乌尔默双代数的一种句法方法
A Syntactic Approach to Ulmer's Bialgebras
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中文总结 AI 辅助
研究乌尔默双代数的句法方法，通过引入签名对和双代数理论，在2 - 范畴中构建内部双代数对象$M^T$，有一般代数诱导函子定理，其构造基于PIE极限等，在适当假设下有多种性质提升。
中文摘要 AI 辅助
乌尔默引入了双代数的语义概念，统一了一大类代数和余代数结构。我们通过引入签名对$(\Sigma,\sigma)$和双代数理论$T$来开发句法对应物，为在2 - 范畴环境中构建内部双代数提供统一语言。对于具有PIE极限的2 - 范畴中的每个双代数理论$T$和$\Sigma$ - 模型$M$，我们构造内部$T$ - 双代数的对象$M^T$。我们的双代数方法承认一个一般的代数诱导函子定理，扩展了宽松幺半函子到内部幺半群范畴的经典提升。由于$M^T$的构造完全根据PIE极限等表达，在适当假设下，沿着$M \mapsto M^T$的构造具有可达性、局部可表示性、正交分解系统、正则性和精确性提升。
英文摘要
Ulmer introduced a semantic notion of bialgebras that unifies a broad class of algebraic and coalgebraic structures. We develop a syntactic counterpart by introducing signature pairs $(Σ,σ)$ and bialgebraic theories $T$, providing a uniform language for constructing internal bialgebras in a $2$-categorical setting. For every bialgebraic theory $T$ and $Σ$-model $M$ within a $2$-category with PIE limits, we construct the object $M^T$ of internal $T$-bialgebras. Our approach to bialgebras admits a general Induced Functor of Algebras Theorem extending the classical lifting of lax monoidal functors to the categories of internal monoids. Since the construction of $M^T$ is expressed entirely in terms of PIE limits, accessibility, local presentability, orthogonal factorization systems, regularity, and exactness lift along the construction $M \mapsto M^T$ under suitable assumptions.
