{"paper":{"title":"Polygonic spectra and TR with coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Achim Krause, Jonas McCandless, Thomas Nikolaus","submitted_at":"2023-02-15T14:23:41Z","abstract_excerpt":"We introduce the notion of a polygonic spectrum which is designed to axiomatize the structure on topological Hochschild homology $\\mathrm{THH}(R,M)$ of an $\\mathbb{E}_1$-ring $R$ with coefficients in an $R$-bimodule $M$. For every polygonic spectrum $X$, we define a spectrum $\\mathrm{TR}(X)$ as the mapping spectrum from the polygonic version of the sphere spectrum $\\mathbb{S}$ to $X$. In particular if applied to $X = \\mathrm{THH}(R,M)$ this gives a conceptual definition of $\\mathrm{TR}(R,M)$.\n  Every cyclotomic spectrum gives rise to a polygonic spectrum and we prove that TR agrees with the cl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.07686","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2302.07686/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}