# Categorical aspects of compact quantum groups

Alexandru Chirvasitu111UC Berkeley<EMAIL_ADDRESS>
###### Abstract

We show that either of the two reasonable choices for the category of compact
quantum groups is nice enough to allow for a plethora of universal
constructions, all obtained “by abstract nonsense” via the adjoint functor
theorem. This approach both recovers constructions which have appeared in the
literature, such as the quantum Bohr compactification of a locally compact
semigroup, and provides new ones, such as the coproduct of a family of compact
quantum groups, and the compact quantum group freely generated by a locally
compact quantum space. In addition, we characterize epimorphisms and
monomorphisms in the category of compact quantum groups.

Keywords: compact quantum group, CQG algebra, presentable category, SAFT
category, adjoint functor theorem

###### Contents

  1. 1 Preliminaries
    1. 1.1 Compact quantum groups
    2. 1.2 CQG algebras
    3. 1.3 Locally compact and algebraic quantum spaces and semigroups
    4. 1.4 SAFT categories and the adjoint functor theorem
    5. 1.5 Finitely presentable categories
  2. 2 The category of CQG algebras is finitely presentable
  3. 3 $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is SAFT
  4. 4 Applications
    1. 4.1 Limits in $\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
    2. 4.2 Quantum groups generated by quantum spaces
    3. 4.3 Variations on the Bohr compactification theme
    4. 4.4 Kac quotients
  5. 5 Monomorphisms

## Introduction

Compact quantum groups were introduced in essentially their present form in
[Wor87] (albeit under a different name), and the area has been expanding
rapidly ever since. The subject can be viewed as part of Connes’ general
program [Con94] to make “classical” notions (spaces, topology, differential
geometry) non-commutative: One recasts compact groups as $C^{*}$-algebras via
their algebras of continuous functions, and then removes the commutativity
assumption from the definition (see § 1.1 for details). Starting with the
relatively simple resulting definition, all manner of compact-group-related
notions and constructions can then be generalized to the non-commutative
setting: Peter-Weyl theory ([Wor87, Wor98]), Tannaka-Krein duality and
reconstruction ([Wor88, Wan97]), Pontryagin duality ([PW90]), actions on
operator algebras ([Boc95, Wan98, Wan99]) and other structures, such as
(classical or quantum) metric spaces ([Ban05, QS10]) or graphs ([Bic03]), and
so on. This list is not (and cannot be) exhaustive.

The goal of this paper is to analyze compact quantum groups from a category-
theoretic perspective, with a view towards universal constructions.

Bits and pieces appear in the literature: In [Wan95], Wang constructs
coproducts in the category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (see §
1.1), opposite to that of compact quantum groups (it consists of
$C^{*}$-algebras, which are morally algebras of functions on the non-existent
quantum groups). More generally, he constructs other types of colimits (e.g.
pushouts) of diagrams with one-to-one connecting morphisms. The fact that this
coproduct can be constructed simply at the level of $C^{*}$-algebras
(forgetting about comultiplications) parallels the fact that classically, the
underlying space of a categorical product $\prod G_{i}$ of compact groups
$G_{i}$ is, as a set, just the ordinary Cartesian product. The category of
compact groups, however, also admits coproducts, and they are slightly more
difficult to construct: One endows the ordinary, discrete coproduct $\coprod
G_{i}$ (i.e. coproduct in the category of discrete groups, also known as the
free product of the $G_{i}$) with the finest topology making the canonical
inclusions $G_{j}\to\coprod G_{i}$ continuous, and then takes the Bohr
compactification of the resulting topological group. It is natural, then, to
ask whether or not coproducts of compact quantum groups exist, or
equivalently, whether the category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
opposite to that of compact quantum groups has products. We will see in § 4.1
that this is indeed the case, and moreover, the category is complete (i.e. it
has all small limits).

Another example of universal construction that fits well within the framework
of this paper is the notion of quantum Bohr compactification [Soł05]. One of
the main results of that paper is, essentially, that the forgetful functor
from compact quantum groups to locally compact quantum semigroups has a left
adjoint; remembering that we are always passing from (semi)groups to algebras
of functions and hence reversing arrows, this amounts to the existence of a
certain right adjoint ([Soł05, 3.1,3.2]). Section 4 recovers this as one among
several right-adjoint-type constructions, such as compact quantum groups
“freely generated by a quantum space” (as opposed to quantum semigroup; see §
4.2).

Most compact-quantum-group-related universal constructions in the literature
seem to be of a “left adjoint flavor”: the already-mentioned colimits in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, the quantum automorphism groups
of, say, [Wan98], which are basically initial objects in the category of
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ objects endowed with a coaction on
a fixed $C^{*}$-algebra, etc. By contrast, apart from the Bohr
compactification mentioned in the previous paragraph, universal constructions
of the right adjoint flavor (limits in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, or right adjoints to functors with
domain $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) appear not to have received
much attention. This is all the more surprising for at least two reasons.
First, they seem to be more likely to exist than the other kind of universal
construction; example: a (unital, say) $C^{*}$-algebra $A$ endowed with a
coassociative map $A\to A\otimes A$ into its minimal tensor square (this would
be the object dual to a compact quantum semigroup) always has a compact
quantum group (meaning its dual object, as in Definition 1.1.1) mapping into
it universally, but does not, in general, have a compact quantum group
receiving a universal arrow from it (Remark 4.3.3). Secondly, the
representation-theoretic interpretation of limits in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is often simpler than that of
colimits; see Proposition 4.1.3 (especially part (a)) and surrounding
discussion.

The structure of the paper is as follows:

Section 1 recalls the machinery that will be used in the sequel and fixes
notations and conventions, introducing the two versions of the category
opposite that of compact quantum groups: $\operatorname{\mathrm{CQG}}$,
consisting of so-called CQG algebras (these are like the algebra of
representative functions on a compact group; see Definition 1.2.2), and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, whose objects are analogous to
algebras of continuous functions on compact groups (Definition 1.1.1).

In Section 2, Theorem 2.0.1 shows that the category
$\operatorname{\mathrm{CQG}}$ is finitely presentable (§ 1.5). This technical
property will later allow us to reduce the existence of right adjoints for
functors defined on $\operatorname{\mathrm{CQG}}$ to checking that these
functors are cocontinuous, i.e. preserve colimits. This is typically an easy
task, as the routine nature of most proofs in Section 4 shows.

Section 3 proves a property slightly weaker than finite presentability for the
category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (Theorem 3.0.1). The nice
features from the previous section are preserved however, and the same types
of results (existence of right adjoints to various functors defined on
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) follow.

In Section 4 we list some of the consequences of the previous two sections.
These include the automatic existence of limits in
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
(§ 4.1), compact quantum groups freely generated by quantum spaces and
semigroups (§§ 4.2 and 4.3 respectively), and universal Kac type compact
quantum groups associated to any given compact quantum group (§ 4.4).

Finally, in Section 5 we characterize monomorphisms in the categories
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
It turns out that in the former they have to be one-to-one, whereas in the
latter being mono is slightly weaker than injectivity (Proposition 5.0.1). The
results are analogous to the fact ([Rei70, Proposition 9]) that epimorphisms
of compact groups are surjective.

### Acknowledgements

This work is part of my PhD dissertation. I would like to thank my advisor
Vera Serganova for all the support, and Piotr Sołtan for helpful discussions
on the contents of [Soł05].

##  1 Preliminaries

All algebraic entities in this paper (algebras, coalgebras, bialgebras, etc.)
are complex. A $*$-algebra is, as usual, a complex algebra endowed with a
conjugate linear, involutive, algebra anti-automorphism ‘$*$’. Unless we are
dealing with non-unital $C^{*}$-algebras as in § 1.3 below, in which case the
reader will be warned, algebras are assumed to be unital (and coalgebras are
always counital).

Our main references for the necessary basics on coalgebra, bialgebra and Hopf
algebra theory are [Abe80, Mon93, Swe69]. The notation pertaining to
coalgebras is standard: $\Delta$ for antipodes and $\varepsilon$ for the
counit, perhaps adorned with the name of the coalgebra if we want to be more
precise (example: $\Delta_{C}$, $\varepsilon_{C}$). The same applies to
antipodes for Hopf algebras, which are usually denoted by $S$. We use Sweedler
notation both for comultiplication, as in $\Delta(c)=c_{1}\otimes c_{2}$, and
for comodule structures: If $\rho:M\to M\otimes C$ is a right $C$-comodule
structure, it will be written as $\rho(m)=m_{0}\otimes m_{1}$. All comodules
are right, and the category of right comodules over a coalgebra $C$ is denoted
by $\mathcal{M}^{C}$.

For any comodule $V$ over any coalgebra $H$ (the notation suggests that it
will become a Hopf algebra soon), there is a largest subcoalgebra $H(V)$ over
which $V$ is a comodule. If the comodule structure map is $\rho:V\to V\otimes
H$ and $(e_{i})_{i\in I}$ is a basis for $V$,then $H(V)$ is simply the span of
the elements $u_{ij}$ defined by

$\rho(e_{j})=\sum_{i}e_{i}\otimes u_{ij}.$

We refer to $u_{ij}$ as the coefficients of the basis $(e_{i})$, to $(u_{ij})$
as the coefficient matrix of the basis, and to $H(V)$ as the coefficient
coalgebra of $V$. The coalgebra structure is particularly simple on
coefficients:

$\Delta(u_{ij})=\sum_{k}u_{ik}\otimes
u_{kj},\quad\varepsilon(u_{ij})=\delta_{ij}.$

Henceforth, the standing assumption whenever we mention coefficients and
coefficient coalgebras is that the comodule in question is finite-dimensional
(that is, $I$ is finite). When $V$ is simple, the coefficients $u_{ij}$ with
respect to some basis are linearly independent, and the coefficient coalgebra
is a matrix coalgebra, in the sense that its dual is a matrix algebra.

######  1.0.1 Remark.

Note that maps $V\to V\otimes C$ are the same as elements of $V\otimes
V^{*}\otimes C=\operatorname{End}(V)\otimes C$. If $u=(u_{ij})$ is the
coefficient matrix of a basis $e_{i}$, $i=\overline{1,n}$ for $V$ and
$\operatorname{End}(V)$ is identified with $M_{n}$ via the same basis $e_{i}$,
then the element of $\operatorname{End}(V)\otimes C\cong M_{n}(C)$
corresponding to the coaction is exactly the coefficient matrix $u$. We will
often blur the distinction between these two points of view, and might refer
to $u$ itself as the comodule structure. $\blacklozenge$

If in the above discussion $H$ is a Hopf algebra, more can be said: The matrix
$S(u_{ij})_{i,j}$ is inverse to $(u_{ij})_{i,j}$. Moreover, giving the dual
$V^{*}$ the usual right $H$-comodule structure

$\langle f_{0},v\rangle f_{1}=\langle f,v_{0}\rangle S(v_{1}),\quad v\in V,\
f\in V^{*},$

the coefficient matrix of the basis dual to $(e_{i})$ is precisely
$(S(u_{ji}))_{i,j}$ (note the flipped indices).

A word on tensor products: In this paper, the symbol ‘$\otimes$’ means at
least three things. When appearing between purely algebraic objects, such as
algebras or just vector spaces, it is the usual, algebraic tensor product.
Between $C^{*}$-algebras it always means the minimal, or injective tensor
product ([Tak02, IV.4]). Finally, on rare occasions, we use the so-called
spatial tensor product (referred to as $W^{*}$-tensor product in [Tak02,
IV.5]) between von Neumann (or $W^{*}$) algebras. It will always be made clear
what the nature of the tensored objects is, so that no confusion is likely to
arise.

###  1.1 Compact quantum groups

This is by now a very rich and well-referenced theory, so we will be very
brief, and will refer the reader to one of the many excellent sources (e.g.
the papers and book cited below and the references therein) for details on the
topic.

