{"id":"https://openalex.org/W4249457474","doi":"https://doi.org/10.1017/9781108636575.007","title":"Monoidal Structures","display_name":"Monoidal Structures","publication_year":2020,"publication_date":"2020-03-09","ids":{"openalex":"https://openalex.org/W4249457474","doi":"https://doi.org/10.1017/9781108636575.007"},"language":"en","primary_location":{"id":"doi:10.1017/9781108636575.007","doi":"https://doi.org/10.1017/9781108636575.007","is_oa":false,"landing_page_url":"https://doi.org/10.1017/9781108636575.007","pdf_url":null,"source":{"id":"https://openalex.org/S4306462995","display_name":"Cambridge University Press eBooks","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"ebook platform"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Stable Homotopy Theory","raw_type":"book-chapter"},"volume":{"id":"https://openalex.org/W2978857425","display_name":"Foundations of Stable Homotopy Theory"},"type":"book-chapter","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024159742","display_name":"David Barnes","orcid":"https://orcid.org/0000-0001-8183-1479","observed_orcids":["https://orcid.org/0000-0001-8183-1479"]},"institutions":[{"id":"https://openalex.org/I126231945","display_name":"Queen's University Belfast","ror":"https://ror.org/00hswnk62","country_code":"GB","type":"education","lineage":["https://openalex.org/I126231945"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"David Barnes","raw_affiliation_strings":["Queen's University Belfast"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Queen's University Belfast","institution_ids":["https://openalex.org/I126231945"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032015603","display_name":"Constanze Roitzheim","orcid":"https://orcid.org/0000-0003-3065-0672","observed_orcids":["https://orcid.org/0000-0003-3065-0672"]},"institutions":[{"id":"https://openalex.org/I20581793","display_name":"University of Kent","ror":"https://ror.org/00xkeyj56","country_code":"GB","type":"education","lineage":["https://openalex.org/I20581793"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Constanze Roitzheim","raw_affiliation_strings":["University of Kent, Canterbury"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Kent, Canterbury","institution_ids":["https://openalex.org/I20581793"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"234","last_page":"316"},"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":1.0,"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":1.0,"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":0.0,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0,"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/model-categories","display_name":"model categories","score":0.9520000219345093},{"id":"https://openalex.org/keywords/weak-equivalence","display_name":"weak equivalence","score":0.8820000290870667},{"id":"https://openalex.org/keywords/homology-isomorphism","display_name":"homology isomorphism","score":0.8920000195503235},{"id":"https://openalex.org/keywords/homotopy-category","display_name":"homotopy category","score":0.8970000147819519},{"id":"https://openalex.org/keywords/p-completion","display_name":"p-completion","score":0.8270000219345093},{"id":"https://openalex.org/keywords/k-theory","display_name":"K-theory","score":0.8899999856948853},{"id":"https://openalex.org/keywords/chromatic-homotopy-theory","display_name":"chromatic homotopy theory","score":0.9559999704360962},{"id":"https://openalex.org/keywords/stable-homotopy-category","display_name":"stable homotopy category","score":0.7649999856948853}],"concepts":[{"id":"https://openalex.org/C131359564","wikidata":"https://www.wikidata.org/wiki/Q1941896","display_name":"Model category","level":4,"score":0.8720071315765381},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8316197991371155},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.7848918437957764},{"id":"https://openalex.org/C79236096","wikidata":"https://www.wikidata.org/wiki/Q14775269","display_name":"Homotopy category","level":3,"score":0.6870784759521484},{"id":"https://openalex.org/C165525559","wikidata":"https://www.wikidata.org/wiki/Q224180","display_name":"Homology (biology)","level":3,"score":0.6649613380432129},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5627433061599731},{"id":"https://openalex.org/C164043645","wikidata":"https://www.wikidata.org/wiki/Q16879377","display_name":"Homotopy hypothesis","level":5,"score":0.5586362481117249},{"id":"https://openalex.org/C150275644","wikidata":"https://www.wikidata.org/wiki/Q1048949","display_name":"Cofibration","level":5,"score":0.4642316699028015},{"id":"https://openalex.org/C191752858","wikidata":"https://www.wikidata.org/wiki/Q6358325","display_name":"n-connected","level":3,"score":0.42695939540863037},{"id":"https://openalex.org/C119720664","wikidata":"https://www.wikidata.org/wiki/Q3527185","display_name":"Whitehead theorem","level":4,"score":0.40834981203079224},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36887049674987793},{"id":"https://openalex.org/C168334404","wikidata":"https://www.wikidata.org/wiki/Q7309601","display_name":"Regular homotopy","level":4,"score":0.18136781454086304},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/9781108636575.007","doi":"https://doi.org/10.1017/9781108636575.007","is_oa":false,"landing_page_url":"https://doi.org/10.1017/9781108636575.007","pdf_url":null,"source":{"id":"https://openalex.org/S4306462995","display_name":"Cambridge University Press eBooks","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"ebook platform"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Stable Homotopy