On the category of Hopf braces

TL;DR

The paper develops a category-theoretic foundation for Hopf braces and their cocommutative equivalents, situating them inside locally presentable and accessible categories. It proves that categories of multi-Hopf structures (e.g., $2$-Hopf algebras and $(I,J)$-Hopf algebras) are locally presentable, with forgetful functors to coalgebras being monadic, and then analyzes the brace category $\textsc{Hbr}$ and its cocommutative version as equifiers, establishing accessibility and monadicity results. The authors provide explicit constructions of colimits in $\textsc{Hbr}^{coc}$, including coequalizers and coproducts, by leveraging intermediate categories like $\textsc{Bialg}$ and $\text{Halg}$, and furnish a free cocommutative Hopf brace on any cocommutative Hopf algebra. These results yield a robust structural framework for generating and classifying solutions to the quantum Yang-Baxter equation via cocommutative Hopf braces and related multi-Hopf structures.

Abstract

Hopf braces are the quantum analogues of skew braces and, as such, their cocommutative counterparts provide solutions to the quantum Yang-Baxter equation. We investigate various properties of categories related to Hopf braces. In particular, we prove that the category of Hopf braces is accessible while the category of cocommutative Hopf braces is even locally presentable. We also show that functors forgetting multiple antipodes and/or multiplications down to coalgebras are monadic. Colimits in the category of cocommutative Hopf braces are described explicitly and a free cocommutative Hopf brace on an arbitrary cocommutative Hopf algebra is constructed.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 2.2
• Proof 1
• Corollary 2.3
• Theorem 3.1
• Proof 2
• Corollary 3.2
• Remark 3.3
• Proposition 4.1