Higher algebra in $t$-structured tensor triangulated $\infty$-categories

[2603.27786] Higher algebra in $t$-structured tensor triangulated $\infty$-categories Skip to main content Search Submit Donate Log in Mathematics > Category Theory arXiv:2603.27786 (math) [Submitted on 29 Mar 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Higher algebra in $t$-structured tensor triangulated $\infty$-categories Authors: Jiacheng Liang View PDF HTML (experimental) Abstract: We generalize fundamental notions of higher algebra, traditionally developed within the $\infty$-category of spectra, to presentably symmetric monoidal stable $\infty$-categories equipped with compatible accessible $t$-structures, which we call $ttt$-$\infty$-categories. Under a natural structural condition, which we call ``projective rigidity'', we establish higher categorical analogues of Lazard's theorem and prove the existence and universal property of Cohn localizations. Furthermore, we generalize higher almost ring theory to the $ttt$-$\infty$-categorical setting, showing that $\pi_0$-epimorphic idempotent algebras are in natural bijection with idempotent ideals. Additionally, by exploiting deformation theory, we establish a general \text{\text{é}}tale rigidity theorem. Finally, we characterize the moduli of such projectively rigid $ttt$-$\infty$-categories and show that the $\infty$-category of $\mathrm{Sp}$-valued presheaves on the $1$-dimensional framed cobordism $\infty$-category is universal among them. Comments: 93 pages. Comments welcome! v3: Revised following the referee's suggestions. Proofs and exposition have been improved; several questions from the previous version have been resolved and incorporated into the text Subjects: Category Theory (math.CT) ; Algebraic Geometry (math.AG); Algebraic Topology (math.AT) MSC classes: 18N60 (Primary), 18G80, 55P43, 18M05 (Secondary) Report number: MPIM-Bonn-2025 Cite as: arXiv:2603.27786 [math.CT] (or arXiv:2603.27786v3 [math.CT] for this version) https://doi.org/10.48550/arXiv.2603.27786 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jiacheng Liang [ view email ] [v1] Sun, 29 Mar 2026 17:42:38 UTC (90 KB) [v2] Wed, 8 Apr 2026 01:15:02 UTC (92 KB) [v3] Thu, 27 Aug 2026 03:16:49 UTC (95 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CT < prev | next > new | recent | 2026-03 Change to browse by: math math.AG math.AT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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