Frobenius templicial 模与 dg-神经
Frobenius templicial modules and the dg-nerve
AI总结：
本文提出 Frobenius templicial 对象，证明正分次 dg-范畴与 Frobenius templicial 模之间的范畴等价，并由此给出经典 dg-神经的自然充实，刻画了模中拟范畴的 Frobenius 结构条件。
摘要：
Templicial 对象在 arXiv:2302.02484v2 中被提出，旨在为充实拟范畴建立一个合适的单纯框架。遵循 Leinster 的思路，这些对象以某些余乘法作为非笛卡尔情形中外侧面的替代。本文考虑 Frobenius templicial 对象，从而将乘法重新引入该框架。当在交换环 k 上以 k-模进行充实时，我们证明了（同调意义下）正分次 dg-范畴与 Frobenius templicial 模之间的范畴等价。这一等价给出了经典 dg-神经的一种自然充实，将 dg-范畴转化为模中的拟范畴。在投射性条件下，我们进一步证明一个 templicial 模恰好在能够配备非结合 Frobenius 结构时成为模中的拟范畴。
英文摘要：
Templicial objects were put forth in arXiv:2302.02484v2 to set up a suitable simplicial framework for enriched quasi-categories. Following Leinster, these objects feature certain comultiplications as a replacement for outer face maps in the non-cartesian case. In the present paper, we consider Frobenius templicial objects, thus re-introducing multiplications into the picture. When enriching over $k$-modules for a commutative ring $k$, we prove an equivalence of categories between (homologically) positively graded dg-categories on the one hand and Frobenius templicial modules on the other hand. This equivalence yields a natural enrichment of the classical dg-nerve, turning dg-categories into quasi-categories in modules. Assuming a projectivity condition, we further prove that a templicial module is a quasi-category in modules precisely when it can be equipped with a nonassociative Frobenius structure.
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