{"id":"https://openalex.org/W2123595703","doi":"https://doi.org/10.1017/s0960129514000553","title":"Sets in homotopy type theory","display_name":"Sets in homotopy type theory","publication_year":2015,"publication_date":"2015-01-30","ids":{"openalex":"https://openalex.org/W2123595703","doi":"https://doi.org/10.1017/s0960129514000553","mag":"2123595703"},"language":"en","primary_location":{"id":"doi:10.1017/s0960129514000553","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0960129514000553","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/12923C0993730DE101029B542D66F736/S0960129514000553a.pdf/div-class-title-sets-in-homotopy-type-theory-a-href-afn1-ref-type-fn-span-class-sup-span-a-div.pdf","source":{"id":"https://openalex.org/S101766102","display_name":"Mathematical Structures in Computer Science","issn_l":"0960-1295","issn":["0960-1295","1469-8072"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Structures in Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/12923C0993730DE101029B542D66F736/S0960129514000553a.pdf/div-class-title-sets-in-homotopy-type-theory-a-href-afn1-ref-type-fn-span-class-sup-span-a-div.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086377801","display_name":"Egbert Rijke","orcid":"https://orcid.org/0000-0002-5272-6175"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"EGBERT RIJKE","raw_affiliation_strings":["Emails:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Emails:","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043123968","display_name":"Bas Spitters","orcid":"https://orcid.org/0000-0002-2802-0973"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"BAS SPITTERS","raw_affiliation_strings":["Emails:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Emails:","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.0867,"has_fulltext":true,"cited_by_count":21,"citation_normalized_percentile":{"value":0.90165277,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"25","issue":"5","first_page":"1172","last_page":"1202"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9901000261306763,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.6480333805084229},{"id":"https://openalex.org/keywords/type-theory","display_name":"Type theory","score":0.5966055393218994},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5690702795982361},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.5457389950752258},{"id":"https://openalex.org/keywords/homotopy-hypothesis","display_name":"Homotopy hypothesis","score":0.5327622294425964},{"id":"https://openalex.org/keywords/cofibration","display_name":"Cofibration","score":0.5073134303092957},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.41659173369407654},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3641566038131714},{"id":"https://openalex.org/keywords/homotopy-category","display_name":"Homotopy category","score":0.3637078106403351},{"id":"https://openalex.org/keywords/n-connected","display_name":"n-connected","score":0.35440951585769653},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.33342140913009644},{"id":"https://openalex.org/keywords/homotopy-sphere","display_name":"Homotopy sphere","score":0.3241681456565857},{"id":"https://openalex.org/keywords/homotopy-lifting-property","display_name":"Homotopy lifting property","score":0.2215624749660492},{"id":"https://openalex.org/keywords/geology","display_name":"Geology","score":0.07010966539382935}],"concepts":[{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.6480333805084229},{"id":"https://openalex.org/C93682546","wikidata":"https://www.wikidata.org/wiki/Q1056428","display_name":"Type theory","level":3,"score":0.5966055393218994},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5690702795982361},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.5457389950752258},{"id":"https://openalex.org/C164043645","wikidata":"https://www.wikidata.org/wiki/Q16879377","display_name":"Homotopy hypothesis","level":5,"score":0.5327622294425964},{"id":"https://openalex.org/C150275644","wikidata":"https://www.wikidata.org/wiki/Q1048949","display_name":"Cofibration","level":5,"score":0.5073134303092957},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.41659173369407654},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3641566038131714},{"id":"https://openalex.org/C79236096","wikidata":"https://www.wikidata.org/wiki/Q14775269","display_name":"Homotopy