Epimorphic quantum subgroups and coalgebra codominions

TL;DR

This paper explores categorical properties of coalgebras, bialgebras, and Hopf algebras, providing new characterizations and invariance results that generalize classical algebraic group theorems.

Contribution

It offers novel categorical characterizations of monomorphisms, epimorphisms, and dominions in coalgebra-related categories, extending classical algebraic results to quantum and categorical contexts.

Findings

Characterization of monomorphisms in coalgebra categories

Invariance of codominions under field extension

Surjections are regular epimorphisms when codomain is cosemisimple

Abstract

We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension $\Bbbk\le \Bbbk'$ is a right adjoint on the category of $\Bbbk$-coalgebras).

Equations

$$\mathcal{C}(c^{\prime },c^{\prime \prime })\stackrel{-\circ f}{\to }\mathcal{C}(c,c^{\prime \prime })$$

$$c^{\prime }$$

$$V\ni v\mapsto v_0\otimes v_1\in V\otimes H$$

$$V^{H^{\prime }}:=\{v\in V|v_0\otimes \pi (v_1)=v\otimes 1\}$$

$$C\ni c$$

$$V\ni v$$

$${\text{textsc}\mathrm{Alg}}_{\bullet },{\text{textsc}\mathrm{Bialg}}_{\bullet },{}_{\square }{\text{textsc}\mathrm{HAlg}}_{\bullet }.$$

$${\text{textsc}\mathrm{Coalg}}_{\bullet },{\text{textsc}\mathrm{Bialg}}_{\bullet },{}_{\square }{\text{textsc}\mathrm{HAlg}}_{\bullet }.$$

$${\text{textsc}\mathrm{HAlg}}_c\stackrel{\text{inclusion}}{\to }{}_2\text{textsc}\mathrm{HAlg}\stackrel{\text{inclusion}}{\to }{}_{bi}\text{textsc}\mathrm{HAlg}\stackrel{\text{inclusion}}{\to }\text{textsc}\mathrm{HAlg}\stackrel{\text{inclusion}}{\to }\text{textsc}\mathrm{Bialg}\stackrel{\text{forget}}{\to }\text{textsc}\mathrm{Coalg}$$

$${\text{textsc}\mathrm{HAlg}}_{c,cc}\stackrel{\text{inclusion}}{\to }{\text{textsc}\mathrm{HAlg}}_{cc}\stackrel{\text{inclusion}}{\to }{\text{textsc}\mathrm{Bialg}}_{cc}\stackrel{\text{forget}}{\to }{\text{textsc}\mathrm{Coalg}}_{cc}.$$

$$H\underset{\cong }{\overset{{\Delta }_H}{\to }}H{\square }_{H^{\prime }}H$$

$$c$$

$$\text{textsc}\mathrm{Alg}\to \text{textsc}Vec\text{and}\text{textsc}\mathrm{Coalg}\to \text{textsc}Vec$$

$$V{\square }_{D}W:=\{x\in V\otimes W|({\rho }_V\otimes {\mathrm{id}}_W)x=({\mathrm{id}}_V\otimes {\rho }_W)x\in V\otimes D\otimes W\}$$

$${\rho }_V:V\to V\otimes C\text{and}{\rho }_W:W\to C\otimes W.$$

$$V^H:=\{v\in V|v_0\otimes v_1=v\otimes 1\in V\otimes H\}$$

$${\Gamma }_f:=\{(v,fv)|v\in V\}\le V\oplus W$$

$$V^H\cong {\mathcal{M}}^H(\mathbb{k},V),$$

$${\mathcal{M}}^H(V,W)\cong (W\otimes V^{\ast }{)}^H,$$

$$H\stackrel{\text{antipode}}{\to }{H}^{op,cop}$$

$$c$$

$$r(\iota ):=H/H{A}^{+},A^{+}=\ker \epsilon {|}_A,\epsilon \text{ being the counit of }H.$$

$$l(\pi ):={}^{C}H:=\{h\in H|\pi (h_1)\otimes h_2=\pi (1)\otimes h\}.$$

$$H_{\le }:=\{\text{right H-coideal subalgebras}\le H\}\text{and}$$

$$H_{\twoheadrightarrow }:=\{\text{left H-module quotient coalgebras of }H\}.$$

$$rA\ge C⟺lC\ge A.$$

$$lr:H_{\le }\to H_{\le }\text{and}rl:H_{\twoheadrightarrow }\to H_{\twoheadrightarrow }$$

$$C{\times }_{D}C$$

$$C{\square }_{D}C$$

$$x_0\otimes f(x_1)=x_0\otimes f(x_{-1})\in X\otimes D⟹x_0\otimes \pi (x_1)=x_0\otimes \pi (x_{-1})\in X\otimes D^{\prime }.$$

Full text

Key words: Hopf algebra; coalgebra; comodule; epimorphism; monomorphism; dominion; codominion; pullback; locally presentable; adjoint functor; algebraic group; cocommutative; CQG algebra

MSC 2020: 18A20; 16T05; 16T15; 20G42; 18A30; 20G05; 20G15; 18C15; 18C20; 16D40

1. 1 Preliminaries
2. 2 Epimorphic quantum subgroups and (co)dominions
3. 3 CQG algebras and compact groups
4. 4 Asides on scalar extension and limits of coalgebras

Introduction

A possible starting point for the considerations below is the familiar problem of determining, for a concrete category (i.e. one consisting of sets and functions [1, Definition 5.1]), whether the epimorphisms are precisely the surjections. Recall [1, Definition 7.3.9] that an epimorphism in a category ${\mathcal{C}}$ is a morphism $f:c\to c^{\prime}$ for which every precomposition map

[TABLE]

is one-to-one. This is one natural category-theoretic generalization of the notion of surjectivity, and for concrete categories, where both concepts make sense, the latter plainly implies the former.

The problem recurs in endless variations throughout a massive amount of literature this introduction cannot possibly do justice. One example would be [54], which addresses the issue for several categories (e.g. compact groups and Lie algebras, most relevantly for the discussion below, but also others, such as $C^{*}$- or von Neumann algebras). Other examples include, say, the earlier [48] (also compact groups), [30] ($C^{*}$-algebras), [6] (Lie algebras, $p$-Lie algebras), [10, 11] (linear algebraic groups), [13] (algebraic groups in general), or the series [34, 31, 35, 36, 33], which systematically studies the broader notion of dominion [1, 14J] in various categories (semigroups, associative algebras): assuming the category being studied is sufficiently well-behaved, the dominion of a morphism $f:c\to c^{\prime}$ is the equalizer of the two structure maps

[TABLE]

into the self-pushout of $f$. The connection between the two notions is that $f:c\to c^{\prime}$ is epic precisely if its dominion is $\operatorname{id}:c^{\prime}\to c^{\prime}$. Dominions make an oblique appearance in [10] too, under a different name: for a closed linear algebraic subgroup $\iota:H\leq G$, the observable envelope [10, §3] of $H$ in $G$ is precisely the dominion of $\iota$ in the category of linear algebraic groups (see the discussion following Remark 2.8).

One of the main results is that the much of the familiar characterization [10, Théorème 1] of epimorphisms in the category of linear algebraic groups goes through for Hopf algebras, taking into account the fact that one must dualize (the category of linear algebraic groups over a field $\Bbbk$ being contravariantly equivalent to that of Hopf $\Bbbk$-algebras [62, §1.4, Theorem]). This is the content of Theorem 2.5 below:

Theorem**.**

For a morphism $\pi:H\to H^{\prime}$ of coalgebras over a field $\Bbbk$ the following conditions are equivalent:

1. (a)

$\pi$* is monic;* 2. (b)

The corestriction functor induced by $\pi$ between categories of (finite-dimensional) comodules is full. 3. (c)

For any (finite-dimensional) $H$-comodule $V$ every idempotent morphism on ${\mathcal{M}}^{H^{\prime}}$ is in ${\mathcal{M}}^{H}$. 4. (d)

For any (finite-dimensional) $H$-comodule $V$ every direct-sum decomposition $V\cong V_{1}\oplus V_{2}$ as $H^{\prime}$-comodules is also one of $H$-comodules.

If furthermore $\pi$ is a morphism of Hopf algebras, these are also equivalent to 5. (e)

For any (finite-dimensional) right $H$-comodule $V$ with comodule structure map

[TABLE]

the space

[TABLE]

of $H^{\prime}$-coinvariants coincides with the analogous space of $H$-coinvariants. 6. (f)

We have $H^{H^{\prime}}=\Bbbk$ (i.e. the only $H^{\prime}$-coinvariants in $H$ are the scalars).

The statement above is somewhat vague on which category (coalgebras or Hopf algebras) the morphisms are supposed to be monic in. The same issue arises in trying to relate the statement back to algebraic groups: the latter form a category dual to that of commutative Hopf algebras, so it is reasonable to wonder whether a morphism contained in two of the categories of interest might be monic in one but not the other. Proposition 2.1 shows that this is not an issue, and Theorem 2.14 expands this to (co)dominions (given a morphism in any of the relevant categories, it does not matter where one computes its codominion):

Theorem**.**

1. (1)

The dominion of an algebra morphism is the same in any of the following categories that happen to contain it:

• •

commutative algebras;

• •

(commutative) bialgebras;

• •

(commutative or cocommutative) Hopf algebras;

• •

Hopf algebras with involutive or bijective antipode.

