{"id":"https://openalex.org/W3122987924","doi":"https://doi.org/10.32408/compositionality-4-3","title":"Structured versus Decorated Cospans","display_name":"Structured versus Decorated Cospans","publication_year":2022,"publication_date":"2022-09-01","ids":{"openalex":"https://openalex.org/W3122987924","doi":"https://doi.org/10.32408/compositionality-4-3","mag":"3122987924"},"language":"lv","primary_location":{"id":"doi:10.32408/compositionality-4-3","is_oa":true,"landing_page_url":"https://doi.org/10.32408/compositionality-4-3","pdf_url":"https://compositionality-journal.org/papers/compositionality-4-3/pdf/","source":{"id":"https://openalex.org/S4210182901","display_name":"Compositionality","issn_l":"2631-4444","issn":["2631-4444"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Compositionality","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://compositionality-journal.org/papers/compositionality-4-3/pdf/","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5056752880","display_name":"John C. Baez","orcid":"https://orcid.org/0000-0002-0609-9836"},"institutions":[{"id":"https://openalex.org/I115455960","display_name":"Centre for Quantum Technologies","ror":"https://ror.org/01mgdzc49","country_code":"SG","type":"facility","lineage":["https://openalex.org/I115455960"]},{"id":"https://openalex.org/I103635307","display_name":"University of California, Riverside","ror":"https://ror.org/03nawhv43","country_code":"US","type":"education","lineage":["https://openalex.org/I103635307"]},{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG","US"],"is_corresponding":true,"raw_author_name":"John C. Baez","raw_affiliation_strings":["Centre for Quantum Technologies, National University of Singapore, Singapore 117543","Department of Mathematics, University of California, Riverside CA, USA 92521"],"affiliations":[{"raw_affiliation_string":"Centre for Quantum Technologies, National University of Singapore, Singapore 117543","institution_ids":["https://openalex.org/I115455960","https://openalex.org/I165932596"]},{"raw_affiliation_string":"Department of Mathematics, University of California, Riverside CA, USA 92521","institution_ids":["https://openalex.org/I103635307"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5077865739","display_name":"Kenny Courser","orcid":null},"institutions":[{"id":"https://openalex.org/I103635307","display_name":"University of California, Riverside","ror":"https://ror.org/03nawhv43","country_code":"US","type":"education","lineage":["https://openalex.org/I103635307"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kenny Courser","raw_affiliation_strings":["Department of Mathematics, University of California, Riverside CA, USA 92521"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California, Riverside CA, USA 92521","institution_ids":["https://openalex.org/I103635307"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5063903373","display_name":"Christina Vasilakopoulou","orcid":"https://orcid.org/0000-0002-1411-934X"},"institutions":[{"id":"https://openalex.org/I174878644","display_name":"University of Patras","ror":"https://ror.org/017wvtq80","country_code":"GR","type":"education","lineage":["https://openalex.org/I174878644"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Christina Vasilakopoulou","raw_affiliation_strings":["Department of Mathematics, University of Patras, Greece 265 04"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Patras, Greece 265 04","institution_ids":["https://openalex.org/I174878644"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5056752880"],"corresponding_institution_ids":["https://openalex.org/I103635307","https://openalex.org/I115455960","https://openalex.org/I165932596"],"apc_list":{"value":0,"currency":"USD","value_usd":0},"apc_paid":null,"fwci":4.6643,"has_fulltext":false,"cited_by_count":13,"citation_normalized_percentile":{"value":0.96005117,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"4","issue":null,"first_page":"3","last_page":"3"},"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":0.9713000059127808,"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":0.9713000059127808,"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/functor","display_name":"Functor","score":0.6404114365577698},{"id":"https://openalex.org/keywords/diagram","display_name":"Diagram","score":0.5946227312088013},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.5399397015571594},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4759778082370758},{"id":"https://openalex.org/keywords/symmetric-monoidal-category","display_name":"Symmetric monoidal