{\prime}_{i}\,\epsilon(h_{i})a^{\prime},$

where on the right hand side we have the counit of $H$. First, we prove that
the map is well-defined with domain the balanced tensor product. By the
definition of the action of $A$ on the cotensor product, we have:

$\displaystyle\Big{(}\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}\Big{)}\otimes_{A}\alpha(a)a^{\prime}-\Big{(}\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}\Big{)}a\otimes_{A}a^{\prime}$ $\displaystyle=\;$
$\displaystyle\Big{(}\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}\Big{)}\otimes_{A}\alpha(a)a^{\prime}-\Big{(}\sum\nolimits_{i}m^{\prime}_{i}\alpha(a_{(0)})\otimes
h_{i}a_{(1)}\Big{)}\otimes_{A}a^{\prime}$ $\displaystyle\mapsto\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}\,\epsilon(h_{i})\alpha(a)a^{\prime}-\sum\nolimits_{i}m^{\prime}_{i}\alpha(a_{(0)})\epsilon(h_{i}a_{(1)})a^{\prime}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}\Big{(}m^{\prime}_{i}\,\epsilon(h_{i})\alpha(a)a^{\prime}-m^{\prime}_{i}\alpha(a_{(0)})\epsilon(h_{i})\epsilon(a_{(1)})a^{\prime}\Big{)}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}\Big{(}m^{\prime}_{i}\,\epsilon(h_{i})\alpha(a)a^{\prime}-m^{\prime}_{i}\alpha\big{(}a_{(0)}\epsilon(a_{(1)})\big{)}\epsilon(h_{i})a^{\prime}\Big{)}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}\Big{(}m^{\prime}_{i}\,\epsilon(h_{i})\alpha(a)a^{\prime}-m^{\prime}_{i}\alpha(a)\epsilon(h_{i})a^{\prime}\Big{)}=0.$

Now, $\epsilon_{M^{\prime}}$ is clearly an $A^{\prime}$-module map. Finally,
we prove that $\epsilon_{M^{\prime}}$ is an $H^{\prime}$-comodule map.
Denoting as above $\mu:=\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i}$, one has

$(\mu\otimes a^{\prime})_{(0)}\otimes(\mu\otimes
a^{\prime})_{(1)}=\sum\nolimits_{i}\big{(}(m^{\prime}_{i}\otimes
h_{i(1)})\otimes_{A}a^{\prime}_{(0)}\big{)}\otimes\chi(h_{i(2)})a^{\prime}_{(1)}.$

Also, by definition of cotensor product

$\sum\nolimits_{i}m^{\prime}_{i(0)}\otimes m^{\prime}_{i(1)}\otimes
h_{i}=\sum\nolimits_{i}m^{\prime}_{i}\otimes\chi(h_{i(1)})\otimes h_{i(2)}$

which implies:

$\sum\nolimits_{i}m^{\prime}_{i(0)}\otimes
m^{\prime}_{i(1)}\epsilon(h_{i})=\sum\nolimits_{i}m^{\prime}_{i}\otimes\chi\big{(}h_{i(1)}\epsilon(h_{i(2)})\big{)}=\sum\nolimits_{i}m^{\prime}_{i}\otimes\chi\big{(}h_{i}).$

Therefore

$\displaystyle\epsilon_{M^{\prime}}\big{(}(\mu\otimes
a^{\prime})_{(0)}\big{)}\otimes(\mu\otimes a^{\prime})_{(1)}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}\,\epsilon(h_{i(1)})a^{\prime}_{(0)}\otimes\chi(h_{i(2)})a^{\prime}_{(1)}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}a^{\prime}_{(0)}\otimes\chi\big{(}\epsilon(h_{i(1)})h_{i(2)}\big{)}a^{\prime}_{(1)}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i}a^{\prime}_{(0)}\otimes\chi(h_{i})a^{\prime}_{(1)}$
$\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}m^{\prime}_{i(0)}a^{\prime}_{(0)}\otimes
m^{\prime}_{i(1)}\epsilon(h_{i})a^{\prime}_{(1)}$ $\displaystyle=\;$
$\displaystyle\sum\nolimits_{i}(m^{\prime}_{i}a^{\prime})_{(0)}\otimes(m^{\prime}_{i}a^{\prime})_{(1)}\epsilon(h_{i})$
$\displaystyle=\;$
$\displaystyle\Big{(}\sum\nolimits_{i}m^{\prime}_{i}a^{\prime}\,\epsilon(h_{i})\Big{)}_{(0)}\otimes\Big{(}\sum\nolimits_{i}m^{\prime}_{i}a^{\prime}\,\epsilon(h_{i})\Big{)}_{(1)}$
$\displaystyle=\;$ $\displaystyle\epsilon_{M^{\prime}}(\mu\otimes
a^{\prime})_{(0)}\otimes\epsilon_{M^{\prime}}(\mu\otimes a^{\prime})_{(1)}$

In order to prove the adjunction we must check that for every objects $M$ of
$\mathcal{M}_{A}^{H}$ and $M^{\prime}$ of
$\mathcal{M}_{A^{\prime}}^{H^{\prime}}$ one has

$\epsilon_{\widetilde{f}^{*}M}\circ\widetilde{f}^{*}(\eta_{M})=1_{\widetilde{f}^{*}M}\qquad\text{and}\qquad\widetilde{f}_{*}(\epsilon_{M^{\prime}})\circ\eta_{\widetilde{f}_{*}M^{\prime}}=1_{\widetilde{f}_{*}M^{\prime}}.$

that more explicitly means

$\epsilon_{M\otimes_{A}A^{\prime}}\circ\widetilde{f}^{*}(\eta_{M})=1_{M\otimes_{A}A^{\prime}}\qquad\text{and}\qquad\widetilde{f}_{*}(\epsilon_{M^{\prime}})\circ\eta_{M^{\prime}\Box^{H^{\prime}}H}=1_{M^{\prime}\Box^{H^{\prime}}H}.$

Using the right $A^{\prime}$-module structure of $M\otimes_{A}A^{\prime}$, we
get

$\displaystyle\widetilde{f}^{*}(\eta_{M}):m\otimes_{A}a^{\prime}$
$\displaystyle\mapsto\big{(}(m_{(0)}\otimes_{A}1)\otimes
m_{(1)}\big{)}\otimes_{A}a^{\prime},$
$\displaystyle\epsilon_{M\otimes_{A}A^{\prime}}\circ\widetilde{f}^{*}(\eta_{M}):m\otimes_{A}a^{\prime}$
$\displaystyle\mapsto(m_{(0)}\otimes_{A}1)\epsilon(m_{(1)})a^{\prime}=m_{(0)}\epsilon(m_{(1)})\otimes_{A}a^{\prime}=m\otimes_{A}a^{\prime}.$

Denoting $\mu:=\sum\nolimits_{i}m^{\prime}_{i}\otimes h_{i}$, we get

$\displaystyle\eta_{M^{\prime}\Box^{H^{\prime}}H}:\mu$
$\displaystyle\mapsto(\mu_{(0)}\otimes_{A}1)\otimes\mu_{(1)}=\sum\nolimits_{i}((m^{\prime}_{i}\otimes
h_{i(1)})\otimes_{A}1)\otimes h_{i(2)},$
$\displaystyle\widetilde{f}_{*}(\epsilon_{M^{\prime}})\circ\eta_{M^{\prime}\Box^{H^{\prime}}H}:\mu$
$\displaystyle\mapsto\sum\nolimits_{i}m^{\prime}_{i}\epsilon(h_{i(1)})\otimes
h_{i(2)}=\sum\nolimits_{i}m^{\prime}_{i}\otimes\epsilon(h_{i(1)})h_{i(2)}=\sum\nolimits_{i}m^{\prime}_{i}\otimes
h_{i}=\mu.$

∎

######  Definition 3.19.

In the adjunction (3.10), replacing the Hopf algebra $H$ by the trivial Hopf
algebra $\Bbbk$, the Hopf algebra map $\chi:H\to H^{\prime}$ by the unit map
$\eta:\Bbbk\to H^{\prime}$, the $H$-module algebra $A$ by the algebra
$B^{\prime}$ with the trivial coaction of $\Bbbk$, and the $H^{\prime}$-module
algebra map $\alpha:A\to A^{\prime}$ by the $H^{\prime}$-invariant map
$\varphi^{\prime}:B^{\prime}\to A^{\prime}$, and using Lemma 3.17, we obtain
an adjunction $\mathcal{q}^{\prime}=(q^{\prime*}\dashv q^{\prime}_{*})$

$\mathcal{M}^{H^{\prime}}_{A^{\prime}}\xrightarrow{\quad\mathcal{q}^{\prime}\quad}\mathcal{M}_{B^{\prime}}$
(3.16)

In the adjunction (3.16), removing all the primes, we obtain an adjunction
$\mathcal{q}=(q^{*}\dashv q_{*})$

$\mathcal{M}^{H}_{A}\xrightarrow{\quad\mathcal{q}\quad}\mathcal{M}_{B}$ (3.17)

In the adjunction (3.10), replacing both the Hopf algebra $H$ and $H^{\prime}$
by the trivial Hopf algebra $\Bbbk$, the Hopf algebra map $\chi:H\to
H^{\prime}$ by the identity, the $H$-module algebra $A$ by the algebra $B$
with the trivial coaction of $\Bbbk$, the $H^{\prime}$-module algebra
$A^{\prime}$ by the algebra $B^{\prime}$ with the trivial coaction of $\Bbbk$,
and the $H^{\prime}$-module algebra map $\alpha:A\to A^{\prime}$ by the map
$\beta:B\to B^{\prime}$, and using the fact that $(-)\Box^{\Bbbk}\Bbbk$ is the
identity functor, we obtain an adjunction $\mathcal{f}=(f^{*}\dashv f_{*})$

$\mathcal{M}_{B^{\prime}}\xrightarrow{\quad\mathcal{f}\quad}\mathcal{M}_{B}$
(3.18)

As a result we obtain the following form of these four adjunctions (3.10),
(3.16), (3.17) and (3.18)

$\begin{array}[]{|c|c|c|}\cline{1-1}\cr\cline{3-3}\cr\mathcal{q}^{\prime}=(q^{\prime*}\dashv
q^{\prime}_{*})&&\mathcal{\widetilde{f}}=(\widetilde{f}^{*}\dashv\widetilde{f}_{*})\\\
\cline{1-1}\cr\cline{3-3}\cr
q^{\prime*}N^{\prime}:=N^{\prime}\otimes_{B^{\prime}}A^{\prime}&&\widetilde{f}^{*}M:=M\otimes_{A}A^{\prime}\\\
q^{\prime}_{*}M^{\prime}:=M^{\prime\textrm{co}H^{\prime}}&&\widetilde{f}_{*}M^{\prime}:=M^{\prime}\;\Box^{H^{\prime}}H\\\
\cline{1-1}\cr\cline{3-3}\cr\vskip
10.0pt\cr\cline{1-1}\cr\cline{3-3}\cr\mathcal{f}=(f^{*}\dashv
f_{*})&&\mathcal{q}=(q^{*}\dashv q_{*})\\\ \cline{1-1}\cr\cline{3-3}\cr
f^{*}N:=N\otimes_{B}B^{\prime}&&q^{*}N:=N\otimes_{B}A\\\
f_{*}N^{\prime}:=N^{\prime}&&q_{*}M:=M^{\textrm{co}H}\\\
\cline{1-1}\cr\cline{3-3}\cr\end{array}$

######  Remark 3.20.

The adjunction $f^{*}\dashv f_{*}$ is well known. The adjunctions $q^{*}\dashv
q_{*}$ and $q^{\prime*}\dashv q^{\prime}_{*}$ are, e.g., in [23] (see also
[6]).

######  Theorem 3.21.

Let us associate with a morphism (3.6) of Hopf-algebra-invariant algebra maps
the diagram

$\mathcal{M}^{H^{\prime}}_{A^{\prime}}$$\mathcal{M}^{H}_{A}$$\mathcal{M}_{B^{\prime}}$$\mathcal{M}_{B}$$\mathcal{f}$$\mathcal{q}$$\mathcal{q}^{\prime}$$\widetilde{\mathcal{f}}$
(3.19)

Then, (3.19) is a weakly commutative square and gives a functor from the
opposite category of Hopf-algebra-invariant algebra maps into the 2-category
$\mathfrak{G}(1)$ of Sect. 2.

