Semi-abelian condition for color Hopf algebras

Recently it was shown that the category of cocommutative Hopf algebras over an arbitrary field $\Bbbk$ is semi-abelian. We extend this result to the category of cocommutative color Hopf algebras, i.e. of cocommutative Hopf monoids in the symmetric monoidal category of $G$-graded vector spaces with $G$ an abelian group, given an arbitrary skew-symmetric bicharacter on $G$, when $G$ is finitely generated and the characteristic of $\Bbbk$ is different from 2 (not needed if $G$ is finite of odd cardinality). We also prove that this category is action representable and locally algebraically cartesian closed, then algebraically coherent. In particular, these results hold for the category of cocommutative super Hopf algebras by taking $G=\mathbb{Z}_{2}$. Furthermore, we prove that, under the same assumptions on $G$ and $\Bbbk$, the abelian category of abelian objects in the category of cocommutative color Hopf algebras is given by those cocommutative color Hopf algebras which are also commutative.

The notion of semi-abelian category was introduced by G. Janelidze et al. in [25] in order to capture typical algebraic properties valid for groups, rings and algebras.As it is said in [25], semiabelian categories provide a good categorical foundation for a meaningful treatment of radical and commutator theory and of (co)homology theory of non-abelian structures.Semi-abelian categories are rich in properties, for instance here the notions of semi-direct product, internal action and of crossed module are natural.Some examples of semi-abelian categories are the categories of groups, Lie algebras, (associative) rings and compact groups.In [21] M. Gran et al. proved that the category of cocommutative Hopf algebras over a field k, denoted by Hopf k,coc , is semi-abelian when k has characteristic 0. Then the result was extended to arbitrary characteristic in [22].Hence it becomes natural to ask if this is true also for the category of cocommutative color Hopf algebras, i.e. of cocommutative Hopf monoids in the category Vec G of G-graded vector spaces, which we denote by Hopf coc (Vec G ), where G is an abelian group since we know that, in this case, Vec G becomes a symmetric monoidal category by using a skew-symmetric bicharacter on G which modifies the braiding of Vec k given by the usual tensor flip.We show that Hopf coc (Vec G ) is semi-abelian if the abelian group G is finitely generated and the characteristic of the field k is not 2 (not needed if G is finite of odd cardinality).This generalizes the result for ordinary cocommutative Hopf algebras since we can recover Hopf k,coc by taking G as the trivial group in which case the symmetric monoidal category Vec G is exactly Vec k .Furthermore, if we consider G = Z 2 we obtain that the category of cocommutative super Hopf algebras, extensively used in Mathematics and Physics, is semi-abelian if chark = 2.
The organization of the paper is the following.After calling back some basic notions and results about monoidal categories and (color) Hopf algebras, we prove the completeness and the cocompleteness of Hopf coc (Vec G ) by showing explicitly limits and colimits and the protomodularity of Hopf coc (Vec G ) by using a categorical result.We also observe that Hopf coc (Vec G ) is locally presentable, which is not guaranteed in general for the category of (cocommutative) Hopf monoids in a symmetric monoidal category.Then we show the regularity of Hopf coc (Vec G ) through the same steps of [22].In particular, we obtain a generalization of a theorem by K. Newman [30,Theorem 4.1] for cocommutative color Hopf algebras in the case chark = 2 and the abelian group G is finitely generated, by using [27,Theorem 3.10 (3)] about cocommutative super Hopf algebras together with a braided strong monoidal functor from the category Vec G to the category Vec Z2 from [7].Then, through an equivalent characterization given in [25], we obtain that Hopf coc (Vec G ) is semi-abelian, still in case the abelian group G is finitely generated and chark = 2. Finally, we also show that, under the same assumptions on G and k, the category Hopf coc (Vec G ) is action representable and locally algebraically cartesian closed (then algebraically coherent) and that the category of abelian objects in Hopf coc (Vec G ) consists of those cocommutative color Hopf algebras which are also commutative and then, as a consequence, this category is abelian.

Preliminaries
2.1.Monoidal categories.First we recall some basic facts about monoidal categories, which can be found in [31,4].Let (M, ⊗, I, a, l, r) be a monoidal category.We write (M, ⊗, I) without the constraints a, l and r if these are clear from the context and we usually omit to write a in the computations since it will be clear when it is needed, in order to have slightly more compact formulas.We know that we can consider the category Mon(M) of monoids in M, whose objects will be denoted as (A, m, u), and the category Comon(M) of comonoids in M, whose objects will be denoted as (C, ∆, ǫ).Recall that a monoid M ′ is a submonoid of a monoid M , provided there exists a monoid morphism i : M ′ → M such that it is a monomorphism in M. Analogously a comonoid C ′ is a subcomonoid of a comonoid C, provided there exists a comonoid morphism i : C ′ → C such that it is a monomorphism in M. In case M has a braiding c the categories Mon(M) and Comon(M) become monoidal with the same constraints a, l, r.In this case, given monoids (M 1 , m 1 , u 1 ) and (M 2 , m 2 , u 2 ) in M, the tensor product ⊗ is such that we have (M 1 , m 1 , u 1 ) ⊗ (M 2 , m 2 , u 2 ) := (M 1 ⊗ M 2 , m, u) where The unit object of Mon(M) is given by (I, r I , Id I ).Similarly, given comonoids (C 1 , ∆ 1 , ǫ 1 ) and (C 2 , ∆ 2 , ǫ 2 ) in M, (C 1 , ∆ 1 , ǫ 1 ) ⊗ (C 2 , ∆ 2 , ǫ 2 ) := (C 1 ⊗ C 2 , ∆, ǫ) is a comonoid where The unit object of Comon(M) is given by (I, r −1 I , Id I ).When Mon(M) and Comon(M) are monoidal we can consider monoids and comonoids in them.Hence we have that (1) Bimon(M) ∼ = Mon(Comon(M)) ∼ = Comon(Mon(M)) where Bimon(M) is the category of bimonoids in M, since for (B, m, u, ∆, ǫ) the fact that m and the u are morphisms of comonoids is equivalent to that ∆ and ǫ are morphisms of monoids (see e.g.[4,Proposition 1.11]) while which are the category of commutative monoids and of cocommutative comonoids in M respectively and this follows from the Eckmann-Hilton argument: ∆ C is a morphism of comonoids if and only if C is cocommutative and m A is a morphism of monoids if and only if A is commutative (see e.g.[4, Section 1.2.7]).We recall that a monoid (A, m, u) Also recall that a bimonoid B ′ is a sub-bimonoid of a bimonoid B, provided there exists a bimonoid morphism i : B ′ → B such that it is a monomorphism in M. Given (C, ∆, ǫ) ∈ Comon(M) and (A, m, u) ∈ Mon(M), Hom M (C, A) is an (ordinary) monoid with convolution product such that, given f, g : C → A in M, the product is f * g := m • (f ⊗ g) • ∆ and the unit is u • ǫ.Hence we can consider the category Hopf(M) of Hopf monoids in M, whose objects are bimonoids B in M equipped with a morphism S : B → B (antipode) which is the convolution inverse of Id B .The monoidal categories Mon(M) and Comon(M) may fail to be braided and then the categories Hopf(M), Bimon(M), Mon c (M) and Comon coc (M) may fail to be monoidal but, when the braided category M is symmetric, i.e. c −1 X,Y = c Y,X for every X and Y in M, these categories are all braided and symmetric with the same braiding c and the same constraints a, l, r of M (see [4,Section 1.2.7]).Indeed, if M is symmetric, given A and B monoids in M, then c A,B : A ⊗ B → B ⊗ A is a morphism of monoids and then Mon(M) is a symmetric monoidal category and, dually, Comon(M) is a symmetric monoidal category.Iterating these results and applying (1) and (2), one can deduce that Bimon(M), Mon c (M) and Comon coc (M) are symmetric monoidal categories as well.Furthermore, if M is symmetric, given (B, S B ) and (B ′ , S B ′ ) in Hopf(M) we have that (B, S B ) ⊗ (B ′ , S B ′ ) := (B ⊗ B ′ , S B ⊗ S B ′ ) is in Hopf(M).The antipode is a bimonoid morphism S : B → B op,cop where (B op,cop , m op , u, ∆ cop , ǫ) is a bimonoid with Since we use several times these facts in the following and, in particular, the fact that Comon coc (M) is a monoidal category is central for our proof of protomodularity, then we will work with a symmetric monoidal category.
Finally, recall that, given monoidal categories (M, ⊗, I, a, l, r) and (M ′ , ⊗, I ′ , a ′ , l ′ , r ′ ) (where we do not use different notations for ⊗ for notation convenience), a monoidal functor (F, φ 0 , φ 2 ) : X,Y are isomorphisms for every X, Y in M and strict if φ 0 and φ 2 X,Y are identities for every X, Y in M. If M and M ′ are (symmetric) braided with braidings c and c ′ respectively, If M is the category Vec k of vector spaces over a field k, we have the usual notions of k-algebras, k-coalgebras, k-bialgebras and k-Hopf algebras, usually denoted without k.In the following we always omit k but it will be understood.For classical results and notions about the theory of Hopf algebras we refer to [35] and [37].

