Central extensions of associative algebras and weakly action representable categories
Central extensions of associative algebras and weakly action representable categories
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关联代数和弱作用可表示范畴的中心扩展
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发表时间：
2022
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通讯作者：
G. Janelidze
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作者：
G. Janelidze
.中心扩张是巴尔正合范畴C中的正则满射，它满足适当的条件，涉及C的给定Birkho正合子范畴（与G. M. Kelly，1994）。本文设C是（酉）交换环上的（非必酉）代数范畴，并考虑了C关于交换代数范畴的中心扩张。我们提出了一种新的方法，避免了由于A。Früohlich证明了α：A → B是中心扩张当且仅当对所有a，a ′ ∈ A，a（a ′）= 0，aa ′ = a ′ a.这种方法促使我们引入我们所谓的弱动作可表征范畴，我们表明，这样的类别总是行动可及的。我们也做评论，我们称之为初始弱表示的行动，并制定了几个开放的问题。
. A central extension is a regular epimorphism in a Barr exact cat- egory C satisfying suitable conditions involving a given Birkhoff subcategory of C (joint work with G. M. Kelly, 1994). In this paper we take 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. Fr¨ohlich in showing that α : A → B is a central extension if and only if aa ′ = a ′ a for all a,a ′ ∈ A with α ( a ′ ) = 0. This approach motivates introducing what we call weakly action representable cat- egories , and we show that such categories are always action accessible. We also make remarks on what we call initial weak representations of actions and formulate several open questions.