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Coalgebraic models of Omega-groups

Maria Bevilacqua

TL;DR

This work develops a unified framework for coalgebraic models of algebraic theories by working in the category of cocommutative coalgebras and introducing $\Omega$-Hopf algebras as coalgebraic models for $\Omega$-group theories. It shows that these categories are semi-abelian, inheriting key exactness properties from cocommutative Hopf algebras, and studies the transfer from set-based models to coalgebraic models via a linearizing process and adjunctions. The paper provides general constructions for algebraic functors, limits and colimits (notably coequalizers and coproducts), and analyzes surjective maps of theories to obtain Birkhoff subcategories. It unifies several known examples (Hopf algebras, braces, diGroups, radical rings) under the Omega-Hopf algebra umbrella and demonstrates how protomodularity and sem-abelianity extend to these coalgebraic settings. Overall, the results lay groundwork for non-abelian homology and broaden the applicability of Lawvere-theoretic methods to coalgebraic contexts, with potential extensions to graded or ring-valued coalgebras and beyond.

Abstract

We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of coalgebraic models of algebraic theories endowed with an underlying structure of cocommutative Hopf algebra, and show that these categories are semi-abelian. We call them ``categories of Omega-Hopf algebras'', since it is possible to characterize them as coalgebraic models of algebraic theories of Omega-groups.

Coalgebraic models of Omega-groups

TL;DR

This work develops a unified framework for coalgebraic models of algebraic theories by working in the category of cocommutative coalgebras and introducing -Hopf algebras as coalgebraic models for -group theories. It shows that these categories are semi-abelian, inheriting key exactness properties from cocommutative Hopf algebras, and studies the transfer from set-based models to coalgebraic models via a linearizing process and adjunctions. The paper provides general constructions for algebraic functors, limits and colimits (notably coequalizers and coproducts), and analyzes surjective maps of theories to obtain Birkhoff subcategories. It unifies several known examples (Hopf algebras, braces, diGroups, radical rings) under the Omega-Hopf algebra umbrella and demonstrates how protomodularity and sem-abelianity extend to these coalgebraic settings. Overall, the results lay groundwork for non-abelian homology and broaden the applicability of Lawvere-theoretic methods to coalgebraic contexts, with potential extensions to graded or ring-valued coalgebras and beyond.

Abstract

We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of coalgebraic models of algebraic theories endowed with an underlying structure of cocommutative Hopf algebra, and show that these categories are semi-abelian. We call them ``categories of Omega-Hopf algebras'', since it is possible to characterize them as coalgebraic models of algebraic theories of Omega-groups.
Paper Structure (12 sections, 23 theorems, 62 equations, 14 figures, 1 table)

This paper contains 12 sections, 23 theorems, 62 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Linearizing process
  3. Examples
  4. From models to coalgebraic models
  5. Protomodularity
  6. Algebraic functors
  7. Colimits
  8. Coequalizers
  9. Coproducts
  10. Surjective maps of theories
  11. $\Omega$-Hopf algebras
  12. Semi-abelianness of $\Omega$-Hopf algebras

Key Result

Theorem 2.3

Let $\Sigma$ be a set of equations and $s\approx t$ a consequence of $\Sigma$. If an $\mathcal{F}$-coalgebra $C$ satisfies all identities in $\Sigma$, then $l_n(s)=l_n(t)$, i.e. it satisfies $s\approx t$.

Figures (14)

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  • ...and 9 more figures

Theorems & Definitions (60)

  • Definition 2.1
  • Remark 2.2
  • Example
  • Theorem 2.3
  • Definition 2.4: GuarnieriVendramin
  • Definition 2.5
  • Definition 2.6: Bourn2000
  • Definition 2.7: Rump
  • Definition 2.8: agore2025categoryhopfbraces
  • Definition 2.9
  • ...and 50 more