category","score":0.4642205238342285},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.37763750553131104},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36439403891563416},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.342909038066864}],"concepts":[{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.6404114365577698},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.5946227312088013},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5399397015571594},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4759778082370758},{"id":"https://openalex.org/C98912367","wikidata":"https://www.wikidata.org/wiki/Q17103860","display_name":"Symmetric monoidal category","level":3,"score":0.4642205238342285},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.37763750553131104},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36439403891563416},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.342909038066864},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.32408/compositionality-4-3","is_oa":true,"landing_page_url":"https://doi.org/10.32408/compositionality-4-3","pdf_url":"https://compositionality-journal.org/papers/compositionality-4-3/pdf/","source":{"id":"https://openalex.org/S4210182901","display_name":"Compositionality","issn_l":"2631-4444","issn":["2631-4444"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Compositionality","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2101.09363","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2101.09363","pdf_url":"https://arxiv.org/pdf/2101.09363","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"}],"best_oa_location":{"id":"doi:10.32408/compositionality-4-3","is_oa":true,"landing_page_url":"https://doi.org/10.32408/compositionality-4-3","pdf_url":"https://compositionality-journal.org/papers/compositionality-4-3/pdf/","source":{"id":"https://openalex.org/S4210182901","display_name":"Compositionality","issn_l":"2631-4444","issn":["2631-4444"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Compositionality","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Good health and well-being","id":"https://metadata.un.org/sdg/3","score":0.6499999761581421}],"awards":[],"funders":[{"id":"https://openalex.org/F4320321917","display_name":"General Secretariat for Research and Technology","ror":"https://ror.org/04yeh8h63"},{"id":"https://openalex.org/F4320327859","display_name":"Hellenic Foundation for Research and Innovation","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3122987924.pdf","grobid_xml":"https://content.openalex.org/works/W3122987924.grobid-xml"},"referenced_works_count":57,"referenced_works":["https://openalex.org/W54701845","https://openalex.org/W75859641","https://openalex.org/W1499965417","https://openalex.org/W1520893111","https://openalex.org/W1527974694","https://openalex.org/W1562513666","https://openalex.org/W1578654632","https://openalex.org/W1635067508","https://openalex.org/W1823163556","https://openalex.org/W1958250673","https://openalex.org/W1964572513","https://openalex.org/W1979791105","https://openalex.org/W1994188141","https://openalex.org/W2039290601","https://openalex.org/W2053145248","https://openalex.org/W2076730925","https://openalex.org/W2077270594","https://openalex.org/W2093064243","https://openalex.org/W2114991530","https://openalex.org/W2148443868","https://openalex.org/W2152476038","https://openalex.org/W2168576232","https://openalex.org/W2182286275","https://openalex.org/W2208563212","https://openalex.org/W2255305994","https://openalex.org/W2397532503","https://openalex.org/W2489166266","https://openalex.org/W2493333586","https://openalex.org/W2502957253","https://openalex.org/W2523361026","https://openalex.org/W2606902047","https://openalex.org/W2737072409","https://openalex.org/W2762464580","https://openalex.org/W2886790899","https://openalex.org/W2890274536","https://openalex.org/W2962868940","https://openalex.org/W2962887654","https://openalex.org/W2963166732","https://openalex.org/W2963886789","https://openalex.org/W2981110147","https://openalex.org/W3025631990","https://openalex.org/W3082960721","https://openalex.org/W3092961706","https://openalex.org/W3099677335","https://openalex.org/W3100557837","https://openalex.org/W3101612166","https://openalex.org/W3104708154","https://openalex.org/W4205814206","https://openalex.org/W4221139768","https://openalex.org/W4295102111","https://openalex.org/W4298067962","https://openalex.org/W4299878724","https://openalex.org/W4301007786","https://openalex.org/W4394643978","https://openalex.org/W4394658783","https://openalex.org/W4394659122","https://openalex.org/