No list of references would be complete without mentioning the seminal papers
[Wor87, Wor88], where Woronowicz laid the foundation of the subject,
introducing the main characters under the name “compact matrix pseudogroups”,
while an exposition of the main features of the theory is given by the same
author in [Wor98]. Other good references are the survey paper [KT99], and
[KS97, 11.4].

As mentioned in the introduction, the main idea is that since one can study
compact groups by means of the algebras of continuous functions on them, which
are commutative, unital $C^{*}$-algebras with some additional structure,
dropping the commutativity assumption but retaining the extra structure should
still lead to interesting objects, which are trying to be “continuous
functions on a quantum group”. The additional structure just alluded to is
captured in the following definition ([KT99, 3.1.1]):

######  1.1.1 Definition.

A compact quantum group is a pair $(A,\Delta)$, where $A$ is a unital
$C^{*}$-algebra, and $\Delta:A\to A\otimes A$ is a morphism of unital
$C^{*}$-algebras satisfying the conditions

  1. 1.

(Coassociativity)
$(\Delta\otimes\operatorname{id})\circ\Delta=(\operatorname{id}\otimes\Delta)\circ\Delta$;

  2. 2.

(Antipode) The subspaces $\Delta(A)(1\otimes A)$ and $\Delta(A)(A\otimes 1)$
are dense in $A\otimes A$.

$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is the category whose objects are
compact quantum groups, and whose morphisms $f:A\to B$ are unital
$C^{*}$-algebra maps preserving the comultiplication in the sense that
$(f\otimes f)\circ\Delta_{A}=\Delta_{B}\circ f$. $\blacklozenge$

The second condition needs some explanation. The space $\Delta(A)(1\otimes A)$
is defined as the linear span of products of the form $\Delta(a)(1\otimes
b)\in A\otimes A$, and similarly for $\Delta(A)(A\otimes 1)$. The condition is
named ‘antipode’ because it has to do with the demand that $A$, regarded as a
kind of bialgebra, have something like an antipode. For comparison, consider
the case of an ordinary, purely algebraic bialgebra $B$. The condition that it
have an antipode, i.e. that it be a Hopf algebra, is equivalent to the
requirement that the map

$B\otimes B\to B\otimes B,\quad x\otimes y\mapsto x_{1}\otimes x_{2}y$

be a bijection.

Since compact quantum groups as defined above are morally functions of
algebras, representations of the group must be comodules of some sort over the
corresponding algebras (endowed with their comultiplication). Some care must
be taken, as Definition 1.1.1 makes no mention of a counit, and so the usual
definition of comodule has to be modified slightly. The solution is (see
[KT99, discussion before Proposition 3.2.1]):

######  1.1.2 Definition.

Let $(A,\Delta)$ be a compact quantum group. A finite-dimensional comodule
over $A$ is a finite-dimensional complex vector space $V$ together with a
coassociative coaction $\rho:V\to V\otimes A$ such that the corresponding
element of $\operatorname{End}(V)\otimes A$ is invertible. $\blacklozenge$

We will often drop the adjective ‘finite-dimensional’. Remark 1.0.1 applies,
and we will often refer to the coefficient comatrix of some basis as being the
comodule structure. [KS97, 11.4.3, Lemma 45] says that any comodule is
unitarizable, in the sense that there is an inner product on $V$ making the
coefficient matrix $u\in M_{n}(A)$ of an orthonormal basis unitary (cf.
Definition 1.2.1).

###  1.2 CQG algebras

These are the algebraic counterparts of compact quantum groups. More
precisely, if a compact quantum group as in Definition 1.1.1 plays the role of
the algebra of continuous functions on a “quantum group”, then the associated
CQG algebra wants to be the algebra of representative functions of the quantum
group, i.e. matrix coefficients of finite-dimensional unitary representations.
The main reference for this subsection is [KS97, 11.1-4].

Recall that a Hopf $*$-algebra $H$ is a Hopf algebra with a $*$-structure
making $H$ into a $*$-algebra, and such that the comultiplication and counit
are morphisms of $*$-algebras. This is the kind of structure that allows one
to define what it means for a representations of a quantum group (i.e. a
comodule over the corresponding “function algebra”) to be unitary.

Let $V$ be an $n$-dimensional comodule over a Hopf $*$-algebra $H$.

######  1.2.1 Definition.

If $(\ \mid\ )$ is an inner product on $V$, the pair $(V,(\ \mid\ ))$ is said
to be unitary provided the coefficients $u_{ij}$ of an orthonormal basis
$e_{i}$, $i=\overline{1,n}$ form a unitary matrix in $H$.

A comodule $V$ is said to be unitarizable if there exists an inner product
making it unitary. This is equivalent to saying that for any basis $(e_{i})$,
the coefficient matrix $(u_{ij})_{i,j}$ can be made unitary by conjugating it
with a scalar $n\times n$ matrix. $\blacklozenge$

This is [KS97, Definition 5], and it is the correct compatibility condition
for a comodule structure and an inner product. See also [KS97, 11.1.5,
Proposition 11] for alternative characterizations of unitary comodules. We are
now ready to recall the main definition of this subsection ([KS97, 11.3.1,
Definition 9]):

######  1.2.2 Definition.

A CQG algebra is a Hopf $*$-algebra which is the linear span of the
coefficient matrices of its unitarizable (or equivalently, unitary) finite-
dimensional comodules.

The category having CQG algebras as objects and Hopf $*$-algebra morphisms as
arrows will be denoted by $\operatorname{\mathrm{CQG}}$. $\blacklozenge$

######  1.2.3 Remark.

It is a simple but useful observation that a quotient Hopf $*$-algebra of a
CQG algebra is automatically CQG. Indeed, a morphism of Hopf $*$-algebras will
turn a unitary coefficient matrix into another such. $\blacklozenge$

Let us recall that CQG algebras are automatically cosemisimple [KS97, 11.2],
i.e. their categories of comodules are semisimple. Another way to say this is
that a CQG algebra is the direct sum of its matrix subcoalgebras.

The following example will play an important role in Section 2. It is a family
of “universal” CQG algebras, in a sense that will be made precise below (see
[KS97, 11.3.1, Example 6], or [VDW96], where these objects were introduced in
their $C^{*}$-algebraic incarnation).

######  1.2.4 Example.

Let $Q\in GL_{n}(\mathbb{C})$ be a positive operator, and denote by $A_{u}(Q)$
the $*$-algebra freely generated by elements $u_{ij}$, $i,j=\overline{1,n}$
subject to the relations making both $u=(u_{ij})_{i,j}$ and $\displaystyle
Q^{\frac{1}{2}}\overline{u}Q^{-\frac{1}{2}}$ unitary, where
$\overline{u}=(u^{*}_{ij})_{i,j}$. Strictly speaking, the main character here
is the pair $(A_{u}(Q),u)$ rather than just $A_{u}(Q)$: We always assume the
$u_{ij}$ are fixed as part of the structure, and refer to them as the standard
generators of $A_{u}(Q)$.

One way to state the universality property mentioned above is: For any CQG
algebra $A$ and any unitary coefficient matrix $v=(v_{ij})$ satisfying
$S^{2}(v)=QvQ^{-1}$, the map $u_{ij}\mapsto v_{ij}$ lifts to a unique CQG
algebra morphism $A_{u}(Q)\to A$.

Note that $A_{u}(Q)$ has a standard $n$-dimensional unitary comodule with
orthonormal basis $(e_{i})_{i=1}^{n}$, with the obvious structure
$e_{j}\mapsto\sum_{i}e_{i}\otimes u_{ij}$. $\blacklozenge$

There are various functors going back and forth between
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
First, since a CQG algebra is generated by elements of unitary matrices, there
is, for any element of the algebra, a uniform bound on the norm that element
can have when acting on any Hilbert space. It follows that any CQG algebra $A$
has an enveloping $C^{*}$-algebra $\overline{A}$. The fact that the
comultiplication and counit of the CQG algebra lift to give $\overline{A}$ a
compact quantum group structure follows from the universality property of this
envelope, as does the functoriality of this construction ([KS97, 11.3.3]).
This functor will be denoted by
$\operatorname{\textsc{univ}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.

On the other hand, for any compact quantum group $B$, the coefficients of all
comodules (Definition 1.1.2) span a sub-Hopf $*$-algebra of $B$ in the obvious
sense, and again, the construction is easily seen to be functorial. We denote
this functor by
$\operatorname{\textsc{alg}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{CQG}}$.
Moreover, $\operatorname{\textsc{alg}}(B)$ is the only dense sub-Hopf
$*$-algebra of $B$ ([BMT01, A.1]).

Functors constructed in some natural way and going in opposite directions are
in the habit of being adjoints, and this situation is no different:
$\operatorname{\textsc{univ}}$ is the left adjoint. As it happens,
$\operatorname{\textsc{alg}}$ almost has right adjoint too. ‘Almost’ because
only its restriction to the category of CQG algebras and one-to-one morphisms
has a right adjoint, $\operatorname{\textsc{red}}$, associating to each CQG
algebra $A$ the so-called reduced [BMT01, $\S$2] compact quantum group having
$A$ as its dense sub-Hopf $*$-algebra. It is the “smallest” such object, in
the sense that any compact quantum group $A$ admits a unique surjective
morphism $A\to\operatorname{\textsc{red}}(\operatorname{\textsc{alg}}(A))$
which restricts to the identity on $\operatorname{\textsc{alg}}(A)$. A
detailed discussion on the interplay between the three functors mentioned in
this paragraph can be found in [BKQ11, 6.2].

It is probably clear by now that (the opposites of)
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are “the two reasonable choices for the category of compact quantum groups” of
the abstract. Which one is most convenient in any given case depends on which
aspects of the theory one wishes to focus on. For representation-theoretic
purposes, $\operatorname{\mathrm{CQG}}$ seems to be the correct choice, since
the CQG algebra $\operatorname{\textsc{alg}}(B)$ discussed above is tailor-
made to capture all information about unitary $B$-comodules. On the other
hand, there are purely analytic concepts (coamenability [BMT01]) whose very
definition requires the use of $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.

In this paper, the $\operatorname{\mathrm{CQG}}$ vs.
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ distinction is a matter of
technical necessity. For various reasons having to do with the topological
aspect of being a $C^{*}$ rather than a $*$-algebra,
$\operatorname{\mathrm{CQG}}$ is the easier category to work with, as should
be apparent from the announced results once we review the necessary category
theory: The title of Section 2 (see § 1.5) is stronger than that of Section 3
(§ 1.4).

###  1.3 Locally compact and algebraic quantum spaces and semigroups

One of the themes that will be explored in Section 4 is, very roughly, the
existence of “compact quantum groups freely generated by quantum objects”.
Here, ‘objects’ can be things like ‘semigroups’ or ‘spaces’. Keeping in mind
that we are placing ourselves in the dual picture, where spaces are explored
through functions on them, we recall in this subsection how non-unital
$C^{*}$-algebras or plain $*$-algebras allow one to formalize such notions.

A good, brief account of more or less everything we need for the locally
compact side of the picture can be found in the ‘Notations and conventions’
section of [KV00] (assuming rudiments on multiplier algebras of
$C^{*}$-algebras [Tak02, III.6]).

Recall that for not-necessarily-unital $C^{*}$-algebras $A$ and $B$, a
morphism from $A$ to $B$ is by definition a continuous $*$-algebra
homomorphism $f:A\to M(B)$ into the multiplier algebra of $B$ which is non-
degenerate, meaning that the space $f(A)B$ is dense in $B$. It is then
explained in [KV00] how two such creatures can be composed, meaning that non-
unital $C^{*}$-algebras together with morphisms as defined above constitute a
category denoted here by $\operatorname{\mathrm{C}^{*}_{0}}$. It is to be
thought of as the category dual to that of locally compact quantum spaces.
$\operatorname{\mathrm{C}^{*}}$ is the subcategory consisting of unital
$C^{*}$-algebras. Note that the non-degeneracy condition on morphisms
automatically makes a $\operatorname{\mathrm{C}^{*}_{0}}$ arrow between
objects of $\operatorname{\mathrm{C}^{*}}$ unital.