###### Proof.

All four arrows in (3.19) depend naturally on the arrows in (3.6). Let
$M^{\prime}$ be an object of $\mathcal{M}^{H^{\prime}}_{A^{\prime}}$. The
$M^{\prime}$-component of the natural transformation

$f_{*}q^{\prime}_{*}\Longrightarrow q_{*}\widetilde{f}_{*}$ (3.20)

is given by the right $B$-module map (3.11). Since by Lemma 3.17 this is an
isomorphism, the natural transformation is an isomorphism, and hence (3.19) is
weakly commutative. ∎

Even if the functor in Theorem 3.21 is not a full embedding, it has the nice
property that a morphism of Hopf-algebra-invariant algebra maps is Cartesian
if and only if the corresponding weakly commutative square is weakly
Cartesian. This is the content of the next theorem.

######  Theorem 3.22.

Consider a morphism (3.6) of Hopf-algebra-invariant algebra maps, and assume
that $\chi:H\to H^{\prime}$ is left coflat. Then, bijectivity of the
generalized canonical map (3.7) is equivalent to the Beck-Chevalley condition
for the diagram (3.19), stating that its mate diagram

$\mathcal{M}_{A^{\prime}}^{H^{\prime}}$$\mathcal{M}_{A}^{H}$$\mathcal{M}_{B^{\prime}}$$\mathcal{M}_{B}$$f_{*}$$q^{*}$$q^{\prime*}$$\widetilde{f}_{*}$
(3.21)

commutes up to an invertible natural transformation
$q^{*}f_{*}\Rightarrow\widetilde{f}_{*}q^{\prime*}$.

###### Proof.

Let us recall that the natural transformation for the diagram (3.21) is a mate
for commutativity of (3.19):

$f_{*}q^{\prime}_{*}\xRightarrow{\;\cong\;}q_{*}\widetilde{f}_{*}$ (3.22)

and decomposes as

$q^{*}f_{*}\xRightarrow{\;q^{*}f_{*}\eta^{q^{\prime}}\;}q^{*}f_{*}q^{\prime}_{*}q^{\prime*}\xRightarrow{\;\cong\;}q^{*}q_{*}\widetilde{f}_{*}q^{\prime*}\xRightarrow{\;\epsilon^{q}\widetilde{f}_{*}q^{\prime*}\;}\widetilde{f}_{*}q^{\prime*}.$
(3.23)

where $\eta^{q}$ and $\epsilon^{q}$ are the unit and the counit of the
adjunction $q^{*}\dashv q_{*}$, respectively, and $\eta^{q^{\prime}}$ and
$\epsilon^{q^{\prime}}$ are the unit and the counit of the adjunction
$q^{\prime*}\dashv q^{\prime}_{*}$. These are given by the following formulas.
For $N\in\mathcal{M}_{B}$ and $M\in\mathcal{M}^{H}_{A}$, their $N$\- and
$M$-components read as

$\displaystyle\eta^{q}_{N}$
$\displaystyle:N\to(N\otimes_{B}A)^{\textrm{co}H},$ $\displaystyle n\mapsto
n\otimes_{B}1,$ $\displaystyle\epsilon^{q}_{M}$
$\displaystyle:M^{\textrm{co}H}\otimes_{B}A\to M,$ $\displaystyle
m\otimes_{B}a\mapsto ma.$

One has similar formulas for the adjunction $q^{\prime*}\dashv
q^{\prime}_{*}$.

Let $N^{\prime}$ be an object of $\mathcal{M}_{B^{\prime}}$. We now compute
the $N^{\prime}$-component of the above natural transformation (3.23). One has

$\displaystyle
q^{*}f_{*}(N^{\prime})=N^{\prime}\otimes_{B}A\xrightarrow{\;q^{*}f_{*}\eta^{q^{\prime}}_{N^{\prime}}}q^{*}f_{*}q^{\prime}_{*}q^{\prime*}(N^{\prime})=(N^{\prime}\otimes_{B^{\prime}}A^{\prime})^{\textrm{co}H^{\prime}}\otimes_{B}A$
$\displaystyle\xrightarrow{\;\cong\;}q^{*}q_{*}\widetilde{f}_{*}q^{\prime*}(N^{\prime})=\big{(}(N^{\prime}\otimes_{B^{\prime}}A^{\prime})\Box^{H^{\prime}}H\big{)}^{\textrm{co}H}\otimes_{B}A$
$\displaystyle\xrightarrow{\;\epsilon_{\widetilde{f}_{*}q^{\prime*}(N^{\prime})}^{q}\;}(N^{\prime}\otimes_{B^{\prime}}A^{\prime})\Box^{H^{\prime}}H=\widetilde{f}_{*}q^{\prime*}(N^{\prime}).$
(3.24)
$n^{\prime}\otimes_{B}a\mapsto(n^{\prime}\otimes_{B^{\prime}}1_{A^{\prime}})\otimes_{B}a\mapsto\big{(}(n^{\prime}\otimes_{B^{\prime}}1_{A^{\prime}})\otimes
1_{H}\big{)}\otimes_{B}a\mapsto\big{(}n^{\prime}\otimes_{B^{\prime}}\alpha(a_{(0)})\big{)}\otimes
a_{(1)}.$

We now use two isomorphisms. One is

$N^{\prime}\otimes_{B^{\prime}}(B^{\prime}\otimes_{B}A)\to
N^{\prime}\otimes_{B}A,\qquad
n^{\prime}\otimes_{B^{\prime}}(b^{\prime}\otimes_{B}a)\mapsto
n^{\prime}b^{\prime}\otimes_{B}a.$ (3.25)

with the obvious inverse

$N^{\prime}\otimes_{B}A\to
N^{\prime}\otimes_{B^{\prime}}(B^{\prime}\otimes_{B}A),\qquad
n^{\prime}\otimes_{B}a\mapsto n^{\prime}\otimes_{B^{\prime}}(1\otimes_{B}a).$

For the second one, let’s start from the defining sequence of a balanced
tensor product

$N^{\prime}\otimes B^{\prime}\otimes A^{\prime}$$N^{\prime}\otimes
A^{\prime}$$N^{\prime}\otimes_{B^{\prime}}A^{\prime}$

Since $\chi$ is left coflat, $(\,\text{-}\,)\;\Box^{H^{\prime}}H$ is right
exact and we get an isomorphism of coequalizers

$(N^{\prime}\otimes B^{\prime}\otimes
A^{\prime})\Box^{H^{\prime}}H$$(N^{\prime}\otimes
A^{\prime})\Box^{H^{\prime}}H$$(N^{\prime}\otimes_{B^{\prime}}A^{\prime})\Box^{H^{\prime}}H$$N^{\prime}\otimes
B^{\prime}\otimes(A^{\prime}\Box^{H^{\prime}}H)$$N^{\prime}\otimes(A^{\prime}\Box^{H^{\prime}}H)$$N^{\prime}\otimes_{B^{\prime}}(A^{\prime}\Box^{H^{\prime}}H)$$\cong$$\cong$
(3.26)

which makes sense since the right $H^{\prime}$-coaction on $A^{\prime}$ is
left $B^{\prime}$-linear.

The induced isomorphism (the dashed arrow) is explicitly given by the formula

$\sum\nolimits_{i}(n^{\prime}_{i}\otimes_{B^{\prime}}a^{\prime}_{i})\otimes
h_{i}\mapsto\sum\nolimits_{i}n^{\prime}_{i}\otimes_{B^{\prime}}(a^{\prime}_{i}\otimes
h_{i})$ (3.27)

The composition of (3.24) with the two isomorphisms (3.25) and (3.27) gives
the map

$\displaystyle N^{\prime}\otimes_{B^{\prime}}(B^{\prime}\otimes_{B}A)$
$\displaystyle\to
N^{\prime}\otimes_{B^{\prime}}(A^{\prime}\Box^{H^{\prime}}H),$ $\displaystyle
n^{\prime}\otimes_{B^{\prime}}(b^{\prime}\otimes_{B}a)$ $\displaystyle\mapsto
n^{\prime}\otimes_{B^{\prime}}\big{(}b^{\prime}\alpha(a_{(0)})\otimes
a_{(1)}\big{)}.$ (3.28)

By naturality in $N^{\prime}$, this is equivalent as a datum to the
generalized canonical map (3.7) $\kappa:B^{\prime}\otimes_{B}A\to
A^{\prime}\Box^{H^{\prime}}H.$ Indeed, (3.28) is induced by $\kappa$, and
taking $N^{\prime}:=B^{\prime}$ in (3.28) we recover $\kappa$. ∎

######  Remark 3.23.

Note that if $H^{\prime}$ is cosemisimple then $\chi$ is automatically left
coflat. Recall also that Peter-Weyl Hopf-algebras of compact quantum groups
are always cosemisimple.

######  Theorem 3.24.

Consider a morphism (3.6) of Hopf-algebra-invariant algebra maps, assume that
$\chi:H\to H^{\prime}$ is left coflat, that $H$ has bijective antipode and
that the generalized canonical map $\kappa$ in (3.7) is bijective. Then, the
following hold.

  1. (1)

The other mate

$\leavevmode\hbox to96.05pt{\vbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{71.1319pt}{8.30632pt}\pgfsys@lineto{71.1319pt}{48.04253pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.33748pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{47.19783pt}{19.15083pt}\pgfsys@lineto{26.42644pt}{35.7676pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.53749pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{47.19783pt}{19.15083pt}\pgfsys@lineto{26.42644pt}{35.7676pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.78087}{0.6247}{-0.6247}{-0.78087}{26.42644pt}{35.76758pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\par \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;,$ (3.29)

of the diagram (3.19), weakly commuting up to a natural transformation
$f^{*}q_{*}\Rightarrow q^{\prime}_{*}\widetilde{f}^{*}$, induces a morphism
$\widetilde{\kappa}:A\otimes_{B}B^{\prime}:\to A^{\prime}\Box^{H^{\prime}}H$
of right $B$-bimodules and right $H$-comodules, and defines a morphism
$\varphi:=\kappa^{-1}\circ\widetilde{\kappa}$ making the diagram

$A\otimes_{B}B^{\prime}$$B^{\prime}\otimes_{B}A$$A^{\prime}\Box^{H^{\prime}}H$$\varphi$$\widetilde{\kappa}\hskip
12.00002pt$$\hskip 12.00002pt\kappa$ (3.30)

commute.

  2. (2)

The morphism $\varphi$ is a distributive law between the monads
$(-)\otimes_{B}A$ and $(-)\otimes_{B}B^{\prime}$ on $\mathcal{M}_{B}$.

  3. (3)

The resulting monad structure on the composition of monads

$(-)\otimes_{B}(B^{\prime}\otimes_{B}A)=((-)\otimes_{B}A)\circ((-)\otimes_{B}B^{\prime})$

gives a monoid structure on the object $B^{\prime}\otimes_{B}A$ in the
monoidal category of $B$-bimodules right $H$-comodules. In this way,
$B^{\prime}\otimes_{B}A$ becomes a right $H$-comodule algebra equipped with an
$H$-invariant algebra map $B\to B^{\prime}\otimes_{B}A$.

  4. (4)

With respect to the latter $H$-comodule algebra structure of
$B^{\prime}\otimes_{B}A$, the generalized canonical map $\kappa$ in (3.7) is
an isomorphism of right $H$-comodule algebras making the following diagram of
right $H$-comodule algebras commute

$A^{\prime}$$A$$B^{\prime}\otimes_{B}A$$B^{\prime}$$B$$A^{\prime}\Box^{H^{\prime}}H$$\alpha$$\kappa$$\beta$
(3.31)

where $A\to B^{\prime}\otimes_{B}A$ is the map $a\mapsto
1_{B^{\prime}}\otimes_{B}a$, $B^{\prime}\to B^{\prime}\otimes_{B}A$ is the map
$b^{\prime}\mapsto b^{\prime}\otimes_{B}1_{A}$, $B^{\prime}\to
A^{\prime}\Box^{H^{\prime}}H$ is the map $b^{\prime}\mapsto
b^{\prime}1_{A^{\prime}}\otimes 1_{H}$.

###### Proof.

The natural transformation for (3.29) decomposes as

$f^{*}q_{*}\xRightarrow{\;f^{*}q_{*}\eta^{\widetilde{f}}\;}f^{*}q_{*}\widetilde{f}_{*}\widetilde{f}^{*}\xRightarrow{\;\cong\;}f^{*}f_{*}q^{\prime}_{*}\widetilde{f}^{*}\xRightarrow{\;\epsilon^{f}q^{\prime}_{*}\widetilde{f}^{*}\;}q^{\prime}_{*}\widetilde{f}^{*},$
(3.32)

where the isomorphism in the middle is induced by the inverse of (3.20).