In particular, a morphism is or is not epic simultaneously in all of these categories containing it. 2. (2)

The codominion of an algebra morphism is the same in any of the following categories that happen to contain it:

• •

cocommutative coalgebras;

• •

(commutative) bialgebras;

• •

(commutative or cocommutative) Hopf algebras;

• •

Hopf algebras with involutive or bijective antipode.

In particular, a morphism is or is not monic simultaneously in all of these categories containing it.

Another theme cropping up in the literature is the independence of the various notions discussed above (dominions, epimorphisms) of the base field: [6, Proposition 5.3] is a case in point (dominions of finite-dimensional Lie-algebra morphisms are preserved by field extensions), as is [13, Theorem 5 (ii)] (where the category is that of algebraic groups instead). In this same spirit, field extensions preserve not only codominions (Proposition 2.17), but also arbitrary limits in categories of objects equipped with ‘coalgebraic structure‘ (Theorem 4.1 2):

Theorem**.**

Let $\Bbbk\leq\Bbbk^{\prime}$ be a field extension.

1. (1)

The functor $\Bbbk^{\prime}\otimes_{\Bbbk}-$ between any of the following categories (of objects linear over $\Bbbk$ and $\Bbbk^{\prime}$ respectively) is a right adjoint and in particular continuous:

• •

(cocommutative) coalgebras;

• •

(commutative or cocommutative) bialgebras;

• •

(commutative or cocommutative Hopf algebras);

• •

Hopf algebras with involutive or bijective antipode. 2. (2)

The same scalar-extension functor $\Bbbk^{\prime}\otimes_{\Bbbk}-$ preserves (co)dominions in all of the categories above, along with those of commutative or plain algebras.

Occasionally, an object will have the property that any monomorphism having it as domain is automatically a dominion (these are, for instance, the absolutely closed algebras of [34, discussion preceding Corollary 1.7]). Examples suggest a general phenomenon whereby “semisimplicity entails absolute closure”: [6, Addenda] reports a remark of Hochschild’s to the effect that embeddings of semisimple Lie algebras are dominions in the category of finite-dimensional Lie algebras, and the analogue for linearly reductive algebraic groups follows from standard GIT theory (e.g. [44, §1.2]).

Dualizing the picture once more, morphisms onto cosemisimple coalgebras are codominions. Several versions of this are recorded in Corollaries 2.22 and 2.24 and Theorem 2.26. Focusing, for simplicity, on Hopf algebras (rather than bialgebras, etc.), a sampling of those results reads as follows:

Theorem**.**

1. (1)

A morphism $\pi:H\to H^{\prime}$ of Hopf algebras is a codominion if either

• •

$H$* is left or right faithfully coflat over $H^{\prime}$;*

• •

or $\pi$ has a right inverse in the category of either right or left $H^{\prime}$-comodules. 2. (2)

In particular, surjections onto cosemisimple Hopf algebras are codominions in the category of Hopf algebras.

These considerations also deliver a non-commutative version of the result that embeddings of compact groups are equalizers [19, Theorem 2.1]. Recall ([23, Definition 2.2], Definition 3.1 below) that CQG algebras are non-commutative analogues of the (Hopf) algebras of representative functions [14, §III.1, Definition 1.1] on compact groups. With this in mind, and paraphrased in dual language, Theorem 3.4 says that embeddings of “compact quantum groups” are equalizers:

Theorem**.**

*Surjections in the category of CQG algebras are coequalizers. *

Acknowledgements

This work was partially supported through NSF grant DMS-2001128.

I am grateful for assorted comments and pointers to the literature from A. Agore, M. Brion, T. Brzezinski, G. Militaru and R. Wisbauer.

1 Preliminaries

Algebras (always associative and unital), coalgebras (coassociative and counital) and the like will typically be assumed linear over a field. Relevant background on coalgebras, bialgebras and Hopf algebras is available in a number of excellent sources such as [58, 42, 24, 53], and the text contains more specific references to those where needed.

Some prominent notation:

• •

As usual (e.g. [42, Definition 1.5.1]), $\Delta$, $\varepsilon$ and $S$ typically denote the comultiplication, counit and respectively antipode of a Hopf algebra (which one will typically be clear from context).

• •

We use Sweedler notation [42, Notation 1.4.2] for comultiplications and (left or right) comodule structures:

[TABLE]

• •

$\operatorname{\textsc{Alg}}$, $\operatorname{\textsc{Coalg}}$, $\operatorname{\textsc{Bialg}}$ and $\operatorname{\textsc{HAlg}}$ denote the categories of algebras, coalgebras, bialgebras and Hopf algebras respectively, over a field $\Bbbk$ fixed beforehand and usually implicit.

We might occasionally adorn the symbols with the field as a subscript, when wishing to emphasize it (e.g. $\operatorname{\textsc{Alg}}_{\Bbbk}$).

• •

Various other subscripts modify those categories to indicate additional properties:

• –

‘c’ stands for ‘commutative’ (e.g. $\operatorname{\textsc{Alg}}_{c}$);

• –

‘cc’ for ‘cocommutative’ [42, Definition 1.1.3] (e.g. $\operatorname{\textsc{Bialg}}_{\Bbbk,cc}$);

• –

while for Hopf algebras, a left-hand subscript might indicate antipode bijectivity or involutivity: ${}_{bi}\operatorname{\textsc{HAlg}}$ is the category of Hopf algebras with bijective antipode, while ${}_{2}\operatorname{\textsc{HAlg}}$ that of Hopf algebras whose antipode squares to the identity.

• –

We will occasionally have to refer to these subscripts in bulk, in which case we use non-alphanumeric symbols such as ‘$\mathrel{\Box}$’ or ‘$\bullet$’. $\operatorname{\textsc{Alg}}_{\bullet}$, for instance, indicates both $\operatorname{\textsc{Alg}}$ and $\operatorname{\textsc{Alg}}_{c}$ in one go, while ${}_{\mathrel{\Box}}\operatorname{\textsc{HAlg}}_{\bullet}$ refers collectively to ${}_{bi}\operatorname{\textsc{HAlg}}$, $\operatorname{\textsc{HAlg}}_{c}$, $\operatorname{\textsc{HAlg}}_{c,cc}$, etc.

• –

As a variation of this, we might need to limit the possibilities for what one of the subscripts might be, in which case we separate the various options by a ‘/’ symbol: $\operatorname{\textsc{HAlg}}_{c/}$ means the category of either commutative or arbitrary Hopf algebras (but not that of cocommutative Hopf algebras).

• •

Categories of modules are depicted as ‘${\mathcal{M}}$’ adorned with subscripts indicating the base algebra, with the handedness of the placement indicating whether they are right or left modules: ${\mathcal{M}}_{A}$, say, means right $A$-modules. This applies also to two-sided structures: ${}^{C}{\mathcal{M}}^{D}$, for instance, is the category of $(C,D)$-bicomodules for coalgebras $C$ and $D$.

One exception will be $\textsc{Vec}=\textsc{Vec}_{\Bbbk}$ (rather than ${\mathcal{M}}_{\Bbbk}$), the category of vector spaces over $\Bbbk$.

• •

Similarly, comodule [42, Definition 1.6.2] categories (over coalgebras, bialgebras, etc.) exhibit the base coalgebra as a superscript, again left or right: ${\mathcal{M}}^{C}$ is the category of $C$-comodules.

• •

Additional ‘$f$’ subscripts indicate finite-dimensionality: ${\mathcal{M}}^{H}_{f}$ would be the category of finite-dimensional $H$-comodules.

Some category-theoretic terminology will feature quite extensively; we refer the reader to, say, [5, 1, 40], again with more specific citations as needed. In general, for objects $c,c^{\prime}\in{\mathcal{C}}$ of a category ${\mathcal{C}}$, we write ${\mathcal{C}}(c,c^{\prime})$ for their space of morphisms.

One remark that is worth making now, as it pervades more or less the entire discussion, is that apart from those imposing size conditions (i.e. the categories of finite-dimensional (co)modules), every one of the categories mentioned above ($\operatorname{\textsc{Alg}}$, $\operatorname{\textsc{Coalg}}$, $\operatorname{\textsc{HAlg}}$, their (co)commutative versions, etc. etc.) is extremely well-behaved. Specifically, they are all locally presentable in the sense of [2, Definition 1.17]:

• •

Module and comodule categories are Grothendieck [49, §2.8, Note 3 following Corollary 8.13] (see [24, Corollary 2.2.8] for categories of comodules), and hence locally presentable [38, Corollary 5.2].

• •

The other categories of “algebraic structures” (i.e. sets equipped with operations of various arities, satisfying various equations; algebras and commutative algebras, say) this is well known and a uniform consequence of, say [2, Theorem 3.28].

• •

For the categories of “coalgebraic” or “mixed” structures (coalgebras, bialgebras, Hopf algebras) this has been worked out in ample detail in a number of papers of Porst’s; see for instance [52, Lemmas 1 and 2, Theorem 6 and Proposition 22].