Semi-abelian categories.
Here we recall some definitions needed for the notion of semiabelian category.For the notions of limits and colimits of a functor, as for other basic notions of category theory, we refer to [10,26].
A finitely complete category C is regular if any arrow of C factors as a regular epimorphism (i.e. the coequalizer of a pair of morphisms of C) followed by a monomorphism and if, moreover, regular epimorphisms are stable under pullbacks along any morphism.A relation on an object X of C is an equivalence class of triples (R, r 1 , r 2 ), where R is an object of C and r 1 , r 2 : R → X is a pair of jointly monic morphisms of C, and two triples (R, r 1 , r 2 ) and (R ′ , r ′ 1 , r ′ 2 ) are identified when they both factor through each other.An equivalence relation in C is a relation R on an object X which is reflexive, symmetric and transitive.A regular category C is (Barr)-exact if any equivalence relation R in C is effective, i.e. it is the kernel pair of some morphism in C. Recall also that a category C is protomodular, in the sense of [11], if it has pullbacks of split epimorphisms along any morphism and all the inverse image functors of the fibration of points reflect isomorphisms.We know that, as it is said for instance in [11,Proposition 3.1.2],if C is pointed (i.e. it has a zero object) and finitely complete, the protomodularity can be expressed by simply asking that the Split Short Five Lemma holds in C. Finally, a category C is semi-abelian if it is pointed, finitely cocomplete, (Barr)-exact and protomodular.Many details and properties about semi-abelian categories can be found in [11].

Color Hopf algebras
In this section we recall what color Hopf algebras are and how they differ from common Hopf algebras.We consider the category Vec G of G-graded vector spaces over an arbitrary field k where G is a group.We add conditions on the group G along the way, to make it clear why these are needed.Objects in Vec G are vector spaces V = g∈G V g where V g is a vector subspace of V for every g ∈ G and the morphisms in Vec G are linear maps f : V → W which preserve gradings, i.e. such that f (V g ) ⊆ W g for every g ∈ G.We know that this category is monoidal with ⊗ the tensor product of Vec k and unit object k ).Also the associativity constraint and left and right unit constraints are the usual ones of Vec k .Remark 3.1.Remember that the category Vec G is isomorphic to the category kG M of left comodules over the group algebra kG with isomorphism given by F : Grothendieck category and then abelian, since this is true in general for C M (and M C ) with C a coalgebra, not always true in general for a coalgebra over a ring (see e.g.[16, 3.13]).So here monomorphisms are exactly the injective maps and epimorphisms the surjective maps.Observe that, given a graded vector space V = g∈G V g and a vector subspace V ′ ⊆ V , we can always consider the graded vector space g∈G V ′ ∩ V g ⊆ V ′ .Furthermore, V ′ is a graded subspace of V if it is a graded vector space such that the inclusion i : V ′ → V is in Vec G and this happens if and only if for every x ∈ V ′ , which is x = g∈G x g with x g ∈ V g , then x g ∈ V ′ for any g ∈ G; in this case V ′ has the induced grading V ′ = g∈G V ′ g where V ′ g = V ′ ∩ V g .Furthermore, we can always consider the graded vector space g∈G V g /(V g ∩ V ′ ) and there is a canonical isomorphism of vector spaces g∈G Vg In this case, we can also consider g∈G (V g + V ′ )/V ′ , where and it is called quotient graded vector space.Remark 3.2.We recall that if f : A → B is in Vec G then ker(f ) and Im(f ) are graded subspaces of A and B respectively.If f is surjective, the grading of B = f (A) is the unique induced by A through f , i.e.B g = f (A g ) for every g ∈ G.
3.1.Graded (co)algebras.The objects of the categories Mon(Vec G ) and Comon(Vec G ) are called G-graded algebras and G-graded coalgebras respectively, which we usually call without G.Many details and properties about graded algebras and graded coalgebras can be found in [28,29].Note that graded algebras and graded coalgebras are often used to denote algebras and coalgebras graded over N, while here gradings will be always over G.
A graded algebra is an algebra (A, m, u) where A = g∈G A g is a graded vector space such that m and u preserve gradings, i.e. for every h, k ∈ G we have A h A k ⊆ A hk and u(k) ⊆ A 1G and a morphism of graded algebras is a morphism of algebras that preserves gradings.Since monomorphisms in Vec G are exactly the injective maps, a submonoid of a graded algebra (A, m, u), called graded subalgebra, is a graded subspace V ⊆ A such that 1 A ∈ V and m(V ⊗V ) ⊆ V .Indeed, in this case, V is a graded vector space with V g = V ∩ A g for every g ∈ G, an algebra and for every g, h ∈ G. Furthermore, if we consider a graded two-sided ideal I of A such that A/I = g∈G (A g + I)/I is a graded vector space, we know that (A/I, u A/I , m A/I ) is an algebra with u A/I = π • u A and m A/I • (π ⊗ π) = π • m A where π : A → A/I is the canonical projection and it is graded since u A/I and m A/I are in Vec G with π, u A and m A in Vec G ; it is called quotient graded algebra.
Similarly, a graded coalgebra is a coalgebra (C, ∆, ǫ) where C = g∈G C g is a graded vector space such that ∆ and ǫ preserve gradings, i.e. ∆(C g ) ⊆ h∈G (C h ⊗ C h −1 g ) and ǫ(C g ) ⊆ δ g,1G k for every g ∈ G and a morphism of graded coalgebras is a morphism of coalgebras that preserves gradings.A subcomonoid of a graded coalgebra (C, ∆, ǫ), called graded subcoalgebra, is a graded vector subspace V ⊆ C such that ∆(V ) ⊆ V ⊗ V (ǫ(V ) ⊆ k is automatic).Indeed, in this case, V is a graded vector space, a coalgebra and for every g ∈ G.In fact observe that, since V is a graded subspace of C, then V ⊗ V is a graded subspace of C ⊗ C. If I is a graded two-sided coideal of C, then C/I is a graded vector space and it is a coalgebra with ∆ C/I • π = (π ⊗ π) • ∆ C and ǫ C/I • π = ǫ C , where π : C → C/I is the canonical projection.Thus C/I is a graded coalgebra because ∆ C/I and ǫ C/I clearly preserve gradings since ǫ C , ∆ C and π are in Vec G and it is called quotient graded coalgebra.