Let $M$ be an object of $\mathcal{M}^{H}_{A}$. We now compute the
$M$-component of the above natural transformation (3.32). One has

$\displaystyle
f^{*}q_{*}(M)=M^{\textrm{co}H}\otimes_{B}B^{\prime}\xrightarrow{\;f^{*}q_{*}\eta^{\widetilde{f}}\;}f^{*}q_{*}\widetilde{f}_{*}\widetilde{f}^{*}(M)=\big{(}(M\otimes_{A}A^{\prime})\Box^{H^{\prime}}H\big{)}^{\textrm{co}H}\otimes_{B}B^{\prime}$
$\displaystyle\xrightarrow{\;\cong\;}f^{*}f_{*}q^{\prime}_{*}\widetilde{f}^{*}(M)=(M\otimes_{A}A^{\prime})^{\text{co}H^{\prime}}\otimes_{B}B^{\prime}$
$\displaystyle\xrightarrow{\;\epsilon^{f}q^{\prime}_{*}\widetilde{f}^{*}\;}(M\otimes_{A}A^{\prime})^{\text{co}H^{\prime}}=q^{\prime}_{*}\widetilde{f}^{*}(M).$
(3.33) $\displaystyle
m\otimes_{B}b^{\prime}\mapsto\big{(}(m_{(0)}\otimes_{A}1)\otimes
m_{(1)}\big{)}\otimes_{B}b^{\prime}\mapsto\big{(}m_{(0)}\epsilon(m_{(1)})\otimes_{A}1\big{)}\otimes_{B}b^{\prime}$
$\displaystyle=(m\otimes_{A}1)\otimes_{B}b^{\prime}\mapsto
m\otimes_{A}b^{\prime}.$

Now, let us take $M:=H\otimes A$ with obvious right $A$-action and diagonal
$H$-coaction. This is an object of $\mathcal{M}^{H}_{A}$. Indeed,

$\displaystyle\hskip 14.0pt(h\otimes aa_{1})_{(0)}\otimes(h\otimes
aa_{1})_{(1)}$ $\displaystyle=h_{(1)}\otimes(aa_{1})_{(0)}\otimes
h_{(2)}(aa_{1})_{(1)}$ $\displaystyle=h_{(1)}\otimes a_{(0)}a_{1(0)}\otimes
h_{(2)}a_{(1)}a_{1(1)}$ $\displaystyle=(h_{(1)}\otimes a_{(0)})a_{1(0)}\otimes
h_{(2)}a_{(1)}a_{1(1)}$ $\displaystyle=(h\otimes
a)_{(0)}a_{1(0)}\otimes(ha)_{(1)}a_{1(1)}$

The right $B$-module map

$A\to(H\otimes A)^{\textrm{co}H},\qquad a\mapsto S^{-1}(a_{(1)})\otimes
a_{(0)},$ (3.34)

is well-defined since

$\displaystyle\hskip 14.0pt\big{(}S^{-1}(a_{(1)})\otimes
a_{(0)}\big{)}_{(0)}\otimes\big{(}S^{-1}(a_{(1)})\otimes a_{(0)}\big{)}_{(1)}$
$\displaystyle=S^{-1}(a_{(1)})_{(1)}\otimes a_{(0)(0)}\otimes
S^{-1}(a_{(1)})_{(2)}a_{(0)(1)}$ $\displaystyle=S^{-1}(a_{(1)(2)})\otimes
a_{(0)(0)}\otimes S^{-1}(a_{(1)(1)})a_{(0)(1)}$
$\displaystyle=S^{-1}(a_{(3)})\otimes a_{(0)}\otimes S^{-1}(a_{(2)})a_{(1)}$
$\displaystyle=S^{-1}(a_{(2)})\otimes a_{(0)}\otimes\epsilon(a_{(1)})$
$\displaystyle=S^{-1}(\epsilon(a_{(1)})a_{(2)})\otimes a_{(0)}\otimes 1_{H}$
$\displaystyle=S^{-1}(a_{(1)})\otimes a_{(0)}\otimes 1_{H},$

and it is an isomorphism with inverse

$(H\otimes A)^{\textrm{co}H}\to A,\qquad\sum\nolimits_{i}h_{i}\otimes
a_{i}\mapsto\sum\nolimits_{i}\epsilon(h_{i})a_{i}.$

The morphism in $\mathcal{M}^{H^{\prime}}_{A^{\prime}}$,

$(H\otimes A)\otimes_{A}A^{\prime}\to H\otimes A^{\prime},\qquad(h\otimes
a)\otimes_{A}a^{\prime}\mapsto h\otimes\alpha(a)a^{\prime},$ (3.35)

is invertible with inverse

$H\otimes A^{\prime}\to(H\otimes A)\otimes_{A}A^{\prime},\qquad h\otimes
a^{\prime}\mapsto(h\otimes 1)\otimes_{A}a^{\prime}.$

The right $B^{\prime}$-module map

$(H\otimes A^{\prime})^{\mathrm{co}H^{\prime}}\to
A^{\prime}\Box^{H^{\prime}}H,\qquad\sum\nolimits_{i}h_{i}\otimes
a^{\prime}_{i}\mapsto\sum\nolimits_{i}a^{\prime}_{i}\otimes S(h_{i}),$ (3.36)

is well-defined. Indeed, let $\sum_{i}h_{i}\otimes a^{\prime}_{i}$ be
invariant under the $H^{\prime}$-coaction, that means

$\sum\nolimits_{i}h_{i(1)}\otimes
a^{\prime}_{i(0)}\otimes\chi(h_{i(2)})a^{\prime}_{i(1)}=\sum\nolimits_{i}h_{i}\otimes
a^{\prime}_{i}\otimes 1_{H^{\prime}}.$

Apply the coproduct to the first leg:

$\sum\nolimits_{i}h_{i(1)}\otimes h_{i(2)}\otimes
a^{\prime}_{i(0)}\otimes\chi(h_{i(3)})a^{\prime}_{i(1)}=\sum\nolimits_{i}h_{i(1)}\otimes
h_{i(2)}\otimes a^{\prime}_{i}\otimes 1_{H^{\prime}}.$

Apply $\chi\circ S$ to the second leg, flip second and third leg and multiply
last two legs:

$\sum\nolimits_{i}h_{i(1)}\otimes
a^{\prime}_{i(0)}\otimes\chi\big{(}S(h_{i(2)})\big{)}\chi(h_{i(3)})a^{\prime}_{i(1)}=\sum\nolimits_{i}h_{i(1)}\otimes
a^{\prime}_{i}\otimes\chi\big{(}S(h_{i(2)})\big{)}.$

Using $\chi\big{(}S(h_{i(2)})\big{)}\chi(h_{i(3)})=\epsilon(h_{i(2)})$ and
permuting the legs again

$\sum\nolimits_{i}a^{\prime}_{i(0)}\otimes a^{\prime}_{i(1)}\otimes
h_{i}=\sum\nolimits_{i}a^{\prime}_{i}\otimes\chi\big{(}S(h_{i(2)})\big{)}\otimes
h_{i(1)}.$

Now we apply $S$ to the third leg and find

$\sum\nolimits_{i}a^{\prime}_{i(0)}\otimes a^{\prime}_{i(1)}\otimes
S(h_{i})=\sum\nolimits_{i}a^{\prime}_{i}\otimes\chi\big{(}S(h_{i})_{(1)}\big{)}\otimes
S(h_{i})_{(2)},$

which proves that the image of (3.36) belongs indeed to the cotensor product
$A^{\prime}\Box^{H^{\prime}}H$.

The inverse of (3.36) is given by

$A^{\prime}\Box^{H^{\prime}}H\to(H\otimes
A^{\prime})^{\mathrm{co}H^{\prime}},\qquad\sum\nolimits_{i}a^{\prime}_{i}\otimes
h_{i}\mapsto\sum\nolimits_{i}S^{-1}(h_{i})\otimes a^{\prime}_{i}.$

The map (3.33), becomes

$(H\otimes A)^{\textrm{co}H}\otimes_{B}B^{\prime}\to((H\otimes
A)\otimes_{A}A^{\prime})^{\text{co}H^{\prime}},\qquad(h\otimes
a)\otimes_{B}b^{\prime}\mapsto(h\otimes a)\otimes_{A}b^{\prime},$

and its composition with the isomorphisms (3.34), (3.35) and (3.36) is

$\displaystyle A\otimes_{B}B^{\prime}$ $\displaystyle\to(H\otimes
A)^{\textrm{co}H}\otimes_{B}B^{\prime}\to((H\otimes
A)\otimes_{A}A^{\prime})^{\text{co}H^{\prime}}\to(H\otimes
A^{\prime})^{\text{co}H^{\prime}}\to A^{\prime}\Box^{H^{\prime}}H$
$\displaystyle a\otimes_{B}b^{\prime}$
$\displaystyle\mapsto\big{(}S^{-1}(a_{(1)})\otimes
a_{(0)}\big{)}\otimes_{B}b^{\prime}\mapsto\big{(}S^{-1}(a_{(1)})\otimes
a_{(0)}\big{)}\otimes_{A}b^{\prime}$ $\displaystyle\mapsto
S^{-1}(a_{(1)})\otimes\alpha(a_{(0)})b^{\prime}\mapsto\alpha(a_{(0)})b^{\prime}\otimes
a_{(1)}$

resulting in a map

$\widetilde{\kappa}:A\otimes_{B}B^{\prime}\to
A^{\prime}\Box^{H^{\prime}}H,\quad\quad
a\otimes_{B}b^{\prime}\mapsto\alpha(a_{(0)})b^{\prime}\otimes a_{(1)}.$ (3.37)

Next, we define
$\varphi:=\kappa^{-1}\circ\widetilde{\kappa}:A\otimes_{B}B^{\prime}\to
B^{\prime}\otimes_{B}A$ and prove that the multiplication in
$B^{\prime}\otimes_{B}A$ defined as a composition

$\textstyle{(B^{\prime}\otimes_{B}A)\otimes_{B}(B^{\prime}\otimes_{B}A)=}$$\textstyle{B^{\prime}\otimes_{B}(A\otimes_{B}B^{\prime})\otimes_{B}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B^{\prime}\otimes_{B}\varphi\otimes_{B}A}$$\textstyle{(B^{\prime}\otimes_{B}B^{\prime})\otimes_{B}(A\otimes_{B}A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{B^{\prime}}\otimes_{B}\mu_{A}}$$\textstyle{B^{\prime}\otimes_{B}A,}$

with the unit $1_{B^{\prime}}\otimes_{B}1_{A}$, is transformed by $\kappa$
into the standard multiplication in $A^{\prime}\Box^{H^{\prime}}H$

$\textstyle{(A^{\prime}\Box^{H^{\prime}}H)\otimes_{B}(A^{\prime}\Box^{H^{\prime}}H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\Box^{H^{\prime}}H,}$
$\textstyle{(\sum_{i}a_{i}^{\prime}\otimes
h_{i})\otimes_{B}(\sum_{j}a_{j}^{\prime}\otimes
h_{j})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sum_{i,j}a_{i}^{\prime}a_{j}^{\prime}\otimes
h_{i}h_{j}}$

with the unit $1_{A^{\prime}}\otimes 1_{H}$.