In particular, by [2, Remark 1.56] all of these categories

• •

are (co)complete (they have arbitrary (co)limits [1, Definition 12.2]);

• •

are well-powered (the isomorphism classes of subobjects [1, Definition 7.77] of any given object form a set [1, Definition 7.82]);

• •

and also co-well-powered (the isomorphism classes of quotient objects [1, Definition 7.84] of any given object form a set [1, Definition 7.87]);

• •

and hence satisfy a host of convenient adjoint functor theorems [1, §18]:

• –

cocontinuous (i.e. colimit-preserving) functors defined on any of these categories are left adjoints [40, §5.8, Corollary on p.130].

• –

and continuous (i.e. limit-preserving) functors having them as domains are right adjoints provided they also preserve $\kappa$-directed colimits [2, Definition 1.13 (1)] for some regular cardinal $\kappa$ [2, Theorem 1.66].

2 Epimorphic quantum subgroups and (co)dominions

As we will be concerned with monomorphisms and epimorphisms of bialgebras, Hopf algebras, etc., it will be worth noting that the notions are invariant under a number of convenient modifications. These are very simple remarks, but it will be helpful to have made them explicitly.

Proposition 2.1**.**

1. (1)

A morphism $f:H\to H^{\prime}$ in $\operatorname{\textsc{Alg}}_{\Bbbk}$ is epic in that category if and only if it is epic in any of the following categories which contain it:

[TABLE] 2. (2)

A morphism $f:H\to H^{\prime}$ in $\operatorname{\textsc{Coalg}}_{\Bbbk}$ is monic in that category if and only if it is monic in any of the following categories which contain it:

[TABLE]

Proof**.**

We focus on 2, to fix ideas; the other half is entirely analogous (and categorically dual).

The functors

[TABLE]

are right adjoints: see [52, diagram (9)] for the inclusions and [51, §2.2] for the last arrow. They thus preserve monomorphisms [1, Proposition 18.6], and also reflect them because they are faithful.

Exactly the same reasoning applies to

• •

the inclusion functors from commutative bialgebras (Hopf algebras) to plain bialgebras (respectively Hopf algebras), which are also faithful right adjoints by **[52, diagram (9)]**;

• •

and the chain of functors

[TABLE]

The only claim left to verify, then, is that a morphism $f:H\to H^{\prime}$ of cocommutative coalgebras is monic in $\operatorname{\textsc{Coalg}}_{cc}$ if and only if it is so in $\operatorname{\textsc{Coalg}}$.

$f$* is monic in $\operatorname{\textsc{Coalg}}$ if and only if the comultiplication $\Delta_{H}$ induces an isomorphism*

[TABLE]

[45, Theorem 3.5]**. Note, though, that that cotensor product is nothing but the self-pullback of $f$ in $\operatorname{\textsc{Coalg}}_{cc}$ (very similar to the remark that the products in $\operatorname{\textsc{Coalg}}_{cc}$ are the tensor products [58, Theorem 6.4.5]), so said condition is also equivalent to $f$ being monic in $\operatorname{\textsc{Coalg}}_{cc}$: it is a formal consequence of the definition that for a morphism $f:c\to c^{\prime}$ in any category ${\mathcal{C}}$ the following conditions are equivalent:

• •

$f$* is monic;*

• •

the diagram

[TABLE]

is a pullback;

• •

the self-pullback $c\times_{c^{\prime}}c$ exists and either one of the two morphisms $c\times_{c^{\prime}}c\to c$ is an isomorphism;

• •

the self-pullback $c\times_{c^{\prime}}c$ exists and the two morphisms $c\times_{c^{\prime}}c\to c$ are equal;

• •

the self-pullback $c\times_{c^{\prime}}c$ exists and the diagonal map $c\to c\times_{c^{\prime}}c$ is an isomorphism.

$\blacksquare$

In reference to the last part of the proof of Proposition 2.1, recall [45, Theorem 3.6] that in $\operatorname{\textsc{Coalg}}_{cc}$ the monomorphisms are precisely the injections. Slightly more is true:

Lemma 2.2**.**

*A $\operatorname{\textsc{Coalg}}$-monomorphism with cocommutative codomain is an embedding. *

Proof**.**

As in the proof of [45, Theorem 3.6], the claim reduces to the case when all coalgebras in sight are finite-dimensional. But then one can dualize the statement to address (epi)morphisms of finite-dimensional algebras, whence the conclusion by [36, §2.11]: an algebra admitting a finite-dimensional epimorphic extension contains a copy of the $2\times 2$ upper-triangular matrices, so cannot be commutative. $\blacksquare$

Lemma 2.2 does indeed imply [45, Theorem 3.6], since by Proposition 2.1 2 a morphism in $\operatorname{\textsc{Coalg}}_{cc}$ is monic if and only if it is so in $\operatorname{\textsc{Coalg}}$.

Remark 2.3**.**

By contrast to the interesting algebra epimorphisms and coalgebra monomorphisms in focus here, the algebra monomorphisms and coalgebra epimorphisms are precisely the injective (respectively surjective) morphisms in the relevant category. This is because the forgetful functors

[TABLE]

are faithful and right (respectively left) adjoint [50, §§2.6, 2.7], so they preserve and reflect monomorphisms (respectively epimorphisms) [1, Proposition 18.6]. $\blacklozenge$

Since it is, furthermore, occasionally convenient to extend scalars, it is worth noting that the properties of interest are invariant under such extensions (or contractions).

Lemma 2.4**.**

*Let $\Bbbk\subset\Bbbk^{\prime}$ be a field extension. The property of a morphism $f$ of algebras, coalgebras, bialgebras or Hopf algebras over $\Bbbk$ of being monic or epic is equivalent to the corresponding property for the scalar extension $f\otimes_{\Bbbk}\Bbbk^{\prime}$. *

Proof**.**

By Proposition 2.1 the claims about bialgebras or Hopf algebras reduce to (co)algebras, so we focus on these.

For algebras being monic is equivalent to injectivity: a surjective morphism $A\to B$ is the coequalizer of the two maps $A\times_{B}A\to A$; dually, for coalgebras being epic is equivalent to surjectivity [45, Theorem 3.1]. We are thus reduced to considering

• •

epimorphisms $A\to B$ of algebras, equivalent to the multiplication map $B\to B\otimes_{A}B$ being isomorphisms **[57, §X.1, Proposition 1.2]**;

• •

and monomorphisms $C\to D$ of coalgebras, equivalent to the coproduct map $C\to C\mathrel{\Box}_{D}C$ being an isomorphism **[45, Theorem 3.5]**, recalling the cotensor product **[42, Definition 8.4.2]**

[TABLE]

of right and left comodules

[TABLE]

Being a faithfully flat [39, §4I] extension, $\Bbbk^{\prime}\otimes_{\Bbbk}-$ preserves and reflects everything in sight: (c)tensor products, exactness, surjectivity, injectivity, etc. $\blacksquare$

In the sequel, for a right comodule $V$ over a bialgebra $H$, we write

[TABLE]

for its space of $H$-coinvariants.

Theorem 2.5**.**

For a morphism $\pi:H\to H^{\prime}$ in $\operatorname{\textsc{Coalg}}$ the following conditions are equivalent:

1. (a)

$\pi$* is monic;* 2. (b)

The corestriction functor induced by $\pi$ between categories of (finite-dimensional) comodules is full. 3. (c)

For any (finite-dimensional) $H$-comodule $V$ every idempotent morphism on ${\mathcal{M}}^{H^{\prime}}$ is in ${\mathcal{M}}^{H}$. 4. (d)

For any (finite-dimensional) $H$-comodule $V$ every direct-sum decomposition $V\cong V_{1}\oplus V_{2}$ as $H^{\prime}$-comodules is also one of $H$-comodules.

If furthermore $\pi$ is a morphism in ${}_{\mathrel{\Box}}\operatorname{\textsc{HAlg}}$, these are also equivalent to 5. (e)

For any (finite-dimensional) right $H$-comodule $V\in{\mathcal{M}}^{H}$, the $H^{\prime}$-coinvariants are also $H$-coinvariants. 6. (f)

The only $H^{\prime}$-coinvariants in $H$ are the scalars.

Proof**.**

1. (Passage to Hopf algebras)* We know from Proposition 2.1 2 that a morphism of Hopf algebras is monic if and only if it is monic in the category of coalgebras.*
2. a** $\Longleftrightarrow$ b*** For arbitrary comodules, this is part of [45, Theorem 3.5] (or its expansion in [3, Theorem 2.1]). Because comodules (of a coalgebra over a field) are unions of finite-dimensional subcomodules [43, Theore 5.1.1], the fullness of ${\mathcal{M}}^{H}\to{\mathcal{M}}^{H^{\prime}}$ is equivalent to the fullness of the analogous functor ${\mathcal{M}}^{H}_{f}\to{\mathcal{M}}^{H^{\prime}}_{f}$ between categories of finite-dimensional comodules.*

This same argument will allow us, in general, to obtain the mutual equivalence between the various statements on arbitrary comodules and their finite-dimensional versions; we will thus not belabor that point. 3. b** $\Longrightarrow$ c*** This is obvious: just apply b to the idempotent morphism in question.* 4. c** $\Longleftrightarrow$ d*** A direct-sum decomposition $V\cong V_{1}\oplus V_{2}$ is nothing but a choice of idempotent endomorphism on $V$ in the relevant category (with range $V_{1}$ and kernel $V_{2}$), so the two are plainly alternative phrasings of a common condition.* 5. d** $\Longrightarrow$ b*** Consider $V,W\in{\mathcal{M}}^{H}$, and a linear map $f:V\to W$, $H^{\prime}$-colinear. The graph*

[TABLE]

is then an $H^{\prime}$-subcomodule, with supplementary summand $W$. Assumption d then ensures that $\Gamma_{f}$ is also an $H$-subcomodule of $V\oplus W$, whence the conclusion that $f$ is also $H$-colinear.

