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We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp.,"},"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":"1101.4803","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2011-01-25T13:07:59Z","cross_cats_sorted":["math.CT","math.RA"],"title_canon_sha256":"4bb690e81b7eac9f1588a56bcfa7769eb8572b01da39209801ec206d267b7e1c","abstract_canon_sha256":"9494804bd4f34fe8fc4fe27779c68dde75ee8a66b4d46f7320ee2c41647278ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:37.848615Z","signature_b64":"nJVOCBAJvZhB/AlO/Aj1rUHaLdfKsD4vtvsFRFRUFOqsZ7Vi3T99IuZHX2X4opr+okD8Fx0iab2dAKhW5o5ZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39cbf58ff6d020e16a38f5dbf0218f926c0245081cc7ff47e3c9b560e87224fe","last_reissued_at":"2026-05-18T03:29:37.847817Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:37.847817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.RA"],"primary_cat":"math.OA","authors_text":"Friedrich Wehrung (LMNO)","submitted_at":"2011-01-25T13:07:59Z","abstract_excerpt":"The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. 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