3.2.
Color bialgebras and color Hopf algebras.We are interested in studying Hopf monoids in Vec G but, in order to do this, first we need that Vec G is braided.One can give to Vec G a braiding by using a bicharacter φ on G (see for example [7]), i.e. a map φ : G × G → k − {0} such that φ(gh, l) = φ(g, l)φ(h, l) and φ(g, hl) = φ(g, h)φ(g, l) for every g, h, l ∈ G.
Y h and g, h ∈ G, defined on the components of the grading and extended by linearity, for every X and Y in Vec G .In order to obtain that the braiding is in Vec G , the group G needs to be abelian as it is said in [7, Section 1.1] or in [5, pag. 193].Hence, from now on, we will always consider G an abelian group.As we said before we also want that the category Vec G is symmetric and then we have to require that φ is a commutation factor on G that is a skew-symmetric bicharacter on G, i.e. that φ satisfies further φ(g, h)φ(h, g) = 1 k for g, h ∈ G.We will usually work on the components of the grading and all maps will be understood to be extended by linearity.For the braiding we use the same notation of [7] and we write c(x ⊗ y) = φ(|x|, |y|)y ⊗ x with x ∈ X and y ∈ Y , intending to work on homogeneous components and extend by linearity.Note that, given X, Y and Z in Vec G , the hexagon relations Also note that, if X and Y are graded coalgebras, then The objects of the categories Bimon(Vec G ) and Hopf(Vec G ) are called color bialgebras and color Hopf algebras, respectively.A color bialgebra is a datum (B, m, u, ∆, ǫ) where (B, m, u) is a graded algebra, (B, ∆, ǫ) is a graded coalgebra, and the two structures are compatible in the sense that ∆ and ǫ are graded algebra maps or, equivalently, m and u are graded coalgebra maps.Hence B = g∈G B g is an ordinary algebra and an ordinary coalgebra with m, u, ∆, ǫ which preserve gradings, but the condition of compatibility between the two structures differs from that in Bialg k , only for the part that involves the braiding.So we have that for every a, b ∈ B. A morphism of color bialgebras is a morphism of algebras and of coalgebras which preserves gradings.Given a color bialgebra B, a sub-bimonoid B ′ ⊆ B, called color subbialgebra, will be a graded subalgebra which is also a graded subcoalgebra (the compatibility between the two structures is that of B).Furthermore, given a color bialgebra B and a graded biideal I (which is a two-sided ideal and two-sided coideal) we know that B/I is a graded algebra and a graded coalgebra and we show that the compatibility between the two structures is automatically maintained.In fact, given π : B → B/I the canonical projection, we have that since c is natural and B is a color bialgebra.Now, since π ⊗ π is surjective, we have that hence B/I is a color bialgebra, called quotient color bialgebra.
Given (C, ∆, ǫ) ∈ Comon(Vec G ) and (A, m, u) ∈ Mon(Vec G ), we have the convolution product of two morphisms f, g : A morphism of color Hopf algebras is simply a morphism of color bialgebras since the compatibility with antipodes is automatically guaranteed (see e.g.[4,Propositon 1.16]).Given a color Hopf algebra H, a color Hopf subalgebra H ′ ⊆ H will be simply a color sub-bialgebra such that S H (H ′ ) ⊆ H ′ .Furthermore, given a graded bi-ideal I such that S H (I) ⊆ I, there is a unique linear map S H/I : H/I → H/I such that S H/I • π = π • S H which preserves gradings since S H and π do.This is clearly the antipode of H/I (which is a color bialgebra), in fact as usual and from the surjectivity of π we obtain m H/I • (S H/I ⊗ Id H/I ) • ∆ H/I = u H/I • ǫ H/I .Analogously for the other equality, so H/I is a color Hopf algebra, called quotient color Hopf algebra.Observe that the properties of the antipode S of a color Hopf algebra H on elements x, y ∈ H are: If H is commutative then S(xy) = S(x)S(y) and S 2 = Id H and if H is cocommutative then ∆(S(x)) = S(x 1 ) ⊗ S(x 2 ) and S 2 = Id H .
Clearly the category of Vec k is exactly Vec G with G = {1 G } the trivial group.Hence, motivated by the fact that Hopf k,coc is a semi-abelian category ([22, Theorem 2.10]), our question is now to establish whether the category Hopf coc (Vec G ) is semi-abelian.

Limits, Colimits and Protomodularity of Hopf coc (Vec G )
In this section we show that Hopf coc (Vec G ) is pointed, finitely complete, cocomplete and protomodular.Clearly k with trivial grading is in Hopf coc (Vec G ) and it is a zero object of the category.In fact, given H in Hopf coc (Vec G ), we have that ǫ is the unique morphism of coalgebras from H to k and it is also of algebras and preserving gradings.Similarly u is the unique morphism of algebras from k to H, also of coalgebras and preserving gradings.Hence k is a terminal and initial object in Hopf coc (Vec G ), so a zero object and Hopf coc (Vec G ) is pointed.Note that this is true also for Hopf(Vec G ) and Bimon(Vec G ) while, with the same reasoning, k is initial in Mon(Vec G ) and terminal in Comon(Vec G ). Now we show the finite completeness of Hopf coc (Vec G ), by constructing equalizers and binary products and by using [10,Proposition 2.8.2].Note that these limits have the same form, as vector spaces, of those of Hopf k,coc , given for instance in [38] (see also [2]).The constructions given for Hopf k,coc fit with this more general context and the naturality of the braiding or the fact that the category is symmetric is often required to check what appears immediate in the Hopf k,coc case.
Since we have not seen these computations in literature for Hopf coc (Vec G ), we give the explicit constructions of these limits, also because they will be used in the following.
Remark 4.1.Recall that, given a color Hopf algebra A and a graded subspace Observe that, as vector space, K = ker(( are morphisms of graded algebras.Indeed, given x, y ∈ K, we have that denoting by • the multiplication m A⊗B , hence K is closed under m A .Furthermore, since by cocommutativity of A we have that since φ is a commutation factor, thus we only have to show by coassociativity and so we have to prove which is exactly (Id A ⊗ g)∆(S A (x)).Thus we have that K is in Hopf coc (Vec G ). Now, since the inclusion i : K → A is in Vec G and it is a morphism of algebras and coalgebras, we obtain that (K, i) is the equalizer in Hopf coc (Vec G ) of the pair (f, g).In fact, with We denote the equalizer of the pair (f, g) in Hopf coc (Vec G ) by (Eq(f, g), i).

