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arXiv:2305.03120v1 [math.CT] 4 May 2023
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES
PAUL GROSSKOPF AND JOOST VERCRUYSSE
Abstract. We show that under mild conditions on the monoidal base category V, the
category V-Hopf of Hopf V-categories is locally presentable and deduce the existence of free
and cofree Hopf categories. We also provide an explicit description of the free and cofree
Hopf categories over a semi-Hopf category. One of the conditions on the base category V,
states that endofunctors obtained by tensoring with a fixed object preserve jointly monic
families, which leads us to the notion of “very flat monoidal product”, which we investigate
in particular for module categories.
Contents
Introduction 1
1. V-graphs and V-categories 4
1.1. Locally presentable categories 4
1.2. The category of V-graphs 5
1.3. The category of V-categories 7
2. Cofree coalgebras 12
2.1. Existence of cofree coalgebras 12
2.2. Very flat monoidal products 12
2.3. The construction of cofree coalgebras 14
3. Existence of free and cofree Hopf categories 17
3.1. Semi-Hopf V-categories 17
3.2. Hopf V-categories 20
4. On the construction of free and cofree Hopf categories 22
4.1. The free Hopf category 22
4.2. The cofree Hopf category 24
5. Conclusions and outlook 25
Acknowledgements 27
References 27
Introduction
Free constructions are a fundamental tool for many algebraic structures. In particular,
the free group over a set or a monoid is a classical construction that allows to describe all
possible symmetries of a given object, in terms of generators and relations. Formally, free
constructions provide left adjoints to forgetful functors. The free monoid over a set can be
recovered combining the construction of a free monoid over a set, and the free group over a
1
2 P. GROSSKOPF AND J. VERCRUYSSE
monoid.
Grp ⊤
⊤//Mon ⊤//
oo
ooSet
oo
In fact, the forgetful functor from groups to monoids also has a right adjoint, which is obtained
by taking all invertible elements in a given monoid. One can think of this construction as the
“cofree group over a monoid”.
Moving from classical to non-commutative geometry, symmetries are no longer described
in terms of groups and their actions, but rather by Hopf algebras (or quantum groups) and
their (co)actions (see e.g. [18]), which leads also to “hidden symmetries” of classical geometric
objects. Replacing groups by Hopf algebras, means that we substitute the Cartesian category
of sets, by the monoidal category of vector spaces. A main consequence of this is that, in
contrast to the Cartesian case of sets where every object has a unique coalgebra structure,
coalgebra structures in the category of vector spaces can be more involved and the above
picture needs to be completed with additional forgetful functors from bialgebras to algebras
and from coalgebras to vector spaces. In contrast to the classical forgetful functors, these
functors usually do no longer have a left adjoint, but rather a right adjoint, providing the
cofree coalgebra over a vector space, and the cofree bialgebra over an algebra, see [24, Section
6.4]. As in the case of groups above, the forgetful functor from Hopf algebras to bialgebras has
both a left and right adjoint. The existence of these adjoints was suggested in [24], although
without providing a proof. The left adjoint, that is the free Hopf algebra or Hopf envelope of
a bialgebra, was first constructed by Manin [18] and generalizes the construction of the free
Hopf algebra over a coalgebra given earlier by Takeuchi [25], as the latter arises by combining
Manin’s construction with the free algebra over a coalgebra (which is just given by the tensor
algebra over the underlying vector space of the coalgebra, see e.g. [24]). A first proof of the
exsistence of the cofree Hopf algebra over a bialgebra was given in [2], based on earlier results
from [20].
Alg
zz✈
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$$
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⊤
Hopf ⊤
⊤//Bialg
::
✈
✈
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$$
❍
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ooVect
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zz✈
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Coalg
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⊤
Closely related to free and cofree constructions, is the question of completeness and cocom-
pleteness of the various categories involved. For the category of Hopf algebras, this was again
already claimed in [24], but a full proof was given only relatively recent, based on the theory
of locally presentable categories [19], or by means of more explicit constructions of limits and
colimits [2,3]. These works culminated in a systematic study of the category of Hopf algebras,
not only over vector spaces but over general symmetric monoidal categories in [21,22].
In this paper, we will investigate a “multi-object” version of the above, motivated by the
fact that in many (geometric) situations, the natural objects that arise are groupoids rather
than groups. With the rise of non-commutative geometry, several algebraic structures have
been introduced to serve as a non-commutative or linear counterpart for groupoids. For
example, weak Hopf algebras [10] and Hopf algebroids [8] both can play this role. More
recently, Hopf categories [6] have been introduced as an alternative approach. A Hopf V-
categories over a symmetric monoidal category V, is a category enriched over the category of
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 3
V-coalgebras, admitting a suitable notion of an antipode. When Vis the category of sets, then
we recover the usual notion of a groupoid. On the other hand, a Hopf V-category with one
object is exactly a Hopf algebra in V. In this sense, the theory of Hopf V-categories naturally
unifies the theory of Hopf algebras with the theory of groupoids. Moreover, if a Hopf category
has only a finite set of objects, the coproduct of all Hom-objects is in a natural way a weak
Hopf algebra, and hence a Hopf algebroid (over a commutative base algebra). However, Hopf
categories with an arbitrary set, and even a class of objects, can also be considered. Such
objects can no longer be described in terms of a weak Hopf algebra or a Hopf algebroid, which
is exactly one of the main advantages of working with Hopf V-categories rather than weak
Hopf algebras or Hopf algebroids. On the other hand, the theory of Hopf categories is in a
sense easier to handle and formally much closer to classical Hopf algebras than weak Hopf
algebras or Hopf algebroids. Hopf categories were also shown to fit in the more general theory
of Hopf monads in monoidal bicategories [9] and alternatively can be described as oplax Hopf
monoids in a braided monoidal bicategory [13]. Galois and descent theory for Hopf categories
has been developed in [14] and a version of the celebrated Larson-Sweedler theorem has been
proven in [12].
Applying the machinery of locally presentable categories and existing results from litera-
ture, we derive that the category of semi-Hopf V-categories is locally presentable if Vis and
some mild conditions on Vare fulfilled, see Proposition 3.3. The case of Hopf V-categories
shows to be more complicated, and in order to prove that the category of Hopf V-categories
is also locally presentable, we will make use of an explicit construction of limits and colimits
in the category of semi-Hopf categories. Especially the case of limits needs particular at-
tentions, as these make use of limits in the category of coalgebras. In order to describe the
latter explicitly, one needs certain exactness properties for the monoidal product. This leads
us to the notion of a “very flat” monoidal product, which requires endofunctors obtained
from tensoring with a fixed object to preserve arbitrary jointly monic families. We study
such monoidal products from a module theoretic point of view in Section 2.2, and believe this
might be of independent interest.
In Section 2.3 we then provide an explicit construction of the cofree coalgebra over an object
in a locally presentable monoidal category with very flat monoidal product. Although the
existence of such objects was already known, an explicit construction only had been given in
case of vector spaces over a field, which is vastly generalized here. Moreover, this construction
allows to provide for a limit of coalgebras in a monoidal category Vwith very flat monoidal
product an explicit family of jointly monic families of morphisms in the underlying monoidal
category V(see Corollary 2.7). These families allow to make explicit computations for such
limits.
Based on the above, we can then deduce (see Section 3.2) that the category of Hopf V-
categories is again locally presentable and deduce the existence of free and cofree Hopf V-
categories, generalizing the picture of the “one-object-case” above. It is important to remark
that these free and cofree constructions leave the set (or class) of objects in the considered
V-category unchanged. We end the paper by providing the explicit description of the free and
cofree Hopf categories over a given semi-Hopf category in Section 4. One can observe that
our construction naturally unifies the description of free Hopf algebras (see [18]) with the
one of free groupoids (see e.g. [11]). As far as the authors are aware, even in the one-object
case, an explicit description of cofree Hopf algebras over a bialgebra has not yet been given
in literature.
4 P. GROSSKOPF AND J. VERCRUYSSE
A particularly motivating example of a semi-Hopf category is the category of algebras,
which is known to be enriched over coalgebras by means of Sweedler’s universal measuring
coalgebras, see [24], [26], [16], [4]. The results of our paper allow to consider the free or
cofree Hopf category over this semi-Hopf category, which then leads then to a Hopf category
structure on the category of algebras. Following [18], one could consider the Hom-objects of
this Hopf category as describing the natural non-commutative (iso)morphisms between the
non-commutative spaces whose coordinate algebras are the objects of this category.
1. V-graphs and V-categories
In this section we recall several notions and results from literature, which we complement
with some new observations.
1.1. Locally presentable categories. Recall (see e.g. [1]) that a category Vis called locally
presentable if and only if Vis cocomplete and there exists a (small) set of objects Sand a
regular cardinal λwith the following two properties:
(i) the representable functor Hom(S, −) : V → Set preserves λ-filtered colimits for any
S∈ S;
(ii) every object in Vis a λ-filtered colimit of elements in S.
Locally presentable categories are known to satisfy a lot of interesting properties: they have
a generator, are complete and cocomplete, well-powered and co-well-powered. In general
they do not have a cogenerator. The dual of a locally presentable category Vis not locally
presentable, unless Vis equivalent to a complete lattice. Among the many examples of locally
presentable categories, let us just mention especially Set, the category of sets, and Modk, the
category of modules over a (commutative) ring k, which will be our main cases of interest in
what follows. However, the theory has much wider applications since many more examples
exist, such as the category of Banach spaces [23] and the category of C∗-algebras, to mention
but a few examples.
The reason why we consider locally presentable categories in this paper, is that they offer
the following powerful adjoint functor theorem.
Proposition 1.1. Let F:V → W be a functor between locally presentable categories.
(1) Fhas a right adjoint if and only if it preserves colimits;
(2) Fhas a left adjoint if and only if it preserves limits and λ-filtered colimits for some
regular cardinal λ.
Proof. (i) follows from the Special Adjoint Functor Theorem in combination with the obser-
vations made above that a locally presentable category is cocomplete, co-well-powered and
has a generator. (ii) is proven in [1, Theorem 1.66].
We will also use the following results.
Proposition 1.2 ( [1, Corollary 2.48]).A full subcategory of a locally presentable category,
which is closed under limits and colimits, is locally presentable.
Proposition 1.3 ( [1, Proposition 1.61]).Every locally presentable category has both (strong
epi, mono)- and (epi, strong mono)-factorizations of morphisms.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 5
1.2. The category of V-graphs.
Definition 1.4. Let Vbe an arbitrary category. A V-graph is a pair A= (A0, A1) where A0
is a setaand A1is a family of objects A1= (Ax,y )x,y ∈A0in Vindexed by A0×A0. The elements
of A0are called the ‘objects’ of A, the V-objects Ax,y are the called the ‘Hom-objects’ of A.
By sheer laziness we also write Ax,y =Axy. If A= (A0, A1) is a V-graph then we also say
that “Ais a V-graph over A0”.
Let Aand Bbe two V-graphs. We define a morphism of V-graphs f:A→Bas a pair
f= (f0, f 1) where f0:A0→B0is a map and f1= (fx,y :Ax,y →Bf x,f y)x,y∈A0is a family of
morphisms in V.
The composition of two morphisms of V-graphs f:A→Band g:B→Cis given by
(g◦f)0=g0◦f0and (g◦f)1= (gfx,f y ◦fx,y :Ax,y →Cgfx,gf y )x,y∈A0This defines the category
of V-graphs that we denote as V-Grph.
Remarks 1.5.(1) One can also define the category V-opGrph as the category (Vop-Grph)op.
Explicitly, this category has the same objects as V-Grph but a morphism f:A→B
is a pair f= (f0, f 1), where f0:B0→A0is a map and f1is a family of morphisms
(fx,y :Afx,f y →Bx,y )x,y∈B0.
(2) Let us remark that the category V-Grph can be viewed as a full subcategory of the
category Fam(V) of “families” over V, which is the free completion of Vunder coprod-
ucts (see e.g. [15, Secdtion 4.2]). In fact, V-Grph can also be interpreted as the free
completion of Vunder “double coproducts”, by which we mean coproducts indexed
over sets of type X×X, where Xis a set. Similarly, the category V-opGrph is a
full subcategory of the category Maf (V), which is the free completion under products.
Recently, in [5] the duality between and pre-rigidity of the categories Fam(V) and
Maf(V) has been investigated. Probably several of the results developed there could
be restricted to the categories V-Grph and V-opGrph.
Definition 1.6. If Ais a graph, then its opposite graph is the graph Aop with the same set
of objects A0and for any x, y ∈A0, we have Aop
x,y =Ay,x.
With the above definition, one observes the following.
Proposition 1.7. There is an involutive endofunctor (−)op :V-Grph → V-Grph, which sends
aV-graph to its opposite V-graph.
The following observation about “base change”, will be useful later on.
Proposition 1.8. Any functor F:V → W induces a functor F-Grph :V-Grph → W-Grph.
Similarly, if G:V → W is another functor, then any natural transformation α:F→G
induces a natural transformation α-Grph :F-Grph →G-Grph. In fact, we obtain a 2-functor
Grph :Cat →Cat. It follows that any adjunction between Vand Walso induces an adjunction
between V-Grph and W-Grph.
Proof. The 2-functor Grph is defined entirely component-wise. More explicitly, for any V-
graph A= (A0, A1), we define F-Grph(A)0=A0and F-Grph(A)1= (F(Axy))x,y∈A0. Similarly,
for any morphism of V-graphs f= (f0, f 1) : A→B, we define F-Grph(f)0=f0and
aIn fact, the definition also makes sense when A0is a class, but to avoid set-theoretical issues we will
mainly consider the case where A0is a set.
6 P. GROSSKOPF AND J. VERCRUYSSE
F-Grph(f)1= (F(fxy))x,y∈A0. Finally, for any object A, let us define α-GrphA= (α0
A, α1
A) by
α0
A=id :A0→A0and (αA)xy =αAxy :F(Axy)→G(Axy).
We leave the verification that this construction provides indeed a 2-functor to reader.
Proposition 1.9. Consider the functor P:V → V-Grph, which sends an object Vin Vto
the V-graph ({∗}, V )over the (fixed) singleton set {∗}. Then we have the following.
(i) Pis a full and faithful functor.
(ii) The functor Phas a left adjoint if and only if Vhas double coproducts (i.e. coproducts
indexed by sets of the form X×Xwhere Xis a set).
Proof. (i). Consider objects Vand Win V. A morphism f:P(V)→P(W) is then defined
as a pair (f0, f 1), where f0:{∗} → {∗} has to be the identity map, and f1consists of one
single morphism f:V→W. This observation shows that Pis fully faithful.
(ii). This follows from V-Grph being the free completion under double coproducts. Explic-
itly, if we denote the left adjoint of Pby L, then for any object A, the object LA ∈ V must
satisfy the property
V-Grph(A, P (V)) ∼
=V(LA, V )
Moreover, for any f= (f0, f1)∈ V-Grph(A, P (V)), we have that f0:A0→ {∗} is the
unique map sending every element of A0to ∗. Henceforth, f1is just a collection of V-
morphisms fx,y :Ax,y →V. The natural bijection above tells that for any such collection,
there exists a unique morphism L(A)→V, which expresses exactly the universal property
of L(A) as the coproduct of the family of objects Ax,y in V. Remark that the canonical maps
Ax,y →`x,y Ax,y =L(A) correspond exactly to the unit of the adjunction.
Proposition 1.10. Consider the forgetful functor U:V-Grph →Set.
(i) Vhas an initial object if and only if Uhas a fully faithful left adjoint.
(ii) Vhas a terminal object if and only if Uhas a fully faithful right adjoint.
In particular, if Vhas a terminal and initial object, then Upreserves all limits and colimits.
Proof. In case Vhas an initial object ⊥, then the forgetful functor Uhas a left adjoint L
which assigns to any set Xthe graph over Xwhich has the initial object in every component,
i.e. L(X) = (X0, X1), where X0=Xand Xx,y =⊥for all x, y ∈X. By construction, we
have that X=UL(X), which is exactly the unit of the adjunction, so Lis fully faithful.
Conversely, suppose that Uhas a fully faithful left adjoint L, then we know that the
underlying set of L(X) is (up to bijection) just X, for any set X. Denote L({∗}) = ({∗},⊥).
Consider any object Vin Vand associate to it the V-graph P(V) = ({∗}, V ) over the singleton
set {∗}. Then we find that (here we use that Pis fully faithful, see Proposition 1.9)
V(⊥, V )∼
=V-Grph(({∗},⊥),({∗}, V )) = V-Grph(L({∗}),({∗}, V )) = Set({∗},{∗}) = {id{∗} }.
This shows that ⊥is indeed an initial object in V.
Similarly, if Vhas a terminal object ⊤, the forgetful functor U:V-Grph →Set has a fully
faithful right adjoint Rwhich assigns to any set Xthe graph over Xwhich has the terminal
object in every component. If a fully faithful right adjoint Rfor Uexists, then we can write
R({∗}) = ({∗},⊤) and we find
V(V, ⊤)∼
=V-Grph(({∗}, V ),({∗},⊤)) = V-Grph(({∗}, V ), R({∗})) = Set({∗},{∗}) = {id{∗}},
which shows that ⊤is a terminal object for V.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 7
Proposition 1.11. A morphism f= (f0, f 1) : A→Bin V-Grph is a
(i) monomorphism if and only if f0:A0→B0is injective and fx,y :Ax,y →Bx,y is a
monomorphism in Vfor all x, y ∈A0.
(ii) epimorphism if and only if f0:A0→B0is surjective and fx,y :Ax,y →Bx,y is an
epimorphism in Vfor all x, y ∈A0.
Proof. It follows from Proposition 1.10 that f0is a monomorphism in Set, hence injective.
Fix any x, y ∈A0and any pair of morphisms g, h :V→Ax,y in Vsuch that fx,y ◦g=fx,y ◦h.
Then consider the V-graph V, with set of objects V0=A0,Vx,y =Vand Vx′,y′=Ax′,y′for
all (x′, y′)∈A0×A0\ {(x, y)}. Then we have morphisms g, h :V→A, where g0and h0
are the identity maps, gx,y =g,hx,y =hand all other components of gand hare identities.
Then clearly, f◦g=f◦hand therefore g=has fis a monomorphism. In particular,
g=gx,y =hx,y =h. We can conclude that fx,y is a monomorphism in V.
The second part is proven in a similar way.
The category of V-graphs inherits many nice properties from V. Many results of this flavour
have been proven throughout literature, but we just mention the following result, which is
sufficient for our needs.
Proposition 1.12 ( [17, Proposition 4.4]).Let Vbe an arbitrary category. If Vis locally
presentable, then so is V-Grph.
Remark 1.13.Recall from [7, Proposition 2] (see also [17, Section 3], [27, Proposition 4.14])
that colimits in V-Grph can be build up explicitly from colimits in Set and V, although not
in a straightforward way. As remarked in [27, Proposition 4.14], the situation of limits in
V-Grph is much easier, and since we will need those further on in the paper, let us describe
these explicitly here. Consider a small category Zand a functor F:Z → V -Grph. For any
Z∈ Z, we denote F Z = (F Z 0, F Z1). Take the limit (L0, λ0
Z:L0→F Z0) in Set of the
composite functor U◦F:Z → Set. We know from Proposition 1.10 that L0will be the set of
objects of lim For more precisely that Ulim F= (L0, λ0). Now fix any x, y ∈L0, and consider
the functor Fx,y :Z → V defined on an object Z∈ Z as Fx,y (Z) = F Zλ0
Z(x),λ0
Z(y)and on a
morphism f:Z→Z′in Zas F fλ0
Z(x),λ0
Z(y). Then consider the limit lim Fx,y = (L1
x,y, λ1
x,y) in
V. From the universal properties of the considered limits, one easily deduces that lim F=
((L0, L1),(λ0, λ1)).
1.3. The category of V-categories. From now on, let us suppose that Vis a monoidal cate-
gory, whose tensor product we denote by ⊗and monoidal unit by I. By Mac Lane’s coherence
theorem we suppose without loss of generality that Vis strict monoidal, and henceforth we
will not write associativity and unitality constraints. Let us recall the (well-known) notion
of a V-enriched category.