For the definition of locally compact quantum semigroups we follow [Soł05],
referring again to the preliminary section of [KV00] for the missing details
on compositions of morphisms in $\operatorname{\mathrm{C}^{*}_{0}}$ (needed to
make sense of the coassociativity condition below).

######  1.3.1 Definition.

A locally compact quantum semigroup is a pair $(A,\Delta)$, where $\Delta:A\to
A\otimes A$ is a morphism in $\operatorname{\mathrm{C}^{*}_{0}}$,
coassociative in the obvious sense.

The category $\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ has locally
compact quantum semigroups as objects, and
$\operatorname{\mathrm{C}^{*}_{0}}$-morphisms compatible with
comultiplications as arrows. $\blacklozenge$

Turning now to the algebraic side, everything just said has a natural
analogue. We again have to deal with multiplier algebras, this time of not-
necessarily-unital $*$-algebras; [VD94, Appendix] provides sufficient
background, and we will freely use the results and terminology therein. All
$*$-algebras are assumed to be non-degenerate, in the sense that $ab=0$,
$\forall b$ implies $a=0$ (the $*$-structure makes this condition symmetric).

For $*$-algebras $A$ and $B$, a morphism $A\to B$ is by definition a
$*$-homomorphism $f:A\to M(B)$, non-degenerate in the sense that $f(A)B$ spans
$B$. Composition goes through essentially as in the $C^{*}$ case, and we thus
get a category $\operatorname{\mathrm{Alg}^{*}_{0}}$. As before, the full
subcategory $\operatorname{\mathrm{Alg}^{*}}$ on unital $*$-algebras has only
unital morphisms as arrows. The algebraic counterpart to Definition 1.3.1 is

######  1.3.2 Definition.

An algebraic quantum semigroup is a pair $(A,\Delta)$, where $\Delta:A\to
A\otimes A$ is a coassociative morphism in
$\operatorname{\mathrm{Alg}^{*}_{0}}$.

The category $\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ has algebraic
quantum semigroups as objects and and
$\operatorname{\mathrm{Alg}^{*}_{0}}$-morphisms compatible with
comultiplications as arrows. $\blacklozenge$

######  1.3.3 Remark.

Whatever results we prove below within the framework of Definition 1.3.2,
close analogues exist for plain complex algebras rather than $*$-algebras. I
believe this one example is sufficient to illustrate how the universal
constructions of Section 4 go through in the algebraic, as well as the
$C^{*}$-algebraic setting. $\blacklozenge$

###  1.4 SAFT categories and the adjoint functor theorem

As Section 4 below is all about showing that certain functors have adjoints,
in this subsection and the next we recall the categorical machinery involved
in this. The main reference here is [ML98]. The set of morphisms $x\to y$ in a
category $\mathcal{C}$ will be denoted by $\mathcal{C}(x,y)$. Recall that
categories with all (co)limits (always small in this paper) are said to be
(co)complete, and functors preserving those (co)limits are called
(co)continuous (so ‘complete’ here means the same thing as Mac Lane’s ‘small-
complete’ [ML98, V]).

A class $S$ of objects in a category is said to be a generator (or a
generating class) if any two distinct parallel arrows $f\neq g:y\to z$ stay
distinct upon composition with an arrow $S\ni x\to y$ ([ML98, V.7]). We call
category generated if there is a generating set (as opposed to a proper
class).

An arrow $f:x\to y$ in a category $\mathcal{C}$ is an epimorphism if arrows
out of $y$ are uniquely determined by their “restriction to $x$” via
composition with $f$ ([ML98, I.5]). The quotient objects of $x$ are the
epimorphisms with source $x$, identified up to isomorphism in the comma
category $x\downarrow\mathcal{C}$ of arrows with source $x$ [ML98, II.6].
Finally, $\mathcal{C}$ is said to be co-wellpowered if for every object $x$,
the class of quotient objects of $x$ is actually a set.

We explained above how the aim is to construct things like “the compact
quantum group freely generated by a quantum semigroup”. What this means,
precisely, remembering that we are working with algebra-of-functions-type
objects, is that we want a right adjoint to, say, the inclusion functor
$\iota:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$
(this is just one example; there is also a $C^{*}$ version). Typically, when
trying to show that a functor $\iota$ is a left adjoint, one needs to check
(1) that $\iota$ is cocontinuous (this is certainly necessary, as left
adjoints are always cocontinuous) and (2) that some kind of solution set
condition is satisfied [ML98, V.6.2]. For some categories, however, (2) is
unnecessary: they are such that any cocontinuous functor out of them is
automatically a left adjoint. One sufficient set of conditions that will
ensure this is provided by the following result, due to Freyd and referred to
in the literature as the special adjoint functor theorem (dual to [ML98,
V.8.2]):

######  1.4.1 Theorem.

Let $\mathcal{C}$ be a cocomplete, generated, and co-wellpowered category.
Then, any cocontinuous functor with domain $\mathcal{C}$ is a left adjoint.

In view of this result, it is natural to isolate the hypotheses:

######  1.4.2 Definition.

A category is SAFT if it is cocomplete, generated, and co-wellpowered.
$\blacklozenge$

######  1.4.3 Remark.

Not-necessarily-cocomplete categories satisfying the adjoint functor theorem
in the sense that functors are only required to preserve those colimits which
exist are called ‘compact’ in [Kel86]. One important property of compact (and
hence SAFT) categories is that they are automatically complete. This is, for
example, the implication (ii) $\Rightarrow$ (v) in [Kel86, Theorem 5.6].
$\blacklozenge$

###  1.5 Finitely presentable categories

One way to be SAFT is to be what in the literature is called ‘locally
presentable’. We review the main features of the theory here, and refer mainly
to [AR94] for details. For brevity, in this paper we drop the word ‘locally’.

A poset $(J,\leq)$ is said to be filtered if every finite subset is majorized
by some element (this is [AR94, 1.4], restricted to posets as opposed to
arbitrary categories). In the sequel, a filtered diagram in a category
$\mathcal{C}$ is a functor a $(J,\leq)\to\mathcal{C}$, and a filtered colimit
is a colimit of such a functor. Then, [AR94, 1.9] is (essentially, via [AR94,
1.5]):

######  1.5.1 Definition.

An object $x\in\mathcal{C}$ is finitely presentable if the functor
$\mathcal{C}(x,-)$ preserves filtered colimits.

The category $\mathcal{C}$ is finitely presentable if it is cocomplete, and
there is a set $S$ of finitely presentable objects such that every object in
$\mathcal{C}$ is a filtered colimit of objects in $S$. $\blacklozenge$

More rigorously, the last condition says that every object is the colimit of a
functor $F:J\to\mathcal{C}$ taking values in $S$, with $(J,\leq)$ filtered.

######  1.5.2 Example.

All categories familiar from algebra, of the form ‘set with this or that kind
of structure’, such as groups, abelian groups, monoids, semigroups, algebras,
$*$-algebras, modules over a ring, etc. are finitely presentable. These are
the so-called finitary varieties of algebras [AR94, 3.A], ‘finitary’ having to
do with ‘finitely presentable’.

The terminology is also inspired by such examples. In the category of modules
over a ring, say, an object is finitely presentable in the above abstract
sense if and only if it has a finite presentation in the usual sense (in full
generality, the result is [AR94, 3.11]). $\blacklozenge$

What matters here is that as mentioned above, finitely presentable implies
SAFT. Indeed, cocompleteness is part of Definition 1.5.1, and the generating
set required for SAFT-ness almost is: $S$ is easily seen to be a generator.
Co-wellpowered-ness, on the other hand, is the difficult result [AR94, 1.58].

The following proposition is the criterion of finite presentability we use in
the proof of the main result of Section 2. It is a consequence of [AR94, 1.11]
(via 0.5, 0.6 of op. cit.), and in order to state it, one more piece of
terminology is needed.

######  1.5.3 Definition.

We will say that a generator $S$ of a cocomplete category $\mathcal{C}$ is
regular if every object of $\mathcal{C}$ is the coequalizer of two parallel
arrows $f,g:y\to z$, where $y$ and $z$ are coproducts of objects in $S$.
$\blacklozenge$

######  1.5.4 Proposition.

A cocomplete category with a regular generator consisting of finitely
presentable objects is finitely presentable.

##  2 The category of CQG algebras is finitely presentable

This section is devoted to proving the result in the title:

######  2.0.1 Theorem.

The category $\operatorname{\mathrm{CQG}}$ is finitely presentable.

The main tool in the proof is Proposition 1.5.4, according to which
cocompleteness (by now well known) is first on the agenda.

######  2.0.2 Proposition.

$\operatorname{\mathrm{CQG}}$ is cocomplete.

######  Proof.

It is enough to show that the category has coproducts and coequalizers of
parallel pairs of arrows [ML98, V.2.1]. Both will be constructed as simply the
colimits of the underlying diagrams of $*$-algebras (i.e. coalgebra structures
play no role in the construction of colimits).

Coproducts are essentially constructed in [Wan95, Theorem 1.1]. That result is
concerned with the $C^{*}$-algebraic version (constructing coproducts in the
category $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$), but as remarked by Wang
at the end of [Wan95, $\S$1], the algebraic version holds as well. Given a set
of CQG agebras, the universal property of the coproduct of underlying algebras
gives this coproduct the extra structure that will make it into a CQG algebra;
we leave the details to the reader.

To construct coequalizers, let $f,g:A\to B$ be morphisms of CQG algebras. The
ideal $I$ generated by the elements $f(a)-g(a)$, $a\in A$ is in fact a
coideal, as well as invariant under $*$. The latter assertion is trivial, so
let us focus on $I$ being a coideal. Compatibiity of $f$ and $g$ with counits
says that $f(a)-g(a)$ is annihilated by $\varepsilon_{B}$. On the other hand,
the familiar computation

$\displaystyle\Delta_{B}(f(a)-g(a))=$ $\displaystyle\ (f\otimes
f)(\Delta_{A}(a))-(g\otimes g)(\Delta_{A}(a))$ $\displaystyle=$
$\displaystyle\ f(a_{1})\otimes f(a_{2})-g(a_{1})\otimes g(a_{2})$
$\displaystyle=$ $\displaystyle\
f(a_{1})\otimes(f-g)(a_{2})+(f-g)(a_{1})\otimes g(a_{2})\in B\otimes
I+I\otimes B$

shows that $I$ plays well with the comultiplication. It follows that the
coequalizer of $f$ and $g$ in $\operatorname{\mathrm{Alg}^{*}}$ is a quotient
Hopf $*$-algebra of $B$, and hence a CQG algebra by Remark 1.2.3.
$\blacksquare$

The plan now is to show that the set $S$ consisting of the CQG algebras
$A_{u}(Q)$ of Example 1.2.4 (for all possible positive operators $Q$, of all
possible sizes) satisfies the hypotheses of Proposition 1.5.4: Every
$A_{u}(Q)$ is finitely presentable in $\operatorname{\mathrm{CQG}}$ in the
sense of Definition 1.5.1, and $S$ is a regular generator. We start with the
former.

######  2.0.3 Proposition.

For any positive $Q\in GL_{n}(\mathbb{C})$, the object
$A=A_{u}(Q)\in\operatorname{\mathrm{CQG}}$ is finitely presentable.

######  Proof.

Let $(J,\leq)$ be a filtered poset, and let $A_{j}$, $j\in J$ implement a
functor $J\to\operatorname{\mathrm{CQG}}$ by means of CQG algebra morphisms
$\iota_{j^{\prime}j}:A_{j}\to A_{j^{\prime}}$ for $j\leq j^{\prime}$. Denote
also by $\displaystyle\iota_{i}:A_{i}\to B=\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}} {}{}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}A_{j}$ the
structural morphisms into the colimit. We have to show that the canonical map

$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}\operatorname{\mathrm{CQG}}(A,A_{j})\to\operatorname{\mathrm{CQG}}(A,B)$
(1)

is a bijection.

