{"id":"https://openalex.org/W3156224329","doi":"https://doi.org/10.1080/10586458.2022.2062073","title":"Simple Type Theory is not too Simple: Grothendieck\u2019s Schemes Without Dependent Types","display_name":"Simple Type Theory is not too Simple: Grothendieck\u2019s Schemes Without Dependent Types","publication_year":2022,"publication_date":"2022-04-03","ids":{"openalex":"https://openalex.org/W3156224329","doi":"https://doi.org/10.1080/10586458.2022.2062073","mag":"3156224329"},"language":"en","primary_location":{"id":"doi:10.1080/10586458.2022.2062073","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2022.2062073","pdf_url":"https://www.tandfonline.com/doi/pdf/10.1080/10586458.2022.2062073?needAccess=true","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.tandfonline.com/doi/pdf/10.1080/10586458.2022.2062073?needAccess=true","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031967042","display_name":"Anthony Bordg","orcid":"https://orcid.org/0000-0003-1694-9467"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Anthony Bordg","raw_affiliation_strings":["University of Cambridge, Cambridge, UK"],"affiliations":[{"raw_affiliation_string":"University of Cambridge, Cambridge, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086565312","display_name":"Lawrence C. Paulson","orcid":"https://orcid.org/0000-0003-0288-4279"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Lawrence Paulson","raw_affiliation_strings":["University of Cambridge, Cambridge, UK"],"affiliations":[{"raw_affiliation_string":"University of Cambridge, Cambridge, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100761702","display_name":"Wenda Li","orcid":"https://orcid.org/0000-0002-9886-9542"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Wenda Li","raw_affiliation_strings":["University of Cambridge, Cambridge, UK"],"affiliations":[{"raw_affiliation_string":"University of Cambridge, Cambridge, UK","institution_ids":["https://openalex.org/I241749"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5031967042"],"corresponding_institution_ids":["https://openalex.org/I241749"],"apc_list":null,"apc_paid":null,"fwci":1.242,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.82132785,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"31","issue":"2","first_page":"364","last_page":"382"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9941999912261963,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9941999912261963,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9886999726295471,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9775000214576721,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.896882951259613},{"id":"https://openalex.org/keywords/hol","display_name":"HOL","score":0.7190269231796265},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.7151415944099426},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6987264752388},{"id":"https://openalex.org/keywords/type-theory","display_name":"Type theory","score":0.6110202670097351},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.5444300770759583},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.43934139609336853},{"id":"https://openalex.org/keywords/proof-assistant","display_name":"Proof assistant","score":0.4319199323654175},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.39045971632003784},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.34049075841903687},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.29234999418258667},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.13363656401634216},{"id":"https://openalex.org/keywords/epistemology","display_name":"Epistemology","score":0.0931348204612732},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07002881169319153},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.05938038229942322}],"concepts":[{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.896882951259613},{"id":"https://openalex.org/C17435882","wikidata":"https://www.wikidata.org/wiki/Q17030435","display_name":"HOL","level":2,"score":0.7190269231796265},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.7151415944099426},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6987264752388},{"id":"https://openalex.org/C93682546","wikidata":"https://www.wikidata.org/wiki/Q1056428","display_name":"Type theory","level":3,"score":0.6110202670097351},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5444300770759583},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.43934139609336853},{"id":"https://openalex.org/C203265346","wikidata":"https://www.wikidata.org/wiki/Q11387554","display_name":"Proof assistant","level":3,"score":0.4319199323654175},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.39045971632003784},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34049075841903687},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29234999418258667},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.13363656401634216},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0931348204612732},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07002881169319153},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.05938038229942322},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1080/10586458.2022.2062073","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2022.2062073","pdf_url":"https://www.tandfonline.com/doi/pdf/10.1080/10586458.2022.2062073?needAccess=true","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2104.09366","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2104.09366","pdf_url":"https://arxiv.org/pdf/2104.09366","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":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:www.repository.cam.ac.uk:1810/342875","is_oa":true,"landing_page_url":"https://www.repository.cam.ac.uk/handle/1810/342875","pdf_url":null,"source":{"id":"https://openalex.org/S4306401777","display_name":"Apollo (University of Cambridge)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I241749","host_organization_name":"University of Cambridge","host_organization_lineage":["https://openalex.org/I241749"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.17863/cam.90290","is_oa":true,"landing_page_url":"https://doi.org/10.17863/cam.90290","pdf_url":null,"source":{"id":"https://openalex.org/S7407050737","display_name":"Apollo","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1080/10586458.2022.2062073","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2022.2062073","pdf_url":"https://www.tandfonline.com/doi/pdf/10.1080/10586458.2022.2062073?needAccess=true","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3156224329.pdf","grobid_xml":"https://content.openalex.org/works/W3156224329.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W1464569014","https://openalex.org/W1532290481","https://openalex.org/W1579074775","https://openalex.org/W1581378641","https://openalex.org/W1755481569","https://openalex.org/W1768814311","https://openalex.org/W1975906542","https://openalex.org/W1986402635","https://openalex.org/W1996404651","https://openalex.org/W2061969020","https://openalex.org/W2106192381","https://openalex.org/W2174838357","https://openalex.org/W2395749688","https://openalex.org/W2596181237","https://openalex.org/W2806477068","https://openalex.org/W2978388704","https://openalex.org/W2993571361","https://openalex.org/W3039393788","https://openalex.org/W3043060850","https://openalex.org/W3155258486","https://openalex.org/W3159761404","https://openalex.org/W3217786894","https://openalex.org/W4248269519","https://openalex.org/W4248797717","https://openalex.org/W4250846042","https://openalex.org/W6646868996","https://openalex.org/W6711936225","https://openalex.org/W6794823504"],"related_works":["https://openalex.org/W4315630564","https://openalex.org/W4295798319","https://openalex.org/W1819460135","https://openalex.org/W3098927713","https://openalex.org/W1516654960","https://openalex.org/W4289109288","https://openalex.org/W183532729","https://openalex.org/W2886069053","https://openalex.org/W4249057168","https://openalex.org/W4386350920"],"abstract_inverted_index":{"Church\u2019s":[0],"simple":[1,8,58],"type":[2,59],"theory":[3,60],"is":[4],"often":[5],"deemed":[6],"too":[7],"for":[9,97],"elaborate":[10],"mathematical":[11,34],"constructions.":[12],"In":[13,45],"particular,":[14],"doubts":[15],"were":[16],"raised":[17],"whether":[18],"schemes":[19,55,84],"could":[20],"be":[21,95],"formalized":[22],"in":[23,36,43,56,79],"this":[24,46,74],"setting":[25],"and":[26,66],"a":[27,51,98],"challenge":[28],"was":[29],"issued.":[30],"Schemes":[31],"are":[32],"sophisticated":[33],"objects":[35],"algebraic":[37],"geometry":[38],"introduced":[39],"by":[40],"Alexander":[41],"Grothendieck":[42],"1960.":[44],"article":[47],"we":[48,67],"report":[49],"on":[50],"successful":[52],"formalization":[53],"of":[54,61,83,90],"the":[57,62,69,80,86],"proof":[63],"assistant":[64],"Isabelle/HOL,":[65],"discuss":[68],"design":[70],"choices":[71],"which":[72],"make":[73],"work":[75],"possible.":[76],"We":[77],"show":[78],"particular":[81],"case":[82],"how":[85],"powerful":[87],"dependent":[88],"types":[89],"Coq":[91],"or":[92],"Lean":[93],"can":[94],"traded":[96],"minimalist":[99],"apparatus":[100],"called":[101],"locales.":[102]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