Definition 1.14. AV-enriched category (or a V-category for short) is a V-graph A= (A0, A1)
endowed with V-morphisms (called the compositionbor multiplication morphisms)
mxyz :Axy ⊗Ayz →Axz jx:I→Axx ,
bAs one can see from the definition of composition, one should interpret Ayx as the object of morphisms
“from yto x”. Traditionally, many authors use the reversed notation, however we believe the notation used
here is more efficient for our needs.
8 P. GROSSKOPF AND J. VERCRUYSSE
for all x, y, z ∈A0satisfying the following axioms.
Axy
id
++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
≃//
≃
Axy ⊗I
id⊗jy//Axy ⊗Ayy
mxyy
Axy ⊗Ayz ⊗Azw
mxyz ⊗id
//
id⊗myzw
Axz ⊗Azw
mxzw
I⊗Axy jx⊗id //Axx ⊗Axy mxxy //Axy Axy ⊗Ayw mxyw //Axw
.
AV-functor is a V-graph morphism that preserves multiplication and unit. Explicitely
f: (A0, A1, m, j)→(B0, B1, m′, j′) consists of a function f0:A0→B0and a family of
morphisms fxy :Axy →Bfx,f y in Vfor all x, y ∈A0, such that the following diagrams
commute for all x, y, z ∈A0.
Ijx//
j′
fx !!
❈
❈
❈
❈
❈
❈
❈
❈
❈Axx
fxx
Axy ⊗Ayz
mxyz //
fxy⊗fy z
Axz
fxz
Bfxf x Bf xf y ⊗Bf yfz m′
fx f yf z
//Bfxf z
.
One observes that V-categories and V-functors form a the category, which we denote by
V-Cat.
Remark 1.15.One can of course also consider V-natural transformations which turn V-Cat
into a 2-category. The same holds for the categories semi-Hopf categories and Hopf-categories
which we consider below.
Clearly, a V-category with one object is exactly an algebra (or monoid) in V. Suppose now
that Vhas an initial object ⊥such that for any object Vin V, we have V⊗ ⊥ ∼
=⊥∼
=⊥ ⊗ V.
We will refer to the property as saying that the initial object is preserved under tensoring.
This is for example satisfied if the endofunctors of the form −⊗ Vand V⊗− preserve (finite)
colimits for all Vin V. Then to any object Vin Vwe can associate a V-category Vin the
following way. As set of objects we take a two-element set V0={0,1}and we define the
Hom-objects Vx,y as follows: V0,0=V11 =I,V0,1=Vand V1,0=⊥. Then the identity on the
monoidal unit Iinduces the unit morphisms j0=j1=idI. Composition is then obtained in
an obvious way from the unit constraints of the monoidal structure on Vand the universal
property of the initial object. These observations lead directly to the following result.
Proposition 1.16. (1) The functor P:V → V-Grph from Proposition 1.9 lifts to a fully
faithful functor P:Mon(V)→ V-Cat such that the following diagram commutes, where
Alg(V)denotes the category of algebras in V, and the unadorned vertical arrows are
the obvious forgetful functors.
Alg(V)P//
V-Cat
VP//V-Grph
(2) If Vhas an initial object that is preserved under tensoring, then there is a fully faithful
functor P′:V → V-Cat, sending Vto the two-object V-category Vdefined above.
Let us start by studying the forgetful functor to Set, similar to Proposition 1.10.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 9
Proposition 1.17. Consider the forgetful functor U:V-Cat →Set.
(i) If Vhas an initial object that is preserved under tensoring, then Uhas a fully faithful
left adjoint.
(ii) If Vhas a terminal object, then Uhas a fully faithful right adjoint.
In particular, if Vhas an initial and terminal object, then Upreserves all limits and colimits,
in particular monomorhpisms and epimorphisms.
Proof. (i) Consider any set X. We define a V-category Lwith L0=Xas set of objects by
putting Lxx =I, the monoidal unit of V, for all x∈Xand Lxy =⊥, the initial object of
V, for all x, y ∈Xwith x6=y. Then Lbecomes in a trivial way a V-category and this
construction gives a fully faithful left adjoint to U.
(ii). If Vhas a terminal object ⊤, then define another V-category Rwith R0=Xas set
of objects by putting Rxy =⊤, the terminal object, for all x, y ∈X. Again Rbecomes in a
trivial way a V-category and this construction provides a fully faithful right adjoint for U.
As for V-Grph, also the category V-Cat inherits many good properties from V.
Proposition 1.18. Let Vbe a monoidal category.
(i) [7, Proposition 3, Theorem 6, Theorem 7] Suppose that Vis cocomplete and for any
object Ain V, endofunctors on Vof the form A⊗ − or − ⊗ Apreserve colimits. Then
V-Cat is cocomplete as well and the forgetful functor U:V-Cat → V-Grph has a left
adjoint and is moreover monadic.
(ii) [17, Theorem 4.5] If Vis closed monoidal and λ-locally presentable, then V-Cat is
λ-locally presentable as well.
Remark 1.19.At first look, it might appear that the condition on the preservation of colimits
by endofunctors A⊗− and −⊗Ain item (i) of the above Proposition is weaker than the closed
monoidality in item (ii). However, in view of Proposition 1.1 both conditions are equivalent
if Vis locally presentable.
The left adjoint T:V-Grph → V-Cat to the forgetful functor U, creates the “free V-
category” over a V-graph. In case we start from a one-object V-graph, that is, just an object
V∈ V, then T(V) is a one-object V-category, that is, just a monoid in V, namely the free
monoid over the object V. In general the functor Tcommutes with the forgetful functor to
Set. Explicitly, for a given V-graph A= (A0, A1), the free Vcategory T(A) has the same set
of objects A0, and for any x, y ∈A0, we have
(1) T(A)x,y =a
n∈N,z1,...,zn∈A0
Ax,z1⊗Az1,z2⊗ · · · ⊗ Azn,y.
In case x=y,T(A)x,x has an additional component of the form I(the monoidal unit), that
induces the unit morphisms jx:I→T(A)x,x. Composition in (or multiplication of) the
V-category T(A) is obtained from the monoidal associativity constraints
(Ax,z1⊗ · · · ⊗ Azn,y )⊗(Ay,u1⊗ · · · ⊗ Aun,z )∼
=//Ax,z1⊗ · · · ⊗ Azn,y ⊗Ay,u1⊗ · · · ⊗ Aun,z
and the universal property of the coproduct (recall that coproducts are preserved under tensor
product by assumption).
10 P. GROSSKOPF AND J. VERCRUYSSE
For any V-graph A, the unit of the adjunction ηA:A→U T (A) is given by the canonical
morphisms
ηA
x,y :Ax,y →T(A)x,y
for all x, y ∈A0, which are induced by the fact that T(A)x,y is defined as a coproduct with
Ax,y as one of the components (see (1)). On the other hand, given a V-category A, the counit
of the adjunction ǫA:T U (A)→Ais induced by the composition in (or multiplication of) A.
Remark that by construction of T(A) as a coproduct and the composition as defined above,
we have that for any x, y ∈A0, the family of morphisms
Ax,z1⊗Az1,z2⊗ · · · ⊗ Azn,y
ηx,z1⊗···⊗ηzn,y
//T(A)x,z1⊗T(A)z1,z2⊗ · · · ⊗ T(A)zn,y
m//T(A)x,y
indexed by all n∈N, z1, . . . , zn∈A0are jointly epic in V, where η:A→U T (A) denotes the
unit of the adjunction (T , U) and mdenotes the obvious morphism induced by the composition
in T(A). In case x=y, one needs add also the unit morphisms jx:I→T(A)x,x to this
family.
We then also obtain the following useful property for colimits in V-Cat. Consider a functor
F:Z → V-Cat and its colimit colim Fin V-Cat together with the canonical morphisms
γZ:F Z →colim Ffor all Z∈ Z. Also consider the composite functor U F :Z → V-Grph and
its colimit colim UF in V-Grph with the canonical morphisms γ′
Z:U F Z →colim U F . Then
colim Fcan be constructed in V-Grph as the “biggest quotient” q:Tcolim UF =colim T U F →
colim F(recall that Tas a left adjoint preserves colimits) endowed with a V-category structure
such that the compositions
F Z ηFZ //T U F Z Tγ ′
Z//colim T U F q//colim F
are morphisms of V-categories for any Z∈ Z. Hence, qbeing an epimorphism in V-Grph,
which implies by Proposition 1.11 that all its components are epimorphisms, and building on
the jointly epic family of morphisms on a free V-category described above, one deduces that
for each x, y ∈colim F0the following family of morphisms in V
(F Z1)x11,x12 ⊗(F Z2)x21 ,x22 · · · ⊗ (F Zn)xn1,xn2
(γZ1)x11x12 ⊗(γZ1)x21x22 ⊗···⊗(γZ1)xn1xn2
(colim F)xx1⊗(colim F)x1x2⊗ · · · ⊗ (colim F)xn−1y
mxx1x2···xn−1y
(colim F)xy
varying over all n∈N0,Z1,...,Zn∈ Z,xi1, xi2∈(U F Z1)0such that γ0
Zi(xi2) = γ0
Zi+1 (xi+1,1) =
xi,γ0
Z1(x11) = xand γ0
Zn(xn2) = yis jointly epic (in V).
Definition 1.20. Let Vbe a monoidal category, then we denote by Vrev the monoidal cat-
egory which has the same underlying category as V, but whose tensor product is reversed.
More precisely, we have that for any two objects A, B in V,
A⊗rev B:= B⊗A.
and similarly for morphisms.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 11
Let Abe a V-category. Then we define the opposite V-category of Aas the Vrev-category
Aop whose underlying V-graph is the opposite graph of A(that is, the set of objects is A0
and for each x, y ∈A0, we have Aop
xy =Ayx) endowed with compositions given by
Aop
x,y ⊗rev Aop
y,z =Az,y ⊗Ay ,x
mzyx //Az,x =Aop
x,z
and the same unit morphisms as A.
Similarly, if f:A→Bis a V-functor, then we define fop :Aop →Bop by (fop)0=f0and
fop
x,y =fy,x.
