{ "artifactId": "artifact.library.mathlib.freyd-mitchell-embedding-theorem.v001", "candidateOnly": false, "category": "Category theory", "claimBoundary": "This target indexes Mathlib's existential Freyd–Mitchell endpoint for a category C equipped with an Abelian instance: there exist a ring R and a functor F : C ⥤ ModuleCat R that is full, faithful, preserves finite limits, and preserves finite colimits. The universe of R and the module category is max u v. The statement is an embedding into a module category, not an equivalence onto that whole category.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem CategoryTheory.Abelian.freyd_mitchell (C : Type u) [Category.{v} C] [Abelian C] :\n ∃ (R : Type (max u v)) (_ : Ring R) (F : C ⥤ ModuleCat.{max u v} R),\n F.Full ∧ F.Faithful ∧ PreservesFiniteLimits F ∧ PreservesFiniteColimits F", "family": "Abelian categories and exact embeddings", "id": "library.mathlib.freyd-mitchell-embedding-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/freyd-mitchell-embedding-theorem/", "entryData": "/data/library-theorems/freyd-mitchell-embedding-theorem.json", "evidence": "/proofs/artifact.library.mathlib.freyd-mitchell-embedding-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.freyd-mitchell-embedding-theorem.v001.evidence.json", "source": "/sources/upstream/freyd-mitchell-embedding-theorem/" }, "landmark": { "completeFormalType": "theorem CategoryTheory.Abelian.freyd_mitchell (C : Type u) [Category.{v} C] [Abelian C] :\n ∃ (R : Type (max u v)) (_ : Ring R) (F : C ⥤ ModuleCat.{max u v} R),\n F.Full ∧ F.Faithful ∧ PreservesFiniteLimits F ∧ PreservesFiniteColimits F", "concepts": [ "abelian category", "module category", "full functor", "faithful functor", "finite limits", "finite colimits", "exact embedding" ], "plainLanguage": "Given any abelian category, Mathlib supplies some ring and a way to represent the category inside that ring's module category. Distinct arrows remain distinct, every arrow between represented objects comes from the original category, and finite limit and finite colimit constructions are preserved.", "poster": { "alt": "A dark green and ivory theorem poster sends an abelian category downward into a highlighted proper subdiagram of modules over some ring and states full, faithful, finite-limit, and finite-colimit preservation.", "byteSize": 2008240, "caption": "The endpoint is an exact embedding into modules over some ring, not an equivalence with every module.", "description": "The poster pairs a one-way categorical embedding diagram with short boundary-safe copy. The source category reappears as a highlighted proper subdiagram inside the larger module field; the exact declared properties are listed below, followed by an explicit non-equivalence boundary.", "derivatives": [ { "byteSize": 53814, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/freyd-mitchell-embedding-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:d7127d2077ba3e1fbbbfdbd74098a76f9d0872ac745edb3c10d81f091bc15d59", "width": 640 }, { "byteSize": 101598, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/freyd-mitchell-embedding-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:ed114af47605e085415130eb36b8f1b68ed849a25987b95b8959b3b206faaa55", "width": 1024 } ], "generatedAt": "2026-07-25T20:23:51.485Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/freyd-mitchell-embedding-theorem/theorem-poster-v1.png", "sha256": "sha256:0739181fe689bb732ef2f79a4ea1003174b70922d8ecf2de72b4af71f23d4595", "title": "Freyd–Mitchell at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "FREYD–MITCHELL\nABELIAN CATEGORY\nMODULES OVER SOME RING\nFULL • FAITHFUL\nFINITE LIMITS • FINITE COLIMITS\nEMBEDDING, NOT EQUIVALENCE" }, "visual": { "alt": "A one-way gold functor sends an abstract abelian-category constellation into a highlighted proper subconfiguration of a larger structured module-category field, while paired finite cone and cocone motifs are preserved.", "byteSize": 1760502, "caption": "A full, faithful functor embeds the abelian category into modules over some ring and preserves finite limits and finite colimits.", "description": "The left side is an abstract category of objects and arrows. A single gold arrow carries it into a highlighted image inside a visibly larger module-category field. Paired cone and cocone motifs show the finite limit and finite colimit structure surviving the passage without suggesting equivalence or essential surjectivity.", "derivatives": [ { "byteSize": 33650, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/freyd-mitchell-embedding-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:88d6b1288f66e3926db37391071e443473ae768cea34055658f9a4ce2e26ca22", "width": 640 }, { "byteSize": 73382, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/freyd-mitchell-embedding-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:90c05221db7c0b32640dc12170f55ddcb205c2c2842fe6d5fae31c6308d1d36d", "width": 1200 } ], "generatedAt": "2026-07-25T20:23:51.485Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/freyd-mitchell-embedding-theorem/theorem-schematic-v1.png", "sha256": "sha256:e57ab2e99e21060b4b21cc05f0cf83e3aeadf61bed62a6d657754129962c7ce0", "title": "Freyd–Mitchell embedding schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "The Freyd–Mitchell theorem connects abstract abelian categories with concrete module categories. It explains why many arguments about kernels, cokernels, exact sequences, and diagram chasing can be understood through module-like models without identifying the entire abstract category with every module over a ring." }, "nonClaims": [ "Proof Atlas did not originate the Freyd–Mitchell theorem or Mathlib's declaration.", "The selected theorem does not assert that C is equivalent to the entire category of modules over R.", "The existentially supplied ring is not claimed to be a fixed familiar ring or canonical for every purpose.", "The endpoint states preservation of finite limits and finite colimits, not arbitrary limits and colimits.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.freyd-mitchell-embedding-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Every abelian category admits a full, faithful functor into modules over some ring that preserves finite limits and finite colimits.", "targetId": "target.library.mathlib.freyd-mitchell-embedding-theorem.v001", "title": "Freyd–Mitchell Embedding Theorem", "upstreamOrigin": { "declarationName": "CategoryTheory.Abelian.freyd_mitchell", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/CategoryTheory/Abelian/FreydMitchell.lean#L168", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:5eaa8e43e2116becda23df95559944990d7ff0b8411c8c517e96e0d54a494c79", "sourceByteLength": 7889, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/CategoryTheory/Abelian/FreydMitchell.lean", "sourceLine": 168, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }