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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.29815 |
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| _version_ | 1866911628878938112 |
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| author | Blom, Thomas Loubaton, Félix Ruit, Jaco |
| author_facet | Blom, Thomas Loubaton, Félix Ruit, Jaco |
| contents | We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29815 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Day convolution for algebraic patterns Blom, Thomas Loubaton, Félix Ruit, Jaco Category Theory Algebraic Topology 18N70 We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations. |
| title | Day convolution for algebraic patterns |
| topic | Category Theory Algebraic Topology 18N70 |
| url | https://arxiv.org/abs/2603.29815 |