Theory","raw_type":"book-chapter"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":93,"referenced_works":["https://openalex.org/W81472618","https://openalex.org/W162494057","https://openalex.org/W235399473","https://openalex.org/W577391182","https://openalex.org/W606247779","https://openalex.org/W621930122","https://openalex.org/W622065658","https://openalex.org/W622277922","https://openalex.org/W838699892","https://openalex.org/W1479919192","https://openalex.org/W1486044528","https://openalex.org/W1507692549","https://openalex.org/W1512401425","https://openalex.org/W1542975694","https://openalex.org/W1547498775","https://openalex.org/W1573436879","https://openalex.org/W1578654632","https://openalex.org/W1583122470","https://openalex.org/W1584027141","https://openalex.org/W1598840153","https://openalex.org/W1602704216","https://openalex.org/W1967316681","https://openalex.org/W1970015879","https://openalex.org/W1970407889","https://openalex.org/W1974096833","https://openalex.org/W1976086282","https://openalex.org/W1980065889","https://openalex.org/W1981171806","https://openalex.org/W1983373140","https://openalex.org/W1987239897","https://openalex.org/W1987649400","https://openalex.org/W1992831671","https://openalex.org/W1997683636","https://openalex.org/W2000046615","https://openalex.org/W2007474602","https://openalex.org/W2009555358","https://openalex.org/W2012099811","https://openalex.org/W2015356281","https://openalex.org/W2016271169","https://openalex.org/W2019605348","https://openalex.org/W2023876595","https://openalex.org/W2028698043","https://openalex.org/W2033614670","https://openalex.org/W2038078135","https://openalex.org/W2050379993","https://openalex.org/W2058000503","https://openalex.org/W2060555613","https://openalex.org/W2067227212","https://openalex.org/W2073909098","https://openalex.org/W2075488415","https://openalex.org/W2079500909","https://openalex.org/W2081332397","https://openalex.org/W2082736512","https://openalex.org/W2086997195","https://openalex.org/W2087218196","https://openalex.org/W2098347702","https://openalex.org/W2105972215","https://openalex.org/W2109547424","https://openalex.org/W2112283096","https://openalex.org/W2117225445","https://openalex.org/W2120713972","https://openalex.org/W2124133718","https://openalex.org/W2142657695","https://openalex.org/W2145777586","https://openalex.org/W2154011799","https://openalex.org/W2156296681","https://openalex.org/W2164656245","https://openalex.org/W2188657366","https://openalex.org/W2207709065","https://openalex.org/W2317025844","https://openalex.org/W2320186043","https://openalex.org/W2321950138","https://openalex.org/W2563542680","https://openalex.org/W2565582626","https://openalex.org/W2788056314","https://openalex.org/W2788929530","https://openalex.org/W2912588745","https://openalex.org/W2963261858","https://openalex.org/W3037999612","https://openalex.org/W3141871949","https://openalex.org/W4205392813","https://openalex.org/W4212887512","https://openalex.org/W4213338265","https://openalex.org/W4229859481","https://openalex.org/W4230955996","https://openalex.org/W4235044808","https://openalex.org/W4236256974","https://openalex.org/W4238037803","https://openalex.org/W4240542376","https://openalex.org/W4242032240","https://openalex.org/W4246784629","https://openalex.org/W4300168388","https://openalex.org/W6780546593"],"related_works":["https://openalex.org/W2951680245","https://openalex.org/W4289536993","https://openalex.org/W2076946668","https://openalex.org/W7097694320","https://openalex.org/W2407571957","https://openalex.org/W2891915807","https://openalex.org/W2497361095","https://openalex.org/W7095392981","https://openalex.org/W4289543930","https://openalex.org/W2519475223"],"abstract_inverted_index":{"Bousfield":[0,6,80,150],"localisation,":[1,7],"or":[2,30],"more":[3,15],"specifically,":[4],"left":[5],"is":[8,26,96],"an":[9,114],"established":[10],"tool":[11],"to":[12,18,34],"formally":[13],"add":[14],"weak":[16,41,59],"equivalences":[17,42],"a":[19,35,50,62,83],"model":[20,51],"category.":[21,74,93],"The":[22],"most":[23],"common":[24],"setting":[25],"localisation":[27,81,151],"of":[28,45,79,116,161],"spaces":[29],"spectra":[31],"with":[32,53,139],"respect":[33],"homology":[36,55,65],"theory:":[37],"rather":[38],"than":[39],"the":[40,54,58,64,71,91,120,127,140,158,162],"being":[43],"isomorphisms":[44,56,66,69],"homotopy":[46,73,92,131,142,164,175],"groups,":[47],"one":[48],"constructs":[49],"structure":[52,160],"as":[57,82],"equivalences.":[60],"As":[61],"consequence,":[63],"become":[67],"strict":[68],"in":[70,90,98,119],"corresponding":[72],"Therefore,":[75],"we":[76,124],"can":[77,152],"think":[78],"good":[84],"formal":[85],"framework":[86],"for":[87],"inverting":[88],"maps":[89],"Typically,":[94],"information":[95],"lost":[97],"this":[99,117],"process,":[100],"but":[101],"some":[102],"specific":[103],"aspects":[104],"may":[105],"stand":[106],"out":[107],"clearer":[108],"after":[109],"localisation.":[110],"We":[111,145],"will":[112,146],"see":[113,148],"example":[115],"behaviour":[118],"final":[121],"section":[122],"when":[123],"show":[125],"that":[126],"p":[128,167,169],"-local":[129],"stable":[130,141,163],"category":[132,143,165],"has":[133],"vast":[134],"computational":[135],"advantages":[136],"over":[137],"working":[138],"itself.":[144],"also":[147],"how":[149],"help":[153],"us":[154],"gain":[155],"insight":[156],"into":[157],"deeper":[159],"via":[166],"-localisation,":[168],"-completion,":[170],"K":[171],"-theory":[172],"and":[173],"chromatic":[174],"theory.":[176]},"counts_by_year":[],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2022-05-12T00:00:00"}