Since the latter multiplication is associative and unital, invertibility of
$\kappa$ will prove the same about the above multiplication in
$B^{\prime}\otimes_{B}A$. To this end, we observe first that $\kappa$ is a
morphism in the category ${}_{B^{\prime}}\mathcal{M}_{A}^{H}$ of left
$B^{\prime}$-modules right relative $(A,H)$-Hopf modules. Here, the domain and
the codomain of $\kappa$ admit the following structure of an object in
${}_{B^{\prime}}\mathcal{M}_{A}^{H}$:

for $B^{\prime}\otimes_{B}A$

$\displaystyle\widetilde{b}^{\prime}(b^{\prime}\otimes_{B}a)\widetilde{a}$
$\displaystyle:=\widetilde{b}^{\prime}b^{\prime}\otimes_{B}a\widetilde{a},$
$\displaystyle\quad(b^{\prime}\otimes_{B}a)_{(0)}\otimes(b^{\prime}\otimes_{B}a)_{(1)}$
$\displaystyle:=b^{\prime}\otimes_{B}a_{(0)}\otimes a_{(1)},$

for $A^{\prime}\Box^{H^{\prime}}H$

$\displaystyle\widetilde{b}^{\prime}(\sum_{i}a^{\prime}_{i}\otimes
h_{i})\widetilde{a}$
$\displaystyle:=\sum_{i}\widetilde{b}^{\prime}a^{\prime}_{i}\widetilde{a}_{(0)}\otimes
h_{i}\widetilde{a}_{(1)},$ $\displaystyle(\sum_{i}a^{\prime}_{i}\otimes
h_{i})_{(0)}\otimes(\sum_{i}a^{\prime}_{i}\otimes h_{i})_{(1)}$
$\displaystyle:=(\sum_{i}a^{\prime}_{i}\otimes h_{i(1)})\otimes h_{i(2)}.$

Therefore we have

$\displaystyle\kappa\big{(}(b^{\prime}\otimes_{B}a)(\widetilde{b}^{\prime}\otimes_{B}\widetilde{a})\big{)}$
$\displaystyle=b^{\prime}\kappa\big{(}(1_{B^{\prime}}\otimes_{B}a)(\widetilde{b}^{\prime}\otimes_{B}1_{A})\big{)}\widetilde{a}$
$\displaystyle=b^{\prime}\kappa\big{(}\varphi(a\otimes_{B}\widetilde{b}^{\prime})\big{)}\widetilde{a}=b^{\prime}\widetilde{\kappa}\big{(}\varphi(a\otimes_{B}\widetilde{b}^{\prime})\big{)}\widetilde{a}$
$\displaystyle=b^{\prime}\big{(}\alpha(a_{(0)})\widetilde{b}^{\prime}\otimes
a_{(1)}\big{)}\widetilde{a}=b^{\prime}\alpha(a_{(0)})\widetilde{b}^{\prime}\widetilde{a}_{(0)}\otimes
a_{(1)}\widetilde{a}_{(1)}$
$\displaystyle=\kappa\big{(}b^{\prime}\otimes_{B}a\big{)}\kappa\big{(}\widetilde{b}^{\prime}\otimes_{B}\widetilde{a}\big{)}$

and

$\displaystyle\kappa\big{(}1_{B^{\prime}}\otimes_{B}1_{A}\big{)}=1_{A^{\prime}}\otimes
1_{H}.$

It is not difficult to check that all above makes $\kappa$ a right
$H$-comodule algebra map and the diagram (3.31) commutes. ∎

######  Example 3.25.

In the commutative case, $\widetilde{\kappa}$ differs from $\kappa$ by the
flip. Hence, when $\kappa$ is invertible, $\varphi$ is simply the flip.

######  Example 3.26.

When $H=H^{\prime}=\Bbbk$, the Beck-Chevalley condition means simply that our
generalized canonical map $B^{\prime}\otimes_{B}A\to A^{\prime}$,
$b^{\prime}\otimes_{B}a\mapsto b^{\prime}\alpha(a)$ is an isomorphism of
$(B^{\prime},A)$-bimodules. In the commutative case it would be an isomorphism
of algebras, meaning that the commutative square of commutative algebras

$A^{\prime}$$A$$B^{\prime}$$B$ (3.38)

is a pushout diagram in a symmetric monoidal category of algebras over $B$,
with symmetry given by the flip. Since in the opposite category (3.38)
corresponds to a Cartesian square of spaces, this justifies understanding the
above Beck-Chevalley condition as a kind of property of being a Cartesian
morphism of families of (quantum) group actions.

######  Example 3.27.

In the context of Example 3.11, the map $\varphi$ in (3.30) becomes the
braiding studied by Đurđević in [10]. If further $A=H$, hence
$B=A^{\text{co}H}=\Bbbk$, the diagram (3.30) becomes

$H\otimes H$$H\otimes H$$H\otimes H$$\varphi$$\widetilde{\kappa}\hskip
12.00002pt$$\hskip 12.00002pt\kappa$

where $\kappa:\widetilde{h}\otimes h\mapsto\widetilde{h}h_{(1)}\otimes
h_{(2)}$ and $\widetilde{\kappa}:h\otimes\widetilde{h}\mapsto
h_{(1)}\widetilde{h}\otimes h_{(2)}$. In this case, the inverse of $\kappa$
can be written explicitly as $\kappa^{-1}:\widetilde{h}\otimes
h\mapsto\widetilde{h}S(h_{(1)})\otimes h_{(2)}$, and hence

$\varphi:h\otimes\widetilde{h}\mapsto h_{(1)}\widetilde{h}S(h_{(2)})\otimes
h_{(3)}.$

This is a special case of Yetter-Drinfeld braiding $V\otimes W\to W\otimes V$,
for the Yetter-Drinfeld modules $V=W=H$, where we think of $H$ as a left-left
Yetter-Drinfeld module over itself, with left adjoint action and left coaction
given by the comultiplication.

######  Remark 3.28.

Summarizing the content of Theorems 3.22 and 3.24, we stress that

  1. (i)

The notion of a Cartesian morphism between Hopf-algebra-invariant algebra maps
translates into the notion of a weakly Cartesian square of Grothendieck
categories in the 2-category $\mathfrak{G}$.

  2. (ii)

Using the distributive law $\varphi$, we can regard the diagram (3.31) of
Hopf-comodule algebras as a noncommutative counterpart of the diagram (3.4) of
families of group actions.

Given a morphism (3.6), we can define three more adjunctions

$\mathcal{h}:\mathcal{M}^{H^{\prime}}\to\mathcal{M}^{H},\qquad\mathcal{p}:\mathcal{M}_{A}^{H}\to\mathcal{M}^{H},\qquad\mathcal{p}^{\prime}:\mathcal{M}_{A^{\prime}}^{H^{\prime}}\to\mathcal{M}^{H^{\prime}}.$

Here the category $\mathcal{M}^{H}$ of right $H$-comodules is regarded as the
category of right relative Hopf-modules $\mathcal{M}^{H}_{\Bbbk}$, where
$\Bbbk$ is the trivial $H$-comodule algebra, and similarly for $H^{\prime}$.
The left adjoint $h^{*}=\mathrm{Res}_{H^{\prime}}^{H}$ is the functor
restricting the comodule structure induced by the Hopf-algebra map $\chi:H\to
H^{\prime}$. The right adjoint $h_{*}=\mathrm{Ind}_{H^{\prime}}^{H}$ is the
functor extending the comodule structure induced by the same map $\chi$, i.e.
$\mathrm{Ind}_{H^{\prime}}^{H}(V^{\prime})=V^{\prime}\Box^{H^{\prime}}H$. The
left adjoint $p^{*}(V)=V\otimes A$ is a relative Hopf module with the obvious
right $A$-module structure and with the diagonal right $H$-coaction

$(v\otimes a)_{(0)}\otimes(v\otimes a)_{(1)}:=(v_{(0)}\otimes a_{(0)})\otimes
v_{(1)}a_{(1)}.$

The right adjoint $p_{*}(M)=M$ is forgetting the right $A$-module structure.
The adjunction $\mathcal{p}^{\prime}=(p^{\prime*}\dashv p^{\prime}_{*})$ is
defined similarly.

######  Proposition 3.29.

There exists a natural transformation making the diagram

$\mathcal{M}^{H^{\prime}}$$\mathcal{M}^{H}$$\mathcal{M}_{A^{\prime}}^{H^{\prime}}$$\mathcal{M}_{A}^{H}$$\mathcal{h}$$\widetilde{\mathcal{f}}$$\mathcal{p}$$\mathcal{p}^{\prime}$
(3.39)

weakly commute.

###### Proof.

One has natural identifications
$h_{*}p^{\prime}_{*}(M^{\prime})=M^{\prime}\Box^{H^{\prime}}H=p_{*}\widetilde{f}_{*}(M^{\prime})$
given by forgetting the $A^{\prime}$-module structure of $M^{\prime}$ and the
$A$-module structure of $\widetilde{f}_{*}(M^{\prime})$. ∎

Since $\mathcal{M}^{H}$ can be regarded as a weak categorical version of the
classifying space of a group, the weakly commutativity of the upper square in
(3.39) can be regarded as a an equivariance condition for
$\widetilde{\mathcal{f}}$ as a generalized map between equivariant
noncommutative spaces (in the sense of Grothendieck categories).

By Remark 3.28, we can extend the fundamental notions of equivariant topology
in the Examples 3.3-3.7 to noncommutative spaces understood as Grothendieck
categories, using the notion of a weakly Cartesian square

$\mathcal{M}_{A^{\prime}}^{H^{\prime}}$$\mathcal{M}^{H}_{A}$$\mathcal{M}_{B^{\prime}}$$\mathcal{M}_{B}$.$\mathcal{f}$$\widetilde{\mathcal{f}}$$\mathcal{q}^{\prime}$$\mathcal{q}$
(3.40)

######  Example 3.30 (Noncommutative orbit spaces).

The diagram (3.40) for the morphism in Example 3.10 reads as

$\mathcal{M}_{A}^{H}$$\mathcal{M}_{\Bbbk}^{\Bbbk}$$\mathcal{M}_{B}$$\mathcal{M}_{\Bbbk}$$\mathcal{q}$

where the arrows that are not labelled are obvious. The condition of being
weakly Cartesian should be regarded as a condition for $\mathcal{q}$ being a
noncommutative quotient map from the equivariant noncommutative space
described by $\mathcal{M}_{A}^{H}$ to the noncommutative orbit space described
by $\mathcal{M}_{B}$.

######  Example 3.31 (Noncommutative slices).

The diagram (3.40) for the morphism in Example 3.11 reads as

$\mathcal{M}_{A}^{H}$$\mathcal{M}^{H}_{H}$$\mathcal{M}_{B}$$\mathcal{M}_{\Bbbk}$$\mathcal{s}$$\mathcal{q}$

where $H$ is regarded as a righ $H$-comodule algebra via the comultiplication
map. The vertical adjunction on the right is the categorical reformulation of
the Fundamental Theorem for Hopf-modules [25, Theorem 4.1.1]. The bottom
horizontal arrow is the equivalence induced by the unit map of the
$\Bbbk$-algebra $B$. The condition of being weakly Cartesian should be
regarded as a condition for $\mathcal{s}$ being a noncommutative slice map.

######  Example 3.32 (Noncommutative principal bundles).

The diagram (3.40) for the morphism in Example 3.12 reads as

$\mathcal{M}_{A\otimes
H}^{H}$$\mathcal{M}^{H}_{A}$$\mathcal{M}_{A}$$\mathcal{M}_{B}$$\mathcal{q}$
(3.41)

where the adjunctions are induced by the maps in Example 3.12 and
$\mathcal{q}$ is a noncommutative quotient map in the sense of Example 3.30.
We call $\mathcal{q}$ a noncommutative principal bundle if the diagram (3.41)
is weakly Cartesian and $\mathcal{q}$ is an equivalence.

Recall that $\mathcal{q}$ is an adjoint equivalence of categories if and only
if $(H,A\leftarrow B)$ is a faithfully flat Hopf-Galois extension, cf. [22,
Theorem 4.10] and [23] (note that invertibility of the antipode is not
required, see [20]).

The left vertical arrow should be regarded as a trivial noncommutative
principal bundle, and the condition of being weakly Cartesian should be
regarded as a noncommutative counterpart of the fact that the pullback of a
principal bundle to its total space is always a trivial principal bundle.

######  Example 3.33 (Change of the structure quantum group).

The diagram (3.40) for the morphism in Example 3.13 reads as

$\mathcal{M}_{A^{\prime}}^{H^{\prime}}$$\mathcal{M}^{H}_{A}$$\mathcal{M}_{B}$$\mathcal{M}_{B}$$=$$\mathcal{q}^{\prime}$$\mathcal{q}$

where the adjunctions are induced by the maps in Example 3.13 and both
$\mathcal{q}$ and $\mathcal{q}^{\prime}$ are noncommutative principal bundles
in the sense of Example 3.32. The condition of being weakly Cartesian should
be regarded as a condition for the top horizontal arrow as being a change of
structure quantum group of a noncommutative principal bundle.

######  Example 3.34 (General morphisms of noncommutative principal bundles).

Combining Example 3.32 with Remark 3.28(i), we argue that a morphism (3.6) of
noncommutative principal bundles translates into the categorical language as
weakly Cartesian square (3.40) with $\mathcal{q}$ and $\mathcal{q}^{\prime}$
being adjoint equivalences. Then, if $\mathcal{f}$ is also an adjoint
equivalence, so is its equivariant lift $\widetilde{\mathcal{f}}$. This is the
proper noncommutative counterpart of the property that every morphism of
compact Hausdorff principal bundles over the same base is necessarily an
isomorphism (cf. Example 3.7).

##  4\. k-Topology

We now start introducing the notions needed to talk about multiplicative
structure of K-theory for arbitrary compact quantum spaces. In this section,
by a maps between topological spaces we shall always mean a continuous map.