category","level":3,"score":0.3637078106403351},{"id":"https://openalex.org/C191752858","wikidata":"https://www.wikidata.org/wiki/Q6358325","display_name":"n-connected","level":3,"score":0.35440951585769653},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.33342140913009644},{"id":"https://openalex.org/C18003601","wikidata":"https://www.wikidata.org/wiki/Q5891833","display_name":"Homotopy sphere","level":4,"score":0.3241681456565857},{"id":"https://openalex.org/C35234519","wikidata":"https://www.wikidata.org/wiki/Q3983973","display_name":"Homotopy lifting property","level":4,"score":0.2215624749660492},{"id":"https://openalex.org/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.07010966539382935},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1017/s0960129514000553","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0960129514000553","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/12923C0993730DE101029B542D66F736/S0960129514000553a.pdf/div-class-title-sets-in-homotopy-type-theory-a-href-afn1-ref-type-fn-span-class-sup-span-a-div.pdf","source":{"id":"https://openalex.org/S101766102","display_name":"Mathematical Structures in Computer Science","issn_l":"0960-1295","issn":["0960-1295","1469-8072"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Structures in Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1305.3835","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1305.3835","pdf_url":"https://arxiv.org/pdf/1305.3835","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.745.82","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.745.82","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1305.3835.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:publications/c5ce9c0a-2ba1-4bfa-a69d-0f3ca587d8b9","is_oa":false,"landing_page_url":"https://pure.au.dk/portal/en/publications/c5ce9c0a-2ba1-4bfa-a69d-0f3ca587d8b9","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Rijke, E & Spitters, B 2015, 'Sets in homotopy type theory', Mathematical Structures in Computer Science, vol. 25, no. 5, pp. 1172-1202. https://doi.org/10.1017/S0960129514000553","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:repository.ubn.ru.nl:2066/141402","is_oa":false,"landing_page_url":"http://hdl.handle.net/2066/141402","pdf_url":null,"source":{"id":"https://openalex.org/S4306401067","display_name":"Radboud Repository (Radboud University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I145872427","host_organization_name":"Radboud University Nijmegen","host_organization_lineage":["https://openalex.org/I145872427"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematical Structures in Computer Science, 25, Special Issue 05, pp. 1172-1202","raw_type":"Article / Letter to editor"}],"best_oa_location":{"id":"doi:10.1017/s0960129514000553","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0960129514000553","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/12923C0993730DE101029B542D66F736/S0960129514000553a.pdf/div-class-title-sets-in-homotopy-type-theory-a-href-afn1-ref-type-fn-span-class-sup-span-a-div.pdf","source":{"id":"https://openalex.org/S101766102","display_name":"Mathematical Structures in Computer Science","issn_l":"0960-1295","issn":["0960-1295","1469-8072"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Structures in Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3356750986","display_name":null,"funder_award_id":"243847","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2123595703.pdf","grobid_xml":"https://content.openalex.org/works/W2123595703.grobid-xml"},"referenced_works_count":61,"referenced_works":["https://openalex.org/W34143232","https://openalex.org/W89881250","https://openalex.org/W200058515","https://openalex.org/W594070437","https://openalex.org/W1481038699","https://openalex.org/W1520893111","https://openalex.org/W1546270920","https://openalex.org/W1560091341","https://openalex.org/W1560253649","https://openalex.org/W1575675278","https://openalex.org/W1579978491","https://openalex.org/W1590132158","https://openalex.org/W1594900487","https://openalex.org/W1647170492","https://openalex.org/W1680171265","https://openalex.org/W1753294933","https://openalex.org/W1773829243","https://openalex.org/W1978777279","https://openalex.org/W1997723626","https://openalex.org/W2029404646","https://openalex.org/W2044712624","https://openalex.org/W2049362364","https://openalex.org/W2054195362","https://openalex.org/W2059204568","https://openalex.org/W2067914884","https://openalex.org/W2097764253","https://openalex.org/W2098691999","https://openalex.org/W2105781220","https://openalex.org/W2111690589","https://openalex.org/W2116438866","https://openalex.org/W2120588128","https://openalex.org/W2120677003","https://openalex.org/W2121213425","https://openalex.org/W2121730419","https://openalex.org/W2130016344","https://openalex.org/W2130779085","https://openalex.org/W2135348665","https://openalex.org/W2148713425","https://openalex.org/W2182177318","https://openalex.org/W2324455841","https://openalex.org/W2395623455","https://openalex.org/W2406975211","https://openalex.org/W2475129850","https://openalex.org/W2490327140","https://openalex.org/W2494316805","https://openalex.org/W2494576483","https://openalex.org/W2502640363","https://openalex.org/W2962780223","https://openalex.org/W3037732692","https://openalex.org/W3102651064","https://openalex.org/W3104826616","https://openalex.org/W3106009785","https://openalex.org/W3149538277","https://openalex.org/W4214621180","https://openalex.org/W4229738979","https://openalex.org/W4230309943","https://openalex.org/W4233061514","https://openalex.org/W4247386353","https://openalex.org/W4249643576","https://openalex.org/W6677302030","https://openalex.org/W6681761935"],"related_works":["https://openalex.org/W121514703","https://openalex.org/W1966906304","https://openalex.org/W3141701452","https://openalex.org/W85903140","https://openalex.org/W2804066429","https://openalex.org/W4214641007","https://openalex.org/W2963362581","https://openalex.org/W4300697885","https://openalex.org/W1936971149","https://openalex.org/W2950590792"],"abstract_inverted_index":{"Homotopy":[0],"type":[1,83,102,137],"theory":[2,84,103],"may":[3,173],"be":[4,88],"seen":[5],"as":[6,30,44,62],"an":[7,117,167],"internal":[8,90],"language":[9,91],"for":[10,34,135,153],"the":[11,20,37,42,57,89,111,114,136,150,175],"\u221e-category":[12],"of":[13,92,94,116,140,144,186],"weak":[14,17,28,45],"\u221e-groupoids.":[15],"Moreover,":[16],"\u221e-groupoids":[18,29,46],"model":[19],"univalence":[21],"axiom.":[22],"Voevodsky":[23],"proposes":[24],"this":[25],"(language":[26],"for)":[27],"a":[31,63,77,105,120,126,130,156,184],"new":[32],"foundation":[33],"Mathematics":[35],"called":[36],"univalent":[38],"foundations.":[39],"It":[40],"includes":[41,54],"sets":[43,99,154],"with":[47],"contractible":[48],"connected":[49],"components,":[50],"and":[51,129,148,179],"thereby":[52],"it":[53],"(much":[55],"of)":[56],"traditional":[58],"set":[59],"theoretical":[60,138],"foundations":[61],"special":[64],"case.":[65],"We":[66,96,122],"thus":[67],"wonder":[68],"whether":[69],"those":[70],"\u2018discrete\u2019":[71],"groupoids":[72],"do":[73],"in":[74,100],"fact":[75,112],"form":[76,104],"(predicative)":[78],"topos.":[79,121],"More":[80],"generally,":[81],"homotopy":[82,101],"is":[85,108,119,155],"conjectured":[86],"to":[87,110],"\u2018elementary\u2019":[93],"\u221e-toposes.":[95],"prove":[97],"that":[98,113,124],"\u03a0W-pretopos.":[106],"This":[107],"similar":[109],"0-truncation":[115],"\u221e-topos":[118],"show":[123],"both":[125,143],"subobject":[127,176],"classifier":[128,132,152,177],"0-object":[131,151],"are":[133,146],"available":[134],"universe":[139],"sets.":[141,165,187],"However,":[142],"these":[145],"large":[147],"moreover":[149],"function":[157],"between":[158,164],"1-types":[159],"(i.e.":[160],"groupoids)":[161],"rather":[162],"than":[163],"Assuming":[166],"impredicative":[168],"propositional":[169],"resizing":[170],"rule":[171],"we":[172,181],"render":[174],"small":[178],"then":[180],"actually":[182],"obtain":[183],"topos":[185]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-18T10:00:31.954636","created_date":"2025-10-10T00:00:00"}