Assume henceforth that $\pi$ is a Hopf-algebra morphism. 6. b** $\Longleftrightarrow$ e*** Since for $H$-comodules $V$ we have a functorial identification*

[TABLE]

fullness implies e.

To verify the converse, note that for finite-dimensional comodules $V,W\in{\mathcal{M}}^{H}_{f}$ we have

[TABLE]

i.e. the $H$-coinvariants of the tensor product of $W$ and the dual of $V$. $H$-comodule morphisms can thus be recast as coinvariants, so e does indeed imply b. 7. e** $\Longleftrightarrow$ f*** The latter is nothing but e applied to the $H$-comodule $H$ and thus formally weaker than that condition. Conversely, it is enough to recall [24, Proposition 2.4.3] that every $H$-comodule can be embedded in a direct sum of copies of $H$.*

$\blacksquare$

Corollary 2.6 confirms, for algebraic groups, some speculative remarks made in the context of studying epimorphisms of Lie algebras on [6, pp.13-14]; see also [10, Théorème 1] for connected algebraic groups and [13, Theorem 1] for the general statement (to which Theorem 2.5 specializes when the Hopf algebras in question are commutative).

Corollary 2.6**.**

*A closed embedding ${\mathbb{H}}\leq{\mathbb{G}}$ is an epimorphism in the category of linear algebraic groups over a field if and only if the only the only ${\mathbb{H}}$-invariants in ${\mathcal{O}}({\mathbb{G}})$ are the scalars. *

Proof**.**

This is an immediate application of Theorem 2.5, noting that a closed embedding ${\mathbb{H}}\leq{\mathbb{G}}$ of linear algebraic groups will be an epimorphism precisely when the dual surjection ${\mathcal{O}}({\mathbb{G}})\to{\mathcal{O}}({\mathbb{H}})$ of Hopf algebras of regular functions is a monomorphism. $\blacksquare$

[20, Corollary 2.2] provides examples of surjective, non-injective monomorphisms of Hopf algebras of the form

[TABLE]

whenever the antipode is surjective but non-injective (e.g. as in [56, Theorem 3.2]), but nothing as exotic as that is needed to exhibit such pathological monomorphisms:

Corollary 2.7**.**

Let ${\mathbb{G}}$ be a connected linear algebraic group over a field and ${\mathbb{P}}\leq{\mathbb{G}}$ a parabolic subgroup.

*The corresponding surjection ${\mathcal{O}}({\mathbb{G}})\to{\mathcal{O}}({\mathbb{P}})$ of algebras of regular functions is a monomorphism in the category of Hopf algebras. *

Proof**.**

By the very definition of parabolic subgroups [12, §11.2], the homogeneous space ${\mathbb{G}}/{\mathbb{P}}$ is projective. It then follows [29, Theorem I.3.4] that it admits no non-constant regular functions, meaning that condition f of Theorem 2.5 is met. $\blacksquare$

Remark 2.8**.**

Note that Corollary 2.7 goes through in positive characteristic; contrast this with the situation for Lie algebras: by [6, Theorems 2.2 and 2.3], in positive characteristic ($p$, say) epimorphisms of both finite-dimensional Lie algebras and finite-dimensional $p$-Lie algebras (the restricted Lie algebras of characteristic $p$ of [37, §V.7, Definition 4]) are surjective. $\blacklozenge$

At the other extreme, so to speak, from epimorphically-embedded (linear algebraic) closed subgroups ${\mathbb{H}}\leq{\mathbb{G}}$ lie the observable subgroups of [8, §4]: those for which the ${\mathbb{H}}$-invariants in ${\mathcal{O}}({\mathbb{G}})$ separate the points of the homogeneous space ${\mathbb{G}}/{\mathbb{H}}$ (as opposed to being constant, as in the epimorphic case).

The connection to epimorphisms was pursued in [10, Proposition 1], with extensions in [13, Theorem 5]: every closed linear algebraic subgroup ${\mathbb{H}}\leq{\mathbb{G}}$ admits a smallest intermediate observable subgroup ${\mathbb{H}}\leq\overline{{\mathbb{H}}}\leq{\mathbb{G}}$, its observable hull (or envelope), which is at the same time the largest subgroup in which ${\mathbb{H}}$ embeds epimorphically.

The notion has been developed further in a number of directions: [13, Theorem 5] in fact applies to possibly non-affine group schemes, [47, Theorem 1] gives a geometric characterization of observability, [28, Theorem 2.1] and [60, Theorem 9] provide numerous other conditions equivalent to observability, etc. etc.

Different fields and contexts suggest different terminology (for observability), which we will adopt here: see for instance [57, §XI.1], or [6], where the term ‘dominion’ is employed extensively, or [1, 14J] for the category-theoretic version we recall and adapt here.

Definition 2.9**.**

Let $f:c\to d$ be a morphism in a category ${\mathcal{C}}$.

1. (1)

Suppose ${\mathcal{C}}$ is complete and well-powered.

• •

The dominion of $f$ is the smallest regular subobject $\overline{c}\lhook\joinrel\xrightarrow{\quad}d$ (i.e. equalizer of a pair of morphisms $d\to\bullet$ [1, Definition 7.56]) through which $f$ factors.

• •

$f$ is a dominion if the map $c\to\overline{c}$ to the dominion is an isomorphism.

• •

$f$ dominates a subobject $c^{\prime}\lhook\joinrel\xrightarrow{\quad}d$ if we have a factorization

[TABLE]

Dually: 2. (2)

Suppose ${\mathcal{C}}$ is cocomplete and co-well-powered.

• •

The codominion of $f$ is the smallest regular quotient object $c\xrightarrow[]{\quad}\mathrel{\mkern-14.0mu}\rightarrow\overline{d}$ (i.e. coequalizer of a pair of arrows $\bullet\to c$ [1, Definition 7.71]) through which $f$ factors.

• •

$f$ (or $d$) is a codominion if the map $\overline{d}\to d$ is an isomorphism.

• •

$f$ dominates (this seems a better choice than ‘codominates’) a quotient object $c\xrightarrow[]{\quad}\mathrel{\mkern-14.0mu}\rightarrow d^{\prime}$ if we have a factorization

[TABLE]

Under the hypotheses on ${\mathcal{C}}$, it can be shown ([1, §14.J] and its categorical dual) that such arrows do exist and are unique up to the obvious notions of isomorphism. $\blacklozenge$

With this in place, note that the observable envelope [10, §3] of a closed linear algebraic subgroup ${\mathbb{H}}\leq{\mathbb{G}}$ is nothing but the dominion of ${\mathbb{H}}\to{\mathbb{G}}$ in the appropriate category (linear algebraic groups in [10], arbitrary algebraic groups in [13, Theorem 5], etc.).

As to Hopf algebras $H$, there is a satisfying correspondence between dominions (in the category $\operatorname{\textsc{Alg}}$) of right coideal subalgebras $A\leq H$ and codominions (in the category $\operatorname{\textsc{Coalg}}$) of left module quotient coalgebras $H\twoheadrightarrow C$. We review some of this material, as worked out partially in [16, §1.1] (the discussion is not present in later iterations of that preprint or in the published version [17]).

Throughout, let $H$ be a Hopf algebra over some field $\Bbbk$. The following constructions are central to [59] and discussed there extensively:

Notation 2.10**.**

1. (1)

For a right $H$-coideal subalgebra $\iota:A\leq H$ (i.e. both a subalgebra and a right coideal [42, discussion preceding Example 1.6.6]), write $r(\iota)$ or slightly abusively $r(A)$ (‘$r$’ for ‘right’) for the quotient

[TABLE]

It is not difficult to see that $r(\iota)$ is a left $H$-module quotient coalgebra of $H$ [59, Proposition 1]. 2. (2)

There is a kind of reciprocal construction: given a left $H$-module quotient coalgebra $\pi:H\to C$, write $l(\pi)$ or $l(C)$ (‘$l$’ meaning ‘left’) for

[TABLE]

The same [59, Proposition 1] shows that $l(\pi)$ is a right coideal subalgebra.

We might occasionally also drop parentheses in applying the operators $l$ and $r$ (e.g. $rA$ in place of $r(A)$). $\blacklozenge$

The operators $l$ and $r$ give back-and-forth maps between

[TABLE]

We regard these as posets in the obvious fashion, ordering them

• •

by inclusion for subalgebras;

• •

and reverse kernel inclusion for quotients.

As observed in [16, §1.1], we have

Lemma 2.11**.**

Let $H$ be a Hopf algebra with its associated posets 2-1.

1. (1)

The maps $l:H_{\twoheadrightarrow}\to H_{\leq}$ and $r:H_{\leq}\to H_{\twoheadrightarrow}$ are order-reversing, and form a Galois connection between the two posets in the sense of **[40, §IV.5, Theorem 1]**:

[TABLE] 2. (2)

The compositions

[TABLE]

are closure operators on the two respective posets in the sense of **[40, §VI.1]** or **[27, Definition 0-3.8]**: order-preserving, idempotent, and such that the image of an element dominates that element.