Binary Products. If we take
In particular (A ⊗ B, m, u, ∆, ǫ, S) is a cocommutative color Hopf algebra since Hopf(Vec G ) and Comon coc (Vec G ) have a monoidal structure with Vec G symmetric and we recall that m ) and S = S A ⊗S B .Furthermore π A and π B are algebra maps and coalgebra maps and they preserve gradings since this is true for r A , l B and ǫ A , ǫ B and then they are morphisms in Hopf coc (Vec G ).We only have to prove that, for every H in Hopf coc (Vec G ), we have a bijection between the set of morphisms in Hopf coc (Vec G ) from H to A ⊗ B and the cartesian product of the set of morphisms from H to A and that of morphisms from H to B in Hopf coc (Vec G ).Given a map f : we can consider the morphism (g ⊗ h) • ∆ H ; this map will be the diagonal morphism of the pair (g, h), usually denoted by g, h .It is in Hopf coc (Vec G ) since ∆ H is a morphism of coalgebras with H cocommutative (and only in this case).Hence it is clear that this construction is specific for the cocommutative case.Clearly, given g : H → A and h : H → B, we have where we only use the naturality of c and the fact that c k,k = Id k,k .Hence (A ⊗ B, π A , π B ) is the binary product of A and B in Hopf coc (Vec G ) and we denote the object by A × B.
We have obtained that Hopf coc (Vec G ) is finitely complete and now we show the cocompleteness.To this aim we prove more generally the cocompleteness of Hopf(Vec G ) by constructing coequalizers and arbitrary coproducts and that the colimits are the same in the cocommutative case.As for limits, also colimits have the same form as vector spaces of those in Hopf k , which are reported for instance in [3].
Remark 4.2.The fact that colimits are the same in the cocommutative case should not surprise us.In fact we recall that, given a symmetric monoidal category M, the forgetful functor U a : Mon(M) → M creates limits and the forgetful functor U c : Comon(M) → M creates colimits and then Mon(M) is closed under limits in M as Comon(M) is closed under colimits in M (see e.g.[31,Fact 10], [32,Fact 4]).Hence also Mon c (M) is closed under limits in Mon(M) and Comon coc (M) is closed under colimits in Comon(M).Furthermore Bimon coc (M) = Comon(Comon(Mon(M))), so Bimon coc (M) is closed under colimits in Bimon(M).We will see that colimits in Hopf(Vec G ) are the same of those in Bimon(Vec G ) and then clearly Hopf coc (Vec G ) is closed under colimits in Hopf(Vec G ). Observe also that Bimon(Vec G )=Comon(Mon(Vec G )) is closed under colimits in Mon(Vec G ) and then colimits in Hopf(Vec G ) will derive from those of Mon(Vec G ). However we show all the details in the sequel.
Remark 4.3.Recall that, given a color Hopf algebra H and a graded bi-ideal I such that S(I) ⊆ I, then H/I is a color Hopf algebra.Observe also that if H is (co)commutative then also H/I is (co)commutative.Indeed, for instance, if H is cocommutative, by naturality of c, we have that Thus, in order to prove that B/I is a color Hopf algebra, we only have to check that I is a two-sided coideal and that S(I) ⊆ I, by Remark 4.3.Given a ∈ A, since f and g are morphisms of coalgebras, we obtain and from this, using that ∆ is a morphism of algebras and that B is a color bialgebra, we have that and then ∆(I) ⊆ I ⊗ B + B ⊗ I. Furthermore ǫ(I) = 0 since ǫ is a morphism of algebras and thus I is a two-sided coideal.Furthermore we have that Hence B/I is a color Hopf algebra and π : we have that I ⊆ ker(h) and then there exists a unique morphism of coalgebras h ′ : B/I → H such that h ′ • π = h which is also of algebras and preserving grading since this is true for π and h, hence it is the unique morphism in Hopf(Vec G ) such that h ′ • π = h.Thus (B/I, π) is the coequalizer in Hopf(Vec G ) of the pair (f, g), which we denote by (Coeq(f, g), π).Observe here that, clearly, this is also the coequalizer for f, g in Hopf coc (Vec G ) since if B is cocommutative also B/I is cocommutative, as said in Remark 4.3.
Remark 4.4.We know that, given for every g ∈ G and then we have so T (V ) is graded as vector space.But T (V ) is also an algebra and it is graded since Coproducts.Let {H l } l∈I be a family of color Hopf algebras, we can take T ( l∈I H l )/L where L is the two-sided ideal in T ( l∈I H l ) generated by the linear span of the set where j t : H t → l∈I H l sends v to the element with v as t-component, the only one not trivial and i : l∈I H l → T ( l∈I H l ) is the canonical inclusion.Now, since H l = g∈G H l,g for every l ∈ I, we have that l∈I H l = l∈I g∈G H l,g = g∈G l∈I H l,g , then l∈I H l is in Vec G and so T ( l∈I H l ) is a graded algebra by Remark 4.4.But now, clearly, L is graded since it is generated by homogeneous elements; indeed i, j l , m l and m T ( H l ) are in Vec G , for every l ∈ I. Thus also T ( l∈I H l )/L is a graded algebra.For all l ∈ I define q l := ν • i • j l , where ν : T ( l∈I H l ) → T ( l∈I H l )/L is the canonical projection.Then q l is a morphism of algebras for every l ∈ I by the relations of J and since ν is an algebra map and it preserves gradings since this is true for the three maps.Now, given a graded algebra C and graded algebra morphisms g l : H l → C for l ∈ I, there exists a unique linear map k : i∈I H i → C such that k • j l = g l for every l ∈ I by the universal property of the coproduct of vector spaces and k also preserves gradings since j l and g l do (it is the universal property of the coproduct in Vec G ).By the universal property of the tensor algebra, there is a unique algebra map s : T ( l∈I H l ) → C such that s • i = k and s also preserves gradings since i and k do.Finally, we have ) since g l and s are algebra maps, for every l ∈ I.So L ⊆ ker(s) and then there exists a unique algebra map p : T ( l∈I H l )/L → C such that p • ν = s which preserves gradings since s and ν do.We have that p • q l = g l and this morphism p is the unique in Mon(Vec G ) such that p • q l = g l for every l ∈ I. Indeed, if there is a morphism p : s and hence p = p.We have shown that (T ( l∈I H l )/L, (q l ) l∈I ) is the coproduct of the family {H l } l∈I in Mon(Vec G ), and we denote T ( l∈I H l )/L by l∈I H l .Now, since H l is a color bialgebra for every l ∈ I, we can show that l∈I H l is a color bialgebra and that it is the coproduct of the family {H l } l∈I in Bimon(Vec G ).The comultiplication and the counit are given by the unique graded algebra maps such that the following diagrams commute (4) by the universal property of the coproduct in Mon(Vec G ). Thus we already have the compatibility and, if we prove that ∆ is coassociative and counitary we will have that ∆ and ǫ make i∈I H i a color bialgebra and the two commutative diagrams (4) will prove that q l is a coalgebra map for every l ∈ I and then a color bialgebra map.In order to obtain is the canonical isomorphism.So, having in mind the two diagrams in (4) and the fact that H l is a coalgebra for l ∈ I, we obtain that Hence l∈I H l is a color bialgebra and q l is a color bialgebra map for every l ∈ I. Now, given a color bialgebra C and color bialgebra maps g l : H l → C, we have a unique graded algebra map p : i∈I H i → C such that p • q l = g l for every l ∈ I by the universal property of the coproduct in Mon(Vec G ).We show that p is also a coalgebra map in order to obtain that ( l∈I H l , (q l ) l∈I ) is the coproduct in Bimon(Vec G ) of the family {H l } l∈I .By the argument used above, it is enough to show that (p ⊗ p) So, since g l is a coalgebra map for every l ∈ I, we have that Now we let H := i∈I H i .Every H l has an antipode S l : H l → H l which is a color bialgebra map from H l to H op,cop l where x• op y : for every x, y ∈ H l .Since q l is a color bialgebra map from H op,cop l to H op,cop , the universal property of the coproduct in Bimon(Vec G ) yields a unique color bialgebra map S : H → H op,cop such that the following diagram commutes for all l ∈ I.
If we prove that S is the antipode of H, then H is a color Hopf algebra and q l is a morphism of color Hopf algebras for every l ∈ I. Furthermore, given C a color Hopf algebra and g l : H l → C a color Hopf algebra map for every l ∈ I, there is a unique color bialgebra map t : H → C (a posteriori the unique color Hopf algebra map) such that t • q l = g l for every l ∈ I. Hence, in this case, (H, (q l ) l∈I ) is the coproduct in Hopf(Vec G ) of the family of color Hopf algebras {H l } l∈I .Thus, in order to conclude, we prove that m cop is a color bialgebra map we only need to prove these on the generators of H as a graded algebra.Indeed, let h, k be generators in H for which the relations hold, we obtain and similarly m(S ⊗ Id)∆(hk) = uǫ(hk), so the relations hold for hk and thus for all the elements in H. So, having in mind that H := T ( l∈I H l )/L, we only need to prove the relations for the elements x = i(x) + L ∈ H with x ∈ l∈I H l whereas the tensor algebra T ( l∈I H l ) is the free algebra on l∈I H l .Moreover, since elements x ∈ l∈I H l are such that x l = 0 for every l ∈ I except for a finite number, by linearity it is enough to show that the relations hold for every x l ∈ H l with l ∈ I. Using the commutativity of the three diagrams before, the fact that H l is a color Hopf algebra and that q l is an algebra map for l ∈ I, we obtain In the same way it can be shown that m•(S ⊗Id)•∆ = u•ǫ.Thus S is the antipode of H and then (H, (q l ) l∈I ) is the coproduct of the family {H l } l∈I in Hopf(Vec G ).It is clear that if we consider H l a cocommutative color Hopf algebra for every l ∈ I then (H, (q l ) l∈I ) will be the coproduct in Hopf coc (Vec G ), since H is cocommutative.In fact, since Vec G is symmetric, c is a braiding for Mon(Vec G ) and, in particular, c H,H is a morphism of graded algebras, so the same is true for c H,H • ∆.Thus from we obtain that c H,H • ∆ = ∆ by universal property of the coproduct.
Hence we have obtained that Hopf coc (Vec G ) (and also Hopf(Vec G )) is cocomplete.Note that, even if only finite cocompleteness is required in the definition of a semi-abelian category, the fact that this category has all small colimits will be used to obtain that it is semi-abelian, through an equivalent characterization.
Remark 4.5.Note that in [33, Proposition 4.1.1]it has been proven that Hopf(M), Hopf coc (M) and Hopf c (M) are always accessible categories for every symmetric monoidal category M. Hence we have that Hopf coc (Vec G ) is accessible and then, since we have shown that it is cocomplete, we obtain that it is complete and locally presentable.In fact we know that, as reported in [1, Corollary 2.47], a category is locally presentable if and only if it is accessible and complete if and only if it is accessible and cocomplete.Observe that, while accessibility is always true for the category of Hopf monoids in a symmetric monoidal category, this is not the same for local presentability.As it is said in [31,Propositions 49,52,53] this is true when the forgetful functor U a : Mon(M) → M is an extremally monadic functor or when the forgetful functor U c : Comon(M) → M is an extremally comonadic functor, since in these cases we have that the category Hopf(M) is closed under colimits and limits in Bimon(M), respectively.4.5.Protomodularity.Recall that if M is a category with binary products, i.e. there exists the binary product A × B for every objects A and B in M, and with terminal object I, the monoidal category (M, ×, I) is called cartesian and the category of internal groups in M, denoted by Grp(M), has objects which are monoids (G, m, u) in M equipped with a morphism i : where t G is the unique morphism from G to I and Id G , i , i, Id G are the diagonal morphisms.
In [14, Propositon 3.24] it is proved that, given a cartesian monoidal category M with finite limits, then the category Grp(M) is protomodular.Note that the same terminal object, equalizers and binary products given before say that Comon coc (Vec G ) is finitely complete.This category is also cartesian since its unit object k is the terminal object and the tensor product is the binary product and then we have that Grp(Comon coc (Vec G )) is protomodular.Furthermore, as it is said for instance in [33,Remark 3.3], for every symmetric monoidal category M we have that Hopf coc (M)=Grp(Comon coc (M)) and then Hopf coc (Vec G ) = Grp(Comon coc (Vec G )).This is easy to show indeed so monoids in Comon coc (Vec G ) are given by cocommutative color bialgebras.Hence an object in Grp(Comon coc (Vec G )) is a cocommutative color bialgebra (B, m, u, ∆, ǫ) equipped with a mor- so i is the antipode of B. Hence we have that Grp(Comon coc (Vec G )) is exactly the category Hopf coc (Vec G ). Thus we have that Hopf coc (Vec G ) is protomodular.Recall that here the fact that Vec G is symmetric ensures that Comon coc (Vec G ) is monoidal.