W6910739090"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2997091376","https://openalex.org/W95605005","https://openalex.org/W1555265147","https://openalex.org/W4293316580","https://openalex.org/W2997389523","https://openalex.org/W2202270624","https://openalex.org/W4301102648","https://openalex.org/W2897139815","https://openalex.org/W2588688245"],"abstract_inverted_index":{"One":[0],"goal":[1],"of":[2,15,47,93,129,140,148,190,226],"applied":[3],"category":[4,88,185,225],"theory":[5],"is":[6,39,83,121,175,180],"to":[7,221,235],"understand":[8],"open":[9,17],"systems.":[10],"We":[11,58,202,229],"compare":[12],"two":[13,209],"ways":[14],"describing":[16],"systems":[18,241],"as":[19],"cospans":[20],"equipped":[21],"with":[22,137,233],"extra":[23],"data.":[24],"First,":[25],"given":[26,106],"a":[27,36,40,60,84,107,118,122,181],"functor":[28],"<mml:math":[29,43,50,65,70,78,94,109,125,132,141,154,162,191,216,227],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi><mml:mo>:</mml:mo><mml:mrow":[30],"class=\"MJX-TeXAtom-ORD\"><mml:mi":[31,34,45,67,72,96,111,114,127,156,165,170,193,218],"mathvariant=\"sans-serif\">A</mml:mi></mml:mrow><mml:mo":[32,112],"stretchy=\"false\">\u2192</mml:mo><mml:mrow":[33,113],"mathvariant=\"sans-serif\">X</mml:mi></mml:mrow></mml:math>,":[35],"`structured":[37],"cospan'":[38,120],"diagram":[41,123],"in":[42,124],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow":[44,66,71,95,126,155,192,217],"mathvariant=\"sans-serif\">X</mml:mi></mml:mrow></mml:math>":[46,73],"the":[48,130,146,223],"form":[49,131],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi><mml:mo":[51],"stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo":[52],"stretchy=\"false\">)</mml:mo><mml:mo":[53,167],"stretchy=\"false\">\u2192</mml:mo><mml:mi>x</mml:mi><mml:mo":[54],"stretchy=\"false\">\u2190</mml:mo><mml:mi>L</mml:mi><mml:mo":[55],"stretchy=\"false\">(</mml:mo><mml:mi>b</mml:mi><mml:mo":[56],"stretchy=\"false\">)</mml:mo></mml:math>.":[57,144],"give":[59],"new":[61],"proof":[62],"that":[63,152,204],"if":[64,153],"mathvariant=\"sans-serif\">A</mml:mi></mml:mrow></mml:math>":[68,97,128,157,194],"and":[69,77,98,161,195,242],"have":[74],"finite":[75,159],"colimits":[76,160],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi></mml:math>":[79],"preserves":[80],"them,":[81],"there":[82,179],"symmetric":[85,176,182],"monoidal":[86,183],"double":[87,184],"whose":[89,99,186,196],"objects":[90,187],"are":[91,102,188,199],"those":[92,189],"horizontal":[100,197],"1-cells":[101,198],"structured":[103],"cospans.":[104,201],"Second,":[105],"pseudofunctor":[108],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>F</mml:mi><mml:mo>:</mml:mo><mml:mrow":[110],"mathvariant=\"bold\">C</mml:mi><mml:mi":[115,171],"mathvariant=\"bold\">a</mml:mi><mml:mi":[116,172],"mathvariant=\"bold\">t</mml:mi></mml:mrow></mml:math>,":[117],"`decorated":[119],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mo":[133],"stretchy=\"false\">\u2192</mml:mo><mml:mi>m</mml:mi><mml:mo":[134],"stretchy=\"false\">\u2190</mml:mo><mml:mi>b</mml:mi></mml:math>":[135],"together":[136],"an":[138],"object":[139],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>F</mml:mi><mml:mo":[142],"stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo":[143],"Generalizing":[145],"work":[147],"Fong,":[149],"we":[150,214],"show":[151],"has":[158],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>F</mml:mi><mml:mo>:</mml:mo><mml:mo":[163],"stretchy=\"false\">(</mml:mo><mml:mrow":[164,169],"mathvariant=\"sans-serif\">A</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mo":[166],"stretchy=\"false\">\u2192</mml:mo><mml:mo":[168],"mathvariant=\"bold\">t</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mo>\u00d7</mml:mo><mml:mo":[173],"stretchy=\"false\">)</mml:mo></mml:math>":[174],"lax":[177],"monoidal,":[178],"decorated":[200],"prove":[203],"under":[205],"certain":[206],"conditions,":[207],"these":[208,231],"constructions":[210],"become":[211],"isomorphic":[212],"when":[213],"take":[215],"mathvariant=\"sans-serif\">X</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mo":[219],"largeop=\"false\">\u222b</mml:mo><mml:mi>F</mml:mi></mml:math>":[220],"be":[222],"Grothendieck":[224],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>F</mml:mi></mml:math>.":[228],"illustrate":[230],"ideas":[232],"applications":[234],"electrical":[236],"circuits,":[237],"Petri":[238],"nets,":[239],"dynamical":[240],"epidemiological":[243],"modeling.":[244]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":4},{"year":2020,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}