###  4.1. The context of compact Hausdorff spaces

In this section we will rephrase the framework of principal bundles and their
pullbacks in a language that is motivated by the notion of Grothendieck fpqc
topology, where faithfully flat morphisms are regarded as covering families.
Since principal bundles of algebraic geometry are by definition faithfully
flat morphisms, they can be used as covering families of fpqc topology. What
we do in the present section is a translation of the thus obtained fpqc sub-
topology to the setting of compact Hausdorff spaces. In the next sections, we
will generalized these notions first to noncommutative algebras, and then to
noncommutative spaces understood as Grothendieck categories.

######  Definition 4.1.

A k-covering family of a compact Hausdorff space $X$ is a compact Hausdorff
principal bundle $(G,U\to X)$. A k-topology on $X$ is a collection of
k-covering families of $X$ satisfying the property that all isomorphisms to
$X$ (i.e. trivial principal bundles over $X$ with trivial structure group) are
contained in this collection. A compact Hausdorff space $X$ equipped with a
k-topology will be called a k-compact Hausdorff space.

######  Example 4.2.

Let $X$ be a compact Hausdorff space. The collection of k-covering families
consisting only of isomorphisms defines what we call the minimal classical
k-topology of $X$. The collection of all compact Hausdorff principal bundles
over $X$ defines what we call the maximal classical k-topology of $X$.

######  Definition 4.3.

A map $X^{\prime}\rightarrow X$ of k-compact Hausdorff space is called
k-continuous if, for every k-covering family $(G,U\to X)$ belonging to the
k-topology on $X$ there exist:

  1. (i)

a k-covering family $(G^{\prime},U^{\prime}\to X^{\prime})$ belonging to the
k-topology on $X^{\prime}$,

  2. (ii)

a Cartesian morphism (3.2) between these two k-covering families.

######  Example 4.4.

Every $G$-equivariant map $U^{\prime}\to U$ between principal $G$-spaces
defines a k-continuous map $U^{\prime}/G\to U/G$ between quotient spaces
equipped with k-topologies defined by the quotient maps $U^{\prime}\to
U^{\prime}/G$ and $U\to U/G$ (since a morphism of principal bundles is
necessarily Cartesian, cf. Example 3.7).

The following proposition shows that, despite the above rephrasing, we do not
change the context of classical topology of compact Hausdorff spaces.

######  Proposition 4.5.

Let $X$ and $X^{\prime}$ be k-compact Hausdorff spaces, and assume that either
the k-topology on $X$ is the minimal classical one or the k-topology on
$X^{\prime}$ is the maximal classical one. Then, every continous map
$X^{\prime}\to X$ is k-continuous.

###### Proof.

Since any continuous map $X^{\prime}\to X$ between bases pulls back principal
bundles and lifts to an equivariant map between their total spaces, it defines
a Cartesian morphism of principal bundles in the sense of Def. 3.1 (cf.
Example 3.7). If the k-topology on $X$ is the minimal classical one, then the
only k-covering families from the k-topology on $X$ are isomorphisms, and
their pullback belongs to the k-topology of $X^{\prime}$ by definition of
k-topology. If the k-topology on $X^{\prime}$ is the maximal classical one,
then the pullback of any k-covering family from the k-topology of $X$ belongs
to the k-topology of $X^{\prime}$. In both cases, continuity implies
k-continuity. ∎

######  Remark 4.6.

Prop. 4.5 is analogous to the statement that, given two topological spaces $X$
and $X^{\prime}$, if either topology on $X$ is the antidiscrete one or the
topology on $X^{\prime}$ is the discrete one, then every map $X^{\prime}\to X$
is continuous. Here the antidiscrete topology plays the role of the minimal
classical k-topology, the discrete one plays the role of the maximal classical
k-topology, maps play the role of continuous maps, and continuous maps play
the role of k-continuous maps.

######  Proposition 4.7.

  1. (1)

k-compact Hausdorff spaces and k-continuous maps form a category, that we
denote $k\text{-}\hskip 1.49994pt\mathcal{C}pct\mathcal{H}aus$.

  2. (2)

The minimal classical k-topology gives a full embedding

$\mathcal{C}pct\mathcal{H}aus\to k\text{-}\hskip
1.49994pt\mathcal{C}pct\mathcal{H}aus,$

where $\mathcal{C}pct\mathcal{H}aus$ is the category of compact Hausdorff
spaces with continuous maps.

  3. (3)

The maximal classical k-topology gives another full embedding
$\mathcal{C}pct\mathcal{H}aus\to k\text{-}\hskip
1.49994pt\mathcal{C}pct\mathcal{H}aus$.

###### Proof.

(1) Let $X,X^{\prime},X^{\prime\prime}$ be k-compact Hausdorff spaces and let
$f^{\prime}:X^{\prime\prime}\to X^{\prime}$ and $f:X^{\prime}\to X$ be two
k-continuous maps. Let $(G,U\to X)$ belong to the k-topology of $X$. Since $f$
is k-continuous, the exists $(G^{\prime},U^{\prime}\to X^{\prime})$ belonging
to the k-topology of $X^{\prime}$ and a morphism $(G^{\prime},U^{\prime}\to
X^{\prime})\to(G,U\to X)$. Since $f^{\prime}$ is k-continuous, the exists
$(G^{\prime\prime},U^{\prime\prime}\to X^{\prime\prime})$ belonging to the
k-topology of $X^{\prime\prime}$ and a morphism
$(G^{\prime\prime},U^{\prime\prime}\to
X^{\prime\prime})\to(G^{\prime},U^{\prime}\to X^{\prime})$. By composition of
morphisms of compact Hausdorff principal bundles

$U^{\prime\prime}$$X^{\prime\prime}$$U^{\prime}$$U$$X^{\prime}$$X$$f$$q^{\prime}$$q$$\widetilde{f}$$q^{\prime\prime}$$\widetilde{f}^{\prime}$$f^{\prime}$

we obtain a morphism $(G^{\prime\prime},U^{\prime\prime}\to
X^{\prime\prime})\to(G,U\to X)$. Since every such a morphism is Cartesian (cf.
Example 3.7), this proves that $f\circ f^{\prime}$ is k-continuous.

(2) and (3) are an obvious consequence of Prop. 4.5. ∎

######  Remark 4.8.

In the spirit of Remark 4.6, the statements (2) and (3) in Prop. 4.7 are
analogous to the statement that there are two full embeddings of the category
of sets into the category of topological spaces, one given by the antidiscrete
topology and one by the discrete topology.

######  Proposition 4.9.

The function $T$ assigning to every k-compact Hausdorff space $X$ its
k-topology is a Cartesian coverage (in the sense of Def. 2.1) on the category
$k\text{-}\hskip 1.49994pt\mathcal{C}pct\mathcal{H}aus$.

###### Proof.

A careful inspection of the definition of k-continuous map shows that $T$ is
indeed a Cartesian coverage. ∎

###  4.2. From spaces to algebras

In this section, we will translate the notion of k-topology of compact
Hausdorff spaces to a purely algebraic setting, parallel to Sect. 3.2. Here,
if $B$ is a unital associative algebra (over a fixed ground field $\Bbbk$), by
$\mathrm{Spec}(B)$ we mean the object in the opposite category corresponding
to $B$.

######  Definition 4.10.

A k-covering family of a compact quantum space $X=\mathrm{Spec}(B)$ is a Hopf-
Galois extension $(H,A\leftarrow B)$. A k-topology on $X$ is a collection of
k-covering families of $X$ satisfying the property that all isomorphisms to
$X$ (i.e. trivial Hopf-Galois extensions with $H\cong\Bbbk$) are contained in
this collection. A compact quantum space $X$ equipped with a k-topology will
be called a k-compact quantum space.

######  Example 4.11.

Let $X=\mathrm{Spec}(B)$ be a compact quantum space. The collection of
k-covering families consisting only of isomorphisms defines what we call the
minimal quantum k-topology of $X$. The collection of all Hopf-Galois
extensions of $B$ could be called maximal quantum k-topology of $X$. However,
we will see that this notion is not especially useful (cf. Prop. 4.7 with
Prop. 4.17).

######  Definition 4.12.

A morphism $X^{\prime}\rightarrow X$ of compact quantum space equipped with a
k-topology is called k-continuous if, for every k-covering family
$(H,A\leftarrow B)$ belonging to the k-topology on $X$ there exist:

  1. (i)

a k-covering family $(H^{\prime},A^{\prime}\leftarrow B^{\prime})$ belonging
to the k-topology on $X^{\prime}=\mathrm{Spec}(B^{\prime})$,

  2. (ii)

a Cartesian morphism (3.6) between these two k-covering families with
$\chi:H\to H^{\prime}$ left coflat.

######  Proposition 4.13.

Let $X=\mathrm{Spec}(B)$ and $X^{\prime}=\mathrm{Spec}(B)$ be k-compact
quantum spaces, and assume that the k-topology on $X$ is the minimal quantum
one. Then, every morphism $X^{\prime}\to X$ is k-continuous.

###### Proof.

The only k-convering families in the k-topology of $X$ are the isomorphisms.
Given an isomorphism $(\Bbbk,A\leftarrow B)$ and an arbitrary morphism
$B^{\prime}\leftarrow B$ we can form the commutative diagram

$\Bbbk$$\Bbbk$$B^{\prime}$$A$$B^{\prime}$$B$$=$$\beta$$\varphi$$\alpha$$=$
(4.1)

where $\alpha=\beta\circ\varphi^{-1}$. The identity on $\Bbbk$ is clearly left
coflat. And the identity map $(\Bbbk,B^{\prime}\leftarrow B^{\prime})$ belongs
to the k-topology of $X^{\prime}$, whatever this k-topology is, by definition
of k-topology. Invertibility of the generalized canonical map $\kappa$ stems
from the following decomposition into isomorphisms

$B^{\prime}\otimes_{B}A$$B^{\prime}\,\Box^{\,\Bbbk}\,\Bbbk$$B^{\prime}\otimes_{B}B$$\kappa$$B^{\prime}\otimes_{B}\varphi^{-1}$$\cong$

where the isomorphism denoted by $\cong$ is the generalized canonical map for
the identity morphism on $(\Bbbk,B^{\prime}\leftarrow B^{\prime})$. This
proves that (4.1) is a Cartesian morphism, and hence the map $X^{\prime}\to X$
is k-continuous. ∎

######  Remark 4.14.

We don’t have a statement analogous to Prop. 4.5 for the maximal k-topology,
since in general given a map $B^{\prime}\leftarrow B$ and a Hopf-Galois
extension of $B$, it is not always possible to fit them into a Cartesian
square (3.6).

######  Remark 4.15.

Observe that even for classical spaces we can define non-classical
k-topologies using (not necessarily commutative) Hopf-Galois extensions
instead of compact principal bundles.

######  Proposition 4.16.

k-compact quantum spaces and k-continuous morphisms form a category that we
denote by $k\text{-}\hskip 1.49994pt\mathcal{C}pct\mathcal{Q}uant$.

