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On the semi-abelianness of cocommutative Hopf monoids

Andrea Sciandra, Zhenbang Zuo

Abstract

By providing a suitable generalization of Newman's bijective correspondence known for cocommutative Hopf algebras, we prove that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, once faithful (co)flatness conditions are satisfied. This result unifies and generalizes the semi-abelianness of cocommutative Hopf algebras and of cocommutative color Hopf algebras known up to now. As a consequence of the semi-abelianness, the category of cocommutative Hopf monoids is also action representable. Finally, we prove that abelian objects in the category of cocommutative Hopf monoids coincide exactly with commutative and cocommutative Hopf monoids, which form so an abelian category.

On the semi-abelianness of cocommutative Hopf monoids

Abstract

By providing a suitable generalization of Newman's bijective correspondence known for cocommutative Hopf algebras, we prove that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, once faithful (co)flatness conditions are satisfied. This result unifies and generalizes the semi-abelianness of cocommutative Hopf algebras and of cocommutative color Hopf algebras known up to now. As a consequence of the semi-abelianness, the category of cocommutative Hopf monoids is also action representable. Finally, we prove that abelian objects in the category of cocommutative Hopf monoids coincide exactly with commutative and cocommutative Hopf monoids, which form so an abelian category.
Paper Structure (16 sections, 135 equations)

This paper contains 16 sections, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pointedness, limits, and protomodularity
  4. Binary products in $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  5. Equalizers in $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  6. Protomodularity
  7. Pullbacks in $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  8. Coequalizers in $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  9. Newman's Theorem for $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  10. An equivalent characterization for kernels in $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  11. Regularity of $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  12. Factorization of morphisms in $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  13. Stability of regular epimorphisms along pullbacks
  14. On the semi-abelianness of $\mathsf{Hopf}_{\mathrm{coc}}(\mathcal{M})$
  15. Abelian objects
  16. ...and 1 more sections

Theorems & Definitions (42)

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