It is clear from the description in Example 1.2.4 that $A$ is finitely
presented as a $*$-algebra, and is hence a finitely presentable object in
$\operatorname{\mathrm{Alg}^{*}}$ ([AR94, 3.11] with $\lambda=\aleph_{0}$). In
conclusion, the canonical map

$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}\operatorname{\mathrm{Alg}^{*}}(A,A_{j})\to\operatorname{\mathrm{Alg}^{*}}(A,B)$
(2)

is bijective, and the injectivity of 1 follows from the commutative square

$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-3.4pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-0.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@moveto{7.14445pt}{-0.2pt}\pgfsys@moveto{6.14445pt}{-3.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@lineto{5.68446pt}{-3.2pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-3.2pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\operatorname{\mathrm{CQG}}(A,A_{j})$$\operatorname{\mathrm{CQG}}(A,B)$$\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-3.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-0.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@moveto{7.14445pt}{-0.2pt}\pgfsys@moveto{6.14445pt}{-3.2pt}\pgfsys@moveto{-6.14445pt}{-3.2pt}\pgfsys@lineto{5.68446pt}{-3.2pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-3.2pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\operatorname{\mathrm{Alg}^{*}}(A,A_{j})$$\operatorname{\mathrm{Alg}^{*}}(A,B)$,

where the vertical arrows are the obvious inclusions.

To prove that 1 is surjective, fix a CQG algebra morphism $f:A\to
B=\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}A_{j}$. It is, in
particular, a morphism in $\operatorname{\mathrm{Alg}^{*}}$, so by the
surjectivity of 2, it factors through a unital $*$-algebra morphism
$f_{i}:A\to A_{i}$ for some $i\in J$, and hence through $f_{j}=\iota_{ji}\circ
f_{i}:A\to A_{j}$ for $j\geq i$. For such $j$, consider the diagram

$A$$A_{j}$$B$$B\otimes B$,$A_{j}\otimes A_{j}$$A\otimes A$$\scriptstyle
f_{j}$$\scriptstyle\iota_{j}$$\scriptstyle f_{j}\otimes
f_{j}$$\scriptstyle\iota_{j}\otimes\iota_{j}$

where the vertical maps are comultiplications. The commutativity of the outer
rectangle is nothing but the preservation of coproducts by $f=\iota_{j}\circ
f_{j}$, while the right hand square commutes because $\iota_{j}:A_{j}\to B$ is
by definition a colimit in $\operatorname{\mathrm{CQG}}$. It follows that the
two $J$-indexed systems of morphisms $\Delta_{A_{j}}\circ f_{j}$ and
$(f_{j}\otimes f_{j})\circ\Delta_{A}$ become equal upon composing further with
$\iota_{j}\otimes\iota_{j}$. The fact that they must then be equal for
sufficiently large $j$ follows from the next lemma, which says essentially
that $\iota_{j}\otimes\iota_{j}$ make $B\otimes B$ the colimit of the diagram
consisting of the maps $\iota_{j^{\prime}j}\otimes\iota_{j^{\prime}j}$,
together with the injectivity of

$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}\operatorname{\mathrm{Alg}^{*}}(A,A_{j}\otimes
A_{j})\longrightarrow\operatorname{\mathrm{Alg}^{*}}(A,\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}A_{j}\otimes
A_{j})$

resulting from the finite presentability of $A$ in
$\operatorname{\mathrm{Alg}^{*}}$. $\blacksquare$

######  2.0.4 Lemma.

The tensor square endofunctor $A\mapsto A\otimes A$ on
$\operatorname{\mathrm{Alg}^{*}}$ preserves filtered colimits.

######  Proof.

Working with $*$-algebras is of little importance here: The forgetful functor
from any finitary variety of algebras to the category Set of sets creates
filtered colimits. We refer again to [AR94, 3.A] for background on varieties
of algebras. The claim just made can be proven either by realizing the variety
as the Eilenberg-Moore category of a finitary monad (i.e. one which preserves
filtered colimits) on Set [AR94, 3.18], or directly. It follows that it is
enough to prove the analogous statement in the category Vec of complex vector
spaces.

Let $(J,\leq)$ be a filtered poset, $\iota_{j^{\prime}j}:V_{j}\to
V_{j^{\prime}}$ a functor from it to Vec, and $\displaystyle
V=\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}V_{j}$. We have
to show that the canonical map $\displaystyle\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}(V_{j}\otimes
V_{j})\to V\otimes V$ is an isomorphism.

Let $(J\times J,\leq)$ be the cartesian square of the category $(J,\leq)$; it
is simply the poset structure on the set $J\times J$ defined by
$(i,j)\leq(i^{\prime},j^{\prime})$ iff $i\leq i^{\prime}$ and $j\leq
j^{\prime}$. Consider the functor $F:(J\times J,\leq)\to\textsc{Vec}$ given by
$(i,j)\mapsto V_{i}\otimes V_{j}$ (with the obvious action on morphisms). For
any vector space $W$, the endofunctor
$W\otimes\bullet:\textsc{Vec}\to\textsc{Vec}$ is left adjoint to
$\operatorname{Hom}(W,\bullet)$, and hence cocontinuous. Applying this
observation first to $V_{i}$ and then to $V$, we get the last two isomorphisms
in the chain

$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i,j}(V_{i}\otimes
V_{j})\cong\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i}\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}(V_{i}\otimes
V_{j})\cong\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-7.14445pt}{-3.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@moveto{7.14445pt}{-3.67221pt}\pgfsys@moveto{6.14445pt}{-6.67221pt}\pgfsys@moveto{-6.14445pt}{-6.67221pt}\pgfsys@lineto{5.68446pt}{-6.67221pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i}(V_{i}\otimes
V)\cong V\otimes V.$ (3)

The first one, on the other hand, is the usual Fubini-type separation of
variables for colimits ([ML98, IX.8]).

The original poset $J$ sits diagonally inside $J\times J$ as the set of pairs
$(j,j)$. Moreover, the fact that $J$ is filtered translates to $J$ being
cofinal in $J\times J$ in the sense that everyone in the latter is majorized
by someone in the former. But it then follows [ML98, IX.3.1] that the
canonical map

$\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{j}(V_{j}\otimes
V_{j})=\operatorname*{{\leavevmode\hbox to14.69pt{\vbox
to10.34pt{\pgfpicture\makeatletter\hbox{\hskip 7.34445pt\lower-6.87221pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}F|_{J}\longrightarrow\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}F=\operatorname*{{\leavevmode\hbox
to14.69pt{\vbox to10.34pt{\pgfpicture\makeatletter\hbox{\hskip
7.34445pt\lower-6.87221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{}{{}}{}{{}{}}{{}}{}{{}}{
{}}{}{{}}{}{{}{}}{{}}{}{{}}{{}}{}{{}}{}{{}}
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.68446pt}{-6.67221pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.94446pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lim$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}_{i,j}(V_{i}\otimes
V_{j})$

is an isomorphism. Composing it with 3 finishes the proof. $\blacksquare$

######  2.0.5 Remark.

We mentioned briefly at the end of § 1.2 that the category
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is not as friendly as
$\operatorname{\mathrm{CQG}}$. Lemma 2.0.4, for example, is somewhat
problematic. The problem with the above proof is that it hinges on functors of
the form
$A\otimes\bullet:\operatorname{\mathrm{Alg}^{*}}\to\operatorname{\mathrm{Alg}^{*}}$
preserving filtered colimits; I do not know whether the analogous result holds
for the category $\operatorname{\mathrm{C}^{*}}$ of unital $C^{*}$-algebras
with the minimal tensor product (which is what would be needed to make the
proof work verbatim for $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$).

Some partial results (which we do not prove here) are that (a) for a
$C^{*}$-algebra $A$, the functor $A\otimes\bullet$ does preserve filtered
colimits of injections, (b) the same functor preserves all filtered colimits
provided $A$ is an exact $C^{*}$-algebra in the sense of [Was94] (this simply
means that minimal tensoring with $A$ preserves short exactness of sequences
in $\operatorname{\mathrm{C}^{*}_{0}}$), and (c) the maximal tensor product
with $A$ does preserve all filtered colimits. This suggests that trying to
adapt Lemma 2.0.4 to $\operatorname{\mathrm{C}^{*}}$ adds a layer of
difficulty, in that one has to deal with issues like nuclearity and exactness.
$\blacklozenge$

The last piece of the puzzle is

######  2.0.6 Proposition.

The set $S$ of all $A_{u}(Q)$ is a regular generator in
$\operatorname{\mathrm{CQG}}$.

######  Proof.

According to Definition 1.5.3, we have to show that an arbitrary CQG algebra
$A$ is the coequalizer of two arrows $f,g:Y\to Z$ between coproducts of
$A_{u}(Q)$’s.

Let $V^{\alpha}$ be representatives for the set $\widehat{A}$ of unitary
simple comodules of $A$. Then, $A$ is the direct sum of the matrix coalgebras
$C^{\alpha}$ spanned by the unitary coefficient matrices
$v^{\alpha}=(v^{\alpha}_{ij})$ with respect to orthonormal bases
$(e^{\alpha}_{i})$ of $V^{\alpha}$.

The squared antipode $S^{2}$ conjugates every matrix $v^{\alpha}$ by some
positive operator $Q^{\alpha}$ [KS97, 11.2.3, Lemma 30], and hence, by the
universality property of the $A_{u}(Q)$’s as cited in Example 1.2.4, the
assignment $u_{ij}\mapsto v^{\alpha}_{ij}$ defines a CQG algebra morphisms
$A_{u}(Q^{\alpha})\to A$. Moreover, the resulting map

$\pi:\coprod_{\widehat{A}}A_{u}(Q^{\alpha})\to A$ (4)

is surjective. The left hand side of this expression will be our $Z$. For
$\alpha\in\widehat{A}$, we denote by $u^{\alpha}$ the standard coefficient
matrix in $A_{u}(Q^{\alpha})$ (earlier in this paragraph, where we reasoned
one $\alpha$ at a time, it was denoted simply by $u$).

The two morphisms $f,g:Y\to Z$ that we are looking for (and whose coequalizer
4 should be) ought to somehow recover the relations of $A$, i.e. the
multiplication table with respect to the basis $(v^{\alpha}_{ij})$ of $A$.

To define $f$, fix $\alpha,\beta\in\widehat{A}$. A simple calculation shows
that $u^{\alpha}u^{\beta}=(u^{\alpha}_{ij}u^{\beta}_{kl})_{ik,jl}$ is a
unitary coefficient matrix of $Z=\coprod A_{u}(Q^{\alpha})$, which the squared
antipode of this CQG algebra conjugates by $Q^{\alpha\beta}=Q^{\alpha}\otimes
Q^{\beta}$ (this is just notation). It follows that there is a unique morphism
$A_{u}(Q^{\alpha\beta})\to Z$ defined by $u_{ik,jl}\mapsto
u^{\alpha}_{ij}u^{\beta}_{kl}$. Putting all of these together for all pairs of
comodules, we get

$f:Y=\coprod_{\widehat{A}\times\widehat{A}}A_{u}(Q^{\alpha\beta})\to\coprod
A_{u}(Q^{\alpha})=Z.$

As before, since we need to distinguish between the various coefficient
matrices in $Y$, we denote them by $u^{\alpha\beta}$ in the obvious way.

We now start on our way towards constructing $g:Y\to Z$. The same game as in
the previous paragraph can be played in $A$: For any
$\alpha,\beta\in\widehat{A}$,
$v^{\alpha}v^{\beta}=(v^{\alpha}_{ij}v^{\beta}_{kl})_{ik,jl}$ is the unitary
coefficient matrix with respect to the tensor product basis
$e^{\alpha}_{i}\otimes e^{\beta}_{k}$ of the tensor product Hilbert space
$V^{\alpha}\otimes V^{\beta}$. But now, since we are in $A$, the elements of
this matrix can be expressed as linear combinations of $v^{\gamma}_{ij}$’s. In
order to avoid cumbersome indices on the coefficients of such linear
combinations, we simply write

$v^{\alpha}_{ij}v^{\beta}_{kl}=\ell^{\alpha\beta}_{ik,jl}\in\bigoplus_{\gamma}C^{\gamma},$

where $C^{\gamma}$ is the matrix coalgebra corresponding to
$\gamma\in\widehat{A}$, and $\gamma$ ranges over the simple comodules
appearing in the decomposition of $V^{\alpha}\otimes V^{\beta}$.

Now, because the restriction of $\pi$ to the direct sum $C\leq Z$ of matrix
coalgebras spanned by $u^{\alpha}\subset Z$ is one-to-one (in fact, this
restriction is by definition an isomorphism onto $A$), the elements
$\ell^{\alpha\beta}_{ik,jl}$ defined above lift uniquely to elements of $C$,
and we slightly abusively denote these lifts by
$\pi^{-1}(\ell^{\alpha\beta}_{ik,jl})$. I claim that for fixed $\alpha$ and
$\beta$, these elements form a unitary coefficient matrix which $S^{2}$
conjugates by $Q^{\alpha}\otimes Q^{\beta}$. Indeed, all of these properties
can be stated inside $C$ (without appealing to multiplication), using only the
antipode and the $*$-structure (being unitary, for example, amounts to the
antipode turning $\pi^{-1}(\ell^{\alpha\beta}_{ik,jl})$ into
$\pi^{-1}(\ell^{\alpha\beta}_{jl,ik})^{*}$); since $\pi$ preserves both the
antipode and the $*$ structure and its restriction to $C$ is a coalgebra
isomorphism, the properties all lift from the $\ell$’s to the
$\pi^{-1}(\ell)$’s.

Finally, the claim just proven allows us to construct $g:Y\to Z$ by sending
$u^{\alpha\beta}_{ik,jl}$ to $\pi^{-1}(\ell^{\alpha\beta}_{ik,jl})$. The
coequalizer of $f$ and $g$ is the quotient of $Z$ by the relations imposing on
$u^{\alpha}_{ij}$ the same multiplication table as that of the
$v^{\alpha}_{ij}$’s, so it is now clear that this coequalizer is precisely
$\pi:Z\to A$. $\blacksquare$

We can now put the last few results together:

######  Proof of Theorem 2.0.1.

We know from Proposition 2.0.2 that $\operatorname{\mathrm{CQG}}$ is
cocomplete, and from Propositions 2.0.3 and 2.0.6 that a set of finitely
presentable objects forms a regular generator. The conclusion follows from
Proposition 1.5.4. $\blacksquare$

######  2.0.7 Remark.

In essentially the same way, we can show that the category
$\operatorname{\mathrm{CQG}_{ab}}$ of commutative CQG algebras is finitely
presentable. In this case, all distinctions between the algebraic and the
$C^{*}$-algbraic vanish: The restriction of $\operatorname{\textsc{univ}}$ to
$\operatorname{\mathrm{CQG}_{ab}}$ is an equivalence onto the full subcategory
$\operatorname{\mathrm{C}^{*}\mathrm{QG}_{ab}}$ of
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ consisting of commutative algebras.
Moreover, $\operatorname{\mathrm{CQG}_{ab}}$ (or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}_{ab}}$) is nothing but the opposite
of the category of compact groups, with a compact group $G$ corresponding to
the CQG algebra of representative functions on it.

The only changes we need to make to the proofs in order to adapt the
presentability result to $\operatorname{\mathrm{CQG}_{ab}}$ are (a) substitute
tensor products (of perhaps infinite families) for coproducts, and (b) use the
set of CQG algebras associated to all unitary groups $U_{n}$ for a generator,
instead of the $A_{u}(Q)$’s. $\blacklozenge$

######  2.0.8 Remark.

Although, strictly speaking, SAFT-ness would have sufficed for the purposes of
Section 4, the finite presentability of $\operatorname{\mathrm{CQG}}$ is
interesting in its own right, as it is somewhat surprising: Given the close
relationship between $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, discussed a little in § 1.2 above,
one might think that the former category should look more or less like “unital
$C^{*}$-algebras with a lot of extra structure”, and hence should be at least
as reluctant to be finitely presentable as the category
$\operatorname{\mathrm{C}^{*}}$ of unital $C^{*}$-algebras. However, this is
not the case.

There is a more general notion of presentability for categories (local
presentability in the literature, e.g. [AR94]) parametrized by a regular
cardinal number, so that the technical term for ‘finitely presentable’ is
‘$\aleph_{0}$-presentable’; the larger the cardinal, the weaker the notion.
Now, it can be shown that $\operatorname{\mathrm{C}^{*}}$ is
$\aleph_{1}$-presentable but not finitely presentable. Worse still, the same
is true in the commutative setting: Although the previous remark notes that
$\operatorname{\mathrm{CQG}_{ab}}$ is finitely presentable, the category of
commutative unital $C^{*}$-algebras is $\aleph_{1}$, but not finitely
presentable. $\blacklozenge$

##  3 $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is SAFT

The main result of the section is the one just stated:

######  3.0.1 Theorem.

The cateory $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is SAFT.

We prove the three properties required for SAFT-ness (Definition 1.4.2)
separately.

######  3.0.2 Proposition.

$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is cocomplete.

######  Proof.

This parallels the proof of Proposition 2.0.2 by constructing coequalizers and
coproducts, so we will be brief.

As noted in the proof just mentioned, coproducts are constructed in [Wan95,
Theorem 1.1].

As for coequalizers, they are constructed as before. The coequalizer of two
morphisms $f,g:A\to B$ in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is the
quotient of $B$ by the closed ideal $I$ generated by $f(a)-g(a)$, and the
argument from the proof of Proposition 2.0.2 can be paraphrased to show this.

Although Sweedler notation is not available anymore (because we are working
with $C^{*}$ tensor products rather than algebraic ones), the computation
carried out there can be written down in a Sweedler-notation-free manner as
saying that $\Delta_{B}\circ(f-g)$ equals $(f\otimes(f-g)+(f-g)\otimes
g)\circ\Delta_{A}$. It follows that $B/I$ inherits a coassociative
comultiplication and $B\to B/I$ respects it, while the (Antipode) condition of
Definition 1.1.1 follows immediately from that of $B$. In conclusion, the
quotient $B\to B/I$ is naturally a map in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$. $\blacksquare$

######  3.0.3 Remark.

Filtered colimits and pushouts in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are constructed in [Wan95, 3.1,3.4] in the case when the morphisms in the
diagram are one-to-one. According to (the proof of) Proposition 3.0.2,
injectivity is not necessary in order to conclude that the colimit of a
diagram in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ in the category of
unital $C^{*}$-algebras is automatically endowed with a compact quantum group
structure. $\blacklozenge$

Next in line is the generation condition of Definition 1.4.2. It turns out
that $A_{u}(Q)$ will once more come in handy. We need them as objects of
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, so recall the enveloping
$C^{*}$-algebra functor
$\operatorname{\textsc{univ}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.

######  3.0.4 Proposition.

The set $\operatorname{\textsc{univ}}(A_{u}(Q))$ for $Q$ ranging over all
positive matrices generates $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.

######  Proof.

That the $A_{u}(Q)$ form a generator in $\operatorname{\mathrm{CQG}}$ is part
of the statement of Theorem 2.0.1. It is a simple exercise that left adjoints,
such as $\operatorname{\textsc{univ}}$, turn generators into generators
provided their right adjoints are faithful. In our case, the faithfulness of
the right adjoint $\operatorname{\textsc{alg}}$ to
$\operatorname{\textsc{univ}}$ follows from the density of the inclusion
$\operatorname{\textsc{alg}}(A)\subset A$: Any arrow $f:A\to B$ in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is the extension by continuity of
$\operatorname{\textsc{alg}}(f):\operatorname{\textsc{alg}}(A)\to\operatorname{\textsc{alg}}(B)$,
and hence $\operatorname{\textsc{alg}}(f)=\operatorname{\textsc{alg}}(g)$
implies $f=g$. $\blacksquare$

The only ingredient of Definition 1.4.2 still to be addressed is co-
wellpoweredness. Recall (§ 1.4) that this meant that every object has only a
set of quotient objects. It will help, then, to know exactly which morphisms
in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ are epimorphisms; this is what
the following result does.

######  3.0.5 Proposition.

A morphism $f:A\to B$ in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is an
epimorphism if and only if it is surjective.

######  Proof.

As usual in categories where objects are sets with some kind of structure and
morphisms are maps preserving that structure, the implication surjective
$\Rightarrow$ epimorphism is immediate.

To prove the other implication, we will show that if $f$ is an epimorphism,
then $\operatorname{\textsc{alg}}(f)$ is surjective (the conclusion follows
from the denseness of $\operatorname{\textsc{alg}}(B)\subset B$). Since we can
always substitute the image of $f$ for $A$, we can (and will) assume that $f$
is injective.

First, recall the construction $\operatorname{\mathrm{CQG}}\ni
X\mapsto\operatorname{\textsc{red}}(X)\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
mentioned in § 1.2. It is functorial when restricted to the category
$\operatorname{\mathrm{CQG}}_{\mathrm{inj}}$ of CQG algebras and injective
morphisms, (this is the essence of [BKQ11, 6.2.4]). In order to keep the
notation manageable, indicate the functors $\operatorname{\textsc{alg}}$ and
$\operatorname{\textsc{red}}$ by superscripts, as in $X^{a}$ for
$\operatorname{\textsc{alg}}(X)$, $X^{r}$ for
$\operatorname{\textsc{red}}(X)$, $X^{ar}$ for
$\operatorname{\textsc{red}}(\operatorname{\textsc{alg}}(X))$, etc.; the same
conventions are in place for morphisms.

Let $\displaystyle\iota:B\to B\coprod_{A}B$ be the left hand canonical
inclusion into the pushout of $f$ along itself in the category
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, or equivalently (by the proof of
Proposition 3.0.2), in the category $\operatorname{\mathrm{C}^{*}}$ of unital
$C^{*}$-algebras. One condition equivalent to $f$ being epimorphic is that
$\iota$ be an isomorphism. Similarly, we denote by $\iota^{\prime}$ the left
hand inclusion $\displaystyle B^{a}\to B^{a}\coprod_{A^{a}}B^{a}$ into the
pushout in $\operatorname{\mathrm{Alg}^{*}}$. Note that $\iota$ and
$\iota^{\prime}$ are both injective, as they have left inverses by the
universality property of pushouts.

Assume now that $f^{a}$ is not surjective. Then, I claim that (a)
$\iota^{\prime}$ cannot be surjective (equivalently, an isomorphism), and
hence (b) neither can $(\iota^{\prime})^{r}$. That (a) does indeed imply (b)
follows from the fact [BKQ11, 6.2.12] that the functor
$\operatorname{\textsc{red}}:\operatorname{\mathrm{CQG}}_{\mathrm{inj}}$
reflects isomorphisms.

To prove (a), recall that an inclusion $K\subseteq H$ of cosemisimple Hopf
algebras always splits as a $K$-$K$-bimodule map (e.g. as argued in the proof
of [Chi11, 2.0.4]). Applied to the inclusion $f^{a}:A^{a}\to B^{a}$, this
observation yields a direct sum decomposition $B^{a}=A^{a}\oplus M$ as
$A^{a}$-$A^{a}$-bimodules for some non-zero $M$, and the pushout
$\displaystyle B^{a}\coprod_{A^{a}}B^{a}$ breaks up as a direct sum of $2^{n}$
copies of $M^{\otimes n}$ for $n\geq 0$ (tensor product of
$A^{a}$-$A^{a}$-bimodules). Moreover, $\iota^{\prime}$ identifies $B^{a}$ with
the summand $A^{a}\oplus M$ therein.