###### Proof.

Let
$X=\mathrm{Spec}(B),X^{\prime}=\mathrm{Spec}(B^{\prime}),X^{\prime\prime}=\mathrm{Spec}(B^{\prime\prime})$
be k-compact quantum spaces and let $f^{\prime}:X^{\prime\prime}\to
X^{\prime}$ and $f:X^{\prime}\to X$ be two k-continuous morphisms. Let
$(H,A\leftarrow B)$ belong to the k-topology of $X$. Since $f$ is
k-continuous, the exists $(H^{\prime},A^{\prime}\leftarrow B^{\prime})$
belonging to the k-topology of $X^{\prime}$ and a Cartesian morphism (3.6)
from $(H^{\prime},A^{\prime}\leftarrow B^{\prime})$ to $(H,A\leftarrow B)$.
Since $f^{\prime}$ is k-continuous, the exists
$(H^{\prime\prime},A^{\prime\prime}\leftarrow B^{\prime\prime})$ belonging to
the k-topology of $X^{\prime\prime}$ and a Cartesian morphism (3.6) from
$(H^{\prime\prime},A^{\prime\prime}\leftarrow B^{\prime\prime})$ to
$(H^{\prime},A^{\prime}\leftarrow B^{\prime})$. By composition of morphisms we
get a commutative diagram

$H^{\prime\prime}$$H^{\prime}$$H$$A^{\prime\prime}$$A^{\prime}$$A$$B^{\prime\prime}$$B^{\prime}$$B$$\chi$$\chi^{\prime}$$\beta$$\beta^{\prime}$$\alpha$$\alpha^{\prime}$$\varphi$$\varphi^{\prime}$$\varphi^{\prime\prime}$

where each square is Cartesian. Since $\chi$ and $\chi^{\prime}$ are left
coflat, their composition $\chi^{\prime\prime}=\chi^{\prime}\circ\chi$ is left
coflat as well. It remains to prove that the outer rectangle is also
Cartesian. Let $\kappa$ be the generalized canonical map of the right square,
$\kappa^{\prime}$ the one of the left square, and $\kappa^{\prime\prime}$ the
one of the outer rectangle. These are given explicitly by

$\displaystyle\kappa:B^{\prime}\otimes_{B}A$ $\displaystyle\rightarrow
A^{\prime}\Box^{H^{\prime}}H,$ $\displaystyle b^{\prime}\otimes_{B}a$
$\displaystyle\mapsto b^{\prime}\alpha(a_{(0)})\otimes a_{(1)},$
$\displaystyle\kappa^{\prime}:B^{\prime\prime}\otimes_{B^{\prime}}A^{\prime}$
$\displaystyle\rightarrow A^{\prime\prime}\Box^{H^{\prime\prime}}H^{\prime},$
$\displaystyle b^{\prime\prime}\otimes_{B^{\prime}}a^{\prime}$
$\displaystyle\mapsto b^{\prime\prime}\alpha^{\prime}(a^{\prime}_{(0)})\otimes
a^{\prime}_{(1)},$
$\displaystyle\kappa^{\prime\prime}:B^{\prime\prime}\otimes_{B}A$
$\displaystyle\rightarrow A^{\prime\prime}\Box^{H^{\prime\prime}}H,$
$\displaystyle b^{\prime\prime}\otimes_{B}a$ $\displaystyle\mapsto
b^{\prime\prime}\alpha^{\prime\prime}(a_{(0)})\otimes a_{(1)},\vskip 6.0pt
plus 2.0pt minus 2.0pt$

where $\alpha^{\prime\prime}=\alpha^{\prime}\circ\alpha$. One can check that
the following diagram is commutative

$B^{\prime\prime}\otimes_{B^{\prime}}(B^{\prime}\otimes_{B}A)\cong
B^{\prime\prime}\otimes_{B}A$$A^{\prime\prime}\Box^{H^{\prime\prime}}H\cong(A^{\prime\prime}\Box^{H^{\prime\prime}}H^{\prime})\Box^{H^{\prime}}H$$B^{\prime\prime}\otimes_{B^{\prime}}(A^{\prime}\Box^{H^{\prime}}H)\cong(B^{\prime\prime}\otimes_{B^{\prime}}A^{\prime})\Box^{H^{\prime}}H$$\kappa^{\prime\prime}$$B^{\prime\prime}\otimes_{B^{\prime}}\kappa$$\kappa^{\prime}\Box^{H^{\prime}}H$

where the top left $\cong$ is the cancellation of $B^{\prime}$ combined with
the associativity of the balanced tensor product, the top right $\cong$ is the
cancellation of $H^{\prime}$ combined with the associativity of the cotensor
product over a field, and the bottom $\cong$ is the isomorphism (3.26) with
$N^{\prime}=B^{\prime\prime}$, which holds since $\chi$ is left coflat. Since
$\kappa$ and $\kappa^{\prime}$ are invertible, $\kappa^{\prime\prime}$ is
invertible as well. ∎

######  Proposition 4.17.

The minimal quantum k-topology gives a full embedding

$\mathcal{C}pct\mathcal{Q}uant\to k\text{-}\hskip
1.49994pt\mathcal{C}pct\mathcal{Q}uant.$ (4.2)

###### Proof.

This is an immediate consequence of Prop. 4.13. ∎

Contrary to the classical case, there is no functor (4.2) analogous to the one
in Prop. 4.17 defined by the maximal quantum k-topology (cf. Remark 4.14).

######  Proposition 4.18.

The function $T$ assigning to every k-compact quantum space $X$ its k-topology
is a Cartesian coverage (in the sense of Def. 2.1) on the category
$k\text{-}\hskip 1.49994pt\mathcal{C}pct\mathcal{Q}uant$.

###### Proof.

As in the classical case, a careful inspection of the definition of
k-continuous map shows that $T$ is indeed a Cartesian coverage. ∎

###  4.3. From algebras to Grothendieck categories

In this section, we translate the notion of k-topology to the setting of
Grothendieck categories. Recall that here by a noncommutative space over
$\mathcal{S}$ we mean a Grothendieck category $\mathcal{X}$.

For the rest of the present section we fix a Hopf algebra $H$ with invertible
antipode and we denote the corresponding quantum affine group scheme by $G$.

By a $G$-equivariant noncommutative space we mean a Grothendieck category
$\mathcal{M}_{A}^{H}$ of right-right relative Hopf modules corresponding to a
right $H$-comodule algebra $A$.

######  Definition 4.19.

A k-covering family of a Grothendieck category $\mathcal{M}_{A}^{H}$ is an
adjoint equivalence $\mathcal{q}:\mathcal{M}_{A}^{H}\to\mathcal{M}_{B}$. A
k-topology on $\mathcal{M}_{B}$ is a collection of k-covering families of
$\mathcal{M}_{B}$ satisfying the property that all equivalences to
$\mathcal{M}_{B}$ are contained in this collection. A noncommutative space
$\mathcal{M}_{B}$ equipped with a k-topology will be called a k-noncommutative
space.

######  Definition 4.20.

A 1-cell $\mathcal{M}_{B}^{\prime}\rightarrow\mathcal{M}_{B}$ of
k-noncommutative spaces is called k-continuous if, for every k-covering family
$\mathcal{M}_{A}^{H}\to\mathcal{M}_{B}$ belonging to the k-topology on $X$
there exist:

  1. (i)

a k-covering family
$\mathcal{M}_{A^{\prime}}^{H^{\prime}}\to\mathcal{M}_{B^{\prime}}$ belonging
to the k-topology on $X^{\prime}$,

  2. (ii)

a weakly Cartesian square (3.40) between these two k-covering families.

The following proposition shows that, despite the above rephrasing of the
algebraic language into the categorical one, we do not change the context of
k-topology of quantum schemes and compact quantum spaces.

######  Proposition 4.21.

The class of k-noncommutative spaces as 0-cells with k-continuous adjunctions
as 1-cells and natural transformations of adjunctions as 2-cells form a
2-category that we denote by $k\text{-}\hskip 1.00006pt\mathcal{NcSp}$.

###### Proof.

Let us consider pasting of two weakly Cartesian squares as follows.

$\mathcal{M}_{A^{\prime\prime}}^{H^{\prime\prime}}$$\mathcal{M}_{B^{\prime\prime}}$$\mathcal{M}_{A^{\prime}}^{H^{\prime}}$$\mathcal{M}_{A}^{H}$$\mathcal{M}_{B^{\prime}}$$\mathcal{M}_{B}$$f_{*}$$q^{\prime}_{*}$$q_{*}$$\widetilde{f}_{*}$$q^{\prime\prime}_{*}$$\widetilde{f}^{\prime}_{*}$$f^{\prime}_{*}$

Since both squares are weakly commutative, we have natural isomorphisms

$f_{*}q^{\prime}_{*}\Longrightarrow q_{*}\widetilde{f}_{*},\ \ \ \ \
f^{\prime}_{*}q^{\prime\prime}_{*}\Longrightarrow
q^{\prime}_{*}\widetilde{f}^{\prime}_{*}.$

Given natural isomorphisms

$f_{*}f^{\prime}_{*}\Longrightarrow f^{\prime\prime}_{*},\ \ \ \ \
\widetilde{f}_{*}\widetilde{f}^{\prime}_{*}\Longrightarrow\widetilde{f}^{\prime\prime}_{*}$

we can form a diagram of induced natural isomorphisms

$f^{\prime\prime}_{*}q^{\prime\prime}_{*}\Longleftarrow
f_{*}f^{\prime\prime}_{*}q^{\prime\prime}_{*}\Longrightarrow
f_{*}q^{\prime}_{*}\widetilde{f}^{\prime}_{*}\Longrightarrow
q_{*}\widetilde{f}_{*}\widetilde{f}^{\prime}_{*}\Longrightarrow
q_{*}\widetilde{f}^{\prime\prime}_{*}$

showing that the composite is weakly commutative as well.

Since both squares are weakly Cartesian, we have natural isomorphisms

$q^{*}f_{*}\Longrightarrow\widetilde{f}_{*}q^{\prime*},\ \ \ \ \
q^{\prime*}f^{\prime}_{*}\Longrightarrow\widetilde{f}^{\prime}_{*}q^{\prime\prime*}.$

which induce another diagram of natural isomorphisms

$q^{*}f^{\prime\prime}_{*}\Longleftarrow
q^{*}f_{*}f^{\prime}_{*}\Longrightarrow\widetilde{f}_{*}q^{\prime}_{*}f^{\prime}_{*}\Longrightarrow\widetilde{f}_{*}\widetilde{f}^{\prime}_{*}q^{\prime\prime}*\Longrightarrow\widetilde{f}^{\prime\prime}_{*}q^{\prime\prime}*$

which shows the composite is weakly Cartesian as well. ∎

######  Proposition 4.22.

The function $T$ assigning to every k-noncommutative space its k-topology is a
Cartesian coverage (in the sense of Def. 2.1) on the category $k\text{-}\hskip
1.00006pt\mathcal{NcSp}$.

###### Proof.

A careful inspection of the definition of k-continuous adjunctions shows that
$T$ is indeed a Cartesian coverage. ∎

##  5\. Associated vector bundles

In the present section we derive the construction of an associated vector
bundle from some dual action of a quantum group scheme induced by a k-covering
family.

######  Definition 5.1.

By a noncommutative space $\widehat{G}$ dual to a quantum group scheme $G$
corresponding to a Hopf algebra $H$, we mean a monoidal Grothendieck category
${}^{H}\\!\\!\\!\mathcal{M}$ of left $H$-comodules.

The following proposition allows us to define the dual action of $\widehat{G}$
on the Grothendieck category of any $G$-equivariant noncommutative space.

######  Proposition 5.2.

Let $H$ be a Hopf algebra over $\Bbbk$ and $A$ a right $H$-comodule
$\Bbbk$-algebra. There is an action of the monoidal category $\widehat{G}$ on
the Grothendieck category category $\mathcal{U}=\mathcal{M}_{A}^{H}$ of
relative $(A,H)$-Hopf modules.

###### Proof.

The action is the following. For $M$ an object of $\mathcal{M}_{A}^{H}$ and
$V$ an object of ${}^{H}\\!\\!\\!\mathcal{M}$, we define $M\triangleleft V$ as
$M\otimes V$ with right $H$-coaction

$(m\otimes v)_{(0)}\otimes(m\otimes v)_{(1)}:=(m_{(0)}\otimes v_{(0)})\otimes
S^{-1}(v_{(-1)})m_{(1)}$

and right $A$-action

$(m\otimes v)a:=ma\otimes v.$

Checking that this defines a relative Hopf-module is an easy calculation, and
is left to the reader (especially one surprised by the appearence of
$S^{-1}$). It is also straightforward to check that the obvious maps

$\displaystyle\mu_{M,V,V^{\prime}}:(M\triangleleft V)\triangleleft
V^{\prime}\longrightarrow M\triangleleft(V\otimes V^{\prime}),$ (5.1a)
$\displaystyle\eta_{M}:M\longrightarrow M\triangleleft\Bbbk,$ (5.1b)

are isomorphisms in $\mathcal{M}_{A}^{H}$. ∎

######  Proposition 5.3.

Let $(H,A\leftarrow B)$ be a faithfully flat Hopf-Galois extension. Then, the
action of $\widehat{G}={}^{H}\\!\\!\\!\mathcal{M}$ on
$\mathscr{U}=\mathcal{M}_{A}^{H}$ descends to an action of $\widehat{G}$ on
$\mathscr{X}=\mathcal{M}_{B}$ induced by the adjoint equivalence $\mathcal{q}$
in (3.17) as follows

$(N,V)\mapstochar\mathrel{\mspace{0.45mu}}\leadsto
q_{*}\big{(}(q^{*}N)\triangleleft V\big{)}.$

###### Proof.

To say that $q_{*}$ is fully faithful is equivalent to say that the counit
$q^{*}q_{*}\Longrightarrow 1$ and the unit $1\Longrightarrow q_{*}q^{*}$ of
the adjunction $\mathcal{q}=(q^{*}\vdash q_{*})$ are natural isomorphism.
Then, we have isomorphisms

