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The functor $M^{GW}_{eff}\\colon Sm_k\\to \\mathbf{DM}^{GW}_{\\mathrm{eff},-}(k)$ of Grothendieck-Witt-motives of smooth varieties is computed and it is proved that for any smooth scheme $X$ and homotopy invariant sheave with GW-transfers $F$ $$ Hom_{\\mathbf{DM}^{GW}_{\\mathrm{eff},-}(k)}(M^{GW}_{ef"},"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":"1709.06273","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-09-19T07:16:39Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"523a62c7eef7b6c6a964b606348a00eb29b68a7e5434a7c658ce1cd85ad9ab82","abstract_canon_sha256":"ab984b778daef13a3306937f43b620eab8d3bf1caf215e10470d2178123668f6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:53.499745Z","signature_b64":"GiMLzLi5xNCtoTR2AJwuW2Cy7fwqrivilz/PhzB6vUbPi/9GWc8WM7BmTcxhf4ZwbmFk05itNc2pQesvn+2FAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9f89a1be45f9b382afc01d0a0233e0291e267a3e5e227495a35af27a861f957","last_reissued_at":"2026-05-18T00:19:53.498963Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:53.498963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Effective Grothendieck-Witt motives of smooth varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AG","authors_text":"Andrei Druzhinin","submitted_at":"2017-09-19T07:16:39Z","abstract_excerpt":"The category of effective Grothendieck-Witt-motives $\\mathbf{DM}^{GW}_{\\mathrm{eff},-}(k)$ (and Witt-motives $\\mathbf{DM}^W_{\\mathrm{eff},-}(k)$) by Voevodsky-Suslin method starting with some category of GW-correspondences (and Witt-correspondences) over a perfect field $k$, $char\\,k\\neq 2$, is defined. 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