{ "model": { "rank": 6930, "code": "thing-q229102", "model_id": "vr.tr.surjective-function", "name": "surjective function", "purpose": "Let an agent define surjective functions, relay properties, examples and theorems, explain their role across mathematics, and distinguish surjections from injections and bijections.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q229102/", "tier": 2, "score": 58, "payload": { "layer": "wikidata", "aliases": [ "surjective homomorphism", "unitary operator", "fibered manifold" ], "aliasCount": 3, "merged": 3, "knownIn": 58, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q229102", "url": "https://www.wikidata.org/wiki/Q229102", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T02:10:35Z", "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": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q229102", "registryId": "vr.tr.surjective-function", "name": "surjective function", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q229102/", "model": { "registry_id": "vr.tr.surjective-function", "name": "surjective function", "purpose": "Let an agent define surjective functions, relay properties, examples and theorems, explain their role across mathematics, and distinguish surjections from injections and bijections.", "definition": "A function from a set to another such that every element of the target is the image of at least one element of the domain, also called onto, one of the basic properties of functions alongside injectivity and bijectivity, appearing in algebra as surjective homomorphisms, in analysis as unitary operators which are bijective isometries, and in geometry as the projections of fibred manifolds; surjectivity is used to define quotients, cardinality comparisons and factorisation of maps.", "what_it_is_for": "Not applicable; a property of functions.", "affordances": [ "define and give examples", "relay properties and theorems", "explain roles across fields", "distinguish from related properties" ], "distinguishing_features": [ "Image equals codomain", "Right inverse under choice", "Quotient constructions", "Epimorphism in categories" ], "appearance": "A mapping diagram in which every target element receives an arrow.", "visual_identification": [ "Function whose image is the whole codomain", "Onto maps, surjective homomorphisms, projections", "Injections map distinct elements to distinct images; bijections are both" ], "physical_properties": [], "families_and_kinds": [ "surjective functions between sets", "surjective homomorphisms in algebra", "surjective linear maps and unitary operators as special cases", "projections of fibred manifolds and bundles", "surjective continuous maps and quotient maps", "epimorphisms in category theory" ], "related_models": [ { "relation": "is a kind of", "target": "function", "why": "in registry terms" }, { "relation": "is a kind of", "target": "surjective relation", "why": "in registry terms" }, { "relation": "is contrasted with", "target": "injective function", "why": "the one-to-one property" }, { "relation": "combines with injectivity to give", "target": "bijection", "why": "the invertible case" } ], "identifiers": [], "standards_and_regulation": [ "No regulation; mathematical conventions" ], "failure_modes_and_hazards": [ "Confusing onto with one-to-one", "Forgetting codomain dependence", "Conflating unrelated examples" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "Definition.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and examples.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a surjective function, and what are standard examples and non-examples?", "kind": "definition" }, { "text": "Is the property surjectivity, injectivity or bijectivity?", "kind": "boundary" } ] } ] }, { "id": "properties", "name": "Properties", "description": "Properties.", "findings": [ { "id": "properties-finding", "name": "Properties", "description": "Properties.", "questions": [ { "text": "What properties do surjections have, including right inverses and composition?", "kind": "definition" }, { "text": "Which entry fits bijection?", "kind": "action" } ] } ] } ] }, { "id": "fields", "name": "Fields", "description": "Across mathematics.", "rationale": "Mathematics.", "layers": [ { "id": "algebra", "name": "Algebra", "description": "Algebra.", "findings": [ { "id": "algebra-finding", "name": "Algebra", "description": "Algebra.", "questions": [ { "text": "How do surjective homomorphisms and the isomorphism theorems relate to quotients?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "geometry", "name": "Geometry", "description": "Analysis and geometry.", "findings": [ { "id": "geometry-finding", "name": "Geometry", "description": "Geometry.", "questions": [ { "text": "How do unitary operators and fibred manifold projections involve surjectivity?", "kind": "provenance" }, { "text": "Which entry fits fibered manifold?", "kind": "action" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Foundations.", "rationale": "Mathematics.", "layers": [ { "id": "cardinality", "name": "Cardinality", "description": "Cardinality.", "findings": [ { "id": "cardinality-finding", "name": "Cardinality", "description": "Cardinality.", "questions": [ { "text": "How are surjections used to compare cardinalities, and how does the axiom of choice enter?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] }, { "id": "category", "name": "Category", "description": "Category theory.", "findings": [ { "id": "category-finding", "name": "Category", "description": "Category.", "questions": [ { "text": "How do epimorphisms generalise surjections, and when do they differ?", "kind": "provenance" }, { "text": "Which entry fits epimorphism?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Teaching and terminology.", "rationale": "Context.", "layers": [ { "id": "teaching", "name": "Teaching", "description": "Teaching.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching.", "questions": [ { "text": "How is surjectivity taught, and what misconceptions arise?", "kind": "provenance" }, { "text": "Which entry fits mathematics education?", "kind": "action" } ] } ] }, { "id": "terminology", "name": "Terminology", "description": "Terminology.", "findings": [ { "id": "terminology-finding", "name": "Terminology", "description": "Terminology.", "questions": [ { "text": "Where do the terms surjective and onto come from, and how did Bourbaki standardise them?", "kind": "provenance" }, { "text": "Which entry fits the history of mathematical notation?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should bijection and epimorphism be the primary linked entries?", "How should reference sources be linked?", "The registry entry has merged aliases naming special cases; should they be split off?" ], "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 surjective function, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent define surjective functions, relay properties, examples and theorems, explain their role across mathematics, and distinguish surjections from injections and bijections.", "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: surjective homomorphism, unitary operator, fibered manifold.", "3 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 58 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 surjective function, 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": [ "surjective homomorphism", "unitary operator", "fibered manifold" ] } ], "standing": "Described in 58 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 3 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 surjective function from another, and a surjective function 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 surjective function, 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 surjective function, 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 surjective function 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 surjective function, 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 surjective function, 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 surjective function 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 surjective function 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 surjective function, 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 surjective function, 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 surjective function 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 surjective function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a surjective function, 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 surjective function, 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 surjective function 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 surjective function 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 surjective function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a surjective function, 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 surjective function, 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 surjective function 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 surjective function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a surjective function 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 surjective function, 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 surjective function, 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 surjective function 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 surjective function 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 surjective function 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 surjective function 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 surjective function 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 surjective function?", "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 surjective function?", "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 surjective function 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 surjective function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a surjective function, 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 surjective function, 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 surjective function 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 surjective function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a surjective function, 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 surjective function, 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 surjective function 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 } } } }