Now consider the commutative diagram

$B$$B\coprod_{A}B$$B^{ar}\coprod_{A^{ar}}B^{ar}$$B^{ar}$$(B^{a}\coprod_{A^{a}}B^{a})^{r}$$\scriptstyle\iota$$\scriptstyle(\iota^{\prime})^{r}$

where the right hand vertical map comes from the universality property of the
pushout applied to the two inclusions $\displaystyle B^{a}\to
B^{a}\coprod_{A^{a}}B^{a}$, and the other two unnamed maps are induced by the
surjection $B\to B^{ar}$. We have just argued that if $f^{a}$ is not
surjective, then the lower left corner path is not surjective. But then the
upper right corner path isn’t either. However, the right hand upper horizontal
arrow is surjective, as is the right hand vertical arrow. In conclusion, the
only morphism in this path which can fail to be surjective (under the
assumption that $f^{a}$ is not surjective) is $\iota$. $\blacksquare$

######  3.0.6 Remark.

The proof makes it clear that the analogous result is true for
$\operatorname{\mathrm{CQG}}$, i.e. epimorphisms are surjective.
$\blacklozenge$

Since for any compact quantum group $A$ there is only a set of quotients of
$\operatorname{\textsc{alg}}(A)$ and hence only a set of compact quantum
groups having such quotients as dense CQG subalgebras, the previous result
implies:

######  3.0.7 Proposition.

$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is co-wellpowered.

######  Proof of Theorem 3.0.1.

Propositions 3.0.2, 3.0.4 and 3.0.7 are precisely what is required by
Definition 1.4.2. $\blacksquare$

##  4 Applications

The goal of this section is to apply Theorems 2.0.1 and 3.0.1, together with
the adjoint functor theorem and abstract properties of presentable or SAFT
categories, to the construction of compact quantum groups with various
universal properties. These constructions fall roughly into two categories:
right adjoints to functors defined on $\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, as direct applications of Theorem
1.4.1, and left adjoints arising in a slightly more roundabout way in § 4.4.

Note that limits in the categories $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, discussed in the next subsection,
fit in this framework as right adjoints: Given a small category $J$ and a
category $\mathcal{C}$, the limits of functors $J\to\mathcal{C}$, if they
exist, can be obtained as images of a right adjoint to the diagonal functor
$\Delta:\mathcal{C}\to\mathcal{C}^{J}$ (the latter is notation for the
category of all functors $J\to\mathcal{C}$) sending an object
$c\in\mathcal{C}$ to the functor $\Delta(c):J\to\mathcal{C}$ constant at $c$.

###  4.1 Limits in $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$

We now know from Theorems 2.0.1 and 3.0.1 that both
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are SAFT. Remembering that SAFT-ness implies completeness (Remark 1.4.3), we
get:

######  4.1.1 Theorem.

The categories $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ are complete.

Limits in these categories are quantum analogues of colimits of compact
groups. It is natural to ask whether functors
$J\to\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ taking commutative values have
commutative limits, or in other words, whether Theorem 4.1.1 recovers ordinary
colimits of compact groups. Since $\operatorname{\mathrm{CQG}_{ab}}$ is
complete (by Remark 2.0.7 or simply constructing coequalizers and coproducts
in the category of compact groups), the next result confirms this:

######  4.1.2 Proposition.

The inclusions
$\operatorname{\mathrm{CQG}_{ab}}\to\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}_{ab}}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
are right adjoints.

######  Proof.

In both cases the left adjoint is abelianization, associating to an object
$A\in\operatorname{\mathrm{CQG}}$ (or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) its largest commutative quotient
$*$-algebra (resp. $C^{*}$-algebra) $A_{ab}$. This is all rather routine, so
we omit most of the details.

The universality property of the canonical quotient map $\pi:A\to A_{ab}$
ensures that the composition

$A$$A\otimes A$$A_{ab}\otimes
A_{ab}$$\scriptstyle\Delta$$\scriptstyle\pi\otimes\pi$

factors through a map $\Delta_{ab}:A_{ab}\to A_{ab}\otimes A_{ab}$. The
coassociativity of $\Delta_{ab}$ follows from the uniqueness of the
factorization of $\pi^{\otimes
3}\circ(\Delta\otimes\operatorname{id})\circ\Delta:A\to A_{ab}^{\otimes 3}$
through $\pi$. By construction, we get a commutative square

$A$$A_{ab}$$A\otimes A$$A_{ab}\otimes
A_{ab}$$\scriptstyle\pi$$\scriptstyle\pi\otimes\pi$$\scriptstyle\Delta$$\scriptstyle\Delta_{ab}$

The rest of the structure and properties (e.g. counit
$\varepsilon_{ab}:A_{ab}\to\mathbb{C}$ and counitality of $\Delta_{ab}$ in the
algebraic case) follow similarly, as do the functoriality and the desired
universality property of $A\mapsto A_{ab}$. To see the latter, for example,
let $f:A\to B$ be a morphism in $\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ with $B$ commutative. Then, $f$
factors uniquely as $f_{ab}\circ\pi$ for an algebra map $f_{ab}:A_{ab}\to B$.
The outer rectangle of

$A$$A_{ab}$$B$$A\otimes A$$A_{ab}\otimes A_{ab}$$B\otimes
B$$\scriptstyle\pi$$\scriptstyle
f_{ab}$$\scriptstyle\pi\otimes\pi$$\scriptstyle f_{ab}\otimes f_{ab}$

is commutative because $f=f_{ab}\circ\pi$ is compatible with
comultiplications, and we have just observed that the left hand square
commutes. It follows that the precomposition of the right hand square with
$\pi$ commutes also, and since $\pi$ is onto, the right hand square must be
commutative. We again skip the entirely similar arguments for compatibility of
$f_{ab}$ with counits and antipodes in the algebraic case. $\blacksquare$

Outside of the general categorical framework provided by Theorems 2.0.1 and
3.0.1, one can also arrive at limits in the categories
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
by means of the Tannakian formalism introduced in [Wor88]
($\operatorname{\mathrm{CQG}}$ is better suited for this, so we focus on it).
There, Woronowicz associates a compact quantum group (or rather a compact
quantum group of the form $\operatorname{\textsc{univ}}(A)$ for
$A\in\operatorname{\mathrm{CQG}}$, so effectively, he recovers a CQG algebra)
from a so-called concrete monoidal $W^{*}$-category with complex conjugation.
These are basically just rigid, monoidal $W^{*}$-categories endowed with a
faithful, monoidal $*$-functor [GLR85] to the category Hilb of finite-
dimensional Hilbert spaces.

This is a version of Tannaka duality for Hopf algebras (e.g. as in [Sch92,
Ver12] and the many references therein): The CQG algebra constructed in
[Wor88, 1.3] given a concrete monoidal $W^{*}$-category $\mathcal{C}$ is what
in [Sch92] would be called the coendomorphism Hopf algebra of the functor
$\mathcal{C}\to\textsc{Hilb}$ that is implicitly part of Woronowicz’s
definition.

Now, if one starts with the category of unitary comodules of a CQG algebra $A$
and performs the above construction, the resulting CQG algebra is again $A$.
In other words, unitary comodules know all there is to know about a CQG
algebra (hence the name ‘Tannaka reconstruction’). It follows from this that
the construction of new CQG algebras out of old (such as, say, the limit of
some functor $F:J\to\operatorname{\mathrm{CQG}}$ out of the values of $F$) has
a chance of being carried out categorically: Identify the category of unitary
comodules, and you know the CQG algebra.

To get some insight into what limits in $\operatorname{\mathrm{CQG}}$ look
like, we do what the previous paragraph suggests, for products (but also state
the result for pullbacks, as it will be useful in Section 5): Given a family
$A_{i}$, $i\in I$ of objects in $\operatorname{\mathrm{CQG}}$, what does the
category of finite-dimensional, unitary comodules of the product
$\displaystyle A=\prod_{i}A_{i}$ in $\operatorname{\mathrm{CQG}}$ look like in
terms of the categories of comodules of the individual $A_{i}$? The answer
turns out to be quite simple; arguably simpler, in fact, than the description
of the category of unitary comodules for the (more familiar, in the
literature) coproduct $\displaystyle\coprod A_{i}$ from [Wan95]. All comodules
below are understood to be finite-dimensional and unitary.

Putting an $A$-comodule structure on an $n$-dimensional Hilbert space $V$ is
the same as giving a CQG algebra morphism $f:A_{u}(Q)\to A$ for some positive
$Q\in GL_{n}(\mathbb{C})$: In one direction, the comodule structure induces a
morphism by sending the standard generators $u_{ij}\in A_{u}(Q)$ (for an
appropriate $Q$) to the coefficients $v_{ij}$ with respect to an orthonormal
basis of $V$; in the opposite direction, make $A$ coact on the standard
$A_{u}(Q)$-comodule by “scalar corestriction” via the coalgebra morphism $f$.
In turn, by the defining property of the categorical product, a morphism
$f:A_{u}(Q)\to A$ means a family of morphisms $f_{i}:A_{u}(Q)\to A_{i}$, $i\in
I$. Going through this comodule structure - morphism correspondence in reverse
for each $i$, the data consisting of the $f_{i}$’s is equivalent to putting an
$A_{i}$-comodule structure on the canonical $n$-dimensional comodule of
$A_{u}(Q)$ for every $i$. A moment’s thought will show how do modify this
argument to take care pullbacks, and all in all, we have the following result:

######  4.1.3 Proposition.

Let $A_{i}\in\operatorname{\mathrm{CQG}}$, $i\in I$ be a set of objects, and
$A=\prod A_{i}$ the product in $\operatorname{\mathrm{CQG}}$. Then, the
category of $A$-comodules has as objects finite-dimensional Hilbert spaces
admitting an $A_{i}$-comodule structure for each $A_{i}$, and as morphisms
linear maps respecting all of these structures.

Let $f:B\to C$ and $f^{\prime}:B^{\prime}\to C$ be morphisms in
$\operatorname{\mathrm{CQG}}$, and $\displaystyle A=B\times_{C}B^{\prime}$ the
pullback in $\operatorname{\mathrm{CQG}}$. The category of $A$-comodules has

  1. (a)

as objects, triples $(V,V^{\prime},\varphi)$ where $V$ and $V^{\prime}$ are
$B$ and $B^{\prime}$-comodules respectively, and $\varphi:V\to V^{\prime}$ is
a unitary map identifying $V$ and $V^{\prime}$ as $C$-comodules;

  2. (b)

as morphisms from $(V,V^{\prime},\varphi)$ to $(W,W^{\prime},\psi)$, pairs
$(\xi,\xi^{\prime})$, where $\xi:V\to W$ and $\xi^{\prime}:V^{\prime}\to
W^{\prime}$ are morphisms in $\mathcal{M}^{B}$ and $\mathcal{M}^{B^{\prime}}$
respectively, making the diagram

$V$$V^{\prime}$$W$$W^{\prime}$$\scriptstyle\varphi$$\scriptstyle\psi$$\scriptstyle\xi$$\scriptstyle\xi^{\prime}$

commutative.

######  4.1.4 Remark.

This statement describes the sought-after categories of unitary comodules very
explicitly. There is a more abstract, but also more elegant way to phrase all
of this. We need some basic 2-categorical notions to say it all (as in
[Lac10]).

Rigid, monoidal $W^{*}$-categories $\mathcal{C}$ endowed with monoidal
$*$-functors $\mathcal{C}\to\textsc{Hilb}$ form a bicategory in a natural way,
while $\operatorname{\mathrm{CQG}}$ can be regarded as a bicategory with only
identity 2-cells. Then, sending a CQG algebra to its category of finite-
dimensional, unitary comodules (together with its forgetful functor to Hilb)
is a pseudofunctor from the latter to the former. The essence of Proposition
4.1.3 is that this pseudofunctor preserves limits. This is a familiar theme in
Tannaka duality: Woronowicz’s construction of a CQG algebra out of a functor
$\mathcal{C}\to\textsc{Hilb}$ is in fact nothing but the left adjoint of the
pseudofunctor mentioned above. This sort of situation is treated in [Sch11],
with a biadjunction analogous to the one just discussed appearing in Theorem
3.1.1. $\blacklozenge$

######  4.1.5 Remark.

The references to [Wor88] in the above discussion are somewhat of a
paraphrase, as Woronowicz works with what are called compact matrix quantum
groups (or CMQG algebras on the algebraic side [DK94, 2.5]). These are
basically compact quantum groups finitely generated as $C^{*}$-algebras, and
are analogous to compact Lie groups (the latter being precisely those compact
groups which embed in some unitary group). He also distinguishes a comodule
whose coefficients generate the algebra, and so works with pairs $(A,u)$,
where $u\in M_{n}(A)$ is a unitary coefficient matrix. Adapting the results of
that paper to the general setting is straightforward.