We leave it to the reader to check that Aop is indeed a Vrev -category. The following result
is immediate
Proposition 1.21. There is an endofunctor (−)op :V-Cat → Vrev -Cat which sends a V-
category to its opposite Vrev-category. This functor is clearly isomorphism of categories,
whose inverse functor is constructed in the same way, since (Vrev )rev =Vas a monoidal
category.
More generally, the 2-functor from Proposition 1.8 induces another 2-functor in our current
setting.
Proposition 1.22. Any lax monoidal functor F:V → W induces a functor F-Cat :V-Cat →
W-Cat. Similarly, if G:V → W is another lax monoidal functor, then any monoidal natural
transformation α:F→Ginduces a natural transformation F-Cat →G-Cat. In fact,
we obtain a 2-functor LMonCat →Cat, where LMonCat denotes the 2-category of monoidal
categories, lax monoidal functors and monoidal natural transformations. It follows that any
lax monoidal adjunction between Vand Walso induces an adjunction between V-Cat and
W-Cat.
Proof. Let us denote by I(resp. J) the monoidal unit of V(resp. W) and the monoidal
product in both cases by ⊗. We also denote the monoidal structure on Fby φ0:J→F I and
φ2:F−⊗F− → F(−⊗−). Consider any V-category A. We already know by Proposition 1.8
that F(A) is a W-graph. It now suffices to observe that the morphisms
F mxyz ◦φ2
Axy,Ay z :F Ax,y ⊗F Ay,z →F Ax,z , F jx◦φ0:J→F Ax,x
endow F A with a W-category structure. Furthermore, given any morphism f:A→Bof
V-categories, one can then check that F-Grph(f) as defined in Proposition 1.8 is a morphism
of W-categories with respect to the W-category structure defined here. This defines the
functor F-Cat. Finally, one observes that if α:F→Gis a monoidal natural transformation,
then α−Grph as defined in Proposition 1.8 is in fact a natural transformation from F-Cat to
G-Cat.
Remark 1.23.Let Vbe a braided monoidal category, whose braiding we denote by σ. Then
the identify functor and braiding induce a strong monoidal functor (id, σ) : Vrev → V. This
functor is moreover an monoidal isomorphism, with inverse (id, σ−1), where σ−1denotes the
inverse braiding. Remark that if Vis symmetric then both functors coincide. Specifying
Proposition 1.22 to this monoidal functor, we find that for a braided monoidal category V,
there is an isomorphism of categories Vr ev -Cat → V-Cat. Combining this isomorphism with
the isomorphism from Proposition 1.21, we obtain a new isomorphism
(−)op :V-Cat → V-Cat
12 P. GROSSKOPF AND J. VERCRUYSSE
which sends a V-category Ato its opposite V-category, which, by abuse of notation, we still
denote as Aop. It has the opposite V-graph as underlying V-graph and compositions defined
by
Aop
x,y ⊗Aop
y,z =Ay,x ⊗Az ,y
σ//Az,y ⊗Ay,x
mzyx //Az,x =Aop
x,z
2. Cofree coalgebras
2.1. Existence of cofree coalgebras. Let now Vbe a symmetric monoidal category, where
we denote the symmetry by σ. Then it is well-known that the category Coalg(V) of coalgebras
in Vinherits in a natural way a (symmetric) monoidal structure from V. Moreover, let us
recall the following result which tells that Coalg(V) also inherits the property of being locally
presentable from V.
Proposition 2.1 ( [19, 2.7, 3.2]).Let Vbe a closed symmetric monoidal and locally pre-
sentable category. Then the following assertions hold.
(i) The category Coalg(V)is closed symmetric monoidal and locally presentable.
(ii) The forgetful functor U:Coalg(V)→ V is strict symmetric monoidal and comonadic, in
particular it has a right adjoint Tc:V → Coalg(V)which creates the “cofree coalgebra”
over an object in V.
Remark 2.2.In contrast to free algebras in V, or more generally, free V-categories (see Remark
1.19), cofree coalgebras are usually more difficult to construct explicitly. Indeed, naively one
would try to make a dual construction of free algebras by considering the product T0(V) :=
Qn∈NV⊗n. However, since usually infinite products are not preserved by tensoring with a
fixed object in V, there is no way in general to endow T0(V) with a suitable comultiplication.
In fact, or at least to the knowledge of the authors, an explicit construction of cofree coalgebra
over an object Vin Vis only known in particular cases, such as V=Vectkwith ka field.
We generalize this construction to the case of a monoidal category whose monoidal product
is sufficiently exact in the sense pointed out in the next subsection.
2.2. Very flat monoidal products. The following terminology of “(very) flat monoidal
product” is inspired by the module-theoretic case and will be essential futher in this paper.
Definition 2.3. Let Vbe a monoidal category and Aan object in V. We say that Ais (left)
flat if the endofunctor − ⊗ Aon Vpreserves monomorphisms.
We say that Ais (left) very flat if the endofunctor − ⊗ Aon Vpreserves arbitrary jointly
monic families. This means, for any jointly monic family of morphisms βi:B→Biin V, we
have that the family βi⊗idA:B⊗A→Bi⊗Ais also jointly monic.
Similarly, we can consider right (very) flat objects. If Vis symmetric (or even braided) left
and right (very) flatness are equivalent. Since the categories we consider are symmetric we
will not distinguish between left and right from now on.
If Vis a monoidal category such that all objects are (very) flat, we say that Vis has a
(very) flat monoidal product.
Although flat modules (that is, flat objects in a category of (bi)modules over a ring) are
a standard notion, what we called “very flat” seems to be unstudied in literature. The next
examples in case Vis a category of modules show that “very flat” modules are strictly in
between flat and projective modules.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 13
Examples 2.4. (1) Obviously, every very flat module is flat.
(2) It is well-known that the Z-module Qis flat, however it is not very flat. Indeed,
consider the jointly monic family of morphisms πp:Z→Zpfor all prime numbers p.
Since for any p, we have that Zp⊗ZQis the zero module, the family πp⊗ZQis not
jointly monic, hence Qis not very flat as Z-module.
(3) Any locally projective module (in the sense of Zimmermann-Huisgen [29], called
weakly locally projective in [28]) is very flat, in particular every projective module
is very flat, but not every very flat module is projectivec. Indeed, let kbe a (commu-
tative) ring and Pa locally projective k-module. Now consider a joinly monic family
of k-module morphisms αj:A→Aj. We claim that P⊗kαj:P⊗kA→P⊗kAj
is again jointly monic. Indeed, consider any element p⊗a∈P⊗kA(summation
understood) such that p⊗αj(a) = 0 for all j. Then for any f∈P∗, we have that
f(p)αj(a) = αj(f(p)a) = 0 for all j. Since the αjwere jointly monic, we find that
f(p)a= 0 for all f∈P∗. Then the local projectivity of Pimplies that p⊗ka= 0
(see e.g. [28, Theorem 2.15]).
The following properties of very flatness will be crucial in what follows.
Lemma 2.5. (1) Let Vbe a monoidal category and Aan object in V. Then Ais very flat
if and only if Ais flat and for any family objects (Bi)i∈Iin V, the canonical mophism
that makes the following diagrams commutative for all i∈I
Qi∈IBi⊗A
πi⊗idA''
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
//Qi∈I(Bi⊗A)
π′
i
ww♣♣♣♣♣♣♣♣♣
♣♣
Bi⊗A
is a monomorphism, Here we denoted by πiand π′
ithe canonical projections on the
ith component of the considered products.
(2) Let Vbe a monoidal category with very flat monoidal product. If (αi:A→Ai)i∈Iand
(βj:B→Bj)j∈Jare jointly monic families of morphisms, then (αi⊗βj:A⊗B→
Ai⊗Bj)(i,j)∈I×Jis also jointly monic. In other words, tensor products of jointly monic
families are again jointly monic.
(3) Let Vbe a monoidal category with very flat monoidal product. Then for any two
families (Ai)i∈Iand (Bj)j∈Jof objects in Vthe canonical morphism ιdefined by the
commutativity of the following diagram for all i∈Iand j∈J, is a monomorphism.
Qi∈IAi⊗Qj∈JBj
πi⊗πj((
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
◗
ι//Q(i,j)∈I×J(Ai⊗Bj)
πi,k
ww♥♥♥♥♥♥♥♥♥♥♥♥♥
Ai⊗Bj
Here πi,πjand πi,j denote the canonical projections of the considered products in V.
Proof. (1). Suppose that Ais very flat, then we know that Ais flat. Furthermore, the
projections πiand π′
iform jointly monic families. Therefore, the commutativity of the above
diagram tell us that is a monomorphism if and only if πi⊗idAis a jointly monic family,
cOne could wonder whether local projectivity is equivalent to very flatness.
14 P. GROSSKOPF AND J. VERCRUYSSE
which is the case since Ais very flat. Conversely, consider a jointly monic family βi:B→Bi.
Then the map β=Qβi:B→Qi∈IBiis a monomorphism and since Ais flat β⊗idAis also
a monomorphism. We then have the following commutative diagram
B⊗A
β⊗idA
βi⊗idA//Bi⊗A
Qi∈IBi⊗A//Qi∈I(Bi⊗A)
π′
i
OO
Since β⊗idAand are monic and π′
iare jointly monic, we find that βi⊗idAare also jointly
monic, hence Ais very flat.
(2). By very flatness, the morphisms (αi⊗idB)i∈Iand (idAi⊗πj)j∈Jare also jointly monic
families. Since αi⊗βj= (αi⊗B)◦(Ai⊗βj), we find that the family (αi⊗βj)(i,j)∈I×Jbeing
a composition of jointly monic families is again jointly monic.
(3). Observe that ιis a monomorphism if and only if (πi⊗πj)(i,j)∈I×Jis a jointly monic
family. Classical properties of a product tell us that (πi)i∈Iand (πj)j∈Jare jointly monic
families hence the statement follows by part (2).
2.3. The construction of cofree coalgebras. In the next proposition, we will use the
following notation. Let (C, ∆, ǫ) be a coalgebra in V. Then we define for all n∈Na
morphism ∆n:C→C⊗nas follows: ∆0=ǫand for all n > 0:
∆n= (∆n−1⊗idC)◦∆,
in particular, ∆1=idCand ∆2= ∆.