$\displaystyle q_{*}\bigg{(}\Big{(}q^{*}q_{*}\big{(}(q^{*}N)\triangleleft
V\big{)}\Big{)}\triangleleft
V^{\prime}\bigg{)}\xrightarrow{\;\;\cong\;\;}q_{*}\Big{(}\big{(}(q^{*}N)\triangleleft
V\big{)}\triangleleft
V^{\prime}\Big{)}\xrightarrow{\;\;\eqref{eq:catactA}\;\;}q_{*}\big{(}(q^{*}N)\triangleleft(V\otimes
V^{\prime})\big{)},$ $\displaystyle
N\xrightarrow{\;\;\cong\;\;}q_{*}q^{*}N\xrightarrow{\;\;\eqref{eq:catactB}\;\;}q_{*}\big{(}(q^{*}N)\triangleleft\Bbbk\big{)},$

proving that the axioms [27][Ch. IV, Def. 4.7] of the action of a monoidal
category are satisfied. ∎

######  Lemma 5.4.

Assume that $\mathcal{q}:\mathcal{M}_{A}^{H}\rightarrow\mathcal{M}_{B}$ comes
from an $H$-invariant algebra map. Then, for every object $M$ of
$\mathcal{M}_{A}^{H}$ and an object $V$ of ${}^{H}\\!\\!\\!\mathcal{M}$, one
has

$q_{*}(M\triangleleft V)=M\Box^{H}V.$

###### Proof.

Assume that the element $\sum_{i}m_{i}\otimes v_{i}\in M\otimes V$ belongs to
$M\Box^{H}V$. Then

$\sum_{i}m_{i}\otimes v_{i(-1)}\otimes v_{i\ (0)}=\sum_{i}m_{i(0)}\otimes
m_{i(1)}\otimes v_{i}.$ (5.2)

After comultiplying the righmost factor, applying $S^{-1}$ to the second,
multiplying the second with the third factor and using one of identities for
$S^{-1}$, next counitality of the coaction on $V$, and finally flipping the
second and the third factors, by the definition of the right $H$-comodule
structure on $M\triangleleft V$ we get

$\Big{(}\sum_{i}m_{i}\otimes v_{i}\Big{)}\otimes
1=\Big{(}\sum_{i}m_{i(0)}\otimes v_{i(0)}\Big{)}\otimes S^{-1}(v_{i\
(-1)})m_{i\ (1)}.$ (5.3)

By definitions of $q_{*}$ and the comodule structure of $M\triangleleft V$,
this means that $M\Box^{H}V\subseteq q_{*}(M\triangleleft V)$. On the other
hand, coacting on the second factor of (5.3), flipping two rightmost factors,
multiplying the second and the third using another identity for $S^{-1}$ on
the right hand side, and finally applying counitality of the coaction on $V$
we get (5.2). By definition of $M\Box^{H}V$ this means that
$q_{*}(M\triangleleft V)\subseteq M\Box^{H}V$. ∎

######  Proposition 5.5.

Assume that $\mathcal{q}:\mathcal{M}_{A}^{H}\rightarrow\mathcal{M}_{B}$ comes
from a faithfully flat $H$-Galois algebra extension. Then, for every object
$N$ of $\mathcal{M}_{B}$ and an object $V$ of ${}^{H}\\!\\!\\!\mathcal{M}$,
one has

$q_{*}((q^{*}N)\triangleleft V)=N\otimes_{B}(A\Box^{H}V).$

######  Remark 5.6.

For completness, we recall the result of Schauenburg saying that a Hopf
algebra-invariant algebra map $(H,A\leftarrow B)$ is a Hopf-Galois extension
if and only if the functor

${}^{H}\\!\\!\\!\mathcal{M}\longrightarrow{}_{B}\mathcal{M}_{B},\ \ \ \
V\hskip 2.5pt\raisebox{0.75pt}{$\shortmid$}\leadsto A\Box^{H}V,$

is strong monoidal.

The following two results are well-known to the experts. However, since their
proofs are not spelled out in the literature, we write them here for the
benefit of the reader.

######  Proposition 5.7.

Under the assumptions, of the previous proposition $A\Box^{H}V$ is a
$B$-bimodule that is finitely generated and projective as a right module.

###### Proof.

By Theorem 4.10 in [22], $A$ is injective as an $H$-comodule. By Theorem 5.6
in [22], this means that $B\subseteq A$ is equivariantly projective. By Lemma
2.2 in [5], this implies the existence of a strong connection, which is used
in [5, Theorem 3.1] to prove that $M$ is finitely generated and projective. ∎

######  Proposition 5.8.

Let us consider objects $M$ of ${}_{B}\mathcal{M}_{A}$, $N$ of
${}_{C}\mathcal{M}_{B}$ and $N\otimes_{B}M$ of ${}_{C}\mathcal{M}_{A}$.

  1. (1)

If $M$ and $N$ are finitely generated from the right, $N\otimes_{B}M$ is
finitely generated from the right as well.

  2. (2)

If $M$ and $N$ are projective from the right, $N\otimes_{B}M$ is projective
from the right as well.

###### Proof.

(1) Let $\\{m^{i}\mid i\in I\\}$ and $\\{n^{j}\mid j\in J\\}$ be finite sets
of generators of $M$ and $N$ over $A$ and $B$, respectively. We show that
$\\{n^{j}\otimes_{B}m^{i}\mid(i,j)\in I\times J\\}$ is a finite set of
generators of $N\otimes_{B}M$ over $A$. Indeed, if $m=\sum_{i}m^{i}a_{i}$ and
$n=\sum_{j}n^{j}b_{j}$, we can use finite combinations

$b_{j}m^{i}=\sum_{k}m^{k}a^{i}_{jk},\ \ \ \ \ a_{jk}=\sum_{i}a^{i}_{jk}a_{i}$

to find a finite presentation

$n\otimes_{B}m=\sum_{j,k}n^{j}\otimes_{B}m^{k}a_{jk}.$

(2) We use the well known facts that projectivity of a given module is
equivalent to exactnes of a Hom functor from that module, and the composition
of exact functors is exact as well, applied to the decomposition coming from
the tensor-hom adjunction

${\rm Hom}_{B}(N\otimes_{B}M,-)={\rm Hom}_{B}(N,{\rm Hom}_{B}(M,-))={\rm
Hom}_{B}(N,-)\circ{\rm Hom}_{B}(M,-).$

∎

Since, by evoking the Serre-Swan theorem, finitely generated modules are
regarded as a noncommutative counterpart of vector bundles, the discussion of
the present section can be subsummed by the following corollary.

######  Corollary 5.9.

Assume that $\mathcal{q}:\mathcal{M}_{A}^{H}\rightarrow\mathcal{M}_{B}$ comes
from a faithfully flat $H$-Galois algebra extension. Then, the dual action of
finite dimensional left $H$-comodules $\widehat{G}$ on $\mathcal{M}_{A}^{H}$
descends to an action on $\mathcal{M}_{B}$ by tensoring by noncommutative
associated vector bundles.

This together with the notion of k-topology allows us to construct in the next
section a functor of multiplicative K-theory of k-noncommutative spaces.

##  6\. From k-topology to multiplicative K-theory

Even in the classical case, when K-theory is an (extraordinary) cohomology
theory, the Mayer–Vietoris principle applies, but not all maps in the
corresponding long (six term, in fact) exact sequence respect the cup product.
The fact that the abelian group structure and the ring structure of K-theory
interact differently with the axioms of cohomology theory, shows that they
manifest as related but distinct.

For a general unital separable C*-algebra $A$, the K-group $K_{0}(A)$ is a
module over the unital ring $KK^{0}(A,A)$, with both ring and module structure
given by the Kasparov product. However, the ring $KK^{0}(A,A)$ cannot be a
replacement for the multiplicative K-theory for two reasons. First, contrary
to the K-theory functor, the association $A\mapsto KK^{0}(A,A)$ is not a
functor, and second, for commutative $A$ it is usually much bigger than
$K_{0}(A)$. (The map is ”acting on bold 1”.)

The following definition proposes a replacement of the ring structure of the
K-theory that takes this phenomenon into account. We will show that this
replacement makes sense even for the non-classical case of compact quantum
spaces, when the cup product cannot be defined in the usual way because the
diagonal map doesn’t make sense.

######  Definition 6.1.

An augmented ring is a triple $(R,M,\mathbb{1}_{M})$ where

  1. (1)

$R$ is a unital ring,

  2. (2)

$M$ is a $R$-module with a distinguished element $\mathbb{1}_{M}\in M^{0}$.

A morphism $(S,N,\mathbb{1}_{N})\rightarrow(R,M,\mathbb{1}_{M})$ of augmented
rings consists of

  1. (1)

a unital ring map $S\rightarrow R$,

  2. (2)

a $S$-module map $N\rightarrow M$ such that
$\mathbb{1}_{N}\mapsto\mathbb{1}_{M}$.

We denote by $\mathcal{A}ug\mathcal{R}ing$ the category of augmented rings.

The above definition is essentially equivalent to Cartan-Eilenberg’s
definition of augmented ring [7, Chap. VIII], except that we don’t assume
surjectivity of the augmentation map and we work with rings and their modules.
Given a distinguished element $\mathbb{1}_{M}\in M$, we can define the
augmentation map $R\to M$, $r\mapsto r\mathbb{1}_{M}$. Conversely, given an
augmentation map $R\to M$ we can define a distinguished element
$\mathbb{1}_{M}$ as image of the unit $1_{R}$ of $R$. For a comment about
surjectivity of the augmentation map, see Remark 6.5.

Recall that compact spaces are reflective in compact quantum spaces (the
coreflector is the subspace of classical points). A similar reflectivity holds
for our extension of rings to augmented rings.

######  Proposition 6.2.

The functor

$\mathscr{L}:\mathcal{R}ing\longrightarrow\mathcal{A}ug\mathcal{R}ing,\ \
R\hskip 2.5pt\raisebox{0.75pt}{$\shortmid$}\leadsto(R,R,1_{R})$

is a full embedding, admitting a right adjoint (a coreflector)

$\mathscr{R}:\mathcal{A}ug\mathcal{R}ing\longrightarrow\mathcal{R}ing,\ \
(R,M,\mathbb{1}_{M})\hskip 2.5pt\raisebox{0.75pt}{$\shortmid$}\leadsto R.$

###### Proof.

It is enough to prove these two functors form an adjunction whose unit natural
transformation is an isomorphism. For the adjunction part we define natural
transformations $\eta:1\longrightarrow\mathscr{R}\mathscr{L},\ \
R\stackrel{{\scriptstyle=}}{{\longrightarrow}}R,$ being an identity on every
$R$, $\varepsilon:\mathscr{R}\mathscr{L}\longrightarrow 1$ defined as
$(R,R,\mathbb{1}_{R})\longrightarrow(R,M,\mathbb{1}_{M})$ consisting of the
identity on the ring $R$, and the morphism of left $R$-modules
$R\longrightarrow M,\ \ r\mapsto r\mathbb{1}_{M}$, and check that the
compositions

$\textstyle{\mathscr{L}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{L}(\eta_{R})}$$\textstyle{\mathscr{L}\mathscr{R}\mathscr{L}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{\mathscr{L}(R)}}$$\textstyle{\mathscr{L}(R),}$
$\textstyle{\mathscr{R}(R,M,\mathbb{1}_{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{\mathscr{R}(R,M,\mathbb{1}_{M})}}$$\textstyle{\mathscr{R}\mathscr{L}\mathscr{R}(R,M,\mathbb{1}_{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{R}(\varepsilon_{\mathscr{R}(R,M,\mathbb{1}_{M})})}$$\textstyle{\mathscr{R}((R,M,\mathbb{1}_{M}))}$

are identities. The latter is obvious, since they read as

$\displaystyle(R,R,1_{R})\xrightarrow{\;\;=\;\;}(R,R,1_{R})\xrightarrow{\;\;=\;\;}(R,R,1_{R}),$
$\displaystyle R\xrightarrow{\;\;=\;\;}R\xrightarrow{\;\;=\;\;}R,$

respectively. ∎

######  Proposition 6.3.

The functor

$\mathcal{A}ug\mathcal{R}ing\longrightarrow\mathcal{A}b,\ \
(R,M,\mathbb{1}_{M})\hskip 2.5pt\raisebox{0.75pt}{$\shortmid$}\leadsto M$

extends forgetting the unital ring structure
$\mathcal{R}ing\rightarrow\mathcal{A}\hskip 0.12497ptb$, i.e. we have a
commutative diagram

${\mathcal{R}ing}$${\mathcal{A}\hskip
0.12497ptb}$${\mathcal{A}ug\mathcal{R}ing}$

.