Proof**.**

1* is immediate from the definitions, whereas 2 is a formal consequence of 1. $\blacksquare$*

The relevance of Lemma 2.11 to the present discussion lies in relation to (co)dominions: these too provide closure operators on the two posets, and the two types of closure coincide (see [16, Proposition 1.1.6]).

Proposition 2.12**.**

Let $H$ be a Hopf algebra and $H_{\leq}$, $H_{\twoheadrightarrow}$ its posets 2-1.

1. (1)

For every right $H$-coideal subalgebra $A\in H_{\leq}$, $lrA$ is precisely the dominion of $A\leq H$ in the category of algebras. 2. (2)

For every left $H$-module quotient coalgebra $C\in H_{\twoheadrightarrow}$, $rlC$ is the codominion of $H\twoheadrightarrow C$ in the category of coalgebras.

The proof of [16, Proposition 1.1.6] relies on having available a number of equivalent characterizations of dominions (and domination) for comodule morphisms, analogous to the algebra-morphism counterparts in [57, §XI.1, Proposition 1.1]. Those characterizations are left unstated in [16], so we revisit the matter here for completeness.

Theorem 2.13**.**

Let $f:C\to D$ and $\pi:C\twoheadrightarrow D^{\prime}$ be two morphisms in $\operatorname{\textsc{Coalg}}_{\Bbbk}$, with $\pi$ onto.

The following conditions are equivalent:

1. (a)

$f$* dominates $\pi$ in the sense of Definition 2.9 2.* 2. (b)

For any pair $f_{i}:C^{\prime}\to C$, $i=1,2$ of coalgebra morphisms with $ff_{1}=ff_{2}$ we have $\pi f_{1}=\pi f_{2}$. 3. (c)

The two compositions

[TABLE]

are equal, where the leftmost term is the pullback in the category of $\Bbbk$-coalgebras and the two parallel left-hand morphisms are the canonical ones. 4. (d)

The two compositions

[TABLE]

are equal, where the leftmost term is the cotensor product **[42, Definition 8.4.2]**. 5. (e)

For any $C$-bicomodule $X$ and element $x\in X$, we have

[TABLE] 6. (f)

The same condition, but only for finite-dimensional bicomodules $X$. 7. (g)

Given a linear map $\varphi:V\to W$ between (right, say) $C$-comodules, if $\varphi$ is a morphism of $D$-comodules via corestriction along $f$ then it is a morphism of $D^{\prime}$-comodules via corestriction along $\pi$:

[TABLE]

where $\rho_{V}:V\to V\otimes C$ and $\rho_{W}:W\to W\otimes C$ are the comodule structure maps. 8. (h)

As above, but only for finite-dimensional $C$-comodules $V$ and $W$.

Proof**.**

1. e** $\iff$ f and g $\iff$ h*** This follows immediately from the fact that comodules are unions of finite-dimensional subcomodules [43, Theorem 5.1.1]. This holds for bicomodules too of course: a $C$-bicomodule is nothing but a comodule over $C\otimes C^{cop}$,where $C^{cop}$ is the coalgebra with “co-opposite” comultiplication*

[TABLE]

We will thus only have to consider finite-dimensional (bi)comodules. 2. a** $\iff$ c*** The codominion of $f$ is the coequalizer of the two parallel left-hand maps in 2-2 [1, 14J (g)], hence the equivalence.* 3. b** $\iff$ c. On the one hand, the former is formally stronger because the two parallel arrows in 2-2 can serve the role of $f_{i}$ in b. On the other, $f_{i}$ as in b will have to factor uniquely through c by the universality property of the pullback. This gives the opposite implication $\lx@cref{creftypecap~refnum}{item:prod}\Rightarrow\lx@cref{creftypecap~refnum}{item:mor}$ and hence the equivalence. 4. c $\iff$ d*** For any pair of morphisms $f_{i}:C^{\prime}\to C$, $i=1,2$ with $ff_{1}=ff_{2}$ we have a map*

[TABLE]

In particular, there is a canonical map $C\times_{D}C\to C\mathrel{\Box}_{D}C$ which together with 2-3 factors 2-2. Note next, as a simple exercise, that the coequalizer (in Vec!) of the two left-hand parallel arrows in 2-3 is already a quotient coalgebra of $C$.

It follows that the codominion of $f$ can also be recovered as the coequalizer in Vec of the two left-hand maps in 2-3, so the conclusion follows. 5. f** $\Longrightarrow$ h*** Given $V,W\in{\mathcal{M}}^{C}_{f}$ (hence finite-dimensional), apply f to the finite-dimensional $C$-bicomodule*

[TABLE]

with its right comodule structure inherited from $W$ and the left structure inherited from that on $V^{*}$ (which in turn results from dualizing the original right comodule structure on $V$). 6. g** $\Longrightarrow$ b*** This will be entirely analogous to the proof of [57, §XI.1, Proposition 1.1, implication (d) $\Rightarrow$ (a)].*

Morphisms $f_{i}:C^{\prime}\to C$ as in b will induce right $C$-comodule structures on $C^{\prime}$ so that $f_{1}$ becomes a $D$-comodule morphism

• •

from $C^{\prime}_{ff_{2}}$, meaning $C^{\prime}$ equipped with the $D$-comodule structure induced by $ff_{2}$ (which is equal to $ff_{1}$ by hypothesis);

• •

to $C_{f}$, meaning $C$ equipped with the $D$-comodule structure induced by $f$.

By assumption, the same goes with $\pi$ in place of $f$:

[TABLE]

This, though, means precisely that $\pi f_{1}=\pi f_{2}$. 7. b** $\Longrightarrow$ f*** The argument is very similar to the proof of [57, §XI.1, Proposition 1.1, implication (a) $\Rightarrow$ (b)].*

Consider a finite-dimensional $C$-bicomodule $X$ with right and left comodule structures denoted as usual, by

[TABLE]

This equips the dual vector space $X^{*}$ with a $C$-bicomodule structure in the only sensible fashion: denoting the generic element of $X^{*}$ by $x^{*}$, we have

[TABLE]

for all $x^{*}\in X$ and $x\in X$. One can then equip the space $C^{\prime}:=C\oplus X^{*}$ with a coalgebra structure by keeping the comultiplication and counit we already have on $C$, and setting

[TABLE]

and $\varepsilon|_{X^{*}}\equiv 0$.

Now fix an $x\in X$ satisfying the hypothesis of e (i.e. the left-hand side of 2-4), and consider the two coalgebra morphisms $f_{i}:C^{\prime}\to C$, $i=1,2$ defined by

• •

$f_{1}:=$* obvious projection of $C^{\prime}=C\oplus X^{*}$ onto $C$;*

• •

$f_{2}|_{C}:=\operatorname{id}_{C}$;

• •

and

[TABLE]

We have $ff_{1}=ff_{2}$ and hence also $\pi f_{1}=\pi f_{2}$ by assumption, whence right-hand side of 2-4.

This completes the proof. $\blacksquare$

The discussion above focuses on dominions of (co)algebra morphisms, whereas we will later be interested in the richer structures mentioned before (bialgebras, Hopf algebras). It is good to know, then, that a result analogous to Proposition 2.1 holds: it doesn’t make much difference where one computes the (co)dominion, so long as it makes sense to do so.

Theorem 2.14**.**

1. (1)

The dominion of an algebra morphism $f:H\to H^{\prime}$ is the same in any of the categories

[TABLE]

that happen to contain it. 2. (2)

The codominion of a coalgebra morphism $f:H\to H^{\prime}$ is the same in any of the categories

[TABLE]

that happen to contain it.

Proof**.**

The two statements are once more mutual duals, with one proof easily adapted into the other, so we focus on 2.

Some of the discussion bifurcates over cocommutativity. For any of the categories in question, the codominion of $f$ is the coequalizer of the two maps

[TABLE]

from the pullback in the corresponding category (by [1, 14J (g)], as in the proof of the equivalence a $\iff$ c in Theorem 2.13). Those pullbacks range over just two possibilities: those computed in $\operatorname{\textsc{Coalg}}_{cc}$ and those computed in $\operatorname{\textsc{Coalg}}$; this is because the relevant forgetful functors to $\operatorname{\textsc{Coalg}}_{\bullet}$ are right adjoints, once more by [52, diagram (9)] and [51, §2.2]. The claim, then, amounts to this:

For $f:H\to H^{\prime}$ in any of the categories of bialgebras or Hopf algebras in 2, the forgetful functor to $\operatorname{\textsc{Coalg}}_{\bullet}$ preserves coequalizers of pairs of the form 2-5.

Even more specifically, $\operatorname{\textsc{Coalg}}_{\bullet}\to\textsc{Vec}$ is a left adjoint and hence cocontinuous [50, §4.3], we are claiming that the coequalizers of the form 2-5 in any of the categories $\operatorname{\textsc{Bialg}}_{\bullet}$ or ${}_{\mathrel{\Box}}\operatorname{\textsc{HAlg}}_{\bullet}$ are computable in Vec.