Theorem 2.6. Let Vbe a closed monoidal and locally presentable category, with very flat
monoidal product. Consider the product Qn∈NV⊗nwhere V⊗0=I, the monoidal unit. Denote
by πn:Qn∈NV⊗n→V⊗nthe canonical projection on the n-th component of the product.
Then the cofree coalgebra C=Tc(V)over an object Vin Vis the biggest subobject c:C→
Qn∈NV⊗nin V, such that there exists a morphism ∆ : C→C⊗Csatisfying (πn⊗πm)◦
(c⊗c)◦∆ = πn+m◦cfor all n, m ∈N.
(2) C
∆
c//Qk∈NV⊗kπn+m//V⊗(n+m)
∼
=
C⊗Cc⊗c//Qk∈NV⊗k⊗Qk∈NV⊗kπn⊗πm
//V⊗n⊗V⊗m
The counit of the forgetful-cofree adjunction is then given by p=π1◦c:C→Vand the
morphisms p⊗n◦∆n:C→V⊗nare jointly monic.
Proof. Consider the family Cof all sub-objects c:C→Qk∈NV⊗kin V, endowed with a
morphism ∆ : C→C⊗Csatisfying (2). We claim that such ∆ is always coassociative and
counital by means of ǫC=π0◦c, i.e. (C, ∆C, ǫC) is a coalgebra. To prove the left counitality
consider (ǫC⊗id)◦∆ : C→I⊗C∼
=Cand compose these with the jointly monic family
(πn◦c)n∈N. Then the commutativity of (2) implies that
πn◦c◦(ǫC⊗id)◦∆ = (π0⊗πn)◦(c⊗c)◦∆ = πn◦c
for any n∈N, hence (ǫC⊗id)◦∆ = idC. The right counitality is proven in the same way. To
prove the coassociativity, consider the parallel pair of morphisms (∆ ⊗id)◦∆,(idC⊗∆) ◦∆ :
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 15
C→C⊗C⊗Cand compose them with (πn⊗πm⊗πp)◦(c⊗c⊗c). Using twice the
commutativity of the diagram (2), we find that both compositions equal πn+m+p◦c. Since
the monoidal product is very flat, the family ((πn⊗πm⊗πp)◦(c⊗c⊗c))n,m,p∈Nis jointly monic
being a tensor product of jointly monic families (see Lemma 2.5), hence ∆ is coassociative.
Now consider any coalgebra (C, ∆, ǫ), and γ:C→Va morphism in V. Then for any
n∈N, we define
γn:C∆n//C⊗nγ⊗n
//V⊗n.
Consequently, we obtain a map γ=Qk∈Nγk:C→Qk∈NV⊗k. Since Vis locally presentable,
we have a strong epi-mono factorization for γ(see Proposition 1.3):
γ:Ce//C′m//Qk∈NV⊗k.
We claim that C′is again a coalgebra and eis a coalgebra morphism. Indeed, since eis a
strong epimorphism and ι◦(m⊗m) is a monomorphism (remark that since the monoidal
product is (very) flat, m⊗mis a monomorphism as mis one and ιin the diagram below
is a monomorphism by Lemma 2.5), we obtain the existence of a morphism ∆′filling the
following commutative diagram.
C
∆
e//C′
∆′
m//Qk∈NV⊗k
D
πp+q//V⊗(p+q)
∼
=
C⊗Ce⊗e//C′⊗C′m⊗m//Qk∈NV⊗k⊗Qk∈NV⊗kι//Qp,q∈N(V⊗p⊗V⊗q)πp,q //V⊗p⊗V⊗q
The morphism Din the above diagram is defined by the commutativity of the rectangle on
the right. Then we see that (C′,∆′) is an element of C, so in particular it is a coalgebra by
the first part of the proof. The morphism eis a coalgebra morphism by the commutativity
of the left square in the above diagram. We conclude that any morphism γ:C→Vwith C
a coalgebra factors through an object of Cvia a coalgebra morphism.
We are now ready to construct the cofree coalgebra over V. Since we assumed that Vis
locally presentable, Vis in particular well-powered, and hence Cis a set (and not a proper
class). Consider the coproduct `CCof all these objects, which is, being a colimit of coalgebras
in V, again a coalgebra in V. Consider the morphism `Cc:`CC→Qk∈NV⊗k, and define
γ=π1◦`Cc. Then we can apply the procedure from the above paragraph. To this end,
consider the morphisms
γn=γ⊗n◦∆n=π⊗n
1◦a
C
c⊗n◦a
C
∆n
C
=a
C
(π⊗n
1◦c⊗n◦∆n
C) = a
C
(πn◦c) = πn◦a
C
c
Here the second equality follows from the definition of γand the commutativity of tensor
product with coproduct. The penultimate equality follows from (2) which is satisfied by all
elements of C. We can conclude that γ=`cand by the above, this morphism admits a
strong epi-mono factorization
`Cc:`CCe′
//C(V)m′
//Qk∈NV⊗k
where C(V) is a coalgebra and e′is a coalgebra morphism.
16 P. GROSSKOPF AND J. VERCRUYSSE
If we now consider again any coalgebra Cwith a morphism γ:C→V, then then the
results above show that γfactors through C(V) via a coalgebra morphism (the morphism
e′◦ιC′◦ein the diagram below, where ιC′is the canonical monomorphism of the coproduct),
so C(V) satisfies the universal property of the cofree coalgebra.
Cγ//
e
❄
❄
❄
❄
❄
❄
❄
❄V
C′
m
44
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
❥
ιC′//QD∈C De′
//C(V)
m′
OO
Since by construction, c:C(V)→Qk∈NV⊗kis a subobject such that πk◦c=p⊗k◦∆k,
these form a jointly monic family indexed by k∈N.
Since the forgetful functor U:Coalg(V)→ V is comonadic, it preserves (and, in fact, cre-
ates) colimits. Limits in a category of coalgebras are, however, difficult to describe, although
it follows from Proposition 2.1 that they exist. In case of a very flat monoidal product, we
can apply the explicit description of the cofree coalgebra given above to give also a more
explicit description of limits of coalgebras.
Corollary 2.7. Let Vbe a closed monoidal and locally presentable category, with very flat
monoidal product. Let F:Z → Coalg(V)be a functor, where Zis a small category and
consider its limit L=lim Fas well as the limit L′=lim U F where U:Coalg(V)→ V is the
forgetful functor together with the canonical projections πZ:L′=lim UF →U F Z for any Z
in Z.
(i) Lcan be constructed as the biggest subobject in Vof the cofree coalgebra Tc(L′), which
has a coalgebra structure such that for any object Zin Zthe composition
πZ:Li//TcL′p//L′π′
Z//U F Z
is a coalgebra map, where iis the monomorphism that turns Linto a subobject of TcL′.
The morphism πZformed this way is exactly the canonical projection of the limit L=
lim Fon the component F Z.
(ii) The family of morphisms
(πZ1⊗ · · · ⊗ πZn)◦∆n
L:L→F Z1⊗ · · · ⊗ F Zn
indexed by n∈Nand objects Z1,...,Zn∈ Z, is jointly monic in V.
Proof. (i). This part can be proven by applying the same technique as in the proof of Theo-
rem 2.6.
(ii). By part (i), we know that there is a monomorphism i:lim F→Tclim U F in V.
From the construction of the cofree coalgebra in Theorem 2.6, we know that the family of
morphisms
Tclim UF ∆n//(Tclim U F )⊗np⊗n
//(lim U F )⊗n
is jointly monic. Moreover, by general properties of categorical limits, the family of projections
π′
Z:lim UF →U F Z over all objects Z∈ Z is jointly monic in Vas well. Finally, by the
very flatness of the monoidal product the last property implies by Lemma 2.5 that also the
family of morphims
π′
Z1⊗ · · · ⊗ π′
Zn: (lim U F )⊗n→UF Z1⊗ · · · ⊗ U F Zn,
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 17
indexed by objects Z1, . . . , Znin Zis jointly monic. Combining the above and since i:
lim F→Tlim UF is a coalgebra map, we obtain that the family of morphisms
lim F
i
∆n//lim F⊗nπZ1⊗···⊗πZn//
i⊗n
U F Z1⊗ · · · ⊗ U F Zn
Tclim UF ∆n//(Tclim U F )⊗np⊗n
//(lim U F )⊗n
π′
Z1⊗···⊗π′
Zn
OO
indexed by n∈Nand objects Z1,...,Zn∈ Z, is jointly monic in V.
3. Existence of free and cofree Hopf categories
3.1. Semi-Hopf V-categories. Recall from [6] that a semi-Hopf category over a symmetric
monoidal category V(where we denote the symmetry by σ) is nothing else than a Coalg(V)-
category. Unwinding this definition, we obtain the following more explicit definition.
Definition 3.1. Asemi-Hopf V-category Aconsists of a collection of objects A0and for
all objects x, y ∈A0we have a (coassociative, counital) coalgebra (Axy , δxy :Axy →Axy ⊗
Axy, ǫxy :Axy →I) in Vtogether with the following morphisms in V, for all x, y, z ∈A0:
mxyz :Axy ⊗Ayz →Axz jx:I→Axx
which turn Ainto a V-category and such that moreover the following axioms hold:
Axy ⊗Ayz
δxy ⊗δxy //
mxyz
Axy ⊗Axy ⊗Ayz ⊗Ayz
Axy⊗σ⊗Ay z
Axy ⊗Ayz ⊗Axy ⊗Ayz
mxyz ⊗mxyz
Axz δxz
//Axz ⊗Axz
I≃//
jx
I⊗I
jx⊗jx
Axy ⊗Ayz
ǫxy⊗ǫxy
//
mxyz
I⊗I
≃
I
jx
=
❅
❅
❅
❅
❅
❅
❅
❅
❅
Axx δxx
//Axx ⊗Axx Axz ǫxz //I Axx ǫxx //I
A morphism between two semi-Hopf categories, called a semi-Hopf V-functor is a morphism
Coalg(V)-graphs that is at the same time a V-functor. Semi-Hopf V-categories form a category
which we denote by V-sHopf.
