拟点精确性与挠理论
Quasi-pointed exactness and torsion
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中文总结 AI 辅助
本文在温和假设下证明每个拟点范畴具有典范近似挠理论，并给出其成为非点挠理论的条件，应用于李-莱因哈特代数、交叉模及有限阶幺半群范畴。
中文摘要 AI 辅助
我们证明，在若干温和假设下，每个拟点范畴 C 都带有一个典范的近似挠理论 (T,F)，其中 T 由 0-支撑对象组成，即那些允许到初始对象 0 的态射的对象，而 F 由子终对象组成。对象 A 的余反射与反射分别由投影 $A \ imes 0 \ o A$ 和 $A \ o 1$ 的强满态射部分给出。利用与近似挠理论相关联的“最大”非点挠理论的一般构造，我们证明 (T,F) 是那些使得典范序列 $t(A) \ o A \ o f(A)$ 短正合（即它既是核图又是余核图）的对象 A 的全子范畴上的（非点）挠理论。当 C 是可序列的，即正则原模且拟点的范畴时，该子范畴就是整个 C。我们描述了李-莱因哈特代数范畴以及固定群上的交叉模范畴中由此产生的挠理论。我们还证明了 (T,F) 是群 G 上所有元素均具有有限阶的幺半群范畴（非原模）中的挠理论。
英文摘要
We show that, under some mild assumptions, every quasi-pointed category C carries a canonical near-torsion theory (T,F), where T consists of the 0-supported objects, that is, those admitting a morphism to the initial object 0, and F consists of the subterminal objects. The coreflection and the reflection of an object A are then given by the projection $A \times 0 \to A$ and by the strong-epimorphic part of $A \to 1$, respectively. Using the general construction of the "largest" non-pointed torsion theory associated with a near-torsion theory, we prove that (T,F) is a (non-pointed) torsion theory in the full subcategory of those objects A for which the canonical sequence $t(A) \to A \to f(A)$ is short exact, in the sense that it is both a kernel and a cokernel diagram. This subcategory is the whole of C whenever C is sequentiable, that is, a regular protomodular and quasi-pointed category. We describe the resulting torsion theories in the category of Lie-Rinehart algebras and in the categories of crossed modules over a fixed group. We also show that (T,F) is a torsion theory in the (non-protomodular) category of monoids over a group G all of whose elements have finite order.
发表机构
- Institut de Recherche en Mathématique et Physique, Université catholique de Louvain（鲁汶天主教大学数学与物理研究 institute）
- Department of Mathematics and Applied Mathematics, University of Cape Town（开普敦大学数学与应用数学系）
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