论文标题
关于函数的晶状体及其同构定理
On functor-quotients and their isomorphism theorems
论文作者
论文摘要
可以将分类商的概念推广，因为其标准分类概念不会在某些类别中恢复预期的商。我们以$ \ mathcal {f} $的形式提出了一个更通用的公式，类别$ \ mathbf {c} $中的商相关地与忠实的函数$ \ Mathcal {f} \ colon \ colon \ colon \ Mathbf {c} \ to \ to \ to \ mathbf {d d} $。通用代数的同构定理将其推广到此设置，我们还找到了一阶结构的具体类别中的$ \ Mathcal {f} $之间的重要链接，以及为模型理论等价类定义的商类别。通过在此分类环境中首先工作，可以自然获得一些与一阶结构的商相关结果。特别是，我们能够直接从模型理论的上下文中证明某些同构定理，直接从其相应的分类同构定理中证明了一些同构定理。
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a category $\mathbf{C}$, which are relativized to a faithful functor $\mathcal{F}\colon \mathbf{C} \to \mathbf{D}$. The isomorphism theorems of universal algebras generalize to this setting, and we additionally find important links between $\mathcal{F}$-quotients in the concrete category of first-order structures, and quotients defined for model-theoretic equivalence classes. By first working in this categorical setting, some quotient-related results for first-order structures can be naturally obtained. In particular, we are able to prove some isomorphism theorems in the context of model theory directly from their corresponding categorical isomorphism theorems.