Examples 3.2. Suppose that V=Set, or any Cartesian category viewed as a monoidal
category, then Coalg(V) = V, as any object Cof Vcan be endowed in a unique way with a
comultiplication C→C×Cby means of the diagonal map. It then follows that a semi-Hopf
V-category is just a V-category.
A semi-Hopf V-category A(for any symmetric monoidal category V) with A0containing
just one element is nothing else than a Hopf algebra in V. More generally, if A0is finite, then
`x,y∈A0Axy is a weak Hopf algebra in V, see [6, Proposition 6.1] for a proof in the k-linear
case.
18 P. GROSSKOPF AND J. VERCRUYSSE
Combining the results from the previous sections, we then immediately arrive at the fol-
lowing.
Proposition 3.3. Let Vbe a closed symmetric monoidal and locally presentable category.
Then the following statements hold.
(i) The category V-sHopf of semi-Hopf V-categories is locally presentable.
(ii) The forgetful functor V-sHopf →Coalg(V)-Grph is monadic. In particular, the forgetful
functor V-sHopf →Coalg(V)-Grph has a left adjoint, which creates the “free semi-Hopf
category” over a Coalg(V)-graph.
(iii) The forgetful functor V-sHopf → V-Cat is comonadic. In particular, the forgetful functor
V-sHopf → V-Cat has a right adjoint, which creates the “cofree semi-Hopf category” over
aV-category.
Proof. Items (i) and (ii) follow directly from combining Proposition 1.18 and Proposition 2.1.
The forgetful-cofree adjunction from (iii) follows by combining Proposition 2.1 with Proposi-
tion 1.22. The comonadicity follows directly from Becks precise tripleablity theorem, as the
forgetful functor V-sHopf → V-Cat preserves colimits and reflects isomorphisms.
Remark 3.4.The construction of the free semi-Hopf category over a Coalg(V)-graph Afollows
directly from the general construction as recalled in Remark 1.19. Moreover, as this construc-
tion is based on coproducts in Coalg(V) and since the forgetful functor Coalg(V)→ V creates
coproducts, the whole construction can be performed simply in V. In paricular, we find that
the free semi-Hopf category has the same set of objects A0as the given Coalg(V)-graph A, and
for each x, y ∈A0, the Hom-object from xto yin the free semi-Hopf category is computed
as the following coproduct in V:
a
n∈N,z1,...,zn∈A0
Ax,z1⊗Az1,z2⊗ · · · ⊗ Azn,y
(with an additional component Iin case x=y), endowed with the natural comultiplication
obtained from the comultiplications of the indvidual Az,z′, and a composition induced by the
associators of the monoidal product in V.
The construction of the cofree semi-Hopf category over a given V-category Acan in a similar
way be deduced from the construction fo cofree coalgebras. Again, the set of objects will be
given by A0, the set of objects of the given V-category A. Then, one considers for any x, y ∈A0
the cofree coalgebra Tc(Ax,y) over the V-object Ax,y (which can be constructed explicitly in
case the monoidal product is very flat following the construction from Theorem 2.6). The
composition (or multiplication) in the cofree semi-Hopf category is obtained from the universal
property of the cofree coalgebra and commutativity of the following diagram.
Tc(Axy)⊗Tc(Ayz )pxy ⊗pyz //
Axy ⊗Ayz
mxyz
Tc(Axz)pxz //Axz
Finally, let us remark that semi-Hopf categories can also be viewed as coalgebra objects in
a monoidal category of V-categories with a fixed set of objects X. For any two V-categories
A= (X, A1) and B= (X, B1) over this set of objects, the monoidal product is given by
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 19
A•B= (X, A1•B1), where
(A•B)xy =Axy ⊗Bxy.
See [6] for this point of view on semi-Hopf categories and [9] for a bicategorical point of view
on this which allows to vary the set of objects. Hence, the above description of the cofree
Hopf category over a V-category can also be obtained directly from Theorem 2.6 applied to
the monoidal category of V-categories with a fixed set of objects.
Corollary 3.5. Consider a small category Zand a functor F:Z → V -sHopf.
(i) The set of objects of the colimit colim Fis computed by taking the colimit of the functor
Fcomposed with the forgetful functor to Set. For each x, y ∈colim F0the following
family of morphisms in V
(F Z1)x11,x12 ⊗(F Z2)x21 ,x22 · · · ⊗ (F Zn)xn1,xn2
(γZ1)x11x12 ⊗(γZ1)x21x22 ⊗···⊗(γZ1)xn1xn2
(colim F)xx1⊗(colim F)x1x2⊗ · · · ⊗ (colim F)xn−1y
mxx1x2···xn−1y
(colim F)xy
varying over all n∈N0,Z1,...,Zn∈ Z,xi1, xi2∈(U F Z1)0such that γ0
Zi(xi2) =
γ0
Zi+1 (xi+1,1) = xi,γ0
Z1(x11) = xand γ0
Zn(xn2) = yis jointly epic (in V).
(ii) The set of objects of the limit lim Fis computed by taking the limit of the functor F
composed with the forgetful functor to Set. If Vhas moreover a very flat monoidal
product, then the family of morphisms
(lim F)x,y
δn
x,y //((lim F)x,y)⊗nπ1
xy⊗···⊗π1
xy //(F Z1)x1,y1⊗ · · · ⊗ (F Zn)xn,yn
indexed by n∈Nand Z1,...,Znin Z, is jointly monic in V. Here for any tuple of
objects Z1,...,Znin Z, we denoted π0
Zi(x) = xiand π0
Zi(y) = yiand (πx,y )Zi=πi
xy.
Proof. (i).Then from Proposition 3.3(iii) we know that the colimit colim Fcan be computed
in V-Cat. The first part of the statement then follows from the fact (see Proposition 1.17)
that the forgetful functor from V-Cat to Set, being a left adjoint, preserves colimits. As
explained in Remark 1.19, we then moreover find on each component of colim F, the family
of morphisms in Vfrom the statement is jointly epic.
(ii). From Proposition 3.3(ii) it follows that the limit lim Fcan be computed in Coalg(V)-
Grph and by Proposition 1.10 we know that the forgetful functor from Coalg(V)-Grph to Set,
being a right adjoint, preserves limits. This already shows the first part of the statement. For
the second part, recall from Remark 1.13 how a limit in Coalg(V)-Grph is build up from the
limit of the underlying sets and limits in Coalg(V). By Corollary 2.7, we know how limits in
the category of V-coalgebras can be constructed explicitly and that there is a jointly monic
family of morphisms in Vas in the statement.
Recall that an op-monoidal functor F:V → W lifts to a functor Coalg(F) : Coalg(V)→
Coalg(W). Moreover, if Vand Ware symmetric monoidal, and Fis symmetric op-monoidal,
then Coalg(F) is a symmetric monoidal functor as well. Applying this in particular to the
strong monoidal functor (id, σ) : V → Vrev , we obtain a strong monoidal functor Coalg(V)→
20 P. GROSSKOPF AND J. VERCRUYSSE
Coalg(V)rev , since indeed Coalg(Vrev) = Coalg(V)rev as monoidal categories. Moreover, us-
ing the symmetry once more, we also have a strong monoidal functor Coalg(V)rev →Coalg(V).
Combing these, we obtain a strong monoidal and involutive autofunctor Coalg(V)→Coalg(V),
which acts as identity on morphisms. The image of a coalgebra (C, ∆, ǫ) under this functor
is called the co-opposite coalgebra of Cand is defined as the coalgebra Ccop having the same
underlying object and same counit, but whose comultiplication is given by
C∆//C⊗Cσ//C⊗C .
By applying the above to base-change from Proposition 1.22 in combination with Proposi-
tion 1.21 and Remark 1.23, we immediately obtain the following.
Proposition 3.6. Let Cbe a symmetric monoidal category, then we have the following invo-
lutive autofunctors
(−)op :V-sHopf → V-sHopf
(−)cop :V-sHopf → V-sHopf
(−)op,cop :V-sHopf → V-sHopf
which send a semi-Hopf V-category respectively to its oppopsite, (locally) coopposite, and
opposite-coopposite semi-Hopf V-category. The functors (−)op and (−)op,cop send morphisms
to their opposite, and the functor (−)cop acts as identity on morphisms.
3.2. Hopf V-categories. Recall from [6] the definition of a Hopf V-category.
Definition 3.7. An antipode for a semi-Hopf V-category Ais a collection of V-morphisms
Sx,y :Axy →Ayx satisfying the following axiom for all x, y ∈A0.
Ax,y ⊗Ax,y
id⊗Sx,y
//Ax,y ⊗Ay,x
mxyx //Axx
Ax,y
ǫx,y //
δxy 99
s
s
s
s
s
s
s
s
s
s
δxy %%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑I
jx
99
s
s
s
s
s
s
s
s
s
s
s
s
jy
%%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑
Ax,y ⊗Ax,y
Sx,y⊗id
//Ay,x ⊗Ax,y
myxy //Ayy
A semi-Hopf V-category that admits an antipode is called a Hopf V-category. We denote by
V-Hopf the category of all Hopf V-categories with semi-Hopf V-functors between them.
Recall from [6] that an antipode for a semi-Hopf category is unique whenever it exists.
Moreover, the antipode defines an “identity on objects” morphism of semi-Hopf categories S:
A→Aop,cop. Furthermore any morphism of (semi-)Hopf categories automatically preserves
the antipode.
The following key-result of this paper shows that in favorable cases, limits and colimits of
Hopf categories can be computed in the category of semi-Hopf categories.
Proposition 3.8. (1) Let Vbe a symmetric monoidal and locally presentable category.
Then V-Hopf is a cocomplete full subcategory of V-sHopf.
(2) Let Vbe a symmetric monoidal, locally presentable category with very flat monoidal
product. Then V-Hopf is a complete full subcategory of V-sHopf.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 21
Proof. By definition, V-Hopf is a full subcategory of V-sHopf. We should show that V-Hopf
is closed in V-sHopf under limits and colimits.
(2). Consider a small category Zand a functor F′:Z → V -Hopf. Let F:Z → V -sHopf
be the composition of F′with the inclusion functor V-Hopf → V-sHopf and consider the limit
lim Fin V-sHopf. We will show that the semi-Hopf V-category lim Fhas an antipode, hence
lim F=lim F′.
