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When $G$ is a compact Lie group, we show that the decomposition is $G$-equivariant with respect to the induced action of conjugation by elements of $G$. The spaces $Hom(L_n,G)$ assemble into a simplicial space $Hom(L,G)$. When $G=U$ we show that its geometric realization $B(L,U)$, has a non-unital $E_\\infty$-ring space structure whenever $Hom(L_0,U(m))$ is path connected for all $m\\geq1$."},"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":"1601.04688","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-18T20:47:46Z","cross_cats_sorted":[],"title_canon_sha256":"f7702c482386785fa98a6d05aed305e2f2f727263b3a944d89628d133307936d","abstract_canon_sha256":"900b4ba8cb1e28ca8a13e3d2917ffcfa240976cf27a0cc7b2d94e5e7cf406bb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:59.889431Z","signature_b64":"FmiN/v9NSvBHxRIqLDOiuaiRYLQst8wKyf1ma2FreUxMCBGjOjf8WSsj9q1UGe4OcN+V27zdVY2XqKwRqvxLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b8bbb0ae72253fe0d32bce7a87a34eb13427fb7dfe795d063ba990a7a650b25","last_reissued_at":"2026-05-18T00:20:59.888824Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:59.888824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cosimplicial Groups and Spaces of Homomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Bernardo Villarreal","submitted_at":"2016-01-18T20:47:46Z","abstract_excerpt":"Let $G$ be a real linear algebraic group and $L$ a finitely generated cosimplicial group. 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