A CMQG algebra always has a countable set of (isomorphism classes of) simple
comodules. As we will see in the next example, abandoning this restriction is
absolutely necessary if we are going to discuss limits in
$\operatorname{\mathrm{CQG}}$, since products, for example, are very unlikely
to satisfy this property. $\blacklozenge$

######  4.1.6 Example.

One does not even have to go “quantum” to give an example of a very large
(that is, non-CMQG) product in $\operatorname{\mathrm{CQG}}$. Indeed, the
smallest possible example will do: a coproduct of two copies of $\mathbb{Z}/2$
in the category of compact groups.

Denote this coproduct by $G$. According to the first part of Proposition
4.1.3, a unitary representation of $G$ consists of a finite-dimensional
Hilbert space $V$ and two involutive unitary operators $x$ and $y$ on $V$. The
projections $p=\frac{1+x}{2}$ and $q=\frac{1+y}{2}$ provide precisely the same
information, so we work with them instead. $V$ is irreducible preciely when
$p$ and $q$ have no common proper, non-zero invariant subspace. By the
discussion carried out prior to the statement of [Tak02, Theorem 1.41], this
is equivalent to the four pairwise infima $p\wedge q$, $p\wedge(1-q)$ etc. all
being zero (where ‘$\wedge$’ means orthogonal projection on the intersection
of the ranges of the two projections).

If $\dim V\geq 2$, the vanishing of all wedges $p\wedge q$, etc. makes it
necessary that $V$ be even-dimensional and that $p$ and $q$ both have rank
$\frac{\dim V}{2}$, but this is it: The set of pairs of projections satisfying
the requirements is open dense in the set of all pairs of projections of rank
$\frac{\dim V}{2}$ on $V$. Hence, if $\dim V=2n$ for some $n\geq 1$, the set
of pairs $(p,q)$ that will make $V$ into an irreducible unitary
$G$-representation is a manifold of dimension $4n^{2}$ (twice the dimension of
the Grassmannian variety of $n$-dimensional subspaces of $V$ as a real
manifold). We now have to quotienting by the equivalence relation
$(p,q)\sim(upu^{*},uqu^{*})$ for unitary $u$, which accounts for isomorphic
$G$-representations induced by different pairs of projections. Since the
unitary group of $V$ has dimension $n(2n-1)$, this still leaves continuum many
isomorphism classes of simples. $\blacklozenge$

We end this subsection with a note on terminology. As observed in [Sol10,
$\S$3], the name ‘free product of compact quantum groups’ from [Wan95] is
somewhat inconsistent with the prevailing point of view that compact quantum
groups form a category opposite to $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
The problem is that in universal algebra, ‘free product’ is often synonymous
to ‘coproduct’. Even though Wang’s construction is a coproduct in
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ rather than its opposite, and hence
‘product of compact quantum groups’ would perhaps be a better fit, ‘free
product’ seems to have been established through use (besides, ‘product’ would
clash with the interpretation of $A\otimes A$ as the Cartesian square of a
compact quantum group, implicit in Definition 1.1.1). On the other hand,
products in $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (whose existence
Theorem 4.1.1 proves) are probably best referred to as ‘coproducts of compact
quantum groups’.

###  4.2 Quantum groups generated by quantum spaces

The idea here is that functors of the form “forget the comultiplication”,
regarded as quantum analogues of forgetting the multiplication on a group,
have right adjoints. As most of the previous discussion, all of this works
both algebraically and $C^{*}$-algebraically:

######  4.2.1 Theorem.

The functors

  1. (a)

$\operatorname{\textsc{forget}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{Alg}^{*}_{0}}$
sending a CQG algebra to its underlying $*$-algebra and

  2. (b)

$\operatorname{\textsc{forget}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}_{0}}$
sending a compact quantum group to its underlying $C^{*}$-algebra

are left adjoints.

######  Proof.

We already know from the proofs of Propositions 2.0.2 and 3.0.2 that the
forgetful functors from $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ to unital $*$-algebras and
$C^{*}$-algebras respectively are cocontinuous. I claim that so are the
inclusions
$\operatorname{\mathrm{Alg}^{*}}\to\operatorname{\mathrm{Alg}^{*}_{0}}$ and
$\operatorname{\mathrm{C}^{*}}\to\operatorname{\mathrm{C}^{*}_{0}}$. Assuming
this for now, (a) and (b) are cocontinuous; that they are left adjoints then
follows from the SAFT-ness of $\operatorname{\mathrm{CQG}}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (Theorems 2.0.1 and 3.0.1) and the
adjoint functor theorem.

Finally, to prove the claim made above that the two inclusions
$\operatorname{\mathrm{Alg}^{*}}\to\operatorname{\mathrm{Alg}^{*}_{0}}$ and
$\operatorname{\mathrm{C}^{*}}\to\operatorname{\mathrm{C}^{*}_{0}}$ are
cocontinuous, note that they are in fact left adjoints: In both cases, the
right adjoint is the multiplier algebra construction $A\mapsto M(A)$.
$\blacksquare$

######  4.2.2 Remark.

Von Neumann algebras would be an alternative way to formalize the idea of
“quantum space”. This is the point of view espoused in [Kor12], where the
category $\operatorname{\mathrm{W}^{*}}$ of unital von Neumann algebras and
unital normal homomorphisms is opposite to the category of so-called quantum
collections. The idea here is that a von Neumann algebra is a quantum analogue
of a set, ordinary sets $X$ corresponding to $\ell^{\infty}(X)$.

Adopting this perspective, the enveloping $W^{*}$-algebra functor
$\operatorname{\textsc{env}}:\operatorname{\mathrm{C}^{*}}\to\operatorname{\mathrm{W}^{*}}$
is a kind of forgetful functor, disregarding the topological side of a compact
quantum space and remembering only the underlying quantum collection;
similarly, the forgetful functor
$\operatorname{\mathrm{W}^{*}}\to\operatorname{\mathrm{C}^{*}}$ (which is
right adjoint to $\operatorname{\textsc{env}}$) is a kind of quantum Stone-
Cech compactification.

Composing (b) of Theorem 4.2.1 further with $\operatorname{\textsc{env}}$ is
again a left adjoint (the composition of two left adjoints), and its right
adjoint could be thought of as the functor associating to every quantum
ollection the compact quantum group freely generated by it. $\blacklozenge$

We refer to the image of a $*$ or $C^{*}$-algebra $A$ through the respective
right adjoint as the cofree CQG algebra or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ object on $A$ (‘co’ because it
universally maps into $A$, as opposed to being mapped into). The notation
$\operatorname{\textsc{cofree}}$ stands for either of the two functors, and we
rely on context to distinguish between the possibilities.

It is to be expected in such cofree-Hopf-algebra-on-an-algebra type
constructions that commutativity will be preserved (as explained, for
instance, in the introduction of [Por11]). In other words, one would like the
right adjoint from part (b), say, when applied to the algebra of functions
vanishing at infinity on a locally compact space $X$, to yield precisely the
compact group freely generated by $X$: Construct the abstract group $G$ freely
generated by $X$, endow it with the strongest topology making the canonical
map $X\to G$ continuous, and take the Bohr compactification of the resulting
topological group. That this is indeed the case is essentially the content of
the following proposition:

######  4.2.3 Proposition.

If $A$ is a commutative $*$ or $C^{*}$-algebra, then
$\operatorname{\textsc{cofree}}(A)$ is commutative.

######  Proof.

To fix ideas, we prove the algebraic statement regarding $*$-algebras, and
leave the simple modifications that will adapt the proof to the other cases to
the reader.

Let $H=\operatorname{\textsc{cofree}}(A)$, and recall the CQG algebra
structure on $H_{ab}$ from Proposition 4.1.2. The universality property of the
abelianization factorizes the left hand diagonal arrow in the following
diagram through the right hand one, while the cofree-ness gives the other
commutative triangle, passing through $\iota$:

$H$$H_{ab}$$A$$\pi$$\iota$

By cofree-ness again, the loop $\iota\circ\pi$ must be the identity; since
$\pi$ was a surjection, it must be an isomorphism. $\blacksquare$

###  4.3 Variations on the Bohr compactification theme

A right adjoint to the inclusion functor
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$
is constructed directly in [Soł05], and this construction is referred to as
the quantum Bohr compactification. For any
$A\in\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$, it provides an object
$H\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ mapping universally into $A$ so
as to preserve the comultiplication; remembering the arrow reversal inherent
to passing from spaces to functions on them, this should indeed be thought of
as a compact quantum group into which the locally compact quantum semigroup
maps universally. Moreover, when $A$ is commutative and hence the algebra of
functions vanishing at infinity on a locally compact semigroup $X$, the
construction returns precisely the algebra of functions on the ordinary Bohr
compactification of $X$ ([Soł05, 4.1]).

We recover these results and their algebraic counterparts below (Theorem 4.3.1
and Proposition 4.3.4), as applications of the categorical machinery already
in place.

######  4.3.1 Theorem.

The inclusion functors

  1. (a)

$\operatorname{\textsc{forget}}:\operatorname{\mathrm{CQG}}\to\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$
sending a CQG algebra to its underlying algebraic quantum semigroup and

  2. (b)

$\operatorname{\textsc{forget}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$
sending a compact quantum group to its underlying compact quantum group

are left adjoints.

######  Proof.

As before, Theorems 2.0.1 and 3.0.1 and the adjoint functor theorem ensure
that we only need to prove the two functors cocontinuous. Equivalently, this
means showing they preserve coequalizers of pairs and coproducts. The four
arguments (coequalizers and coproducts, (a) and (b)) follow essentially the
same path, so let us focus on coproducts for part (a).

Let $I$ be a set, and $A_{i}$, $i\in I$ objects in
$\operatorname{\mathrm{CQG}}$. Let also $f_{i}:A_{i}\to B$ be morphisms in
$\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ (strictly speaking, they are
morphisms $\operatorname{\textsc{forget}}(A_{i})\to B$, but since
$\operatorname{\textsc{forget}}$ really is just an inclusion, we omit it in
the rest of the proof). Since forgetting further to
$\operatorname{\mathrm{Alg}^{*}_{0}}$ (i.e. disregarding comultiplications)
is, according to part (a) of Theorem 4.2.1, cocontinuous, the $f_{i}$
aggregate into a unique $*$-algebra morphism $f:A=\coprod A_{i}\to B$. We are
done if we can show that $f$ preserves comultiplications. To see this,
consider the diagram

$A_{i}$$A$$B$$A_{i}\otimes A_{i}$$A\otimes A$$B\otimes
B$$\scriptstyle\iota_{i}$$\scriptstyle
f$$\scriptstyle\iota_{i}\otimes\iota_{i}$$\scriptstyle f\otimes f$

where the vertical arrows are comultiplications, and $\iota_{i}:A_{i}\to A$
are the structure maps of the coproduct. The commutativity of the outer
rectangle is the preservation of comultiplications by the
$f_{i}=f\circ\iota_{i}$, while the left hand square commutes because $A$ was
defined as the coproduct of the $A_{i}$ in $\operatorname{\mathrm{CQG}}$. It
follows that precomposing the two possible ways to get from $A$ to $B\otimes
B$ with $\iota_{i}$ yields the same morphism $A_{i}\to B\otimes B$; then, by
the universality in $\operatorname{\mathrm{Alg}^{*}_{0}}$ of the coproduct
$A$, the right hand square must also be commutative. $\blacksquare$

######  4.3.2 Remark.

Part (b) of the proposition can be tweaked slightly in the spirit of Remark
4.2.2.