Regularity of Hopf coc (Vec G )
The most delicate point is the regularity, as in the case of Hopf k,coc .Following [22] the regularity will be shown through the following characterization: Lemma 5.1.Let C be a finitely complete category.Then C is a regular category if and only if (1) any arrow in C factors as a regular epimorphism followed by a monomorphism; (2) given any regular epimorphism f : A → B in C and any object E in C, the induced arrow (3) regular epimorphisms are stable under pullbacks along split monomorphisms.
Since the zero morphism in Hopf coc (Vec G ) between A and B is u B • ǫ A , the categorical kernel of f : A → B in Hopf coc (Vec G ), i.e. the equalizer of the pair (f, u B • ǫ A ), is given by (Hker(f ), j : Hker(f ) → A) with and j is the canonical inclusion, while the categorical cokernel of f in Hopf coc (Vec G ), i.e. the coequalizer of the pair (f, u B • ǫ A ), is given by (B/I, π : B → B/I) where and π is the canonical projection and where, for any coalgebra C, we write and the other inclusion is trivial.Now, given f : A → B in Hopf coc (Vec G ), we can consider the categorical cokernel of its categorical kernel in Hopf coc (Vec G ) that is given, as a map, by p : A → A/A(Hker(f )) + A. Since j is the kernel of f we have that f , thus, by the universal property of the cokernel, there exists a unique morphism i in Hopf coc (Vec G ) such that f = i • p.

Hker(f )
If we show that i is a monomorphism we obtain the decomposition regular epimorphism-monomorphism of f in Hopf coc (Vec G ).In the case of Hopf k,coc , Newman's Theorem [30, Theorem 4.1] tells us that for a cocommutative Hopf algebra H there is a bijective correspondence between the set of Hopf subalgebras of H and that of left ideals which are also two-sided coideals of H: given a Hopf subalgebra K of H and a left ideal, two-sided coideal I of H the two maps are , where π : H → H/I is the canonical projection and this result is used in [22] to deduce that the vector space ker(f ) is exactly A(Hker(f )) + A and then that the morphism i of the previous factorization is injective and so a monomorphism.We would like to obtain the same fact in the graded case.
Remark 5.2.Recall that given a graded algebra A = g∈G A g , i.e. an object in Mon(Vec G ), we can consider the category A Vec G , whose objects are graded vector spaces V = g∈G V g that are also left A-modules such that the left A-action µ : A⊗V → V is in Vec G and then µ(A g ⊗V h ) ⊆ V gh for every g, h ∈ G and whose morphisms are linear maps preserving grading which are also left A-linear.If A is in Bimon(Vec G ), i.e. it is a color bialgebra, then the category A Vec G is monoidal with the same tensor product, unit object and constraints of Vec G and then of Vec k .Here the unit object k has left A-action such that a • k = ǫ(a)k for a ∈ A and k ∈ k and, given V and W in A Vec G , V ⊗ W has left A-action given by a With quotient color left A-module coalgebras we mean quotient objects in Comon( A Vec G ), thus quotient graded vector spaces which are left A-modules with left A-action in Vec G , which are also coalgebras with ∆ and ǫ in A Vec G ; in particular, as coalgebras, they are quotients of a graded coalgebra with a graded two-sided coideal.
Given A in Hopf coc (Vec G ) we define in Vec G the morphism , with c the braiding of Vec G .By analogy with Theorem 5.6, we say that a color Hopf subalgebra we show some properties of the map ξ A .
In fact, by Remark 3.2, we know that f (A) is a graded subspace of B and, as in the usual case, it contains Similarly ker(f ), which is graded by Remark 3.2, is a two-sided ideal of A (since f is an algebra map), a two-sided coideal of A (since f is a coalgebra map) and it is closed under S A , so that A/ker(f ) is in Hopf coc (Vec G ) by Remark 4.3.
Lemma 5.4.Let A and B in Hopf coc (Vec G ). Then the following properties hold: 1) ξ A is a morphism of coalgebras.

2) Given
As a consequence with p surjective, if D is a normal color Hopf subalgebra of A then p(D) is a normal color Hopf subalgebra of B.

3) A is commutative if and only if ξ
Proof.In order to prove 1) we have to show that since ∆ is a morphism of graded algebras and A is cocommutative, we have that where ( * ) follows since A is cocommutative and then ∆ A is of graded coalgebras, i.e.Proof.Suppose that B = Hker(f ) for some morphism f : A → C in Hopf coc (Vec G ).We already know that B is a color Hopf subalgebra of A, we have to prove that it is normal, i.e. that, given x ∈ Hker(f ) and a ∈ A, then ξ A (a ⊗ x) ∈ Hker(f ), i.e.
and, since f is a morphism of graded algebras, we obtain and then, using 1) of Lemma 5.4, we have that A generalization of Newman's Theorem for the category Hopf coc (Vec Z2 ) of cocommutative super Hopf algebras is proved by A. Masuoka in the case chark = 2.The result is the following: Theorem 5.6.(c.f.[27, Theorem 3.10 (3)]) Let H be a cocommutative super Hopf algebra.Then the super Hopf subalgebras K ⊆ H and the quotient super left H-module coalgebras Q of H are in 1-1 correspondence, under K → H/HK + , Q → coQ H(= H coQ ).This restricts to a 1-1 correspondence between those super Hopf subalgebras K which are normal in the sense that (−1) |h2||x| h 1 xS(h 2 ) ∈ K for every h ∈ H and x ∈ K and the quotient super Hopf algebras.
We call the bijections in analogy with those given for Newman's Theorem in [22], i.e.
where H coQ is defined as before.Observe that last statement in Theorem 5.6 is a generalization of the equivalence between ( 1) and ( 2) of [22,Corollary 2.3].Here we obtain immediately a complete generalization of [22,Corollary 2.3] for cocommutative super Hopf algebras.
Corollary 5.7.For a super Hopf subalgebra B ⊆ A of a cocommutative super Hopf algebra A, the following conditions are equivalent: (1) B is a normal super Hopf subalgebra; (2) A/AB + is a quotient super Hopf algebra; (3) the inclusion morphism B → A is the categorical kernel of some morphism in Hopf coc (Vec Z2 ).
Proof.We already know that (1) and ( 2) are equivalent by Theorem 5.6.
(2) =⇒ (3).Since A/AB + is a quotient super Hopf algebra, the canonical projection π : A → A/AB + is a morphism of cocommutative super Hopf algebras and then clearly A co A AB + is exactly Hker(π) since x ⊗ π(1 A ) = x ⊗ 1 A/AB + , for x ∈ A. But now, using Theorem 5.6, we obtain Hence (B, j) is the kernel of π in Hopf coc (Vec Z2 ), where j : B → A is the canonical inclusion.We already know that (3) =⇒ (1) by Lemma 5.5 and then we are done.
We will obtain a generalization of Theorem 5.6 and of Corollary 5.7 for Hopf coc (Vec G ) that will be used for the regularity and the semi-abelian condition of Hopf coc (Vec G ).

