{"id":"https://openalex.org/W4414402247","doi":"https://doi.org/10.4204/eptcs.429.3","title":"Learners are Almost Free Compact Closed","display_name":"Learners are Almost Free Compact Closed","publication_year":2025,"publication_date":"2025-09-22","ids":{"openalex":"https://openalex.org/W4414402247","doi":"https://doi.org/10.4204/eptcs.429.3"},"language":"en","primary_location":{"id":"doi:10.4204/eptcs.429.3","is_oa":true,"landing_page_url":"https://doi.org/10.4204/eptcs.429.3","pdf_url":"https://eptcs.web.cse.unsw.edu.au/paper.cgi?ACT2024.3.pdf","source":{"id":"https://openalex.org/S4210216316","display_name":"Electronic Proceedings in Theoretical Computer Science","issn_l":"2075-2180","issn":["2075-2180"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310318029","host_organization_name":"Open Publishing Association","host_organization_lineage":["https://openalex.org/P4310318029"],"host_organization_lineage_names":["Open Publishing Association"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Electronic Proceedings in Theoretical Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://eptcs.web.cse.unsw.edu.au/paper.cgi?ACT2024.3.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Mitchell Riley","orcid":null},"institutions":[{"id":"https://openalex.org/I120250893","display_name":"New York University Abu Dhabi","ror":"https://ror.org/00e5k0821","country_code":"AE","type":"education","lineage":["https://openalex.org/I120250893","https://openalex.org/I57206974"]}],"countries":["AE"],"is_corresponding":true,"raw_author_name":"Mitchell Riley","raw_affiliation_strings":["Mathematics , Division of Science New York University Abu Dhabi"],"affiliations":[{"raw_affiliation_string":"Mathematics , Division of Science New York University Abu Dhabi","institution_ids":["https://openalex.org/I120250893"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I120250893"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.31253071,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"429","issue":null,"first_page":"49","last_page":"62"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.14399999380111694,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.14399999380111694,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/morphism","display_name":"Morphism","score":0.7465000152587891},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5386999845504761},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.4715000092983246},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4578000009059906},{"id":"https://openalex.org/keywords/category-of-groups","display_name":"Category of groups","score":0.4507000148296356},{"id":"https://openalex.org/keywords/enriched-category","display_name":"Enriched category","score":0.42419999837875366},{"id":"https://openalex.org/keywords/equivalence-of-categories","display_name":"Equivalence of categories","score":0.4205999970436096},{"id":"https://openalex.org/keywords/cartesian-closed-category","display_name":"Cartesian closed category","score":0.3912000060081482}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8034999966621399},{"id":"https://openalex.org/C137212723","wikidata":"https://www.wikidata.org/wiki/Q1948412","display_name":"Morphism","level":2,"score":0.7465000152587891},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5386999845504761},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.492900013923645},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.4715000092983246},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4578000009059906},{"id":"https://openalex.org/C153491289","wikidata":"https://www.wikidata.org/wiki/Q3912012","display_name":"Category of groups","level":4,"score":0.4507000148296356},{"id":"https://openalex.org/C156163052","wikidata":"https://www.wikidata.org/wiki/Q5379515","display_name":"Enriched category","level":3,"score":0.42419999837875366},{"id":"https://openalex.org/C11846945","wikidata":"https://www.wikidata.org/wiki/Q1082992","display_name":"Equivalence of categories","level":3,"score":0.4205999970436096},{"id":"https://openalex.org/C171032842","wikidata":"https://www.wikidata.org/wiki/Q1725874","display_name":"Cartesian closed category","level":2,"score":0.3912000060081482},{"id":"https://openalex.org/C46263353","wikidata":"https://www.wikidata.org/wiki/Q4596935","display_name":"2-category","level":4,"score":0.35580000281333923},{"id":"https://openalex.org/C2779904274","wikidata":"https://www.wikidata.org/wiki/Q1945014","display_name":"Monoidal category","level":2,"score":0.3264999985694885},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.31439998745918274},{"id":"https://openalex.org/C162205778","wikidata":"https://www.wikidata.org/wiki/Q4915506","display_name":"Biproduct","level":4,"score":0.3095000088214874},{"id":"https://openalex.org/C83177873","wikidata":"https://www.wikidata.org/wiki/Q2000555","display_name":"Category of topological spaces","level":5,"score":0.2808000147342682},{"id":"https://openalex.org/C2777044963","wikidata":"https://www.wikidata.org/wiki/Q5384707","display_name":"Equivalence class (music)","level":2,"score":0.2773999869823456},{"id":"https://openalex.org/C7879355","wikidata":"https://www.wikidata.org/wiki/Q5135329","display_name":"Closed