Left exact monoidal localizations from tidy maps

[2606.04263] Left exact monoidal localizations from tidy maps Skip to main content Search Submit Donate Log in Mathematics > Algebraic Topology arXiv:2606.04263 (math) [Submitted on 2 Jun 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Left exact monoidal localizations from tidy maps Authors: Mathieu Anel , Georg Biedermann , Eric Finster , André Joyal View PDF HTML (experimental) Abstract: We put Goodwillie's calculus of functors and Weiss' orthogonal calculus in a unified framework. We do so in two ways. On the one hand, the relevant categories are all symmetric monoidal and controlled by their compact objects. We introduce the notion of tidy map as a means to generate symmetric monoidal localizations in this setting. These localizations are always left exact. Then we show that both the Goodwillie and Weiss towers are generated by such maps. On the other hand, the relevant categories are also topoi, for which there is a general theory of completion towers of left exact localizations. We had shown in a previous work that the Goodwillie tower is an instance a such a tower. We show here that the Weiss tower is a completion tower as well, and therefore that the general theory applies to orthogonal calculus. Comments: v2: submitted version Subjects: Algebraic Topology (math.AT) ; Category Theory (math.CT) Cite as: arXiv:2606.04263 [math.AT] (or arXiv:2606.04263v2 [math.AT] for this version) https://doi.org/10.48550/arXiv.2606.04263 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mathieu Anel [ view email ] [v1] Tue, 2 Jun 2026 22:30:21 UTC (41 KB) [v2] Thu, 27 Aug 2026 10:11:35 UTC (41 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AT < prev | next > new | recent | 2026-06 Change to browse by: math math.CT References & Citations SciX 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)

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