{ "model": { "rank": 9526, "code": "thing-q215111", "model_id": "vr.tr.homomorphism", "name": "homomorphism", "purpose": "Let an agent define homomorphisms for various structures, explain kernels, images and isomorphism theorems, verify whether a map is a homomorphism, and relate the concept to category theory.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q215111/", "tier": 2, "score": 47, "payload": { "layer": "wikidata", "aliases": [ "endomorphism", "quotient mapping", "ring homomorphism", "rng homomorphism", "chain map", "affine map", "bimodule homomorphism", "field homomorphism", "augmentation", "module homomorphism", "additive map", "graph homomorphism", "algebra homomorphism", "group homomorphism", "group representation", "nilpotent endomorphism", "ring endomorphism", "Frobenius endomorphism", "formally smooth map", "genus of a multiplicative sequence", "semilinear map", "Boustrophedon transform", "additive function", "One-parameter group", "isogeny", "multiplicative character", "J-homomorphism", "Lie group homomorphism", "projective representation", "group isomorphism", "unitary representation", "reducible representation", "undecomposable representation", "spin representation", "representation on coordinate rings", "dual representation", "Induced representation", "representation of the quaternion group", "cuspidal representation", "subrepresentation" ], "aliasCount": 58, "merged": 58, "knownIn": 47, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q215111", "url": "https://www.wikidata.org/wiki/Q215111", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T21:39:36Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q215111", "registryId": "vr.tr.homomorphism", "name": "homomorphism", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q215111/", "model": { "registry_id": "vr.tr.homomorphism", "name": "homomorphism", "purpose": "Let an agent define homomorphisms for various structures, explain kernels, images and isomorphism theorems, verify whether a map is a homomorphism, and relate the concept to category theory.", "definition": "A structure-preserving map between two algebraic structures of the same kind, such as groups, rings, vector spaces or modules, sending operations in one to the corresponding operations in the other, including special cases such as endomorphisms, isomorphisms, quotient maps, ring homomorphisms and chain maps; homomorphisms are the morphisms of algebraic categories and underpin much of abstract algebra.", "what_it_is_for": "Structure-preserving maps in algebra.", "affordances": [ "define homomorphisms for a given structure", "check whether a map is a homomorphism", "explain kernels, images and isomorphism theorems", "relate to category theory" ], "distinguishing_features": [ "Structure-preserving", "Composable", "Has kernel and image", "Category-theoretic morphism" ], "appearance": "Not physical; a function between algebraic structures.", "visual_identification": [ "Preserves operations", "Between structures of the same kind", "An arbitrary function need not preserve structure" ], "physical_properties": [], "families_and_kinds": [ "group homomorphisms", "ring and rng homomorphisms", "linear and affine maps", "module and algebra homomorphisms", "endomorphisms, isomorphisms and quotient maps" ], "related_models": [ { "relation": "is a kind of", "target": "function", "why": "category" }, { "relation": "is related to", "target": "group", "why": "a source structure" }, { "relation": "is related to", "target": "ordered pair", "why": "functions as sets of pairs" }, { "relation": "is related to", "target": "complex conjugate", "why": "an example of a ring automorphism" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical notation conventions" ], "failure_modes_and_hazards": [ "Forgetting to check preservation of all operations", "Confusing homomorphism and homeomorphism", "Ignoring identity preservation for rings" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "What a homomorphism is.", "rationale": "Definitions.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition by structure.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What must a map preserve to be a homomorphism of groups, rings, vector spaces or modules?", "kind": "definition" }, { "text": "What are endomorphisms, automorphisms and isomorphisms?", "kind": "definition" } ] } ] }, { "id": "check", "name": "Check", "description": "Checking a map.", "findings": [ { "id": "check-finding", "name": "Check", "description": "Checking.", "questions": [ { "text": "Is this given map a homomorphism, and what is its kernel and image?", "kind": "action" }, { "text": "Is it injective, surjective or bijective?", "kind": "action" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Key results.", "rationale": "Theorems.", "layers": [ { "id": "theorems", "name": "Theorems", "description": "Isomorphism theorems.", "findings": [ { "id": "theorems-finding", "name": "Theorems", "description": "Theorems.", "questions": [ { "text": "What do the isomorphism theorems say, and how do quotient maps arise?", "kind": "definition" }, { "text": "How is a quotient structure constructed from a kernel?", "kind": "definition" } ] } ] }, { "id": "category", "name": "Category", "description": "Category theory view.", "findings": [ { "id": "category-finding", "name": "Category", "description": "Category theory.", "questions": [ { "text": "How are homomorphisms the morphisms of algebraic categories, and what are functors between them?", "kind": "definition" }, { "text": "What are chain maps?", "kind": "definition" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Uses.", "rationale": "Applications.", "layers": [ { "id": "examples", "name": "Examples", "description": "Important examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "Which homomorphisms are fundamental, such as determinant, exponential map or evaluation maps?", "kind": "definition" }, { "text": "How are affine maps related to linear maps?", "kind": "definition" } ] } ] }, { "id": "computing", "name": "Computing", "description": "Homomorphisms in computing.", "findings": [ { "id": "computing-finding", "name": "Computing", "description": "Computing.", "questions": [ { "text": "How do homomorphisms appear in cryptography, such as homomorphic encryption, and in coding theory?", "kind": "definition" }, { "text": "Which entry fits those topics?", "kind": "action" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Teaching.", "rationale": "Education.", "layers": [ { "id": "teach", "name": "Teach", "description": "Teaching homomorphisms.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can homomorphisms be taught with concrete examples before abstraction?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] }, { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did the concept of homomorphism develop in nineteenth and twentieth century algebra?", "kind": "provenance" }, { "text": "Who introduced the terminology?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should each structure homomorphism be a separate entry?", "How should textbooks be linked?", "How should proof assistants be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a homomorphism, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent define homomorphisms for various structures, explain kernels, images and isomorphism theorems, verify whether a map is a homomorphism, and relate the concept to category theory.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: endomorphism, quotient mapping, ring homomorphism, rng homomorphism, chain map, affine map, bimodule homomorphism, field homomorphism, augmentation, module homomorphism, additive map, graph homomorphism.", "58 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 47 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a homomorphism, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "endomorphism", "quotient mapping", "ring homomorphism", "rng homomorphism", "chain map", "affine map", "bimodule homomorphism", "field homomorphism", "augmentation", "module homomorphism", "additive map", "graph homomorphism" ] } ], "standing": "Described in 47 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 58 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one homomorphism from another, and a homomorphism from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a homomorphism sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a homomorphism from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a homomorphism passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a homomorphism between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a homomorphism, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a homomorphism is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What homomorphism is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of homomorphism are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is homomorphism operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of homomorphism and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of homomorphism?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as homomorphism?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a homomorphism came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a homomorphism and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a homomorphism is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a homomorphism, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a homomorphism, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a homomorphism is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }