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An application of this result is the $\\textsf{fp}$-dual formula which states that for any covariant finitely presented functor $F$, $F^*\\cong \\big(\\hspace{0.05cm}\\underline{\\ \\ }\\hspace{0.1cm} , w(F)\\big)$. The defect sequence is shown to be isomorphic to both the double dual sequence $$0\\long"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.0054","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2012-10-31T23:26:43Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"7fef46ae258355705ebd6e79ba06269c3a2eeaa8a2c82da92005be43ed3e135f","abstract_canon_sha256":"3f58919d3691091c6dc1103817ee7bd5ad2a04f5e456478e2a4661d38938fa44"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:10.078036Z","signature_b64":"DhdJKlUVoIEbs3X5ucsBoftwkqzl71vcwzMrBATnJlfGhYsHYRCJ3Esn0dLApUN98uL85qin2xcukIZ7rCY5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48a7fb4dacdd613cda667abf14413fdbf804c0c873d299c66101e3a9d136fe80","last_reissued_at":"2026-05-18T01:13:10.077675Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:10.077675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Applications of the Defect of a Finitely Presented Functor","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CT","authors_text":"Jeremy Russell","submitted_at":"2012-10-31T23:26:43Z","abstract_excerpt":"For an abelian category $\\mathcal{A}$, the defect sequence $$0\\longrightarrow F_0\\longrightarrow F\\overset{\\varphi}{\\longrightarrow} \\big(w(F),\\hspace{0.05cm}\\underline{\\ \\ }\\hspace{0.1cm} \\big)\\longrightarrow F_1\\longrightarrow 0$$ of a finitely presented functor is used to establish the CoYoneda Lemma. An application of this result is the $\\textsf{fp}$-dual formula which states that for any covariant finitely presented functor $F$, $F^*\\cong \\big(\\hspace{0.05cm}\\underline{\\ \\ }\\hspace{0.1cm} , w(F)\\big)$. 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