{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.indexed-family","code":"thing-q856215","url":"https://ver.cy/models/thing/q856215/","name":"indexed family","alternateNames":["Stochastic Flow","natural transformation","parametric family","family of sets","natural isomorphism","cohomology operation","parametrized function family","laminar set family","cumulative hierarchy","set system","greedoid","pointclass"],"kind":"thing","status":"research-draft","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"In mathematics, a collection of objects each labelled by an element of an index set, formally a function from the index set to the objects, written as x sub i for i in I; indexed families generalise sequences and tuples, allow repeated members, and are the standard way to write families of sets, parametric families of functions and distributions, and collections indexed by arbitrary sets.","purpose":"Let an agent define indexed families precisely, distinguish them from sets, sequences and tuples, explain notation and operations such as unions and products over families, and route specialised uses such as natural transformations to their own entries.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Indexed by a set","Repetition allowed","Generalises sequences and tuples","Standard in set theory and analysis"],"structure":{"bundles":[{"id":"define","name":"Define","description":"What an indexed family is.","layers":[{"id":"definition","name":"Definition","description":"Definition and notation.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is an indexed family, and how is it written and read?","kind":"definition"},{"text":"Is the question about a family, a set, a sequence or a tuple?","kind":"boundary"}]}]},{"id":"distinctions","name":"Distinctions","description":"Distinctions.","findings":[{"id":"distinctions-finding","name":"Distinctions","description":"Distinctions.","questions":[{"text":"How does a family differ from the set of its members and from a sequence?","kind":"definition"},{"text":"Which entry fits sequences?","kind":"action"}]}]}]},{"id":"operate","name":"Operate","description":"Operations over families.","layers":[{"id":"operations","name":"Operations","description":"Unions, products, sums.","findings":[{"id":"operations-finding","name":"Operations","description":"Operations.","questions":[{"text":"How are unions, intersections, products and sums taken over families?","kind":"definition"},{"text":"Which entry fits Cartesian product?","kind":"action"}]}]},{"id":"choice","name":"Choice","description":"Axiom of choice.","findings":[{"id":"choice-finding","name":"Choice","description":"Choice.","questions":[{"text":"How do families of sets relate to the axiom of choice?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"use","name":"Use","description":"Uses across mathematics.","layers":[{"id":"analysis","name":"Analysis","description":"Parametric families.","findings":[{"id":"analysis-finding","name":"Analysis","description":"Analysis.","questions":[{"text":"How are parametric families of functions and distributions used?","kind":"provenance"},{"text":"Which entry fits parametric statistics?","kind":"action"}]}]},{"id":"category","name":"Category","description":"Category theory.","findings":[{"id":"category-finding","name":"Category","description":"Category.","questions":[{"text":"How do indexed families appear in category theory, and how do natural transformations differ?","kind":"provenance"},{"text":"Which entry fits natural transformation?","kind":"action"}]}]}]},{"id":"learn","name":"Learn","description":"Teaching.","layers":[{"id":"teach","name":"Teach","description":"Teaching.","findings":[{"id":"teach-finding","name":"Teach","description":"Teaching.","questions":[{"text":"How are indexed families introduced in courses, and which misconceptions arise?","kind":"provenance"},{"text":"Which entry fits set theory?","kind":"action"}]}]},{"id":"notation","name":"Notation","description":"Notation.","findings":[{"id":"notation-finding","name":"Notation","description":"Notation.","questions":[{"text":"What notational conventions apply, and where do they vary?","kind":"provenance"},{"text":"Which entry fits mathematical notation?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Define indexed families and work examples with stated index sets.","Explain how families differ from sets, sequences and tuples."],"mustNot":["Confuse a family with the set of its members.","Ignore repeated members when counting or comparing.","Mix up the index set and the set of values.","Present notation from one text as universal."],"requiresHuman":["Approving assessed mathematical work where marks have consequences."]},"ethics":{"considerations":["Clear definitions prevent errors that propagate into proofs and software."],"affectedParties":["Learners","Mathematicians"]},"owners":{"steward":"Nobody owns the concept; it belongs to the shared body of mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q11348","type":"parent","note":"registry parent class"},{"target":"function","type":"related","note":"from an index set"},{"target":"sequence","type":"parent","note":"indexed by natural numbers"},{"target":"set theory","type":"related","note":"families of sets"},{"target":"parametric model","type":"related","note":"parametric families in statistics"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.indexed-family","Wikidata: Q856215 (https://www.wikidata.org/wiki/Q856215)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Function from an index set to objects","Members may repeat; order comes from the index","A set has no indices; a sequence is indexed by natural numbers","Not visible; notation with subscripts over an index set."]},"capabilities":{"applicability":"required","items":["define and use the notation","distinguish families from sets and sequences","explain operations over families","route specialised uses"]},"hazards":{"applicability":"required","items":["Confusing a family with the set of its members","Ignoring repeated members","Mixing the index set with the members"]},"interfaces":{"applicability":"required","items":["No regulation; notation follows ISO 80000-2 and standard texts"]},"context":{"applicability":"required","items":["Organising collections labelled by an index.","families of sets","sequences and tuples as special cases","parametric families of functions and distributions","families indexed by arbitrary sets","families in category theory such as natural transformations, a related use"]}},"sources":[{"title":"Wikidata item Q856215: indexed family","url":"https://www.wikidata.org/wiki/Q856215","note":"identity and sense of the item"},{"title":"Wikipedia: Indexed family","url":"https://en.wikipedia.org/wiki/Indexed_family","note":"general description of the item"}],"openQuestions":["Should family of sets be a separate entry?","How should notation standards be linked?","The registry entry has merged aliases for natural transformations and stochastic flows, which are distinct concepts; should they be split off?"],"resources":{"spec":"/models/things/publications/thing-q856215/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q856215/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T20:57:04Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}