Abstract:We introduce rigid algebras, a generalization of rigid categories to arbitrary symmetric monoidal $(\infty,2)$-categories. We develop their general theory, showing in particular that the a priori $(\infty,2)$-category of rigid algebras is in fact an $(\infty,1)$-category. For the $(\infty,2)$-category of cospans in an $(\infty,1)$-category $\mathcal{C}$, we show that the $(\infty,1)$-category of rigid commutative algebras is canonically identified with $\mathcal{C}$. This identification is used to construct an adjunction between the cospan construction and the functor assigning to a symmetric monoidal $(\infty,2)$-category its $(\infty,1)$-category of rigid commutative algebras.
| Comments: | After referee comments |
| Subjects: | Category Theory (math.CT); Algebraic Topology (math.AT) |
| Cite as: | arXiv:2506.22072 [math.CT] |
| (or arXiv:2506.22072v2 [math.CT] for this version) | |
| https://doi.org/10.48550/arXiv.2506.22072 arXiv-issued DOI via DataCite |
|
| Journal reference: | International Mathematics Research Notices, Volume 2026, Issue 10, May 2026, rnag095 |
| Related DOI: | https://doi.org/10.1093/imrn/rnag095
DOI(s) linking to related resources |
From: Leor Neuhauser [view email]
[v1]
Fri, 27 Jun 2025 10:01:13 UTC (208 KB)
[v2]
Fri, 22 May 2026 07:31:28 UTC (208 KB)
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