{"id":"https://openalex.org/W3120033270","doi":"https://doi.org/10.1007/s10485-021-09671-9","title":"Kan Extensions are Partial Colimits","display_name":"Kan Extensions are Partial Colimits","publication_year":2022,"publication_date":"2022-01-31","ids":{"openalex":"https://openalex.org/W3120033270","doi":"https://doi.org/10.1007/s10485-021-09671-9","mag":"3120033270","pmid":"https://pubmed.ncbi.nlm.nih.gov/35875343"},"language":"en","primary_location":{"id":"doi:10.1007/s10485-021-09671-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10485-021-09671-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10485-021-09671-9.pdf","source":{"id":"https://openalex.org/S148906134","display_name":"Applied Categorical Structures","issn_l":"0927-2852","issn":["0927-2852","1572-9095"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Applied Categorical Structures","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10485-021-09671-9.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005788902","display_name":"Paolo Perrone","orcid":"https://orcid.org/0000-0002-9123-9089"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Paolo Perrone","raw_affiliation_strings":["Department of Computer Science, University of Oxford, Oxford, England, UK"],"raw_orcid":"https://orcid.org/0000-0002-9123-9089","affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Oxford, Oxford, England, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024663910","display_name":"Walter Tholen","orcid":"https://orcid.org/0000-0003-2167-9449"},"institutions":[{"id":"https://openalex.org/I192455969","display_name":"York University","ror":"https://ror.org/05fq50484","country_code":"CA","type":"education","lineage":["https://openalex.org/I192455969"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Walter Tholen","raw_affiliation_strings":["Department of Mathematics and Statistics, York University, Toronto, ON Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, York University, Toronto, ON Canada","institution_ids":["https://openalex.org/I192455969"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5005788902"],"corresponding_institution_ids":["https://openalex.org/I40120149"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.03473849,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"30","issue":"4","first_page":"685","last_page":"753"},"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":0.9984999895095825,"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":0.9984999895095825,"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":0.9869999885559082,"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/T10420","display_name":"Intracranial Aneurysms: Treatment and Complications","score":0.9312000274658203,"subfield":{"id":"https://openalex.org/subfields/2728","display_name":"Neurology"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/morphism","display_name":"Morphism","score":0.6946041584014893},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6001826524734497},{"id":"https://openalex.org/keywords/monad","display_name":"Monad (category theory)","score":0.5218436121940613},{"id":"https://openalex.org/keywords/extensionality","display_name":"Extensionality","score":0.5074920058250427},{"id":"https://openalex.org/keywords/distributive-property","display_name":"Distributive property","score":0.46327945590019226},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.43676117062568665},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.38931116461753845},{"id":"https://openalex.org/keywords/functor","display_name":"Functor","score":0.310737669467926},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.24304506182670593}],"concepts":[{"id":"https://openalex.org/C137212723","wikidata":"https://www.wikidata.org/wiki/Q1948412","display_name":"Morphism","level":2,"score":0.6946041584014893},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6001826524734497},{"id":"https://openalex.org/C33959348","wikidata":"https://www.wikidata.org/wiki/Q1630568","display_name":"Monad (category theory)","level":3,"score":0.5218436121940613},{"id":"https://openalex.org/C150355047","wikidata":"https://www.wikidata.org/wiki/Q965432","display_name":"Extensionality","level":2,"score":0.5074920058250427},{"id":"https://openalex.org/C11821877","wikidata":"https://www.wikidata.org/wiki/Q187959","display_name":"Distributive property","level":2,"score":0.46327945590019226},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.43676117062568665},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.38931116461753845},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.310737669467926},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.24304506182670593}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1007/s10485-021-09671-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10485-021-09671-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10485-021-09671-9.pdf","source":{"id":"https://openalex.org/S148906134","display_name":"Applied Categorical Structures","issn_l":"0927-2852","issn":["0927-2852","1572-9095"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Applied Categorical Structures","raw_type":"journal-article"},{"id":"pmid:35875343","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/35875343","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Applied categorical structures","raw_type":null},{"id":"pmh:oai:arXiv.org:2101.04531","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2101.04531","pdf_url":"https://arxiv.org/pdf/2101.04531","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":"mag:3120033270","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2101.04531.pdf","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:ora.ox.ac.uk:uuid:b9be0e40-be40-45d4-8115-e32c63d57a9b","is_oa":true,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:b9be0e40-be40-45d4-8115-e32c63d57a9b","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symplectic Elements","raw_type":"Journal article"},{"id":"pmh:oai:pubmedcentral.nih.gov:9296069","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/9296069","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Appl Categ Struct","raw_type":"Text"},{"id":"doi:10.48550/arxiv.2101.04531","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2101.04531","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s10485-021-09671-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10485-021-09671-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10485-021-09671-9.pdf","source":{"id":"https://openalex.org/S148906134","display_name":"Applied Categorical Structures","issn_l":"0927-2852","issn":["0927-2852","1572-9095"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Applied Categorical Structures","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.5799999833106995,"id":"https://metadata.un.org/sdg/8","display_name":"Decent work and economic growth"}],"awards":[{"id":"https://openalex.org/G1072749480","display_name":null,"funder_award_id":"FA9550-17","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G1523888516","display_name":null,"funder_award_id":"FA9550-","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G260750634","display_name":null,"funder_award_id":"FA9550-17-1","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G3293285712","display_name":null,"funder_award_id":"501260","funder_id":"https://openalex.org/F4320321487","funder_display_name":"Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G5065251311","display_name":null,"funder_award_id":"FA9550-19-1-0113","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G5809100787","display_name":null,"funder_award_id":"FA9550","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G73798499","display_name":null,"funder_award_id":"FA9550-17-1-0058","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"}],"funders":[{"id":"https://openalex.org/F4320309369","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44"},{"id":"https://openalex.org/F4320319877","display_name":"Fields Institute for Research in Mathematical Sciences","ror":"https://ror.org/03zzj3f20"},{"id":"https://openalex.org/F4320321487","display_name":"Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3120033270.pdf","grobid_xml":"https://content.openalex.org/works/W3120033270.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W342302233","https://openalex.org/W583126019","https://openalex.org/W602280613","https://openalex.org/W633412478","https://openalex.org/W1560018030","https://openalex.org/W1578654632","https://openalex.org/W1604392671","https://openalex.org/W1980822824","https://openalex.org/W2012265449","https://openalex.org/W2050434204","https://openalex.org/W2074676877","https://openalex.org/W2078551041","https://openalex.org/W2093328479","https://openalex.org/W2129186906","https://openalex.org/W2430256791","https://openalex.org/W2486963111","https://openalex.org/W2896200016","https://openalex.org/W3036458910","https://openalex.org/W3086667163","https://openalex.org/W3087537363","https://openalex.org/W4205308139","https://openalex.org/W4237365407","https://openalex.org/W4244129601"],"related_works":["https://openalex.org/W3121177248","https://openalex.org/W1819430339","https://openalex.org/W1778582051","https://openalex.org/W2161616972","https://openalex.org/W2949299139","https://openalex.org/W2497414066","https://openalex.org/W2769923424","https://openalex.org/W2951923673","https://openalex.org/W2079655821","https://openalex.org/W2942282180","https://openalex.org/W2164268874","https://openalex.org/W3043991061","https://openalex.org/W1572021354","https://openalex.org/W2236935938","https://openalex.org/W2339107952","https://openalex.org/W2604066929","https://openalex.org/W3097151221","https://openalex.org/W2962912206","https://openalex.org/W2962963437","https://openalex.org/W2151448345"],"abstract_inverted_index":{"Abstract":[0],"One":[1],"way":[2],"of":[3,14,21,34,44,48,56,61,70,90,103,113,118,124,132,150,155,170,180,191,200,214,221,230,233],"interpreting":[4],"a":[5,12,22,109,130,174,186,192,197,218,222,227],"left":[6,188],"Kan":[7,189],"extension":[8,190],"is":[9,77,206],"as":[10,173],"taking":[11],"kind":[13],"\u201cpartial":[15],"colimit\u201d,":[16],"whereby":[17],"one":[18],"replaces":[19],"parts":[20],"diagram":[23,193],"by":[24,32],"their":[25],"colimits.":[26],"We":[27,93],"make":[28],"this":[29,181],"intuition":[30],"precise":[31],"means":[33],"the":[35,40,54,59,66,88,100,115,122,148,168,212],"partial":[36,198,228],"evaluations":[37],"sitting":[38],"in":[39,87,144,161,211],"so-called":[41],"bar":[42],"construction":[43,117],"monads.":[45],"The":[46,177],"(pseudo)monads":[47],"interest":[49],"for":[50],"forming":[51],"colimits":[52,159],"are":[53,84],"monad":[55,60],"diagrams":[57],"and":[58],"small":[62,72],"presheaves,":[63],"both":[64],"on":[65,108],"(huge)":[67],"category":[68],"CAT":[69],"locally":[71],"categories.":[73],"Throughout,":[74],"particular":[75,145],"care":[76],"taken":[78],"to":[79,121,146,196,208,226],"handle":[80],"size":[81],"issues,":[82],"which":[83,136,157],"notoriously":[85],"delicate":[86],"context":[89],"free":[91],"cocompletion.":[92],"spell":[94],"out,":[95],"with":[96],"all":[97],"2-dimensional":[98],"details,":[99],"structure":[101],"maps":[102],"these":[104],"pseudomonads.":[105],"Then,":[106],"based":[107],"detailed":[110],"general":[111],"proof":[112],"how":[114,185],"restriction-of-scalars":[116],"monads":[119,133],"extends":[120],"case":[123,213],"pseudoalgebras":[125],"over":[126],"pseudomonads,":[127],"we":[128,137],"consider":[129],"morphism":[131,142],"between":[134],"them,":[135],"call":[138],"image":[139],".":[140],"This":[141,165,203],"allows":[143],"generalize":[147],"idea":[149],"confinal":[151],"functors":[152,156],",":[153],"i.e.":[154],"leave":[158],"invariant":[160],"an":[162],"absolute":[163,171],"way.":[164],"generalization":[166],"includes":[167],"concept":[169],"colimit":[172],"special":[175],"case.":[176],"main":[178],"result":[179,205],"paper":[182],"spells":[183],"out":[184],"pointwise":[187],"corresponds":[194,225],"precisely":[195],"evaluation":[199,229],"its":[201,231],"colimit.":[202],"categorical":[204],"analogous":[207],"what":[209],"happens":[210],"probability":[215],"monads,":[216],"where":[217],"conditional":[219],"expectation":[220],"random":[223],"variable":[224],"center":[232],"mass.":[234]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-16T08:24:45.110214","created_date":"2025-10-10T00:00:00"}