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We relate these equivariant coarse homology theories to coarse algebraic $K$-theory $\\mathcal{X} K^G_{k}$ and to coarse ordinary homology $\\mathcal{X} H^G$ by constructing a trace-like natural transformation $\\mathcal{X} K_{k}^G\\to"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.02849","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2019-07-05T14:24:20Z","cross_cats_sorted":["math.AT","math.MG"],"title_canon_sha256":"2f80c67317271857380879b5f3ac13aec89b929f26e0d3b15fda6d1708252843","abstract_canon_sha256":"8bbf210ea8ec50118ad7a0adad2178c470708ef4aa940a9c4b80f76ab694a874"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:42:52.037416Z","signature_b64":"13wmkOPIjuhCO2ZxvBVRvTy0XgipcrdYRUE/ThYIG7qD/8aQ2koLlUzVxoLSfIndR/NtVG63Dm8L/ZiOekq7Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d0408c656b40a487a913bc3b55ba2c1cab91db0b47818613b5e05ea262f6706c","last_reissued_at":"2026-07-05T01:42:52.036838Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:42:52.036838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cyclic homology for bornological coarse spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.MG"],"primary_cat":"math.KT","authors_text":"Luigi Caputi","submitted_at":"2019-07-05T14:24:20Z","abstract_excerpt":"We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\\mathcal{X}HH_{k}^G$ and $\\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse spaces $G\\mathbf{BornCoarse}$ to the cocomplete stable $\\infty$-category of chain complexes $\\mathbf{Ch}_\\infty$. 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