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Categorical isomorphisms for Hopf braces

José Manuel Fernández Vilaboa, Ramón González Rodríguez, Brais Ramos Pérez

TL;DR

We address the classification of Hopf braces in a braided monoidal category by introducing brace triples and their corresponding s-Hopf braces, then prove a categorical isomorphism between these two frameworks. We extend the Li–Sheng–Tang equivalence from vector spaces to braided settings and show cocommutative Hopf braces correspond to cocommutative brace triples; we then develop the post-Hopf algebra perspective and connect it to Hopf braces via explicit functors, culminating in a unified isomorphism among cocPost-Hopf, cocBT^f, and cocHBr^f. The results provide a structural, category-theoretic realization of Hopf braces in braided contexts and generalize known cocommutative equivalences. Practically, this yields new ways to interpret and construct Yang-Baxter solutions within braided monoidal categories. The work thus offers a cohesive bridge between brace theory, Hopf-algebraic structures, and the post-Hopf framework in a broad categorical setting.

Abstract

In this paper, we introduce the category of brace triples in a braided monoidal setting and prove that it is isomorphic to the category of s-Hopf braces, which are a generalization of cocommutative Hopf braces. After that, we obtain a categorical isomorphism between finite cocommutative Hopf braces and a certain subcategory of cocommutative post-Hopf algebras, which suppose an expansion to the braided monoidal setting of the equivalence obtained by Y. Li, Y. Sheng and R. Tang for the category of vector spaces over a field $\mathbb{K}$.

Categorical isomorphisms for Hopf braces

TL;DR

We address the classification of Hopf braces in a braided monoidal category by introducing brace triples and their corresponding s-Hopf braces, then prove a categorical isomorphism between these two frameworks. We extend the Li–Sheng–Tang equivalence from vector spaces to braided settings and show cocommutative Hopf braces correspond to cocommutative brace triples; we then develop the post-Hopf algebra perspective and connect it to Hopf braces via explicit functors, culminating in a unified isomorphism among cocPost-Hopf, cocBT^f, and cocHBr^f. The results provide a structural, category-theoretic realization of Hopf braces in braided contexts and generalize known cocommutative equivalences. Practically, this yields new ways to interpret and construct Yang-Baxter solutions within braided monoidal categories. The work thus offers a cohesive bridge between brace theory, Hopf-algebraic structures, and the post-Hopf framework in a broad categorical setting.

Abstract

In this paper, we introduce the category of brace triples in a braided monoidal setting and prove that it is isomorphic to the category of s-Hopf braces, which are a generalization of cocommutative Hopf braces. After that, we obtain a categorical isomorphism between finite cocommutative Hopf braces and a certain subcategory of cocommutative post-Hopf algebras, which suppose an expansion to the braided monoidal setting of the equivalence obtained by Y. Li, Y. Sheng and R. Tang for the category of vector spaces over a field .
Paper Structure (3 sections, 26 theorems, 109 equations, 1 figure)

This paper contains 3 sections, 26 theorems, 109 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Brace triples and Hopf braces
  3. Post-Hopf algebras and Hopf braces

Key Result

Theorem 1.8

Let $X=(X,\eta_{X},\mu_{X},\varepsilon_{X},\delta_{X},\lambda_{X})$ and $H=(H,\eta_{H},\mu_{H},\varepsilon_{H},\delta_{H},\lambda_{H})$ be Hopf algebras in $\sf{C}$ such that there exists a morphism $\varphi_{H}\colon X\otimes H\rightarrow H$ satisfying the following conditions: Then, $\varphi_{H}\circ(X\otimes\eta_{H})=\varepsilon_{X}\otimes\eta_{H}$ holds.

Figures (1)

  • Figure 1: Categorical relationships between $\sf{HBr}$, $\sf{BT}$ and $\sf{Post}$-${\sf Hopf}$.

Theorems & Definitions (76)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • proof
  • Corollary 1.9
  • ...and 66 more