发表机构
Hefei University of Technology; Southern University of Science and Technology（合肥工业大学; 南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在对称幺半范畴中为 Lie-Rinehart 幺半群及其弱模建立范畴框架，通过交叉同态构造 Shen-Larsson 函子的范畴版本，并发展对偶理论及双模结构，最后特化到超向量空间恢复超代数构造。
AI 中文摘要
我们在对称幺半范畴中为 Lie-Rinehart 幺半群及其弱模建立了一个范畴框架。利用交叉同态，我们在 Lie 幺半群的模范畴上构造了弱 Lie-Rinehart 模范畴的自然作用，从而得到了 Shen-Larsson 函子的范畴版本。我们进一步刻画了弱模范畴具有幺半结构的条件，并确定了相应函子为严格幺半的条件。利用余交叉同态，我们发展了 Lie-Rinehart 余幺半群与弱余模的对偶理论。将模与余模构造相结合，我们在弱模范畴上得到了一个双模范畴结构。最后，我们将一般框架特化到超向量空间的对称幺半范畴，恢复了 Lie-Rinehart 超代数及其相关的 Shen-Larsson 型构造。
英文摘要
We develop a categorical framework for Lie-Rinehart monoids and their weak modules in a symmetric monoidal category. Using crossed homomorphisms, we construct a natural action of the monoidal category of modules over a Lie monoid on the category of weak Lie-Rinehart modules, thereby obtaining categorical versions of the Shen-Larsson functors. We further characterize the conditions under which the category of weak modules admits a monoidal structure and identify the corresponding condition for the associated functors to be strict monoidal. A dual theory for Lie- Rinehart comonoids and weak comodules is developed using cocrossed homomorphisms. Combining the module and comodule constructions, we obtain a bimodule category structure on the category of weak modules. Finally, we specialize the general framework to the symmetric monoidal category of super vector spaces, recovering Lie-Rinehart superalgebras and their associated Shen-Larsson-type constructions.
