The kernel of a Hopf algebra map $\phi:H\_1 \to H\_2$ is in general not a Hopf 
sub-algebra of $H\_1$. Is there some replacement or alteration of the notion 
of a kernel in the Hopf algebra setting. Same question for cokernels. So can we construct an abelian category from the category of Hopf algebras (over a fixed field $k$). 



Abelian category from the category of Hopf algebras

> 
> $\DeclareMathOperator\Hker{Hker}\DeclareMathOperator\Hcoker{Hcoker}\DeclareMathOperator\Im{Im}\DeclareMathOperator\coIm{coIm}\DeclareMathOperator\Id{Id}$The category $\mathcal{H}$ of finite dimensional, commutative, cocommutative hopf algebras is an abelian category.  
> 
> The set $\mathcal{H}(F,G)$ of all hopf algebra maps $F\to G$ is an abelian group with sum given by the convolution product. (The convolution inverse for $f\in F$ is: $f\!\circ\! S\_F=S\_G\!\circ\! f$ and the neutral element is $\eta\_G\!\circ\!\varepsilon\_F$).  
> 
> We have also product of maps, given by composition (and [composition distributes over convolution](https://math.stackexchange.com/a/2756755/195021)).
> 
> 
> 


This is a classic result. I have been told it was first shown by Grothendieck. I know you can find details in Sweedler's book, ch. XVI, p. 314 (basically the whole chapter, p.303–315 presents a detailed proof of the above).


**Remark 1:** You are right to mention that in the general case the definitions of the kernels and the cokernels seem to pose a problem when one tries to figure out which hopf algebras would be good to make an abelian category from.  

For $F$, $G$, f.d., commut, cocommut, hopf algebras and $\omega:F\to G$ a hopf algebra map, then in $\mathcal{H}$, kernels are defined through: $$\Hker\omega=\{f\in F|(I\otimes\omega)\Delta (f)=f\otimes 1\}$$
which -due to cocommutativity- is the same with $\{f\in F|(\omega\otimes I)\Delta (f)=1\otimes f\}$,  

and cokernels through:
$$\Hcoker\omega=G\big/\big(\omega (F^+)G\big)$$
where $F^+$ is the [augmentation ideal](https://mathoverflow.net/a/331635/85967) and $\omega (F^+)G$ denotes the right ideal (which by commutativity is a two-sided ideal), of $G$ generated by $\omega(F^+)$.  

Under the above defs, it can be shown that $\Hker\omega$ is a sub-hopf algebra of $F$, $\Hcoker\omega$ is a quotient hopf algebra of $G$ (i.e. $\omega (F^+)G$ is a hopf ideal) and any hopf algebra map $\omega\in\mathcal{H}(F,G)$ "factorizes" as:
$$
\Hker\omega \overset{i\_{\omega}}{\hookrightarrow} A
\overset{\pi\_{i\_{\omega}}}{\twoheadrightarrow} \coIm\omega \cong \Im\omega
\overset{i\_{\pi\_{\omega}}}{\hookrightarrow} B
\overset{\pi\_{\omega}}{\twoheadrightarrow} \Hcoker\omega
$$
where:


* $\Hker\omega \overset{i\_{\omega}}{\hookrightarrow} A$ is the kernel of $A\overset{\omega}{\to}B$,
* $B\overset{\pi\_{\omega}}{\twoheadrightarrow} \Hcoker\omega$ is the cokernel of $A\overset{\omega}{\to}B$,
* $A\overset{\pi\_{i\_{\omega}}}{\twoheadrightarrow} \coIm\omega$ is the cokernel of $\Hker\omega \overset{i\_{\omega}}{\hookrightarrow} A$ and
* $\Im\omega\overset{i\_{\pi\_{\omega}}}{\hookrightarrow} B$ is the kernel of $B
\overset{\pi\_{\omega}}{\twoheadrightarrow} \Hcoker\omega$.


**Remark 2:** There are even more detailed things to be said about the category $\mathcal{H}$ of finite dimensional, commutative, cocommutative hopf algebras over a field: We can construct a functor $\mathcal{G} : \mathcal{H} \Rrightarrow \mathcal{Ab}\_{\text{fin}}$ from $\mathcal{H}$ to the category $\mathcal{Ab}\_{\text{fin}}$ of finite, abelian groups (assigning to each hopf algebra of $\mathcal{H}$ the group of its grouplikes) and another functor $\mathcal{F} : \mathcal{Ab}\_{\text{fin}} \Rrightarrow \mathcal{H}$ (assigning to each fin, abelian group its group hopf algebra). It is not difficult to show that these functors satisfy
$$
\begin{array}{cccc}
 \mathcal{G} \mathcal{F} = \Id\_{\mathcal{Ab}\_{\text{fin}}} & & & \mathcal{F} \mathcal{G} \cong
 \Id\_{\mathcal{H}}\\
\end{array}
$$
constituting thus an equivalence of the categories $\mathcal{H}$, $\mathcal{Ab}\_{\text{fin}}$.


