{"id":{"repo_id":"calgary","oai_identifier":"oai:ucalgary.scholaris.ca:1880/114807"},"canonical_url":"https://search.dev.ndltd.org/etd/calgary/oai:ucalgary.scholaris.ca:1880/114807","repository":{"repo_id":"calgary","name":"University of Calgary","base_url":"https://ucalgary.scholaris.ca/server/oai/request"},"display":{"title":"The functorial semantics of Lie theory","abstract":"Ehresmann’s introduction of differentiable groupoids in the 1950s may be seen as a starting point for two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids) and sketch theory. This thesis uses tangent categories to build a bridge between these two lines of research, providing a structural account of Lie algebroids and the Lie functor. To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of involution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classifying category of an involution algebroid. This exhibits the category of Lie algebroids as a tangent-categorical functor category, and the Lie functor via precomposition with a functor ∂ : Weil1 → TGpd, bringing Lie algebroids and the Lie functor into the realm of functorial semantics.","abstract_html":"Ehresmann’s introduction of differentiable groupoids in the 1950s may be seen as a starting point for two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids) and sketch theory. This thesis uses tangent categories to build a bridge between these two lines of research, providing a structural account of Lie algebroids and the Lie functor. To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of involution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classifying category of an involution algebroid. This exhibits the category of Lie algebroids as a tangent-categorical functor category, and the Lie functor via precomposition with a functor ∂ : Weil1 → TGpd, bringing Lie algebroids and the Lie functor into the realm of functorial semantics.","abstract_has_math":false,"creators":["MacAdam, Benjamin"],"institution":"Science","degree_name":"Doctor of Philosophy (PhD)","degree_level":null,"degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":["Cockett, J. Robin B.","Scheidler, Renate"],"committee_chairs":[],"committee_members":["Bauer, Kristine","Ching, Michael","Bitoun, Thomas"],"year":2022,"date_issued":"2022-06","date_published":"2022-06","updated_at":"2026-07-24T01:30:18Z","subjects":["Category Theory","Lie theory","Enriched category theory","Synthetic Differential Geometry","Lie Algebroids","Lie groupoids"],"languages":["eng"],"rights":["University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["http://dx.doi.org/10.11575/PRISM/39877"],"render_values":[{"text":"http://dx.doi.org/10.11575/PRISM/39877","href":"http://dx.doi.org/10.11575/PRISM/39877","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1880/114807","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Cockett, J. 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This thesis uses tangent categories to build a bridge between these two lines of research, providing a structural account of Lie algebroids and the Lie functor. To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of involution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classifying category of an involution algebroid. This exhibits the category of Lie algebroids as a tangent-categorical functor category, and the Lie functor via precomposition with a functor ∂ : Weil1 → TGpd, bringing Lie algebroids and the Lie functor into the realm of functorial semantics."]},{"key":"dc:title","label":"Title","values":["The functorial semantics of Lie theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Cockett, J. Robin B.","Scheidler, Renate"],"dc:contributor.committeemember":["Bauer, Kristine","Ching, Michael","Bitoun, Thomas"],"dc:creator":["MacAdam, Benjamin"],"dc:date":["2022-11"],"dc:date.accessioned":["2022-07-05T17:41:17Z"],"dc:date.available":["2022-07-05T17:41:17Z"],"dc:date.issued":["2022-06"],"dc:description.abstract":["Ehresmann’s introduction of differentiable groupoids in the 1950s may be seen as a starting point for two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids) and sketch theory. This thesis uses tangent categories to build a bridge between these two lines of research, providing a structural account of Lie algebroids and the Lie functor. To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of involution algebroids in smooth manifolds, and that the category of Weil algebras is precisely the classifying category of an involution algebroid. This exhibits the category of Lie algebroids as a tangent-categorical functor category, and the Lie functor via precomposition with a functor ∂ : Weil1 → TGpd, bringing Lie algebroids and the Lie functor into the realm of functorial semantics."],"dc:identifier.doi":["http://dx.doi.org/10.11575/PRISM/39877"],"dc:identifier.uri":["http://hdl.handle.net/1880/114807"],"dc:language.iso":["eng"],"dc:publisher.institution":["University of Calgary"],"dc:rights":["University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission."],"dc:subject":["Category Theory","Lie theory","Enriched category theory","Synthetic Differential Geometry","Lie Algebroids","Lie groupoids"],"dc:title":["The functorial semantics of Lie theory"],"dc:type":["doctoral thesis"],"thesis:degree_discipline":["Computer Science"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["University of Calgary"]},"updated_at":"2026-07-24T01:30:18Z"}