Denote lim F=L= (L0, L1) and denote for any object Z∈ Z the canonical projections of
the limit by πZ:L→F Z. Since the functor (−)op,cop :V-sHopf → V-sHopf is an isomorphism
of categories (see Proposition 3.6), it commutes with colimits. Hence, denoting (−)op,c op ◦F=
Fop,cop, we find that lim Fop,cop = (lim F)op,cop , and the projections πop
Z:lim Fop,cop →FZare
the opposite of the projections πZ(i.e. (πop
Z)xy = (πZ)yx). The morphisms SF Z ◦πZ:lim F→
F Zop,cop defines a cone on Fop,cop (in the category of semi-Hopf categories). Therefore we
obtain a unique morphism of semi-Hopf categories S:lim F→lim Fop,cop = (lim F)op,cop such
that πop
Z◦S=SF Z ◦πZ. We claim that Sis an antipode for lim F.
Now fix x, y ∈L0. By Corollary 3.5d, we know that (using the same notation as in
Corollary 3.5) the family of morphisms (π1
yy ⊗ · · · ⊗ πn
yy )◦δn
yy indexed over all n∈Nand
Z1,...,Zn∈ Z is jointly monic in V. Hence, to check the antipode property myxy ◦(Sxy ⊗id)◦
δxy it is enough to check it post-composed with these morphisms. As a diagrammatic proof
would be unreadible, we present the proof of this property by means of Sweedler notation
on generalized elements. To this end consider any morphism a:A→Lxy, where A∈ V is
an arbitrary object and denote δn
x,y ◦a=a(1) ⊗ · · · ⊗ a(n):A→L⊗n
xy . Any permutation of
the indices then corresponds to postcomposing δn
x,y ◦awith the corresponding composition
of symmetry morphisms in V. Then we find
(π1
yy ⊗π2
yy ⊗ · · · ⊗ πn
yy )◦δn
yy ◦myxy ◦(Sxy ⊗id)◦δxy (a)
= (π1
yy ⊗π2
yy ⊗ · · · ⊗ πn
yy )◦δn
yy ◦myxy (Sxy (a(1) )⊗a(2)))
=π1
yy ◦myxy Sxy (a(n))⊗a(n+1) ⊗π2
yy ◦myxy Sxy (a(n−1) )⊗a(n+2))⊗ · · ·
· · · ⊗ πn
yy ◦myxy Sxy (a1)⊗a(2n))
=m1
y1x1y1S1
x1y1(π1
xy(a(n))(1))⊗π1
xy(an)(2)⊗π2
yy ◦myxy S(a(n−1) )⊗a(n+1)⊗ · · ·
· · · ⊗ πn
yy ◦myxy S(a1)⊗a(2n−1)
=j1
y1y1ǫ1
x1y1π1
xy(a(n))⊗π2
yy ◦myxy S(a(n−1))⊗a(n+1)⊗ · · ·
· · · ⊗ πn
yy ◦myxy S(a1)⊗a(2n−1)
=π1◦jyǫxy(a(n))⊗π2
yy ◦myxy S(a(n−1))⊗a(n+1)⊗ · · ·
· · · ⊗ πn
yy ◦myxy S(a1)⊗a(2n−1)
=π1
yy ◦jy⊗π2
yy ◦myxy S(a(n−1))⊗a(n)⊗ · · · ⊗ πn
yy ◦myxy S(a(1))⊗a(2n−2)
=···
=π1
yy ◦jy⊗π2
yy ◦jy⊗ · · · ⊗ πn
yy ◦jyǫxy (a)
= (π1⊗ · · · ⊗ πn)◦δn
yy ◦jyǫxy (a)
dRemark that at this point we use the very flat monoidal product.
22 P. GROSSKOPF AND J. VERCRUYSSE
Here we used in the second equality the compatibility between multiplication (composition)
and comultiplication in L, together with coassociativity and the fact that the antipode mor-
phism Sxy is an anticoalgebra morphism. The third equality follows from the fact that π1
is a semi-Hopf category morphism, combined with the construction of Swhich tells us that
π1
yx ◦Sxy =S1
xy ◦π1
xy. The forth equality is the antipode property for the Hopf category
F Z1. The fifth equality follows from the fact that π1is a morphism of semi-Hopf categories,
hence it commutes with the units jand counits ǫ. The sixth equality follows from the counit
property of the coalgebra Lxy . Then we use an induction step. The last equality follows from
the compatibility between comultiplicaiton and unit in the semi-Hopf category L.
Since (π1
yy ⊗ · · · ⊗ πn
yy )◦δn
yy is a jointly monic family and a:A→Lxy is a jointly epic
family, we can conclude from the above calculation that myxy ◦(Sxy ⊗id)◦δxy =jyǫxy for
all x, y ∈L0. Similarly, one shows that Lsatisfies the right antipode property, hence Lis a
Hopf category.
(1). Consider again a functor F′:Z → V-Hopf and compose it with the forgetful functor
to semi-Hopf categories to obtain the functor F:Z → V-sHopf. As the proof of part (2),
we consider colim Fin V-sHopf and build a candidate antipode on this object. Corollary 3.5
provides a jointly epic family in Von each Hom-object of colim Fe. By a similar computation
as in part (1), one then proves the antipode property by pre-composing it with this jointly
epic family.
Theorem 3.9. Let Vbe a closed symmetric monoidal and locally presentable category with
very flat monoidal product. Then V-Hopf is locally presentable and the fully faithful inclusion
functor V-Hopf → V-sHopf has a left adjoint Hand right adjoint Hc. That is, for any Hopf
category Hand any semi-Hopf category A, we have that
V-sHopf(A, H) = V-Hopf(HA, H)(3)
V-sHopf(H, A) = V-Hopf(H, H cA)(4)
Proof. By Proposition 3.3, the category V-sHopf is locally presentable and since V-Hopf is
a complete and cocomplete full subcategory of V-sHopf by Proposition 3.8, it follows from
Proposition 1.2 that V-Hopf is itself locally presentable. Moreover, as the inclusion functor
V-Hopf → V-sHopf preserves all limits an colimits, it has both a left and right adjoint by
Proposition 1.1.
4. On the construction of free and cofree Hopf categories
In the previous section, we proved the existence of free and co-free Hopf categories. The
aim of this section is to sketch explicitly the construction of the free Hopf category H A and
the cofree Hopf category HcAover a given semi-Hopf category A. As we already observed
before, the free and cofree Hopf categories over Ahas the same set of objects as A, so
HA0=A0=HcA0. The Hom-objects of the free and cofree Hopf categories are described in
the following two subsections.
4.1. The free Hopf category. Let us now describe the construction of Hom-objects for the
free Hopf category, HA1. This construction goes in 3 steps.
eIn this case, the monoidal product is not required to be very flat.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 23
Step 1. Consider any x, y ∈A0. We will transform the coalgebra Axy into a bigger V-
coalgebra, to have enough room to freely contain the image of the antipode. To this end, we
define for any i∈N, a coalgebra
A(i)
xy := Axy if 2 |i
Acop
yx if 2 ∤i
Then we consider the V-coalgebra A′
xy := `i∈NA(i)
xy . In this way, we obtain a new Coalg(V)-Grph
A′= (A0, A′
xy). Moreover, there is an identity-on-objects morphism of Coalg(V)-graphs
ι:A→A′, defined for all x, y ∈A0by the inclusion Axy =A(0)
xy →A′
xy. Furthermore,
A′is naturally equipped with an anti-Coalg(V)-Grph endomorphism s:A′→A′op , induced by
the coalgebra morphisms s(i)
xy :A(i)
xy →(A(i+1)
yx )cop, which are all given by identities. Moreover,
for each even i∈Nand each odd j∈Nwe have a morphism of Coalg(V)-graphs
ι(i):A→A′, ι(i)
xy :Axy →A(i)
xy
ι(j):Aop,cop →A′, ι(j)
xy :Acop
yx →A(j)
xy
Step 2. Now we consider the free semi-Hopf category over A′. That is, we apply the left
adjoint Tto the forgetful functor Coalg(V)-Cat →Coalg(V)-Grph from Proposition 1.18 and
as described in Remark 1.19. Explicitly, we obtain at this stage for any x, y ∈(T A′)0=A0
(T A′)xy =a
n∈N,z1,...,zn∈A0
A′
xz1⊗ · · · ⊗ A′
zny,
adding an additional component of the monoidal unit Ito this coproduct if x=y.
We can then endow T A′with a morphism of semi-Hopf V-categories S′:T A′→(T A′)op,cop.
In each component, S′is given by an anticoalgebra morphism S′
xy : (T A′)xy →(T A′)yx
obtained by reversing the “path” from xto yand applying swz :A′
wz →A′
zw for all pairs
(w, z) of consecutive indices in the path. Since Vis a symmetric monodial category, we
can realize this morphism by a composition a suitable symmetry isomorphism with a tensor
product of swz:
A′
x,x1⊗ · · · ⊗ A′
xn,y ∼
=A′
xn,y ⊗ · · · ⊗ A′
x,x1
sxn,y ⊗···⊗sx,x1//A′
y,xn⊗ · · · ⊗ A′
x1,x
And furthermore S′
xx acts as the identity on the additional component I. Composing the
morphisms ι(i)defined in Step 1 with the canonical morphism of Coalg(V)-graphs A′→T A′
we obtain morphism of Coalg(V)-graphs
ι(i)′:A→T A′, ι(i)′
xy :Axy →T A(i)
xy
ι(j)′:Aop,cop →T A′, ι(j)′
xy :Acop
yx →T A(j)
xy
Step 3. We “factor out” certain relations by means of a suitable multiple coequalizer
q:T A′→HA in Coalg(V)-Grph, such that HA becomes the universal Hopf category we are
looking for. In other words, we want q:T A′→HA to be the “biggest quotient” in the
category of V-categories such that the following conditions hold.
(a) The morphisms q◦ι(i)′are morphisms of V-categories. Then HA naturally also has a
Coalg(V)-graph structure, turning it into a semi-Hopf category, and q◦ι(i)′become mor-
phisms of semi-Hopf categories. Moreover one obtains a morphism S:HA →(HA)op,cop
such that S◦q=qop ◦S′.
