标题： 关于霍普夫括号的范畴 显示英文标题

标题： On the category of Hopf braces

摘要： Hopf括号是斜括号的量子类似物，作为其余交换对应物，它们为量子Yang-Baxter方程提供了解。 我们研究与Hopf括号相关的范畴的各种性质。 特别是，我们证明了Hopf括号的范畴是可访问的，而余交换Hopf括号的范畴甚至还是局部可表示的。 我们还证明了将多个反元素和/或乘法遗忘到余代数上的函子是单子的。 余交换Hopf括号范畴中的余极限被显式描述，并构造了一个任意余交换Hopf代数上的自由余交换Hopf括号。 显示英文摘要

摘要： 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.