5.1.
From color Hopf algebras to super Hopf algebras.In order to use Theorem 5.6 we are interested in obtaining a braided strong monoidal functor from the category Vec G to the category Vec Z2 .In this subsection G and L will denote arbitrary abelian groups.
Remark 5.8.As it is said in [19,Example 2.5.2],given f : G → L a morphism of groups, any Ggraded vector space is naturally L-graded (by pushforward of grading) and we have a natural strict monoidal functor (F, φ 0 , φ 2 ) : Vec G → Vec L (also denoted by f * ).The functor F : Vec G → Vec L is defined, given V = g∈G V g and f in Vec G , such that is the direct sum of all the V g ′ 's such that f (g ′ ) is the same element f (g) in L and then F (V ) is still V as vector space but with a grading over L in which V l = {0} if l / ∈ Im(f ).Observe that F (k) = k with k 1L = k and k l = 0 if l = 1 L , so that one can define φ 0 := Id k and, given V, W in Vec G , we have which is F (V ) ⊗ F (W ), so F (V ⊗ W ) and F (V ) ⊗ F (W ) are the same L-graded vector space and then one can define φ 2 V,W := Id for every V, W in Vec G .Clearly this remark is true also for groups G and L not abelian in which case Vec G and Vec L are not braided.
In [7,Remark 1.2] it is said how to obtain a braided strong monoidal functor from Vec G to Vec L when G and L are finite abelian groups and it is not difficult to see that this works also in the case G and L are not necessarily finite.We recall here how to do it.Clearly, if we define φ 2 V,W := Id for every V and W in Vec G , we can not obtain in general a braided monoidal functor from Vec G to Vec L since the braiding of Vec G and that of Vec L are different.Thus we define φ 0 := Id k but we modify the morphisms φ 2 V,W that we want to be isomorphisms in Vec L in order to have a strong monoidal functor and we recall that and w ∈ W h and g, h ∈ G, defined on the components of the grading and extended by linearity.We define φ 2 V,W := F (f V,W ) = f V,W , which are isomorphisms in Vec L for every V and W in Vec G .In order to obtain a monoidal functor we need that γ is a 2-cocycle on G, i.e. that it satisfies (5) γ(gh, k)γ(g, h) = γ(g, hk)γ(h, k) for every g, h, k ∈ G.
Thus, from now on, we suppose that the abelian group G is finitely generated.We know that a braided strong monoidal functor preserves Hopf monoids (see [4,Propositions 3.46,3.50]),thus, via (F, φ 0 , φ 2 ), every color Hopf algebra becomes a super Hopf algebra and every morphism of color Hopf algebras becomes a morphism of super Hopf algebras (we already know that it is automatically in Vec Z2 , so it will be also a morphism of algebras and of coalgebras with respect to new products and new coproducts).Given a color Hopf algebra (H := g∈G H g , m, u, ∆, ǫ, S), the super Hopf algebra will be given by Lemma 5.12.Given a faithful braided strong monoidal functor then, from the previous two computations, we obtain that A is a (co)commutative (co)monoid in M by using that F is faithful.
In the following we will often refer to the functor F restricted to the category of cocommutative color Hopf algebras, still calling it F .Remark 5.14.In order to avoid confusion, here we denote Vec φ G by indicating the bicharacter associated to the braiding at the top.As it is said in [7, Section 1.5], the normalized 2-cocycle γ : G × G → k − {0} induces an equivalence of braided monoidal categories from Vec φκ G to Vec t G , where t : G × G → k − {0} is the trivial bicharacter such that t(g, h) = 1 k for every g, h ∈ G. Indeed, if we consider the morphism of groups Id G and γ, we have ) for every g, h ∈ G and then a braided strong monoidal functor Vec φκ G → Vec t G by Lemma 5.9.But now, clearly, we can consider the normalized 2-cocycle γ −1 : G × G → k − {0}, (g, h) → γ(g, h) −1 and then t(g, h) = φκ(g, h) γ(h, g) γ(g, h) = φκ(g, h) γ −1 (g, h) γ −1 (h, g) for every g, h ∈ G so that we have a braided strong monoidal functor Vec t G → Vec φκ G , again by Lemma 5.9.These two functors give an equivalence of (symmetric) braided monoidal categories between Vec φκ G and Vec t G and now the possibilities are two.If we have that φ(g, g) = 1 k for every g ∈ G, then clearly κ(g, h) = 1 k for every g, h ∈ G and then φκ = φ, so that we have an equivalence of symmetric monoidal categories between Vec φ G and Vec t G .Indeed observe that, in this case, G 0 = G and then, given V in Vec φ G , we have V = V 0 and V 1 = 0 and η(0, 0) = 1 k .The objects of Hopf coc (Vec t G ), the category of G-graded cocommutative Hopf algebras, are ordinary cocommutative Hopf algebras graded over G as vector spaces and with m, u, ∆, ǫ, S which preserve gradings (thus G-graded algebras and coalgebras) and morphisms are algebra and coalgebra maps which preserve gradings.In particular from a cocommutative color Hopf algebra we can obtain an ordinary cocommutative Hopf algebra and vice versa.Otherwise if φ(g, g) = −1 k for some g ∈ G, we can return to the braided strong monoidal functor (F, φ 0 , φ 2 ) : Vec φ G → Vec η Z2 of before and, given H and f in Hopf coc (Vec φ G ), we obtain F (H) and F (f ) in Hopf coc (Vec η G ), the category of G-graded cocommutative super Hopf algebras, where objects are G-graded algebras and coalgebras (since also φ 2 H,H is in Vec G ), then also Z 2 -graded algebras and coalgebras by considering the new grading, with respect to which, considering the braiding of super vector spaces, they are cocommutative super Hopf algebras and morphisms are algebra and coalgebra maps which preserve the G-grading (and then that over Z 2 ).
Hence every H in Hopf coc (Vec G ) can be seen as a cocommutative super Hopf algebra.If φ(g, g) = 1 k for every g ∈ G we have that this is effectively an ordinary cocommutative Hopf algebra and Newman's Theorem holds true; this always happens if we have a finite group G of odd cardinality by Remark 5.10, for example.If φ(g, g) = −1 k for some g ∈ G we can use the more general Theorem 5.6 for cocommutative super Hopf algebras, where chark = 2 is needed, which allows us to deal with the more general case.5.2.Generalized Newman's theorem for color Hopf algebras.Now we can generalize Theorem 5.6 and Corollary 5.7 to the case of cocommutative color Hopf algebras by using the functor F : Vec G → Vec Z2 , in the case chark = 2 and G is a finitely generated abelian group.
Lemma 5.15.The forgetful functor K : Vec G → Vec k is injective on subobjects and on quotients of the same object.As a consequence, the same holds true if K is restricted to the categories Mon(Vec G ), Comon(Vec G ), Bimon(Vec G ) and Hopf(Vec G ).
Proof.Given A in Vec G and B, C graded subspaces of A, then B g = B ∩ A g and C g = C ∩ A g for every g ∈ G. Thus, if K(B) = K(C), i.e.B and C are the same vector space, then they must be the same object in Vec G .Furthermore, if we consider A in Mon(Vec G ), Comon(Vec G ), Bimon(Vec G ) or Hopf(Vec G ) and B, C subobjects of A in these categories, we have that B and C are the same object in these categories if and only if they are the same object in Vec G because their operations are the restrictions of those of A and then this happens if K(B) = K(C).Moreover, given A/B and A/C in Vec G such that K(A/B) = K(A/C), i.e.A/B and A/C are the same vector space, then B = 0 A/B = 0 A/C = C.As a consequence, (A/B) g = (A g + B)/B = (A g + C)/C = (A/C) g for every g ∈ G and then A/B and A/C are the same object in Vec G .The same result holds true when A/B and A/C are in Mon(Vec G ), Comon(Vec G ), Bimon(Vec G ) or Hopf(Vec G ), since A/B and A/C are the same object in these categories if and only if they are the same object in Vec G because their operations are induced by those of A through the canonical projection.
Proof.We consider the case of Hopf submonoids which includes all the others in itself.We already know that if C is a color Hopf subalgebra of A, i.e. the inclusion i : C → A is in Hopf coc (Vec G ), then F (i) : F (C) → F (A) is in Hopf coc (Vec Z2 ), i.e.F (C) is a super Hopf subalgebra of F (A), so we show the other direction, assuming A in Hopf coc (Vec G ) and C ⊆ A a graded subspace such that F (C) is a super Hopf subalgebra of F (A).But now we only have to observe that ) so that, since F (C) is a super Hopf subalgebra of F (A) (i.e.F (C) is closed under the operations of F (A)), then C is a color Hopf subalgebra of A (i.e.C is closed under the operations of A), since F does not change the structure of vector space.
Remark 5.17.Observe that, given π : A → A/I in Vec G , then F (π) : F (A) → F (A/I) is in Vec Z2 and it is still surjective, then by Remark 3.2 F (A/I) has the unique grading induced by F (A) through the surjection, i.e.F (A/I) i = F (π)(F (A) i ) = (F (A) i + I)/I = (F (A) i + F (I))/F (I) for i = 0, 1 and this is exactly the grading in Vec Z2 of the quotient of F (A) with its super subspace F (I). Thus F (π) : ) and F (A/I) has the unique structure in Hopf coc (Vec Z2 ) induced by F (A).
Proof.If we take f, g : A → B in Hopf coc (Vec G ) and j : Eq(f, g) → A the equalizer of the pair (f, g) in Hopf coc (Vec G ), then we can consider F (j) : F (Eq(f, g)) → F (A) in Hopf coc (Vec Z2 ) and we can show that F (Eq(f, g)) = Eq(F (f ), F (g)).We know that Eq(F (f ), F (g)) in Hopf coc (Vec Z2 ) is given by those x ∈ F (A) such that (Id . But now we have that and these are exactly the elements of F (Eq(f, g)).Hence we have that (F (Eq(f, g)), F (j)) is the equalizer of the pair (F (f ), F (g)) in Hopf coc (Vec Z2 ), so F preserves equalizers.The fact that F reflects equalizers follows using that F preserves equalizers and that F reflects isomorphisms (see [10,Proposition 2.9.7]).
Clearly the previous result holds true by considering F : Comon coc (Vec G ) → Comon coc (Vec Z2 ).
Lemma 5.19.Given a graded algebra H and a color left H-module A, then F (A) is a super left F (H)-module.
Proof.Given the action µ : Theorem 5.20.Let H be a cocommutative color Hopf algebra.Then the color Hopf subalgebras K ⊆ H and the quotient color left H-module coalgebras Q of H are in 1-1 correspondence, under Proof.First of all we show that the two maps are well-defined.So, given a color Hopf subalgebra K ⊆ H, we know that HK + is a graded left ideal of H and also a two-sided coideal of H since , where i : K → H is the inclusion (see [37, and thus H/HK + is a quotient color left H-module coalgebra of H. Furthermore, let Q be a quotient color left H-module coalgebra of H; we show that In addition, by Proposition 5.18, we know that F (H coQ ) is the equalizer of the pair (F (π), Q) .But now F (H) is a super Hopf algebra and F (Q) is a quotient super coalgebra and a quotient super left F (H)-module of F (H) by Lemma 5.19.Furthermore, ∆ F (Q) and ǫ F (Q) are morphisms of left F (H)-modules.Indeed, recalling the hexagon relation which holds true since F is a braided strong monoidal functor, we obtain that ).Thus F (Q) is a quotient super left F (H)-module coalgebra of F (H) and then, by Theorem 5.6, we have that F (H) coF (Q) = F (H coQ ) is a super Hopf subalgebra of F (H); thus H coQ is a color Hopf subalgebra of H by Lemma 5.16.Thus the two maps are well-defined and now we want to prove that they are inverse to each other.So we compute since F (K) is a super Hopf subalgebra of F (H) by Lemma 5.16.Thus, since K and H co H HK + are color Hopf subalgebras of H and they are the same vector space by the previous equality because F does not change the structure of vector space, they must be the same object in Hopf coc (Vec G ) by Lemma 5.15.Furthermore we compute We know that Q and H/H(H coQ ) + are quotient color left H-module coalgebras of H and so, since they are the same vector space by the previous equality, they must be the same quotient color left H-module coalgebra of H by Lemma 5.15.
We call the two maps φ H : K → H/HK + and ψ H : Q → H coQ as for Theorem 5.6.The bijection restricts to a 1-1 correspondence between normal color Hopf subalgebras and quotient color Hopf algebras as it is shown in the following result.Thus we extend Theorem 5.6 and Corollary 5.7 to the case of cocommutative color Hopf algebras.(2) A/AB + is a quotient color Hopf algebra; (3) the inclusion morphism B → A is the categorical kernel of some morphism in Hopf coc (Vec G ).
Proof.(1) =⇒ (2).Let B be a normal color Hopf subalgebra of A and consider the quotient color left A-module coalgebra A/AB + .In order to show that this is a quotient color Hopf algebra we have to prove that AB + is a right ideal of A and that it is closed under the antipode of A. First If we have a morphism p : A → B in Hopf coc (Vec G ) and we consider C ⊆ B a color Hopf subalgebra of B, then the subspace p −1 (C) of A defined as in [24] by since the maps preserve gradings.Hence p −1 (C) = g∈G P g is a graded vector space where By Remark 4.1, we only have to show that p −1 (C) is closed under ∆ A , m A and S A and that it contains 1 A .Clearly it contains 1 A and it is closed under m It is also easy to see closure under antipode since, in the cocommutative case, we have that ∆(S(x)) = (S ⊗ S)∆(x), so with x ∈ p −1 (C) we have Finally we have to show that and then ∆(x) ∈ p −1 (C) ⊗ A, so that p −1 (C) is closed under ∆ A and hence it is a color Hopf subalgebra of A. Now here we show some results which generalize those given in [22] for the case of Hopf k,coc .The following Lemma 5.23 and Lemma 5.24 correspond to [22, Lemma 2.5] and [22,Lemma 2.6] respectively and they have the same proof, which we report for the sake of completeness and in order to show that there are no problems with the respective generalizations.
3) For all color Hopf subalgebras C ⊆ B, then C = p(p −1 (C)) if and only if C = p(D), for some D ⊆ A color Hopf subalgebra.
Proof.Thus we show that α(T ) ⊆ p −1 (C).Given t ∈ T , since α is a morphism of coalgebras, we have then the diagram is a pullback.so that, by applying ψ B and by using Theorem 5.20 again, we obtain that C = p(D).
We have shown the stability of surjective morphisms (i.e.regular epimorphisms by Lemma 5.22) along inclusions under pullbacks in Hopf coc (Vec G ).But now every injective morphism f : C → B in Hopf coc (Vec G ) can be decomposed as i • φ with φ an isomorphism between C and f (C) and i the inclusion of f (C) into B. Now, if we consider the pullback of p along f we have that, since the inner right square is a pullback too by Lemma 5.24, also the left square is a pullback by [10, Proposition 2.5.9].

