原子幺半群的范畴代数：可表示性、正则性和预挠理论
Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories
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中文总结 AI 辅助
研究原子幺半群范畴$\mathsf{AtoMon}$，证明其局部有限可表示，指出其允许特定分解但非正则范畴，建立单位群伴随关系并构造余自由原子幺半群，还展示了将$\mathsf{Grp}$挠理论提升到$\mathsf{AtoMon}$预挠理论的方法。
中文摘要 AI 辅助
我们研究原子幺半群和保原子同态的范畴$\mathsf{AtoMon}$。通过展示由紧对象组成的强生成元，证明了$\mathsf{AtoMon}$是局部有限可表示的。表明$\mathsf{AtoMon}$允许（正则满射，单射）分解，但不是正则范畴：构造了一个非拉回稳定的正则满射。还建立了单位群的伴随关系，并明确构造了任意幺半群上的“余自由原子幺半群”。最后，展示了一种将$\mathsf{Grp}$的挠理论提升到$\mathsf{AtoMon}$的预挠理论的方法，并将此构造扩展到更一般的情形。
英文摘要
We study the category $\mathsf{AtoMon}$ of atomic monoids and atom-preserving homomorphisms. We prove that $\mathsf{AtoMon}$ is locally finitely presentable by exhibiting a strong generator consisting of compact objects. We show that $\mathsf{AtoMon}$ admits (regular epi, mono)-factorizations but that it is not a regular category: we construct a regular epimorphism which is not pullback-stable. We also establish adjunctions for the group of units and explicitly construct the ``cofree atomic monoid'' over an arbitrary monoid. Finally, we exhibit a way to lift torsion theories of $\mathsf{Grp}$ to pretorsion theories of $\mathsf{AtoMon}$ and extend this construction to a more general setting.