The enveloping von Neumann algebra functor
$\operatorname{\textsc{env}}:\operatorname{\mathrm{C}^{*}_{0}}\to\operatorname{\mathrm{W}^{*}}$
can be lifted to a functor (again called $\operatorname{\textsc{env}}$) from
$\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ to the category
$\operatorname{\mathrm{W}^{*}\mathrm{QS}}$ of von Neumann quantum semigroups,
consisting of von neumann algebras $M$ endowed with a coassociative morphism
$\Delta:M\to M\otimes M$ in $\operatorname{\mathrm{W}^{*}}$ (remember that the
tensor product of von Neumann algebras here is the spatial one) and with
$\operatorname{\mathrm{W}^{*}}$ maps which preserve these comultiplications as
morphisms. It is to be thought of as a kind of forgetful functor, ignoring the
topology of a locally compact quantum semigroup and remembering only the
underlying quantum collection, together with the “multiplication”.

It can be shown further that
$\operatorname{\textsc{env}}\circ\operatorname{\textsc{forget}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{W}^{*}\mathrm{QS}}$
is cocontinuous, and hence a left adjoint. In other words, every $W^{*}$
quantum semigroup has a quantum Bohr compactification. $\blacklozenge$

######  4.3.3 Remark.

In a $C^{*}$-algebraic variant of the Tambara construction [Tam90], (a
particular case of) [Soł09, Theorem 3.3] constructs, for every finite-
dimensional $C^{*}$-algebra $A$, an object $B$ of
$\operatorname{\mathrm{C}^{*}\mathrm{QS}}$ (the full subcategory of
$\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ consisting of unital algebras)
coacting universally on $A$. In other words, there is a coassociative
$C^{*}$-algebra map $A\to A\otimes B$ making $B$ an initial object in the
category of objects of $\operatorname{\mathrm{C}^{*}\mathrm{QS}}$ endowed with
such maps.

If $B$ were to map universally into an object
$B^{\prime}\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$, the latter would be
the quantum automorphism group of $A$ in the sense of [Wan98]. However, we
know from [Wan98, Theorem 6.1 (1)] that finite-dimensional $C^{*}$-algebras do
not have compact quantum automorphism groups, in general. In conclusion,
although
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{C}^{*}\mathrm{QS}}$
is a left adjoint (by a variant of Theorem 4.3.1), it is not a right adjoint.
$\blacklozenge$

As in § 4.2, we denote the right adjoints of Theorem 4.3.1 by
$\operatorname{\textsc{cofree}}$. Once more, it turns out that they preserve
commutativity.

######  4.3.4 Proposition.

If the object $A$ of $\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ or
$\operatorname{\mathrm{C}^{*}_{0}\mathrm{QS}}$ is commutative, so is
$\operatorname{\textsc{cofree}}(A)$.

######  Proof.

We focus on the $\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$ case.

Setting $H=\operatorname{\textsc{cofree}}(A)$, the canonical map $f:H\to A$
factors as

$H$$H_{ab}$$A$$\scriptstyle\pi$$\scriptstyle f$$\scriptstyle f_{ab}$

for some morphism $f_{ab}$ in $\operatorname{\mathrm{Alg}^{*}_{0}}$.
Essentially the same argument as the one used in the proof of Proposition
4.1.2 shows that $f_{ab}$ is actually a morphism in
$\operatorname{\mathrm{A}^{*}_{0}\mathrm{QS}}$. Finally, we can now repeat the
proof of Proposition 4.2.3 to conclude that $\pi$ is in fact an isomorphism,
and hence $H$ is commutative. $\blacksquare$

######  4.3.5 Remark.

Proposition 4.3.4 goes through in the setting of Remark 4.3.2: If $M$ is a
commutative von Neumann algebra, then it can be shown in much the same way as
above that the quantum Bohr compactification is commutative. $\blacklozenge$

###  4.4 Kac quotients

Recall that a compact quantum group
$A\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is said to be of Kac type if
the antipode on $\operatorname{\textsc{alg}}(A)$ lifts to a continuous map
$A\to A$. Equivalently, the antipode of $\operatorname{\textsc{alg}}(A)$ is
involutive ($S^{2}=\operatorname{id}$), or commutes with the $*$ operation.
This definition extends in the obvious way to CQG algebras; in that case, ‘of
Kac type’ or simply ‘Kac’ will be synonymous to ‘having involutive antipode’.
Alternate terms are ‘Kac algebra’ (under which these objects and their
relatives were introduced; e.g. [ES92] and the references therein) or
sometimes ‘Woronowicz-Kac algebra’ (as in [Ban99]). For brevity, we will
sometimes simply use ‘Kac’ as an adjective. A $k$ subscript on either
$\operatorname{\mathrm{CQG}}$ of $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
indicates the full subcategory on objects of Kac type.

In [Soł05, Appendix], Sołtan constructs what in that paper is called the
canonical Kac quotient of an object
$A\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ (notion originally due to of
Stefaan Vaes). It is obtained by quotienting out all elements of $A$ killed by
some trace (meaning, as usual, that the trace sends $x^{*}x$ to zero). This,
however, seems to be a bit of a misnomer: While it is shown in [Soł05, A.1]
that the result is indeed a compact quantum group of Kac type, it is not clear
that a Kac compact quantum group is its own canonical Kac quotient222I am
grateful to Piotr Sołtan for pointing this out.! A more appropriate term might
be, perhaps, canonical tracial quotient: One quotients out as much as one
needs to in order to ensure that the result has enough traces.

The notion has also received attention in [Tom07], where Tomatsu shows in
Theorem 4.8 that a compact quantum group $A$ has a largest quotient of Kac
type (he uses dual phrasing, thinking of the latter as a largest compact
quantum subgroup of Kac type) provided $A$ is coamenable and the Grothendieck
ring of its category of comodules (its so-called fusion algebra) is
commutative. Regarding the terminology problem from the previous paragraph,
note that for the reason pointed out there, it is not clear, a priori, that
Tomatsu’s quotient is the same as Sołtan’s. The two do coincide, however, if
the Haar measure of the compact quantum group is faithful (which is the
standing assumption of [Tom07]), hence [Tom07, Remark 4.9].

In conclusion, the question of whether or not every
$A\in\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ has a largest Kac quotient seems to
be an interesting one. One of the aims of this subsection is to show that this
is indeed the case: The desired quotient map is precisely the reflection of
$A$ in the subcategory $\operatorname{\mathrm{CQG}}_{k}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}$ (i.e. the image of $A$ through
the left adjoint to the inclusion of the subcategory into
$\operatorname{\mathrm{CQG}}$ or $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$,
respectively).

######  4.4.1 Theorem.

The inclusions $\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}$
and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
each have a left, as well as a right adjoint.

######  Proof.

That the inclusions are both left adjoints is shown in much the same way in
which we have proven all results asserting the existence of various right
adjoints so far. The arguments of Sections 2 and 3 can be repeated to show
that $\operatorname{\mathrm{CQG}}_{k}$ is finitely presentable and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}$ is SAFT. The only difference
that is even remotely significant is the fact that now the regular generator
to go into an analogue of Proposition 2.0.6 consists of $A_{u}(I_{n})$, $n\geq
1$ rather than all $A_{u}(Q)$. Colimits are again constructed simply at the
level of $*$ or $C^{*}$-algebras, making it clear that the inclusions are
cocontinuous and hence left adjoints.

The interesting problem, then, is the one discussed before the statement of
the theorem: constructing left adjoints to the two inclusions. For each
$A\in\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ we want an arrow $\kappa:A\to
A_{k}$ into a Kac type object, universal in the sense that any morphism from
$A$ into a Kac object factors uniquely through $\kappa$. In other words, we
have to show that the comma category $A\downarrow\operatorname{\mathrm{CQG}}$
(resp. $A\downarrow\operatorname{\mathrm{C}^{*}\mathrm{QG}}$) consisting of
arrows from $A$ into Kac objects has an initial object. To do this, we apply
Freyd’s initial object theorem [ML98, V.6.1]. It says that a complete category
has an initial object as soon as it has a weakly initial set of objects; this
simply means a set $S$ of objects such that any object $y$ admits at least one
arrow $S\ni x\to y$ (not necessarily unique).

In our case, a weakly initial set is easy to come by: All surjections $A\to B$
for Kac type $B$ will do, since any map of $A$ into a Kac type object will
certainly factor through the image of that map. On the other hand completeness
follows quickly if we show that $\operatorname{\mathrm{CQG}}_{k}$ and
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}$ are closed under limits in
$\operatorname{\mathrm{CQG}}$ and $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$
respectively: Limits would then be created by the forgetful functor
$A\downarrow\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}$
(and similarly for $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$).

In conclusion, it is enough to show that products of Kac objects in
$\operatorname{\mathrm{CQG}}$ (or $\operatorname{\mathrm{C}^{*}\mathrm{QG}}$)
are again Kac, and similarly, equalizers of parallel pairs of arrows between
Kac objects are Kac. Since (a)
$\operatorname{\textsc{alg}}:\operatorname{\mathrm{C}^{*}\mathrm{QG}}\to\operatorname{\mathrm{CQG}}$
is a right adjoint and hence preserves limits, and (b) by definition, an
object $B\in\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ is Kac if and only if
$\operatorname{\textsc{alg}}(B)$ is, it is enough to restrict ourselves to
$\operatorname{\mathrm{CQG}}$.

Equalizers are easy: If $f,g:B\to C$ are arrows between Kac CQG algebras, the
equalizer injects into $B$, so it is again Kac. To prove the statement about
products, let $B_{i}$, $i\in I$ be a set of Kac CQG algebras, and denote the
structure maps of their product in $\operatorname{\mathrm{CQG}}$ by
$\pi_{i}:B\to B_{i}$. Throughout the rest of this proof, for a CQG algebra
$C$, we denote by $C^{\prime}$ the CQG algebra with the same underlying set,
but reversed multiplication and comultiplication. Note that the product of the
objects $B_{i}^{\prime}$ is precisely $B^{\prime}$. Let $S_{i}$ be the
antipodes of $B_{i}$, and $S=\prod S_{i}:B\to B^{\prime}$ the map obtained
from the functoriality of products. By this same functoriality, $S$ is
involutive (strictly speaking, this means $S^{\prime}\circ
S=\operatorname{id}$, where $S^{\prime}:B^{\prime}\to B$ is $S$ as a map, but
we have switched the domain and codomain). If we show that $S$ is the antipode
$S_{B}$ of $B$, we are done (Kac means involutive). To prove this, note that
the two squares in the diagram

$B$$B^{\prime}$$B_{i}$$B_{i}^{\prime}$$\scriptstyle S$$\scriptstyle
S_{B}$$\scriptstyle S_{i}$$\scriptstyle\pi_{i}$$\scriptstyle\pi_{i}$

are both commutative (the $S$-square by the definition of $S$ as the product
of the $S_{i}$, and the $S_{B}$-square because $\pi_{i}$ are Hopf algebra
morphisms and hence preserve antipodes). $\blacksquare$

Denote by $A\mapsto A_{k}$ the left adjoint to either of the inclusions
$\operatorname{\mathrm{CQG}}_{k}\to\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}_{k}\to\operatorname{\mathrm{C}^{*}\mathrm{QG}}$.
It is a simple observation now that the canonical map $A\to A_{k}$ is a
surjection, and hence warrants the name ‘Kac quotient’:

######  4.4.2 Proposition.

Let $A\in\operatorname{\mathrm{CQG}}$ or
$\operatorname{\mathrm{C}^{*}\mathrm{QG}}$ and $\kappa:A\to A_{k}$ the
universal arrow resulting from the unit of the adjunction that the Kac
quotient functor is part of. Then, $\kappa$ is onto.

######  Proof.

To keep things streamlined, let us focus on $\operatorname{\mathrm{CQG}}$. We
have seen this sort of argument before, in a dual form, in Proposition 4.2.3.
Let $\iota:A_{k}^{\prime}\to A_{k}$ be the inclusion of
$A_{k}^{\prime}=\mathrm{Im}(\kappa)$, and denote the corestriction by