On the Semi-Abelianness of Affine Group Schemes

We prove that the category of commutative Hopf algebras over a field $k$ is co-semi-abelian. Consequently, the category of affine group $k$-schemes is semi-abelian. We establish coregularity by identifying the orthogonal factorization system of surjections and faithfully flat injections, and we deduce coexactness from Takeuchi’s correspondence between normal Hopf ideals and Hopf subalgebras of commutative Hopf $k$-algebras.