论文标题
联想代数和弱势代表类别的中心扩展
Central extensions of associative algebras and weakly action representable categories
论文作者
论文摘要
中央扩展是在Barr确切类别中的常规表达性$ \ MATHSCR {C} $满足适当条件的涉及$ \ Mathscr {C} $的给定Birkhoff子类别（与G. M. Kelly，1994年联合工作）。在本文中，我们将$ \ mathscr {c} $作为（UNITAL）通勤环上的（不必要的）代数（不必要的）类别，并考虑有关交换代数类别的中心扩展。我们提出了一种新方法，该方法避免了由于A.fröhlich而避免中央扩展的中间概念，表明$α：a \ to b $是当$ a'$ a的$ a'= a'a'，a $ a'in $ a $ a $ a $α（a''）= 0 $时的$ aa'= a'a $。这种方法激发了我们所谓的$ \ textit {弱操作代表类别} $，我们证明此类类别始终是可以访问的。我们还对所谓的$ \ textit {Action} $的$ \ textit {初始弱表示} $发表评论，并提出了几个开放问题。
A central extension is a regular epimorphism in a Barr exact category $\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper we take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebras over a (unital) commutative ring and consider central extensions with respect to the category of commutative algebras. We propose a new approach that avoids the intermediate notion of central extension due to A. Fröhlich in showing that $α:A\to B$ is a central extension if and only if $aa'=a'a$ for all $a,a'\in A$ with $α(a')=0$. This approach motivates introducing what we call $\textit{weakly action representable categories}$, and we show that such categories are always action accessible. We also make remarks on what we call $\textit{initial weak representations of actions}$ and formulate several open questions.
