标题： 关于Hopf braces范畴的研究 显示英文标题

标题： On the category of Hopf braces

摘要： Hopf braces 是斜 braces 的量子类比，因此它们的余交换版本为量子Yang-Baxter方程提供了解。 我们研究了与Hopf braces相关的范畴的各种性质。 特别是，我们证明了Hopf braces范畴是可接近的，而余交换Hopf braces范畴甚至是局部可表示的。 我们还表明，遗忘多个反极和/或多乘法到余代数的函子是单子的。 余交换Hopf braces范畴中的余极限被明确描述，并且在任意余交换Hopf代数上构造了一个自由的余交换Hopf braces。 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.