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Frobenius templicial modules and the dg-nerve

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arxiv 2005.04778 v4 pith:YOMYDNCA submitted 2020-05-10 math.CT

Wendy Lowen , Arne Mertens This is my paper · ORCID
classification math.CT
keywords modulestemplicialfrobeniusobjectsdg-categoriesdg-nerveequivalencehand
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abstract

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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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The category of necklaces is a test category

    math.CT 2026-07 accept novelty 6.0 of 10

    The category of necklaces is a test category, so its presheaf category is a model for homotopy types.

  2. Templicial nerve of an A-infinity category

    math.CT 2024-11 conditional novelty 6.0 of 10

    The authors define a templicial A-infinity nerve functor that lifts Faonte's simplicial A-infinity nerve to vector-space-enriched templicial objects and prove it is a quasi-category in vector spaces.

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