{"id":"https://openalex.org/W2963062026","doi":"https://doi.org/10.1007/s10485-019-09572-y","title":"Differential Categories Revisited","display_name":"Differential Categories Revisited","publication_year":2019,"publication_date":"2019-07-04","ids":{"openalex":"https://openalex.org/W2963062026","doi":"https://doi.org/10.1007/s10485-019-09572-y","mag":"2963062026"},"language":"en","primary_location":{"id":"doi:10.1007/s10485-019-09572-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10485-019-09572-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10485-019-09572-y.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":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10485-019-09572-y.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087503903","display_name":"Richard Blute","orcid":null},"institutions":[{"id":"https://openalex.org/I153718931","display_name":"University of Ottawa","ror":"https://ror.org/03c4mmv16","country_code":"CA","type":"education","lineage":["https://openalex.org/I153718931"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"R. F. Blute","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada","institution_ids":["https://openalex.org/I153718931"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053636420","display_name":"J.R.B. Cockett","orcid":null},"institutions":[{"id":"https://openalex.org/I168635309","display_name":"University of Calgary","ror":"https://ror.org/03yjb2x39","country_code":"CA","type":"education","lineage":["https://openalex.org/I168635309"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"J. R. B. Cockett","raw_affiliation_strings":["Department of Computer Science, University of Calgary, Calgary, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Calgary, Calgary, Canada","institution_ids":["https://openalex.org/I168635309"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5064711276","display_name":"Jean-Simon Pacaud Lemay","orcid":"https://orcid.org/0000-0003-4124-3722"},"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":"J.-S. P. Lemay","raw_affiliation_strings":["Department of Computer Science, University of Oxford, Oxford, UK"],"raw_orcid":"https://orcid.org/0000-0003-4124-3722","affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Oxford, Oxford, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084038892","display_name":"R. A. G. Seely","orcid":null},"institutions":[{"id":"https://openalex.org/I216472412","display_name":"John Abbott College","ror":"https://ror.org/0560ksp05","country_code":"CA","type":"education","lineage":["https://openalex.org/I216472412"]},{"id":"https://openalex.org/I3133086287","display_name":"Cegep de Thetford","ror":"https://ror.org/04k5sn095","country_code":"CA","type":"education","lineage":["https://openalex.org/I3133086287"]},{"id":"https://openalex.org/I323585489","display_name":"Cegep de Trois-Rivieres","ror":"https://ror.org/00fjnjn69","country_code":"CA","type":"education","lineage":["https://openalex.org/I323585489"]},{"id":"https://openalex.org/I4047665","display_name":"Dawson College","ror":"https://ror.org/04pmj7516","country_code":"CA","type":"education","lineage":["https://openalex.org/I4047665"]},{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"R. A. G. Seely","raw_affiliation_strings":["Departments of Mathematics, CEGEP John Abbott College, McGill University, Montreal, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departments of Mathematics, CEGEP John Abbott College, McGill University, Montreal, Canada","institution_ids":["https://openalex.org/I216472412","https://openalex.org/I5023651","https://openalex.org/I4047665","https://openalex.org/I3133086287","https://openalex.org/I323585489"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5064711276"],"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":5.7031,"has_fulltext":true,"cited_by_count":23,"citation_normalized_percentile":{"value":0.96863237,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"28","issue":"2","first_page":"171","last_page":"235"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9980999827384949,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9980999827384949,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T10126","display_name":"Logic, programming, and type systems","score":0.9916999936103821,"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":0.9901999831199646,"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/coalgebra","display_name":"Coalgebra","score":0.9305533766746521},{"id":"https://openalex.org/keywords/bialgebra","display_name":"Bialgebra","score":0.6876246333122253},{"id":"https://openalex.org/keywords/differential","display_name":"Differential (mechanical device)","score":0.6082904934883118},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6042356491088867},{"id":"https://openalex.org/keywords/symmetric-monoidal-category","display_name":"Symmetric monoidal category","score":0.5889554023742676},{"id":"https://openalex.org/keywords/modality","display_name":"Modality (human\u2013computer interaction)","score":0.5083686709403992},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.45192500948905945},{"id":"https://openalex.org/keywords/category-theory","display_name":"Category theory","score":0.4472188651561737},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4207736849784851},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.22106480598449707},{"id":"https://openalex.org/keywords/hopf-algebra","display_name":"Hopf