###### Proof.

The proof is a straightforward application of the definition
$\mathcal{A}ug\mathcal{R}ing$ and of the functors involved. ∎

Note that both categories $\mathcal{C}pct\mathcal{H}aus$ and $k\text{-}\hskip
1.49994pt\mathcal{C}pct\mathcal{Q}uant$ are cw-Waldhausen categories, whose
cofibrations are monomorphisms. Now we are ready to claim the main theorem of
the present section.

######  Theorem 6.4.

Let $Q$ be the full embedding of $\mathcal{C}pct\mathcal{H}aus$ into
$k\text{-}\hskip 1.49994pt\mathcal{C}pct\mathcal{Q}uant.$ There exists a
commutative square

${\mathcal{C}pct\mathcal{H}aus}$${\mathcal{R}ing}$${k\text{-}\hskip
1.49994pt\mathcal{C}pct\mathcal{Q}uant}$${\mathcal{A}ug\mathcal{R}ing}$$\scriptstyle{K}$$\scriptstyle{Q}$$\scriptstyle{\bm{K}}$

with vertical embeddings onto coreflective subcategories such that

  1. (1)

the weak equivalences of both cw-Waldhausen categories are $K$-equivalences
and $\bm{K}$-equivalences, respectively,

  2. (2)

the composition of $\bm{K}$ with
$\mathcal{A}ug\mathcal{R}ing\rightarrow\mathcal{A}\hskip 0.12497ptb$ factors
through K-theory of unital C*-algebras.

###### Proof.

Let $G$ be any quantum affine group scheme corresponding to the Hopf algebra
$H:=\mathcal{O}(G)$, and $R(G)$ its representation ring, i.e. the Grothendieck
group of the monoid of isomorphism classes of finite-dimensional left
$H$-comodules, with operations induced by direct sum and tensor product of
comodules. Let $(H,A\leftarrow B)$ be a faithfully flat Hopf-Galois extension.
By Cor. 5.9, there is an action of $R(G)$ on the Grothendieck group of the
monoid of isomorphism classes of finitely generated projective right
$B$-modules. Indeed, if $N$ is a finitely generated projective right
$B$-module and $V$ a finite-dimensional left $H$-comodule, by Prop. 5.7,
$A\Box^{H}V$ is a $B$-bimodule that is finitely-generated and projective as a
right module, and $N\otimes_{B}(A\Box^{H}V)$ is a finitely generated
projective right $B$-module.

For $X=\mathrm{Spec}(B)$, we consider a category of indices with objects given
by Hopf-Galois extensions $(\mathrm{O}(G),A\leftarrow B)$ from the k-topology
of $X$ and with Cartesian morphisms (3.6). Given a morphisms (3.6) of Hopf-
Galois extensions of $B$, there is an associated restriction functor

$\mathrm{Res}_{G^{\prime}}^{G}:{}^{\mathcal{O}(G)\\!}\mathcal{M}\to{}^{\mathcal{O}(G^{\prime})\\!}\mathcal{M}$

preserving finite dimension of a comodule, and hence inducing a ring
homomorphism $\mathrm{res}_{G^{\prime}}^{G}:R(G)\to R(G^{\prime})$. We can
then define

$R^{0}(X):={\rm colim}_{(G,U\rightarrow X)\in T(X)}R(G).$

Since all actions of all rings $R(G)$ on $K^{0}(X)=K_{0}(B)$ are compatible
thanks to the $B$-bimodule isomorphism

$A\Box^{H}V\longrightarrow(A^{\prime}\Box^{H^{\prime}}H)\Box^{H}V\cong
A^{\prime}\Box^{H^{\prime}}\mathrm{Res}_{H^{\prime}}^{H}(V).$

induced by the canonical map $\kappa:A=B\otimes_{B}A\rightarrow
A^{\prime}\Box^{H^{\prime}}H$, the ring $R^{0}(X)$ acts on $K^{0}(X)$ as well.
The triple

$\bm{K}(X):=(R^{0}(X),K^{0}(X),\mathbb{1}_{K(X)})$

is an augmented ring, $\mathbb{1}_{K(X)}$ is the $K^{0}$-class of the right
module $B$. In this way, we get the desired functor $\bm{K}:k\text{-}\hskip
1.49994pt\mathcal{C}pct\mathcal{Q}uant\to\mathcal{A}ug\mathcal{R}ing$. ∎

######  Remark 6.5.

In the classical case every vector bundle is associated with a compact
Hausdorff principal bundle (for example, the bundle of orthonormal frames),
which implies that the augmentation map for the triple
$(R^{0}(X),K^{0}(X),\mathbb{1}_{K(X)})$ is surjective. In the quantum case, to
the best of our knowledge, it is an open problem to understand when a given
noncommutative vector bundle is associated to a noncommutative principal
bundle.

##  7\. The Atiyah–Todd picture

###  7.1. The classical picture revisited

The classical result of Atiyah–Todd says that the $K_{0}$-group
$K^{0}(\mathbb{CP}^{n})=K_{0}(C(\mathbb{CP}^{n}))$ equipped with the ring
structure defined via the tensor product of vector bundles over
$\mathbb{CP}^{n}$ (or, equivalently, the tensor product of finitely generated
projective left $C(\mathbb{CP}^{n})$-modules, which can be regarded as
symmetric $C(\mathbb{CP}^{n})$-bimodules) fits into the following commutative
square of rings:

$\mathbb{Z}[t,t^{-1}]$$R(U(1))$$\mathbb{Z}[x]/(x^{n+1})$$K^{0}(\mathbb{CP}^{n})$$\cong$$\cong$
(7.1)

Here the left vertical arrow is given by $t\mapsto 1+x$, the right vertical
arrow is induced by the associated vector bundle construction, the top
isomorphism maps $t$ into the fundamental representation of $U(1)$ in the
representation ring $R(U(1))$, and the bottom isomorphism maps $x$ to the
K-theory element $[\mathrm{L}_{1}]-[1]$, where $\mathrm{L}_{1}$ denotes the
Hopf line bundle on $\mathbb{CP}^{n}$ associated with the fundamental
representation of $U(1)$. Below, for any $k\in\mathbb{Z}$, we denote by
$\mathrm{L}_{k}$ the $k$-th tensor power of $\mathrm{L}_{1}$ when $k$ is non-
negative, and the $|k|$-th tensor power of $\mathrm{L}_{-1}$ when $k$ is
negative and where $\mathrm{L}_{-1}$ is the Hopf line bundle on
$\mathbb{CP}^{n}$ associated with the dual of the fundamental representation
of $U(1)$. Equivalently, $\mathrm{L}_{k}$ is the Hopf line bundle on
$\mathbb{CP}^{n}$ associated with the $k$-th tensor power of the fundamental
representation of $U(1)$, where negative tensor powers refer to tensor powers
of the dual of the fundamental representation of $U(1)$.

Since the elements $(1+x)^{k}$, $k=0,\ldots,n$, form a basis of the free
$\mathbb{Z}$-module $\mathbb{Z}[x]/(x^{n+1})$ and the assignment
$(1+x)\mapsto[\mathrm{L}_{1}]$ gives an isomorphism of rings, the classes

$[\mathrm{L}_{0}],\ldots,[\mathrm{L}_{n}]$ (7.2)

form a basis of the free $\mathbb{Z}$-module $K^{0}(\mathbb{CP}^{n})$. We call
this basis the shifted Atiyah–Todd basis of $K^{0}(\mathbb{CP}^{n})$ whose
name refers to the standard Atiyah–Todd basis $x^{k}$, $k=0,\ldots,n$ replaced
by the basis $(1+x)^{k}$, $k=0,\ldots,n$.

Our next step is to unravel how the classes $[\mathrm{L}_{k}]$, for $k=-1$ or
$k=n+1$, can be expressed in the shifted Atiyah–Todd basis.

Note first that the equality

$0=x^{n+1}=((1+x)-1)^{n+1}=\sum_{k=0}^{n+1}(-1)^{n+1-k}\left({{n+1}\atop{k}}\right)(1+x)^{k}$
(7.3)

in $\mathbb{Z}[x]/(x^{n+1})$ translates to the equality

$\sum_{k=0}^{n+1}(-1)^{n+1-k}\left({{n+1}\atop{k}}\right)[\mathrm{L}_{k}]=0$
(7.4)

in $K^{0}\left(\mathbb{CP}^{n}\right)$. Thus we obtain

$[\mathrm{L}_{n+1}]=\sum_{k=0}^{n}(-1)^{n-k}\left({{n+1}\atop{k}}\right)[\mathrm{L}_{k}],$
(7.5)

which we will refer to as the primary shifted Atiyah–Todd identity.

Furthermore, since $(1+x)$ is invertible in $\mathbb{Z}[x]/(x^{n+1})$ and the
initial equality (7.3) can be rewritten as

$(1+x)\sum_{k=1}^{n+1}(-1)^{1-k}\left({{n+1}\atop{k}}\right)(1+x)^{k-1}=1,$

we obtain

$(1+x)^{-1}=\sum_{k=1}^{n+1}(-1)^{1-k}\left({{n+1}\atop{k}}\right)(1+x)^{k-1}=\sum_{k=0}^{n}(-1)^{k}\left({{n+1}\atop{k+1}}\right)(1+x)^{k}$
(7.6)

in $\mathbb{Z}[x]/(x^{n+1})$. This equality translates to
$K^{0}(\mathbb{CP}^{n})$ as

$[\mathrm{L}_{-1}]=\sum_{k=0}^{n}(-1)^{k}\left({{n+1}\atop{k+1}}\right)[\mathrm{L}_{k}].$
(7.7)

We will refer to (7.7) as the secondary shifted Atiyah–Todd identity.

###  7.2. The quantum picture

In [1, Prop. 3.3 and 3.4], only the additive version of the bottom isomorphism
in the diagram (7.1) was established for the Vaksman–Soibelman quantum complex
projective spaces $\mathbb{CP}^{n}_{q}$. It yields a noncommutative version of
the Atiyah–Todd basis (7.2). All this seems interesting because Atiyah–Todd’s
method to prove the existence of the commutative diagram (7.1) uses the ring
structure of K-theory, which is missing in the noncommutative setting.

Analogues of the Atiyah–Todd identities (7.5) and (7.7) were proved in [8] for
$\mathbb{CP}^{n}_{q}$ and multipullback quantum projective spaces
$\mathbb{CP}^{n}_{H}$. In each of these two examples one has a $U(1)$-quantum
principal bundle $\mathbb{S}^{2n+1}_{q}\to\mathbb{CP}^{n}_{q}$ and
$\mathbb{S}^{2n+1}_{H}\to\mathbb{CP}^{n}_{H}$, and for each $k$ a module over
the algebra of the base space associated to the representation $z\mapsto
z^{k}$ of $U(1)$. This module will be denoted by $L_{k}$ for both quantum
projective spaces.

######  Theorem 7.1 ([8]).

If $X$ is either $\mathbb{CP}^{n}_{q}$ or $\mathbb{CP}^{n}_{H}$, for every
$n\in\mathbb{N}$ we have noncommutative analogues of the shifted Atiyah–Todd
basis and identities:

$\displaystyle K^{0}(X)$ $\displaystyle=\bigoplus_{k=0}^{n}\mathbb{Z}[L_{k}],$
$\displaystyle[L_{n+1}]$
$\displaystyle=\sum_{k=0}^{n}(-1)^{n-k}\left({{n+1}\atop{k}}\right)[L_{k}],$
$\displaystyle[L_{-1}]$
$\displaystyle=\sum_{k=0}^{n}(-1)^{k}\left({{n+1}\atop{k+1}}\right)[L_{k}].$

If $X$ one of the two quantum projective spaces above, we will consider the
k-topology on $X$ consisting of the single compact quantum principal bundle
$U\to X$, where $U$ is either $\mathbb{S}^{2n+1}_{q}$ or
$\mathbb{S}^{2n+1}_{H}$, and $G=U(1)$. Using the previous theorem, we can
prove the following interesting property of these k-compact quantum spaces.

######  Proposition 7.2.

If $X$ one of the two k-compact quantum spaces above, the augmentation map of
$\bm{K}(X)$ is surjective.

###### Proof.

By definition of $R^{0}(X)$, it is enough to show that the map $R(U(1))\to
K^{0}(X)$ given by the action on $\mathbb{1}_{K(X)}$ is surjective.

Each $L_{k}$ is a noncommutative associated vector (line) bundle. Its class
$[L_{k}]$ in K-theory comes from an element of
$R(U(1))\cong\mathbb{Z}[t,t^{-1}]$. Since such elements generate K-theory, the
map under consideration is surjective.

∎

## Acknowledgements

This work is partially supported by NCN grant UMO2021/41/B/ST1/03387 as a part
of the project ”Applications of graph algebras and higher-rank graph algebras
in noncommutative geometry”.

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