A ×
Then, since φ is an isomorphism so is φ and from p • φ = φ • α, we obtain that α = φ −1 Let G be a finitely generated abelian group and chark = 2.In [25, 3.7] an equivalent characterization for semi-abelian categories is given.It is required that C satisfies the following properties: 1) C has binary products and coproducts and a zero object; 2) C has pullbacks of (split) monomorphisms; 3) C has cokernels of kernels and every morphism with zero kernel is a monomorphism; 4) the Split Short Five Lemma holds true in C; 5) cokernels are stable under pullback; 6) images of kernels by cokernels are kernels.
For the second part of 3) we observe that, since the categorical kernel of a morphism f : A → B in Hopf coc (Vec G ) is given by the inclusion i : Hker(f ) → A, if this is the zero morphism u A • ǫ Hker(f ) then, given x ∈ Hker(f ), we have x = ǫ(x)1 A and again Hker(f ) = k1 A and then Hker(f ) + = 0. Hence, since we know that the vector space ker(f ) = A(Hker(f )) + A, then f is injective or equivalently a monomorphism in Hopf coc (Vec G ) by Lemma 5.22.Since we have shown that Hopf coc (Vec G ) is pointed, finitely complete (also complete by Remark 4.5), cocomplete, protomodular and regular, properties 1)-5) follow (recall that with C a pointed and finitely complete category 4) is equivalent to the protomodularity of C) and then it remains only to prove that the image of a kernel by a cokernel is a kernel.Precisely we want to show that, given j : Hker(g) → X a kernel of a mophism g : X → Z in Hopf coc (Vec G ) and µ : X → X/Xf (A) + X a cokernel of a morphism f : A → X in Hopf coc (Vec G ), there exist a morphism p : Hker(g) → H in Hopf coc (Vec G ) and a kernel ι : H → X/Xf (A) + X in Hopf coc (Vec G ) such that the following diagram commutes.Now, if we consider the morphism µ • j we know that it has a factorization regular epimorphismmonomorphism in Hopf coc (Vec G ) since this category is regular, i.e. there exist a regular epimorphism p and a monomorphism ι in Hopf coc (Vec G ) such that µ • j = ι • p.But now it is not true in general that every monomorphism is a kernel and then we do not have that ι is a kernel automatically.We know that p is surjective and ι is injective by Lemma 5.22, then we have ι = i • ι ′ where i is an inclusion and ι ′ is an isomorphism between p(Hker(g)) and ι(p(Hker(g))) = µ(j(Hker(g))) = µ(Hker(g)).
Hker(g) X µ(Hker(g)) p(Hker(g))  The notion of semi-abelian category was introduced to capture typical algebraic properties of groups but it was noted that there are many significant aspects of groups which are not captured in this more general context, then reinforcements of this notion were born.We recall that a category with finite limits C is called algebraically coherent if for each morphism f : X → Y in C the change-of-base functor f * : Pt Y (C) → Pt X (C) is coherent, i.e. it preserves finite limits and jointly strongly epimorphic pairs (see [18,Definition 3.1]).In [18,Theorems 6.18 and 6.24] it is shown that semi-abelian categories which are algebraically coherent satisfy both the condition (SH) and (NH) and are peri-abelian and strongly protomodular, thus they are significantly stronger than general semi-abelian categories.So it is interesting to understand if the category Hopf coc (Vec G ) is algebraically coherent, still with G a finitely generated abelian group and chark = 2.We recall from [15] that a finitely complete category C is said to be locally algebraically cartesian closed when, for every f : X → Y in C, the change-of-base functor f * : Pt Y (C) → Pt X (C) is a left adjoint and that if C is locally algebraically cartesian closed then it is algebraically coherent by [18,Theorem 4.5].We conclude with the following result: Proposition 6.3.The category of cocommutative color Hopf algebras is action representable and locally algebraically cartesian closed.
Proof.By [34, Proposition 3.2] (see also [8]), the category Comon coc (Vec G ) is cartesian closed since Vec G is a symmetric monoidally closed category (see e.g.[17]).Thus, since Hopf coc (Vec G ) = Grp(Comon coc (Vec G )), we have that Hopf coc (Vec G ) is locally algebraically cartesian closed by [23,Proposition 5.3].Furthermore, the category of internal groups in a cartesian closed category is always action representable, provided it is semi-abelian, as it is shown in [12,Theorem 4.4] and then Hopf coc (Vec G ) is also action representable.
and then, since π : H → H/I is surjective, we obtain c H/I,H/I • ∆ H/I = ∆ H/I .4.3.Coequalizers.Let f, g : A → B in Hopf(Vec G ), we can consider I = B((f −g)(A))B, the twosided ideal of B generated by the graded subspace of B given by (f −g)(A) := {f (a)−g(a) | a ∈ A}, which is graded by Remark 3.2, since using the universal property, since (Id ⊗ ∆) • ∆ and (∆ ⊗ Id) • ∆ are both graded algebra maps and, for the same argument, if we show l