To see this, note that the pair 2-5 is reflexive [5, §3.3, preceding Proposition 4]: the two arrows have a common right inverse $H^{\prime}\to H^{\prime}\times_{H}H^{\prime}$ whose components are both $\operatorname{id}_{H^{\prime}}$. The conclusion then follows from Lemma 2.15. $\blacksquare$

Lemma 2.15**.**

1. (1)

The forgetful functors from $\operatorname{\textsc{Bialg}}_{\bullet}$ or ${}_{\mathrel{\Box}}\operatorname{\textsc{HAlg}}_{\bullet}$ to Vec preserve reflexive equalizers, with ‘$\bullet$’ meaning any combination of ‘c’ or ‘cc’ and ‘$\mathrel{\Box}$’ either blank, ‘bi’ or ‘2’. 2. (2)

Dually, the forgetful functors from $\operatorname{\textsc{Bialg}}_{\bullet}$ or ${}_{\mathrel{\Box}}\operatorname{\textsc{HAlg}}_{\bullet}$ to Vec preserve reflexive coequalizers.

Proof**.**

Singling out one of the mutually-dual claims again, say 2:

Let $f_{i}:X\to Y$, $i=1,2$ be a reflexive pair of parallel morphisms in any of the categories in question, with a common right inverse $s:Y\to X$: $f_{1}s=f_{2}s=\operatorname{id}_{Y}$. Note now that the kernel

[TABLE]

of the vector-space coequalizer $Y\to\overline{Y}$ of $(f_{i})$ is both a coideal (naturally, since colimits in $\operatorname{\textsc{Coalg}}$ coincide with those in Vec) and, by the reflexivity assumption, an ideal:

[TABLE]

and similarly for left multiplication. It follows that the vector-space coequalizer is in fact a quotient bialgebra, and invariance of 2-6 under antipodes (in $\operatorname{\textsc{HAlg}}$) or inverse antipodes (in ${}_{bi}\operatorname{\textsc{HAlg}}$, etc.) is immediate. $\blacksquare$

Remark 2.16**.**

The proof of Theorem 2.14 touches on the issue of cokernel preservation for the various forgetful functors

[TABLE]

Since these functors are in fact monadic (or tripleable [5, §3.3]) by [52, Theorem 10 and Proposition 28] and [50, §4.3]. This entails the preservation of some reflexive coequalizers (Beck’s Theorem: [5, §3.3, Theorem 10]) but not, generally, all of them.

As a matter of fact, Lemma 2.15 could have been obtained in a more roundabout (and conceptual) manner by tracing through the monadicity proof in [50]; the more direct route seemed preferable in this specific case. $\blacklozenge$

There is also an analogue, for (co)dominions, of Lemma 2.4: field extensions preserve the relevant constructions.

Proposition 2.17**.**

*The scalar-extension functor $\Bbbk^{\prime}\otimes_{\Bbbk}-$ along a field extension preserves (co)dominions in all of the categories $\operatorname{\textsc{Alg}}_{\bullet}$, $\operatorname{\textsc{Coalg}}_{\bullet}$, $\operatorname{\textsc{Bialg}}_{\bullet}$ and ${}_{\mathrel{\Box}}\operatorname{\textsc{HAlg}}_{\bullet}$. *

Proof**.**

Theorem 2.14* reduces the problem to $\operatorname{\textsc{Alg}}$ and $\operatorname{\textsc{Coalg}}$. Moreover, since codominions of algebra morphisms and dominions of coalgebra morphisms are uninteresting (they are just the images of the respective morphisms), we discuss dominions in $\operatorname{\textsc{Alg}}$ and codominions in $\operatorname{\textsc{Coalg}}$.*

We saw in the course of the proof of Theorem 2.13 that the codominion of $C\to D$ is the coequalizer (in $\operatorname{\textsc{Vec}}$) of the two maps $C\mathrel{\Box}_{D}C\to C$. Dually, the dominion of an algebra morphism $A\to B$ is the equalizer of the two maps $B\to B\otimes_{A}B$ [57, §XI.1, Proposition 1.1]. As in the proof of Lemma 2.4, field scalar extensions preserve all of these constructions. $\blacksquare$

Remarks 2.18**.**

1. (1)

See [6, Proposition 5.3] for an analogue of Proposition 2.17 for finite-dimensional Lie algebras. 2. (2)

A version of Proposition 2.17 holds for algebraic groups [13, Theorem 5 (ii)]: the latter’s specialization to linear algebraic groups amounts precisely to the $\operatorname{\textsc{HAlg}}_{c}$-codominion instance of Proposition 2.17.

$\blacklozenge$

Theorem 2.13 helps relate codominions to the notion of coflatness. Recall [15, §10.8] that a left $C$-comodule $M$ is

• •

coflat if the functor ${\mathcal{M}}^{C}\xrightarrow{\quad\mathrel{\Box}_{C}M\quad}\operatorname{\textsc{Vec}}$ is exact;

• •

and faithfully coflat if that functor is in addition faithful.

We apply the term to coalgebra morphisms $C\to D$: they are right (faithfully) coflat if $C\in{\mathcal{M}}^{D}$ is (faithfully) coflat, and similarly on the left.

Remark 2.19**.**

As it happens, for coalgebras coflatness and injectivity interact very conveniently:

• •

coflatness is equivalent to injectivity in the relevant category of comodules ([24, Theorem 2.4.17] or [15, §10.12, (ii) (1)]);

• •

and faithful coflatness is equivalent [15, §10.12, (ii) (2)] to the comodule in question being an injective cogenerator (recall [40, §V.7] that a cogenerator in a category ${\mathcal{C}}$ is an object $d$ such that whenever ${\mathcal{C}}$-morphisms $f_{i}:c\to c^{\prime}$, $i=1,2$ differ so do $gf_{i}$ for some $g:c^{\prime}\to d$).

$\blacklozenge$

One can now replicate (and dualize) the usual discussion of faithfully flat descent ([21, Teorema], [46, Theorem 3.8]) for categories of modules, in its formulation in terms of categories of comodules over corings [15, §25.4].

Construction 2.20**.**

For any coalgebra $C$

• •

the category ${}^{C}{\mathcal{M}}^{C}$ of $C$-bicomodules is monoidal ([40, §VII.1], [25, Definition 2.1.1]), with $\mathrel{\Box}_{C}$ as the tensor bifunctor and $C$ as the monoidal unit.

• •

It thus makes sense to speak of algebras internal to ${}^{C}{\mathcal{M}}^{C}$ [25, Definition 7.8.1] and categories of modules over them [25, Definition 7.8.5]

• •

Similarly, for a unital internal algebra $A\in{}^{C}{\mathcal{M}}^{C}$, we denote by ${\mathcal{M}}_{A}$ the category (of right $A$-modules) consisting of right $C$-comodules $V\in{\mathcal{M}}^{C}$ equipped with a morphism

[TABLE]

associative and unital in the obvious sense.

The difference to [25, Definition 7.8.5] lies in $V$ being only a right $C$-comodule rather than a $C$-bicomodule.

Consider, now, a coalgebra morphism $C\to D$.

• •

The cotensor product $C\mathrel{\Box}_{D}C$ is a unital algebra in ${}^{C}{\mathcal{M}}^{C}$, with multiplication

[TABLE]

and unit

[TABLE]

• •

For any (right) $D$-comodule $W$, the cotensor product $W\mathrel{\Box}_{D}C$ is not only a right $C$-comodule, but also a right $C\mathrel{\Box}_{D}C$-module with multiplication

[TABLE]

• •

This gives a functor

[TABLE]

the right half of an adjunction

[TABLE]

where the space $N_{C\mathrel{\Box}_{D}C}$ of coinvariants (by analogy to the group-homology term [26, §1.1]) of the $C\mathrel{\Box}_{D}C$-module $N$ is defined as the (vector-space) coequalizer

[TABLE]

$\blacklozenge$

Having set up all of this, Theorem 2.21 is a straightforward dualization of the standard arguments for faithfully flat descent [21, Teorema]. We give a proof for completeness; as noted in [22, §4.2] though, such results are nowadays easily available as consequences of Beck’s [5, §3.3, Theorem 10] on (co)monadic functors.

Theorem 2.21**.**

let $f:C\to D$ be a coalgebra morphism. The adjunction 2-7 is an equivalence in either of the following conditions:

• •

$f$* is left faithfully coflat;*

• •

or $f$ is a split epimorphism (i.e. has a right inverse) in ${}^{D}{\mathcal{M}}^{D}$.

Proof**.**

Consider the commutative diagram

[TABLE]

Now observe that

• •

the left-hand arrow is right adjoint to the scalar corestriction functor ${\mathcal{M}}^{C}\to{\mathcal{M}}^{D}$;

• •

the monad on ${\mathcal{M}}^{C}$ associated to that adjunction **[5, §3.1, Theorem 1]** is nothing but $-\mathrel{\Box}_{D}C:{\mathcal{M}}^{C}\to{\mathcal{M}}^{C}$;

• •

and the horizontal arrow is the comparison functor **[5, §3.2, following Proposition 2]** associated to that selfsame adjunction (and monad).

The condition that the horizontal arrow be an equivalence is now equivalent to the monadicity of $-\mathrel{\Box}_{D}C:{\mathcal{M}}^{D}\to{\mathcal{M}}^{C}$, this, in turn holds in the two flagged cases:

• •

If $C\in{}^{D}{\mathcal{M}}$ is faithfully coflat then the conclusion follows from Beck’s theorem (**[5, §3.3, Theorem 10]**): we have already observed it is a right adjoint, and it reflects isomorphisms and preserves all coequalizers (not just suitable reflexive ones) by faithful coflatness.