algebra","score":0.15562507510185242},{"id":"https://openalex.org/keywords/functor","display_name":"Functor","score":0.14035820960998535},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.12232646346092224}],"concepts":[{"id":"https://openalex.org/C2778249326","wikidata":"https://www.wikidata.org/wiki/Q1777803","display_name":"Coalgebra","level":3,"score":0.9305533766746521},{"id":"https://openalex.org/C2778676360","wikidata":"https://www.wikidata.org/wiki/Q494141","display_name":"Bialgebra","level":4,"score":0.6876246333122253},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.6082904934883118},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6042356491088867},{"id":"https://openalex.org/C98912367","wikidata":"https://www.wikidata.org/wiki/Q17103860","display_name":"Symmetric monoidal category","level":3,"score":0.5889554023742676},{"id":"https://openalex.org/C2780226545","wikidata":"https://www.wikidata.org/wiki/Q6888030","display_name":"Modality (human\u2013computer interaction)","level":2,"score":0.5083686709403992},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.45192500948905945},{"id":"https://openalex.org/C54884031","wikidata":"https://www.wikidata.org/wiki/Q217413","display_name":"Category theory","level":2,"score":0.4472188651561737},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4207736849784851},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22106480598449707},{"id":"https://openalex.org/C29712632","wikidata":"https://www.wikidata.org/wiki/Q1627597","display_name":"Hopf algebra","level":3,"score":0.15562507510185242},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.14035820960998535},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.12232646346092224},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10485-019-09572-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10485-019-09572-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10485-019-09572-y.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"}],"best_oa_location":{"id":"doi:10.1007/s10485-019-09572-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10485-019-09572-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10485-019-09572-y.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":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320317153","display_name":"DeepMind","ror":"https://ror.org/00971b260"},{"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/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"},{"id":"https://openalex.org/F4320334841","display_name":"Fonds de recherche du Qu\u00e9bec \u2013 Nature et technologies","ror":"https://ror.org/00b9f9778"},{"id":"https://openalex.org/F4320336183","display_name":"Clarendon Fund","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2963062026.pdf","grobid_xml":"https://content.openalex.org/works/W2963062026.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W584090173","https://openalex.org/W848641472","https://openalex.org/W1505195721","https://openalex.org/W1605806361","https://openalex.org/W1644475622","https://openalex.org/W1761392971","https://openalex.org/W1820790138","https://openalex.org/W1973799895","https://openalex.org/W2088376608","https://openalex.org/W2092216512","https://openalex.org/W2093561140","https://openalex.org/W2120713972","https://openalex.org/W2522785819","https://openalex.org/W2913248536","https://openalex.org/W3105507374","https://openalex.org/W4237109299","https://openalex.org/W4256453086","https://openalex.org/W4285719527","https://openalex.org/W6600745895"],"related_works":["https://openalex.org/W1634154041","https://openalex.org/W2108680998","https://openalex.org/W2974208691","https://openalex.org/W2950316645","https://openalex.org/W2041819297","https://openalex.org/W2361695990","https://openalex.org/W2737177195","https://openalex.org/W2807927563","https://openalex.org/W2963062026","https://openalex.org/W3039716079"],"abstract_inverted_index":{"Differential":[0],"categories":[1],"were":[2,31],"introduced":[3],"to":[4,27,61,77],"provide":[5,162],"a":[6,37,41,49,53,78,95,118,157,163,181,186],"minimal":[7],"categorical":[8],"doctrine":[9],"for":[10,103,123],"differential":[11,144,158,182],"linear":[12,62,104],"logic.":[13],"Here":[14],"we":[15,82],"revisit":[16],"the":[17,23,46,66,72,92,150,168],"formalism":[18],"and,":[19],"in":[20,45,91],"particular,":[21],"examine":[22],"two":[24],"different":[25],"approaches":[26],"defining":[28],"differentiation":[29],"which":[30,128,138,184],"introduced.":[32],"The":[33,55],"basic":[34],"approach":[35,57],"used":[36,52],"deriving":[38],"transformation,":[39],"while":[40],"more":[42],"refined":[43],"approach,":[44],"presence":[47,93],"of":[48,89,94,112,120,180],"bialgebra":[50,79],"modality,":[51,98],"codereliction.":[54],"latter":[56],"is":[58,69,108,149],"particularly":[59],"relevant":[60],"logic":[63,105],"settings,":[64,106],"where":[65],"coalgebra":[67,97,124,165,188],"modality":[68],"monoidal":[70,96,164],"and":[71,130,140],"Seely":[73],"isomorphisms":[74],"give":[75],"rise":[76],"modality.":[80,166,189],"Here,":[81],"prove":[83],"that":[84],"these":[85],"apparently":[86],"distinct":[87],"notions":[88],"differentiation,":[90],"are":[99,129,131,173],"completely":[100],"equivalent.":[101],"Thus,":[102],"there":[107],"only":[109],"one":[110],"notion":[111],"differentiation.":[113],"This":[114],"paper":[115],"also":[116],"presents":[117],"number":[119],"separating":[121],"examples":[122,127,137],"modalities":[125],"including":[126],"not":[132,142],"monoidal,":[133],"as":[134,136],"well":[135],"do":[139,141],"support":[143],"structure.":[145],"Of":[146],"particular":[147],"interest":[148],"observation":[151],"that\u2014somewhat":[152],"counter-intuitively\u2014differential":[153],"algebras":[154],"never":[155],"induce":[156],"category":[159,183],"although":[160],"they":[161],"On":[167],"other":[169],"hand,":[170],"Rota\u2013Baxter":[171],"algebras\u2014which":[172],"usually":[174],"associated":[175],"with":[176],"integration\u2014provide":[177],"an":[178],"example":[179],"has":[185],"non-monoidal":[187]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}