Corollary 5 . 21 .
For a color Hopf subalgebra B ⊆ A of a cocommutative color Hopf algebra A, the following conditions are equivalent:(1) B is a normal color Hopf subalgebra, i.e. φ(|a 2 |, |b|)a 1 bS(a 2 ) ∈ B for every a ∈ A and b ∈ B;

1 )Lemma 5 . 24 .
is shown.Recall that if D is a color Hopf subalgebra of A then p(D) is a color Hopf subalgebra of B by Remark 5.3.If d ∈ D we have that (p ⊗ Id A )∆(d) ∈ p(D) ⊗ D ⊆ p(D) ⊗ A and then also 2) is proved.Finally if C = p(p −1 (C)) clearly we can take D = p −1 (C) while if C = p(D) for some color Hopf subalgebra D of A then D ⊆ p −1 (p(D)) = p −1 (C) by 2) and by applying p one gets C = p(D) ⊆ p(p −1 (C)) and, since p(p −1 (C)) ⊆ C by 1), we have C = p(p −1 (C)) and we obtain 3).Given p : A → B in Hopf coc (Vec G ) and an inclusion i : C → B in Hopf coc (Vec G ), then the diagram p −1 (C) in Hopf coc (Vec G ), where j is the inclusion and p is the restriction of p to p −1 (C).Proof.By 1) of Lemma 5.23 the diagram is commutative.To check the universal property, consider two morphisms α : T → A and β : T → C in Hopf coc (Vec G ) such that p • α = i • β and let us show that α(T ) ⊆ p −1 (C).Then, taken c : T → p −1 (C) as α with codomain p −1 (C) we have j • c = α and i • p • c = p • j • c = p • α = i • β, hence p • c = β since i injective; clearly this c is unique since we must have j • c = α.

Proposition 5 . 25 .
Consider a surjective morphism p : A → B in Hopf coc (Vec G ) and an inclusion i : C → B in Hopf coc (Vec G ). Then the morphism p in the pullback of Lemma 5.24 is also surjective.Proof.If we compute the pullback of the pair (p, i) in Hopf coc (Vec G ) we obtain (p −1 (C), j, p) as in Lemma 5.24 and we want to show that p is surjective if p is surjective.Since p is just given by the restriction of p, we have that p is surjective if and only if C = p(p −1 (C)) and this is equivalent, with C a color Hopf subalgebra of B, to prove that C = p(D) for some color Hopf subalgebra D of A by 3) of Lemma 5.23.We know that the canonical projection π : B → B/BC + is a quotient color left B-module coalgebra and, since p is a morphism of color Hopf algebras, we have that π • p is a morphism of color left A-module coalgebras, so that A/ker(π • p) is a quotient color left A-module coalgebra.We setD := A co A ker(π•p) = ψ A (A/ker(π • p)),which is a color Hopf subalgebra of A by Theorem 5.20.Then we obtain A/AD + = φ A (D) = φ A (ψ A (A/ker(π • p))) = A/ker(π • p) by Theorem 5.20, hence AD + = ker(π • p).Thus, since p is a surjective morphism of algebras, we obtain Bp(D) + = p(A)p(D + ) = p(AD + ) = p(ker(π • p)) = ker(π) = BC + and then φ B (C) = B/BC + = B/Bp(D) + = φ B (p(D))