• •

On the other hand, if $f:C\to D$ has a right inverse in ${}^{D}{\mathcal{M}}^{D}$, then the counit

[TABLE]

of the adjunction between ${\mathcal{M}}^{D}$ and ${\mathcal{M}}^{C}$ has a right inverse, hence the monadicity claim by (the categorical dual of) **[41, Proposition 3.16]**.

$\blacksquare$

To circle back to codominions:

Corollary 2.22**.**

A morphism $C\to D$ in any of the categories

[TABLE]

that

• •

is left or right faithfully coflat;

• •

or is a split epimorphism in ${}^{D}{\mathcal{M}}^{D}$

*is a codominion. *

Proof**.**

Per Theorem 2.14 2, it is enough to handle $\operatorname{\textsc{Coalg}}$. That claim, though, follows immediately from the description of the codominion of $C\to D$ as the coequalizer 2-3 and from Theorem 2.21: for

[TABLE]

the two parallel left-hand arrows of 2-8 specialize precisely to those of 2-3. $\blacksquare$

Remark 2.23**.**

In the context of left module quotient coalgebras of Hopf algebras, fitting into the framework of Proposition 2.12, Corollary 2.22 specializes back to (a particular case of) [59, Theorem 2]. $\blacklozenge$

Recall that the cosemisimple coalgebras ([42, Definition 2.4.1 and Lemma 2.4.3], [24, Theorem 3.1.5]) are those whose (left or right) comodules are all injective (or equivalently, projective).

Corollary 2.24**.**

*A surjection in any of the categories 2-9 onto a cosemisimple object is a codominion. *

Proof**.**

This follows from Corollary 2.22 once we observe that for cosemisimple $C$ a (right, say) $C$-comodule $M$ surjecting onto $C$ is faithfully coflat. Indeed, $M$ coflat by assumption, and, $C$ being projective in ${\mathcal{M}}^{C}$ [24, Theorem 3.1.5], $M$ will also contain it as a summand. But every coalgebra is an injective cogenerator in its own category right comodules [15, §9.1], hence so is $M\in{\mathcal{M}}^{C}$ (so that it is also faithfully coflat [15, §10.12, (ii) (2)]). $\blacksquare$

Remark 2.25**.**

Compare Corollary 2.24 to the remark made in passing in [6, Addenda] (and attributed to Hochschild) that every semisimple Lie subalgebra of any finite-dimensional Lie algebra is a dominion in the category of finite-dimensional Lie algebras.

Since by [6, Theorems 2.2 and 2.3] that question is only interesting in characteristic zero, where semisimple Lie algebras have semisimple categories of finite-dimensional representations [32, §6.3, Theorem], the connection to cosemisimplicity starts to become apparent. $\blacklozenge$

As a matter of fact, the second bullet point in Corollary 2.22 is sub-optimal: one-sided splitting suffices.

Theorem 2.26**.**

*The conclusion of Corollary 2.22 holds provided $C\to D$ has a right inverse as a either a left or right $D$-comodule morphism. *

Proof**.**

The proof of Corollary 2.22 makes it apparent that one does not quite need the adjunction 2-7 to be an equivalence: it is enough that its counit be an equivalence (in other words [40, §IV.3, Theorem 1]: that the right adjoint $-\mathrel{\Box}_{D}C$ be fully faithful).

In other words, fora coalgebra morphism $f:C\to D$ to be a codominion all we need is that

[TABLE]

be a coequalizer in $\operatorname{\textsc{Vec}}$. In the presence of a left $D$-comodule right inverse $s:D\to C$ to $f$, the maps

[TABLE]

make 2-10 into a split (or contractible) coequalizer [5, §3.3, following Proposition 1] in $\operatorname{\textsc{Vec}}$, so in particular (as the name suggests) a $\operatorname{\textsc{Vec}}$-coequalizer [5, §3.3, Proposition 2]. $\blacksquare$

Remark 2.27**.**

The remark behind the proof of Theorem 2.26 is precisely dual to [9, Proposition 2.2], concerned with algebra morphisms $A\to B$ split as right $A$-module maps. $\blacklozenge$

3 CQG algebras and compact groups

One class of Hopf algebras particularly well-suited for generalizing compact-group representation theory is that of CQG algebras [23, Definition 2.2]. One version of that definition is as follows:

Definition 3.1**.**

A CQG algebra is

• •

a complex Hopf $*$-algebra $H$ ([23, §1]: complex Hopf algebra equipped with a conjugate-linear multiplication-reversing involution ‘$*$’ such that both $\Delta$ and $\varepsilon$ are $*$-morphisms);

• •

all of whose finite-dimensional comodules

[TABLE]

are unitarizable: admitting an inner product $\braket{-\mid-}$ (linear in the second variable, say), compatible with the comodule structure in the sense that

[TABLE]

$\operatorname{\textsc{CQG}}$ will denote the category of CQG algebras, with Hopf *-algebra morphisms as its arrows. $\blacklozenge$

Remark 3.2**.**

Cosemisimplicity is a consequence of Definition 3.1: the orthogonal complement of a subcomodule is again a subcomodule, hence the semisimplicity of the category ${\mathcal{M}}^{H}$. $\blacklozenge$

The motivation stems from the fact that the commutative CQG algebras are precisely the algebras of representative functions [14, §III.1, Definition 1.1] on compact groups, and the usual Tannakian duality machinery [14, §III.7] can phrased as a contravariant equivalence between the category $\operatorname{\textsc{CQG}}_{c}$ of commutative CQG algebras and the category of compact groups [61, Theorems 2.6 and 2.8]. For that reason, one thinks of the category $\operatorname{\textsc{CQG}}$ as dual to that of compact quantum groups (this being one possible definition of the latter category, though not the only one: see e.g. [61, Definition 2.3 and subsequent discussion]).

Recall, in this context, that embeddings of compact groups are equalizers [19, Theorem 2.1], whence

Corollary 3.3**.**

The dominion of a compact-group morphism is the embedding of its image in its codomain.

*Dually, the codominion of a morphism in the category $\operatorname{\textsc{CQG}}_{c}$ of commutative CQG algebras is the surjection of its domain onto its image. *

This also implies that epimorphisms of compact groups are onto ([48, Theorem], [54, Proposition 9]) or equivalently, monomorphisms in $\operatorname{\textsc{CQG}}_{c}$ are embeddings. This latter result transports over to $\operatorname{\textsc{CQG}}$: the monomorphisms in that category are precisely the injective morphisms [18, Proposition 6.1].

In view of all of the above, it seems reasonable to ask whether one can strengthen [18, Proposition 6.1] in the same direction, by showing that codominions (rather than just monomorphisms) behave “as expected”:

Theorem 3.4**.**

*A surjection in $\operatorname{\textsc{CQG}}$ is a coequalizer in that category. *

Proof**.**

Recall [18, Theorem 3.1] that $\operatorname{\textsc{CQG}}$ is locally presentable, so much of the discussion in Section 1 applies: the category is complete and cocomplete, etc. In particular, we can freely refer to pullbacks therein.

Let $f:H\to H^{\prime}$ be a surjective morphism of CQG algebras. As in the proof of the equivalence c $\iff$ d in Theorem 2.13, the vector-space coequalizer of the two maps

[TABLE]

is already a CQG algebra, and hence also the coequalizer of the two maps 2-5 from the pullback in $\operatorname{\textsc{CQG}}$. The latter is the codominion in $\operatorname{\textsc{CQG}}$ [1, 14J (g)], so that codominion is also computable in $\operatorname{\textsc{Coalg}}$ (or $\operatorname{\textsc{Bialg}}$, etc.). But $H^{\prime}$ is cosemisimple, so the codominion is $H\to H^{\prime}$ itself by Corollary 2.24. $\blacksquare$

The non-commutative analogue of Corollary 3.3 is now a consequence (or rather rephrasing):

Corollary 3.5**.**

*the codominion of a morphism in the category $\operatorname{\textsc{CQG}}$ of CQG algebras is the surjection of its domain onto its image. *

Remark 3.6**.**

In fact, Corollary 3.3 follows from Corollary 3.5 (thus giving an alternative proof of [19, Theorem 2.1], on which we need not rely): the inclusion functor $\operatorname{\textsc{CQG}}_{c}\to\operatorname{\textsc{CQG}}$

• •

is a right adjoint so on the one hand it preserves pullbacks and hence diagrams of the form 2-5;

• •

and also preserves split coequalizers, as in the proof of Lemma 2.15 2.

This means in particular that said inclusion functor preserves codominions. $\blacklozenge$

4 Asides on scalar extension and limits of coalgebras

In light of the proof of Proposition 2.17, one might wonder whether it could have been phrased in terms of the other description of a coalgebra codominion: as a coequalizer of the two coalgebra (rather than vector space) morphisms on the left-hand side of 2-2, with the symbol $\times_{D}$ denoting the pullback in $\operatorname{\textsc{Coalg}}$.

