{"id":"https://openalex.org/W7214670964","doi":"https://doi.org/10.48550/arxiv.2609.31445","title":"Geometric morphisms of virtual ultracategories","display_name":"Geometric morphisms of virtual ultracategories","publication_year":2026,"publication_date":"2026-09-25","ids":{"openalex":"https://openalex.org/W7214670964","doi":"https://doi.org/10.48550/arxiv.2609.31445"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.31445","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.31445","pdf_url":"https://arxiv.org/pdf/2609.31445","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.31445","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015792268","display_name":"Saadia Gabriel","orcid":"https://orcid.org/0009-0001-9353-951X","observed_orcids":["https://orcid.org/0009-0001-9353-951X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gabriel, Saadia","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5136826149","display_name":"Errol Yuksel","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Errol, Yuksel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"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":0.33719998598098755,"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":0.33719998598098755,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.29589998722076416,"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/T11727","display_name":"Advanced Algebra and Logic","score":0.16689999401569366,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/geometric-morphisms","display_name":"geometric morphisms","score":0.9070000052452087},{"id":"https://openalex.org/keywords/embeddings","display_name":"embeddings","score":0.7630000114440918},{"id":"https://openalex.org/keywords/filtered-colimits","display_name":"filtered colimits","score":0.5929999947547913}],"concepts":[{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.7904000282287598},{"id":"https://openalex.org/C137212723","wikidata":"https://www.wikidata.org/wiki/Q1948412","display_name":"Morphism","level":2,"score":0.7379999756813049},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6532999873161316},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.42590001225471497},{"id":"https://openalex.org/C136520226","wikidata":"https://www.wikidata.org/wiki/Q302814","display_name":"Geometric data analysis","level":2,"score":0.41940000653266907},{"id":"https://openalex.org/C2776477805","wikidata":"https://www.wikidata.org/wiki/Q4460773","display_name":"Topological data analysis","level":2,"score":0.38019999861717224},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3537999987602234},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3292999863624573},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32339999079704285},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3125},{"id":"https://openalex.org/C182141236","wikidata":"https://www.wikidata.org/wiki/Q2143621","display_name":"Topos theory","level":2,"score":0.31220000982284546},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30469998717308044},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.28839999437332153},{"id":"https://openalex.org/C104065381","wikidata":"https://www.wikidata.org/wiki/Q1002535","display_name":"Geometric modeling","level":2,"score":0.2856000065803528},{"id":"https://openalex.org/C7305733","wikidata":"https://www.wikidata.org/wiki/Q207961","display_name":"Geometric shape","level":2,"score":0.2809999883174896},{"id":"https://openalex.org/C181095308","wikidata":"https://www.wikidata.org/wiki/Q1541599","display_name":"Geometric primitive","level":2,"score":0.272599995136261},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.2515000104904175}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.31445","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.31445","pdf_url":"https://arxiv.org/pdf/2609.31445","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"doi:10.48550/arxiv.2609.31445","doi":"https://doi.org/10.48550/arxiv.2609.31445","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.31445","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.31445","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.31445","pdf_url":"https://arxiv.org/pdf/2609.31445","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214670964.pdf","grobid_xml":"https://content.openalex.org/works/W7214670964.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W36355091","https://openalex.org/W950096932","https://openalex.org/W2060621887","https://openalex.org/W2150369467","https://openalex.org/W2197451100","https://openalex.org/W3091315172","https://openalex.org/W4206671793","https://openalex.org/W4399794824","https://openalex.org/W4414687944","https://openalex.org/W4416524137"],"related_works":["https://openalex.org/W4416524137","https://openalex.org/W4414687944","https://openalex.org/W6979222443","https://openalex.org/W4308613640","https://openalex.org/W7162221825","https://openalex.org/W7099144970","https://openalex.org/W3088986164","https://openalex.org/W4388254206","https://openalex.org/W184605745","https://openalex.org/W1541510438"],"abstract_inverted_index":{"We":[0],"build":[1],"on":[2,53],"the":[3,47],"recent":[4],"development":[5],"that":[6],"toposes":[7],"with":[8,19],"enough":[9],"points":[10,17,83],"can":[11],"be":[12],"represented":[13],"as":[14],"categories":[15],"of":[16,28,42,49,56,69,79,82,84],"equipped":[18],"ultraconvergence":[20],"data":[21],"(a.k.a.":[22],"virtual":[23],"ultracategories),":[24],"characterizing":[25],"usual":[26,61],"classes":[27,81],"geometric":[29,50],"morphisms":[30],"---":[31,39],"surjections,":[32],"embeddings,":[33],"hyperconnected":[34],"maps,":[35],"and":[36,63,74],"localic":[37],"maps":[38],"in":[40],"terms":[41],"their":[43],"points.":[44],"In":[45],"particular,":[46],"characterisation":[48,78],"surjections":[51],"relies":[52],"a":[54,76,85],"notion":[55,68],"\"virtual":[57],"ultraretractions\",":[58],"which":[59],"generalises":[60],"retractions":[62],"filtered":[64],"colimits,":[65],"categorifies":[66],"Johnstone's":[67],"subclosure":[70],"for":[71],"topological":[72],"spaces,":[73],"provides":[75],"new":[77],"separating":[80],"topos.":[86]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-09-29T00:00:00"}