24 P. GROSSKOPF AND J. VERCRUYSSE
(b) The morphism Sshould behave as an antipode on H A, turning it into a Hopf V-category.
Although it is quite clear which relations are to be factored out, it is quite involved to verify
in detail that every step is justified and this construction gives indeed the free Hopf category
over A.
From the construction above, one sees that for any x, y ∈A0we have the following jointly
epic family of morphisms in V
A(xz1)⊗ · · · ⊗ A(zny)
ι(i1)
xz1⊗···⊗ι(in+1)
zny//HAxy
indexed by n, i1,...,in+1 ∈Nand z1,...,zn∈A0. We have put the indices in the first tensor
product between brackets, to indicate that their order needs to be reversed whenever the
corresponding ikis odd.
A given morphism of semi-Hopf categories f:A→H, where His a Hopf category, then
induces a unique morphism of Hopf categories ˜
f:HA →Hunder the bijection (3). By the
above jointly epic family, ˜
fis completely determined by means of its pre-composition with
the morphisms in this family. In this way, ˜
fis defined by the following morphisms
A(xz1)⊗ · · · ⊗ A(zny)
f⊗···⊗f//H(xz1)⊗ · · · ⊗ H(zny)
Si1
xz1⊗...⊗Sin+1
zny//Hxz1⊗ · · · ⊗ Hzny
m//Hxy
4.2. The cofree Hopf category. As in the case of the free Hopf category, we construct the
Hom-objects of the cofree Hopf category in three steps.
Step 1. For any x, y ∈A0, we set for any i∈N
A(i)
xy := Axy if 2 |i
Ayx if 2 ∤i
Then define A′
xy =Qi∈NA(i)
xy , which defines a new V-graph A′over A0. Moreover this V-graph
is a V-category by means of the compositions m′
xyz :A′
xy ⊗A′
yz →A′
xz and units j′
xx :I→A′
xx
defined by the commutativity of the following diagrams
A′
xy ⊗A′
yz
m′
xyz //
π(i)
xy ⊗π(i)
yz
A′
xz
π(i)
xz
A(i)
xy ⊗A(i)
yz
m(i)
xyz //A(i)
xz
I
jxx ''
P
P
P
P
P
P
P
P
P
P
P
P
P
j′
xx //A′
xx
π(i)
xx
A(i)
xx =Axx
In the above diagram, the morphism πdenote the obvious projection morphisms from the
product in V,m(i)
xyz is just the composition mxy z of Aif iis even and it is the opposite
composition mzy x ◦σin case iis odd. Moreover consider for each i∈Nand x, y ∈A0the
morphism s(i)
xy :A(i)
xy →A(i+1)
yx given by identity. Then these induce a morphism of V-categories
s:A′→A′op. Moreover, for each even i∈Nand each odd j∈Nwe have a morphism of
V-categories
π(i):A′→A, π(i)
xy :A(i)
xy →Axy
π(j):A′→Aop, π(j)
xy :A(j)
xy →Ayx
which are defined as the identity in each component.
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 25
Step 2. Now we take the cofree semi-Hopf V-category over the V-category A′. In other
words, we replace each Hom-object A′
xy by the cofree coalgebra over it TcA′
xy. The morphism
of V-categories sinduces then a morphism of semi-Hopf categories S′:TcA′→(TcA′)op,cop .
Precomposing the morphisms π(i)from Step 1 with the cofree projection morphism p:TcA′→
A, we obtain the following morphisms for all i∈Neven and j∈Nodd
π(i)′:TcA′→A, π(i)′
xy :TcA(i)
xy →Axy
π(j)′:TcA′→Aop, π(j)′
xy :TcA(j)
xy →Ayx
Step 3. Finally, following an approach similar to the constructions in Theorem 2.6 and
Corollary 2.7, one constructs the biggest Coalg(V)-subgraph ι:HcA→TcA′such that the
following conditions hold.
(a) Each of the compositions π(i)′◦ιis a morphism of Coalg(V)-graphs. Then Hccan be en-
dowed naturally with a V-category structure, turning it into a semi-Hopf V-category such
that π(i)′◦ιis a morphism of semi-Hopf V-categories. Moreover, we obtain a morphism
of semi-Hopf categories S:HcA→(HcA)op,cop such that ιop ◦S=S′◦ι.
(b) The morphism Sshould behave as an antipode for HcA.
Again, verifying that the above construction indeed yields the cofree Hopf category over Ais
rather involved.
From the above construction, and the construction of the cofree coalgebra as given in
Theorem 2.6, we see that for any pair x, y ∈A0, we have the following jointly monic family
of morphisms in V.
HcAxy
π(i1)
xy ⊗···⊗π(in+1)
xy //A(xy)⊗ · · · ⊗ A(xy)
indexed by n, i1,...,in+1 ∈Nand z1,...,zn∈A0. Again, we have put the indices in latter
tensor product between brackets, to indicate that their order needs to be reversed whenever
the corresponding ikis odd.
Given a morphism of semi-Hopf categories f:H→A, where His a Hopf category then
induces a unique morphism of Hopf categories ˜
f:H→HcAunder the bijection (4). By the
above jointly monic family, ˜
fis completely determined by means of its composition with the
morphisms in this family. In this way, ˜
fis defined by the following morphisms
Hxy
δn
xy //Hxy ⊗ · · · ⊗ Hxy
Si1
xy⊗...⊗Sin+1
xy //H(xy)⊗ · · · ⊗ H(xy )
f⊗···⊗f//A(xy)⊗ · · · ⊗ A(xy)
5. Conclusions and outlook
The results of this paper can now be summarized in the following diagram. The arrows
going from left to right in this diagram are ‘forgetful’ functors, the arrows from right to left
are the free and cofree constructions, so left and right adjoints (as indicated by the adjunction
symbol), which we constructed in this paper. The diagram ‘below’ is the classical one-object
situation which we mentioned in the introduction. The diagram shows how we indeed lifted
this to the multi-object setting, since the vertical arrows indicate the functor that send internal
structures in Vto the one-object versions above. As the diagram commutes, our multi-object
26 P. GROSSKOPF AND J. VERCRUYSSE
construction indeed strictly generalizes the one-object constructions.
V-Hopf
**
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
V-sHopf
jj❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯
jj❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯
⊤
⊤
⊤//
%%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑Coalg(V)-Grph
''
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
oo
Hopf(V)
**
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
❯
OO
V-Cat ⊤//
ee❑
❑
❑❑
❑❑
❑
❑❑
❑
⊤
V-Grph
oo
gg❖❖❖❖❖❖❖❖❖❖❖❖
⊤
Bialg(V)
jj❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯
jj❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯
⊤
⊤
⊤//
OO
%%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑Coalg(V)
OO
''
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
❖
oo
Alg(V)⊤//
OO
ee❑
❑
❑❑
❑❑
❑
❑❑
❑
⊤
V
OO
oo
gg❖❖❖❖❖❖❖❖❖❖❖❖
⊤
Moreover, all functors in the upper part of the above diagram commute with the forgetful func-
tors to Set. Hence, if we consider V-categories, semi-Hopf V-categories of Hopf V-categories
with a fixed set of objects, then these form complete and cocomplete subcategories. This
holds in particular for the one-object case as we remarked just above.
When we consider the particular instance V=Set, we recover another well-known case.
Firstly, observe that since Set is Cartesian, Coalg(Set) = Set. Hence Set-categories and semi-
Hopf Set-categories coincide, and are just small categories. Hopf Set-categories are groupoids.
The right adjoint of the forgetful functor Set-sHopf →Set-Cat associates to category Athe
groupoid of all isomorphims in A. The left adjoint is more involved and constructs the free
groupoid over a category, see e.g. [11].
Another case of particular interest, is given by the semi-Hopf category of algebras, which are
known to be enriched over coalgebras by means of Sweedler’s universal measuring coalgebras
[4,24,26]. Our construction of the (co)free Hopf category over a semi-Hopf category allow
to turn the category of algebras into a Hopf category. This construction is then similar to
what is done in [18] to obtain general quantum symmetry groups of non-commutative spaces.
However, an important remark should be made here. In the Hopf category of algebras which
we would obtain as explained above, the endo-Hom-object of an algebra Awould naturally
act on A. In [18] however, a dual situation is considered, where the Hopf category would
“coact” on A. However, in order to obtain such a universal coacting object, a finiteness
condition on Ais required. Indeed, in [18], Ais supposed to be a graded algebra that is
finite dimensional in each degree. This finiteness condition has been refined in [4]. In fact,
following the constructions of [18] and [4], one would obtain a Hopf opcategory, rather then
a Hopf category. Recall that in a Hopf V-opcategory A, the Hom-objects Axy are algebras in
V, which are endowed with “cocomposition morphisms” ∆xyz :Axz →Axy ⊗Ayz and counits
ǫx:Axx →I, satisfying dual conditions as those of a Hopf-category. The natural question
arises whether the free and cofree construction of the present paper can also be obtained in
the setting of Hopf-opcategories. Let us remark that in [27], that co-categories rather then
opcategories are considered, and these are shown to be comonadic over V-graph and inheriting
locally presentability from V. Whether opcategories and Hopf-opcategories also are locally
FREE AND CO-FREE CONSTRUCTIONS FOR HOPF CATEGORIES 27
presentable remains a question for future investigations and should be studied along with
(Sweedler) duality between Hopf-categories and Hopf-opcategories.
Acknowledgements. The authors would like to thank warmly Christina Vasilakopoulou,
for interesting discussions and useful explanations of locally presentable categories. For this
research, JV would like to thank the FWB (f´ed´eration Wallonie-Bruxelles) for support through
the ARC project “From algebra to combinatorics, and back”. PG thanks the FNRS (Fonds
de la Recherche Scientifique) for support through a FRIA fellowship.
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Paul Großkopf, D´
epartement de Math´
ematiques, Universit´
e Libre de Bruxelles, Belgium
Email address:paul.grosskopf@gmx.at
Joost Vercruysse, D´
epartement de Math´
ematiques, Universit´
e Libre de Bruxelles, Bel-
gium
Email address:joost.vercruysse@ulb.be
