Coherence for 3-dualizable objects
<p>A fully extended framed topological field theory with target in a symmetric monoidal <em>n</em>-catgeory 𝐶 is a symmetric monoidal functor <em>Z</em> from Bord(n) to 𝐶, where Bord(n) is the symmetric monoidal <em>n</em>-category of <em>n</em>-...
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| Format: | Thesis |
| Language: | English |
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2017
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| _version_ | 1826289073951604736 |
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| author | Araújo, M |
| author2 | Douglas, C |
| author_facet | Douglas, C Araújo, M |
| author_sort | Araújo, M |
| collection | OXFORD |
| description | <p>A fully extended framed topological field theory with target in a symmetric monoidal <em>n</em>-catgeory 𝐶 is a symmetric monoidal functor <em>Z</em> from Bord(n) to 𝐶, where Bord(n) is the symmetric monoidal <em>n</em>-category of <em>n</em>-framed bordisms. The cobordism hypothesis says that such field theories are classified by fully dualizable objects in 𝐶.</p> <p>Given a fully dualizable object <em>X</em> in 𝐶, we are interested in computing the values of the corresponding field theory on specific framed bordisms. This leads to the question of finding a presentation for Bord(n). In view of the cobordism hypothesis, this can be rephrased in terms of finding <em>coherence data</em> for fully dualizable objects in a symmetric monoidal <em>n</em>-category. </p> <p>We prove a characterization of full dualizability of an object <em>X</em> in terms of existence of a dual of <em>X</em> and existence of adjoints for a finite number of higher morphisms. This reduces the problem of finding coherence data for fully dualizable objects to that of finding coherence data for duals and adjoints. For <em>n</em>=3, and in the setting of strict symmetric monoidal 3-categories, we find this coherence data, and we prove the corresponding coherence theorems. The proofs rely on extensive use of a graphical calculus for strict monoidal 3-categories.</p> |
| first_indexed | 2024-03-07T02:23:21Z |
| format | Thesis |
| id | oxford-uuid:a4b8f8de-a8e3-48c3-a742-82316a7bd8eb |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T02:23:21Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | oxford-uuid:a4b8f8de-a8e3-48c3-a742-82316a7bd8eb2022-03-27T02:35:41ZCoherence for 3-dualizable objectsThesishttp://purl.org/coar/resource_type/c_db06uuid:a4b8f8de-a8e3-48c3-a742-82316a7bd8ebTopologyManifoldsTopological Field TheoryHigher CategoriesEnglishORA Deposit2017Araújo, MDouglas, CBarrett, JKremnitzer, Y<p>A fully extended framed topological field theory with target in a symmetric monoidal <em>n</em>-catgeory 𝐶 is a symmetric monoidal functor <em>Z</em> from Bord(n) to 𝐶, where Bord(n) is the symmetric monoidal <em>n</em>-category of <em>n</em>-framed bordisms. The cobordism hypothesis says that such field theories are classified by fully dualizable objects in 𝐶.</p> <p>Given a fully dualizable object <em>X</em> in 𝐶, we are interested in computing the values of the corresponding field theory on specific framed bordisms. This leads to the question of finding a presentation for Bord(n). In view of the cobordism hypothesis, this can be rephrased in terms of finding <em>coherence data</em> for fully dualizable objects in a symmetric monoidal <em>n</em>-category. </p> <p>We prove a characterization of full dualizability of an object <em>X</em> in terms of existence of a dual of <em>X</em> and existence of adjoints for a finite number of higher morphisms. This reduces the problem of finding coherence data for fully dualizable objects to that of finding coherence data for duals and adjoints. For <em>n</em>=3, and in the setting of strict symmetric monoidal 3-categories, we find this coherence data, and we prove the corresponding coherence theorems. The proofs rely on extensive use of a graphical calculus for strict monoidal 3-categories.</p> |
| spellingShingle | Topology Manifolds Topological Field Theory Higher Categories Araújo, M Coherence for 3-dualizable objects |
| title | Coherence for 3-dualizable objects |
| title_full | Coherence for 3-dualizable objects |
| title_fullStr | Coherence for 3-dualizable objects |
| title_full_unstemmed | Coherence for 3-dualizable objects |
| title_short | Coherence for 3-dualizable objects |
| title_sort | coherence for 3 dualizable objects |
| topic | Topology Manifolds Topological Field Theory Higher Categories |
| work_keys_str_mv | AT araujom coherencefor3dualizableobjects |