category","level":3,"score":0.273499995470047},{"id":"https://openalex.org/C31498916","wikidata":"https://www.wikidata.org/wiki/Q583034","display_name":"Locally compact space","level":2,"score":0.265500009059906},{"id":"https://openalex.org/C98912367","wikidata":"https://www.wikidata.org/wiki/Q17103860","display_name":"Symmetric monoidal category","level":3,"score":0.25920000672340393},{"id":"https://openalex.org/C128520899","wikidata":"https://www.wikidata.org/wiki/Q1575634","display_name":"Concrete category","level":3,"score":0.25040000677108765}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.4204/eptcs.429.3","is_oa":true,"landing_page_url":"https://doi.org/10.4204/eptcs.429.3","pdf_url":"https://eptcs.web.cse.unsw.edu.au/paper.cgi?ACT2024.3.pdf","source":{"id":"https://openalex.org/S4210216316","display_name":"Electronic Proceedings in Theoretical Computer Science","issn_l":"2075-2180","issn":["2075-2180"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310318029","host_organization_name":"Open Publishing Association","host_organization_lineage":["https://openalex.org/P4310318029"],"host_organization_lineage_names":["Open Publishing Association"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Electronic Proceedings in Theoretical Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2509.20930","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2509.20930","pdf_url":"https://arxiv.org/pdf/2509.20930","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"doi:10.4204/eptcs.429.3","is_oa":true,"landing_page_url":"https://doi.org/10.4204/eptcs.429.3","pdf_url":"https://eptcs.web.cse.unsw.edu.au/paper.cgi?ACT2024.3.pdf","source":{"id":"https://openalex.org/S4210216316","display_name":"Electronic Proceedings in Theoretical Computer Science","issn_l":"2075-2180","issn":["2075-2180"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310318029","host_organization_name":"Open Publishing Association","host_organization_lineage":["https://openalex.org/P4310318029"],"host_organization_lineage_names":["Open Publishing Association"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Electronic Proceedings in Theoretical Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4414402247.pdf","grobid_xml":"https://content.openalex.org/works/W4414402247.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W2008699979","https://openalex.org/W2022595765","https://openalex.org/W2089773888","https://openalex.org/W2102357532","https://openalex.org/W2107835116","https://openalex.org/W2116877111","https://openalex.org/W2153731375","https://openalex.org/W3001712757","https://openalex.org/W3134406394","https://openalex.org/W4385342863"],"related_works":[],"abstract_inverted_index":{"The":[0],"category":[1,34,44,99,142,171],"of":[2,31,39,61,70,232],"learners":[3,79,203],"has":[4,116],"a":[5,16,36,46,71,84,90,108,117,151,181,208,233],"pleasant":[6],"symmetric":[7,47,110,175,188],"formulation":[8],"when":[9],"the":[10,20,25,40,59,68,94,130,141,145,158,167,173,230],"morphisms":[11],"are":[12,80,137],"considered":[13,81],"up":[14],"to":[15,103,124,143,147,198,213,222],"coarser":[17,85],"equivalence":[18,86],"than":[19,88,186],"one":[21],"originally":[22],"described":[23],"in":[24,93,150,165],"paper":[26],"\"Backprop":[27],"as":[28],"Functor\".A":[29],"quotient":[30],"this":[32,50,128,192],"modified":[33],"gives":[35],"new":[37,98],"construction":[38],"free":[41,168],"compact":[42,105,152,169],"closed":[43,153,170,178],"on":[45,54,172],"monoidal":[48,111,176,189],"category.In":[49],"note":[51,159],"I":[52],"expand":[53],"an":[55],"observation":[56],"made":[57],"at":[58],"end":[60,157],"[15]":[62],"and":[63,113,119,134,155,200,218],"further":[64],"discussed":[65],"online":[66],"[16].If":[67],"definition":[69,95],"learner":[72,209],"[7]":[73],"is":[74,96,107,164,220,229],"tweaked":[75],"slightly":[76],"so":[77,135,191],"that":[78,162,206],"identical":[82],"under":[83,127],"relation":[87],"before,":[89],"striking":[91],"symmetry":[92],"revealed.This":[97],"comes":[100],"quite":[101],"close":[102],"being":[104],"closed.There":[106],"strict":[109],"involution":[112],"each":[114],"object":[115],"cup":[118],"cap":[120],"morphism":[121],"connecting":[122],"it":[123,163,196],"its":[125],"\"dual\"":[126],"involution.However,":[129],"snake":[131],"equations":[132,146],"fail":[133],"these":[136],"not":[138,211],"true":[139],"duals.Quotienting":[140],"force":[144],"hold":[148],"results":[149],"category,":[154],"we":[156],"by":[160,225],"showing":[161],"fact":[166],"starting":[174],"category.Compact":[177],"categories":[179],"support":[180],"more":[182],"powerful":[183],"diagrammatic":[184],"calculus":[185],"bare":[187],"categories,":[190],"work":[193],"may":[194],"make":[195],"easier":[197],"design":[199],"reason":[201],"about":[202],"compositionally.The":[204],"operation":[205],"dualises":[207],"does":[210],"appear":[212],"have":[214],"been":[215],"previously":[216],"noticed":[217],"there":[219],"much":[221],"investigate.As":[223],"asked":[224],"David":[226],"Spivak,":[227],"what":[228],"dual":[231],"neuron?":[234]},"counts_by_year":[],"updated_date":"2026-04-17T18:11:37.981687","created_date":"2025-10-10T00:00:00"}