The algebra analogue, whereby the dominion of $f:A\to B$ is the equalizer of the two morphisms

[TABLE]

(pushout in $\operatorname{\textsc{Alg}}$) is obvious enough:

[TABLE]

is a left adjoint so preserves pushouts, as it does (co)equalizers. Note, though an asymmetry in the behavior of $\Bbbk^{\prime}\otimes_{\Bbbk}-$ for algebras vs. coalgebras: in both cases the functor $\Bbbk^{\prime}\otimes_{\Bbbk}-$ is a left adjoint, so the customary dualization does not obtain. Indeed, as observed in Section 1, it is enough to argue that the functor is (in both cases) cocontinuous:

• •

For coalgebras this is obvious, since the colimits in $\operatorname{\textsc{Coalg}}$ are just those in $\operatorname{\textsc{Vec}}$ (Remark 2.3).

• •

While for algebras one can argue directly that coequalizers and coproducts are preserved, e.g. using the convenient description of coproducts [7, Corollary 8.1] as direct sums of tensor products.

That nevertheless the variant of the proof of Proposition 2.17 alluded to above would have gone through follows from Theorem 4.1: even though it is a left adjoint and not a right one, it does preserve the $\operatorname{\textsc{Coalg}}$-equalizer of

[TABLE]

for a coalgebra morphism $C\to D$.

Theorem 4.1**.**

Let $\Bbbk\leq\Bbbk^{\prime}$ be a field extension, and consider the left adjoint $(-)_{\Bbbk^{\prime}}:=\Bbbk^{\prime}\otimes_{\Bbbk}-$ between the two versions (of $\Bbbk$- and $\Bbbk^{\prime}$-linear objects respectively) of any of the categories

[TABLE]

1. (1)

For $\operatorname{\textsc{Alg}}_{\bullet}$, $(-)_{\Bbbk^{\prime}}$ preserves infinite products (and hence is also right adjoint) precisely when $\Bbbk\leq\Bbbk^{\prime}$ is a finite extension.

Otherwise, the functor does not preserve any infinite product of non-zero algebras (and hence is not right adjoint). 2. (2)

For all other categories $(-)_{\Bbbk^{\prime}}$ preserves equalizers of arbitrary families and arbitrary products, and hence is a right adjoint.

Proof**.**

As noted repeatedly (in various formulations; e.g. [52, diagram (9)] and [51, §2.2]), the forgetful functors

[TABLE]

are right and left adjoints respectively (with the ‘$\bullet$’ symbols understood to match: for algebras, if ‘c’ appears on the left it does so on the right, and similarly for coalgebras and ‘cc’). It is thus enough to treat $\operatorname{\textsc{Alg}}_{\bullet}$ (algebras, commutative or plain) and $\operatorname{\textsc{Coalg}}_{\bullet}$ (coalgebras, cocommutative or plain).

It is mentioned in Section 1 that functions defined on any of the categories are right adjoint as soon as they are continuous and preserve certain filtered colimits, so in particular, for a functor that is already know to be left adjoint, continuity is equivalent to being a right adjoint. Additionally, continuity is also equivalent to preservation of products and pair equalizers [1, Proposition 13.4], hence the claims about right adjointness.

We now address the two claims.

1. (1)

In $\operatorname{\textsc{Alg}}_{\bullet}$ the limits are just the ordinary ones form $\operatorname{\textsc{Vec}}$, the forgetful functor $\operatorname{\textsc{Alg}}_{\bullet}\to\operatorname{\textsc{Vec}}$ being right adjoint. Part 1, then, follows immediately from the remark that the canonical map

[TABLE]

is

• •

an isomorphism if $\Bbbk\leq\Bbbk^{\prime}$ is a finite extension;

• •

and a proper embedding if the field extension is infinite and infinitely many $V_{i}$ are non-zero. 2. (2)

We will discuss the various types of limits separately.

1. (a)

(Equalizers)* In both $\operatorname{\textsc{Coalg}}$ and $\operatorname{\textsc{Coalg}}_{cc}$ these are computed identically: given a family of morphisms $f_{i}:C\to D$, their equalizer is the largest subcoalgebra of $C$ contained in the vector-space equalizer*

[TABLE]

(see for instance **[4, first paragraph of the proof of Theorem 1.1]** for pairs of morphisms; the argument works generally). That $\Bbbk^{\prime}\otimes_{\Bbbk}-$ preserves vector-space equalizers 4-2 is straightforward linear algebra, so the conclusion follows from Lemma 4.2 below. 2. (b)

(Products in $\operatorname{\textsc{Coalg}}$)* Let $C_{i}$, $i\in I$ be a family of $\Bbbk$-coalgebras, and consider the canonical morphism*

[TABLE]

where the products are taken in the categories of $\Bbbk$- and $\Bbbk^{\prime}$-coalgebras respectively.

Consider a finite-dimensional $\Bbbk$-vector space $V$. By Lemma 4.3, the morphism 4-3 induces a bijection between the comodule structures on $V_{\Bbbk^{\prime}}$ over the two sides of that morphism. The same result also implies that for finite-dimensional $\Bbbk$-spaces $V$ and $W$ equipped with such comodule structures we have

[TABLE]

Given that every finite-dimensional $\Bbbk^{\prime}$-vector space is of the form $V_{\Bbbk^{\prime}}$ for some $V\in\operatorname{\textsc{Vec}}_{\Bbbk,f}$, the Tannakian reconstruction of the coalgebras 4-3 from their respective finite-dimensional comodules **[55, Theorem 2.1.12 and Lemma 2.2.1]** now makes it clear that 4-3 is an isomorphism. 3. (c)

(Products in $\operatorname{\textsc{Coalg}}_{cc}$)* Consider an arbitrary family of cocommutative coalgebras $C_{i}\in\operatorname{\textsc{Coalg}}_{cc}$, $i\in I$. For any finite subfamily thereof the product in $\operatorname{\textsc{Coalg}}$ is nothing but the tensor product [58, Theorem 6.4.5], so in general we have*

[TABLE]

where

• •

$F$* ranges over the finite subsets of the full index set $I$, ordered by inclusion*

• •

For $F\subseteq F^{\prime}$ the connecting morphism

[TABLE]

acts as the identity for $i\in F$ and as the counit for $i\in F^{\prime}\setminus F$.

As we already know from points 2a and 2b above that $(-)_{\Bbbk^{\prime}}$ preserves arbitrary limits in $\operatorname{\textsc{Coalg}}$, and since it of course preserves tensor products, it will also preserve limits of the form 4-4.

This concludes the proof. $\blacksquare$

Lemma 4.2**.**

For a linear subspace $V\leq C$, denote by $C_{V}\leq V$ the largest subcoalgebra of $C$ contained in $V$.

For any field extension $\Bbbk\leq\Bbbk^{\prime}$ and linear subspace $V\leq C$ of a $\Bbbk$-coalgebra the canonical inclusion

[TABLE]

*is an equality. *

Proof**.**

In words, scalar extensions along field inclusions preserve the largest-subcoalgebra-contained-in-a-subspace construction.

To see this, observe that

[TABLE]

where

[TABLE]

is the iterated comultiplication (so that $\Delta_{1}=\Delta_{C}$ and $\Delta_{0}=\operatorname{id}_{C}$). We thus have

[TABLE]

and we conclude by noting that $\Bbbk^{\prime}\otimes_{\Bbbk}-$ preserves intersections of arbitrary families of subspaces (e.g. by choosing a basis for $\Bbbk^{\prime}$ over $\Bbbk$, etc.). $\blacksquare$

Lemma 4.3**.**

Let $\Bbbk\leq\Bbbk^{\prime}$ be a field extension, $C$ a $\Bbbk$-coalgebra, $V$ a $\Bbbk$-vector space, and denote by $(-)_{\Bbbk^{\prime}}$ the functor $\Bbbk^{\prime}\otimes_{\Bbbk}-$.

*A right $C_{\Bbbk^{\prime}}$-comodule structure on $V_{\Bbbk^{\prime}}$ arises by scalar extension from a unique $C$-comodule structure on $V$. *

Proof**.**

Per the adjunction

[TABLE]

a $\Bbbk^{\prime}$-linear map $V_{\Bbbk^{\prime}}\to V_{\Bbbk^{\prime}}\otimes_{\Bbbk^{\prime}}C_{\Bbbk^{\prime}}$ is (the same thing as) a $\Bbbk$-linear map $V\to V\otimes C\otimes\Bbbk^{\prime}$ (unadorned tensor products being over $\Bbbk$). Choosing a basis $e_{i}$, $i\in I$ for $\Bbbk^{\prime}/\Bbbk$ with $1=e_{i_{0}}$, such a map is in turn uniquely determined by its components

[TABLE]

The condition that the original map be a $C_{\Bbbk^{\prime}}$-comodule structure then amounts to

• •

the map

[TABLE]

being a right $C$-comodule structure;

• •

while

[TABLE]

are coassociative and satisfy

[TABLE]

(by contrast to the usual counitality condition, that would equate the left-hand side with $v$).

Coassociativity together with 4-6 are easily seen to imply the vanishing of all 4-5, proving the claim: the $C_{\Bbbk^{\prime}}$-comodule structures on $V_{\Bbbk^{\prime}}$ are precisely those of the form

[TABLE]

for a $C$-comodule structure $v\mapsto v_{0}\otimes v_{1}$ on $V$. $\blacksquare$