内禀Schreier特殊天体
Intrinsic Schreier special objects
摘要：
受幺半群分裂满态的范畴代数分析的启发，我们研究了在具有二元余积、余单子复盖和自然虚分裂的正则酉范畴的背景下，由内禀Schreier分裂满态诱导的特殊对象的概念[Intrinsic Schreier split extensions，Appl. Categ.结构28（2020），517- 538]。在这种情况下，每个物体都自然地配备了一个想象中的岩浆结构。我们分析了内在的Schreier分裂满态在这种情况下，显示出它们的属性改善时，想象的岩浆结构发生关联。我们比较了内禀Schreier特殊天体和原模天体，并根据假想的岩浆结构对它们进行了描述。我们还将它们与群和李代数的恩格尔性质联系起来。
英文摘要：
Motivated by the categorical-algebraic analysis of split epimorphisms of monoids, we study the concept of a special object induced by the intrinsic Schreier split epimorphisms in the context of a regular unital category with binary coproducts, comonadic covers and a natural imaginary splitting in the sense of our article [Intrinsic Schreier split extensions, Appl. Categ. Structures 28 (2020), 517--538]. In this context, each object comes naturally equipped with an imaginary magma structure. We analyse the intrinsic Schreier split epimorphisms in this setting, showing that their properties improve when the imaginary magma structures happen to be associative. We compare the intrinsic Schreier special objects with the protomodular objects, and characterise them in terms of the imaginary magma structure. We furthermore relate them to the Engel property in the case of groups and Lie algebras.
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