{"doi":"10.1017/s0960129508007184","title":"On traced monoidal closed categories","abstract":"<jats:p>The structure theorem of Joyal, Street and Verity says that every traced monoidal category <jats:private-char><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129508007184_char1\" /></jats:private-char> arises as a monoidal full subcategory of the tortile monoidal category <jats:bold>Int</jats:bold><jats:private-char><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129508007184_char1\" /></jats:private-char>. In this paper we focus on a simple observation that a traced monoidal category <jats:private-char><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129508007184_char1\" /></jats:private-char> is <jats:italic>closed</jats:italic> if and only if the canonical inclusion from <jats:private-char><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129508007184_char1\" /></jats:private-char> into <jats:bold>Int</jats:bold><jats:private-char><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129508007184_char1\" /></jats:private-char> has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper more self-contained, we also include various background results for traced monoidal categories.</jats:p>","journal":"Mathematical Structures in Computer Science","year":2009,"id":37723,"datarank":1.8523365178083444,"base_score":3.9318256327243257,"endowment":3.9318256327243257,"self_citation_contribution":0.5897738449086489,"citation_network_contribution":1.2625626728996955,"self_endowment_contribution":0.5897738449086489,"citer_contribution":1.2625626728996955,"corpus_percentile":null,"corpus_rank":null,"citation_count":50,"citer_count":43,"citers_with_citation_signal":24,"citers_with_endowment":24,"datacite_reuse_total":3,"is_dataset":false,"is_dataset_confidence":null,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":null,"fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":188455,"name":"MASAHITO HASEGAWA","orcid":null,"position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"base_score":3.9318256327243257,"endowment":3.9318256327243257,"datacite_reuse_total":3,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"24523987","pmcid":null,"openalex_id":"https://openalex.org/W2167575691","authors":[],"funders":[],"total_grants":0,"fwci":6.4616,"citation_percentile":0.96368548,"influential_citations":4,"citation_trend":[{"year":2012,"count":6},{"year":2013,"count":2},{"year":2014,"count":2},{"year":2016,"count":5},{"year":2019,"count":15},{"year":2020,"count":2},{"year":2021,"count":2},{"year":2022,"count":1},{"year":2023,"count":2},{"year":2024,"count":2}],"oa_status":"closed","license":"https://www.cambridge.org/core/terms","oa_locations":[{"url":"http://www.kurims.kyoto-u.ac.jp/~hassei/papers/mscs2009.pdf","host_type":"GREEN"},{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0960129508007184","host_type":"publisher"},{"url":"https://doi.org/10.1017/s0960129508007184","host_type":"journal"},{"url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.218.9569","host_type":""},{"url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.297.2042","host_type":""},{"url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.865.2037","host_type":""}],"fields_of_study":["Logic, programming, and type systems","Homotopy and Cohomology in Algebraic Topology","Algebraic structures and combinatorial models","Computer Science","Mathematics"],"mesh_terms":[],"keywords":["Symmetric monoidal category","Subcategory","Closed monoidal category","Enriched category","Cartesian closed category","Mathematics","Monoidal category","Pure mathematics","Higher category theory","Algebra over a field","Functor"],"sdg_mappings":[{"sdg_number":0,"sdg_label":"Reduced inequalities"}],"linked_datasets":[{"doi":"10.4230/lipics.calco.2019.8","title":"Coinduction in Flow: The Later Modality in Fibrations","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","resource_type":"ConferencePaper"},{"doi":"10.4230/lipics.fscd.2023.20","title":"On the Lattice of Program Metrics","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","resource_type":"ConferencePaper"},{"doi":"10.4230/lipics.fscd.2023.14","title":"Rewriting Modulo Traced Comonoid Structure","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","resource_type":"ConferencePaper"}],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-06-10T20:46:54.097550Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}