Morita homotopy theory of C*-categories
In this article we establish the foundations of the Morita homotopy theory of C*-categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure (denoted by M[subscript Mor]) on the category C1*cat of small unital C*-categories. The weak equivalence...
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| Format: | Article |
| Language: | en_US |
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Elsevier
2016
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| Online Access: | http://hdl.handle.net/1721.1/103964 https://orcid.org/0000-0001-5558-9236 |
| _version_ | 1826216635517632512 |
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| author | DellʼAmbrogio, Ivo Trigo Neri Tabuada, Goncalo Jo |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics DellʼAmbrogio, Ivo Trigo Neri Tabuada, Goncalo Jo |
| author_sort | DellʼAmbrogio, Ivo |
| collection | MIT |
| description | In this article we establish the foundations of the Morita homotopy theory of C*-categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure (denoted by M[subscript Mor]) on the category C1*cat of small unital C*-categories. The weak equivalences are the Morita equivalences and the cofibrations are the *-functors which are injective on objects. As an application, we obtain an elegant description of Brown–Green–Rieffelʼs Picard group in the associated homotopy category Ho(M[subscript Mor]). We then prove that Ho(M[subscript Mor]) is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category Ho(M[subscript Mor])[superscript −1] and a composite functor C1*cat→Ho(M[subscript Mor][superscript −1] which is characterized by two simple properties: inversion of Morita equivalences and preservation of all finite products. Finally, we prove that the classical Grothendieck group functor becomes co-represented in Ho(M[subscript Mor])[superscript −1] by the tensor unit object. |
| first_indexed | 2024-09-23T16:50:17Z |
| format | Article |
| id | mit-1721.1/103964 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:50:17Z |
| publishDate | 2016 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/1039642022-10-03T08:41:06Z Morita homotopy theory of C*-categories DellʼAmbrogio, Ivo Trigo Neri Tabuada, Goncalo Jo Massachusetts Institute of Technology. Department of Mathematics Trigo Neri Tabuada, Goncalo Jo In this article we establish the foundations of the Morita homotopy theory of C*-categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure (denoted by M[subscript Mor]) on the category C1*cat of small unital C*-categories. The weak equivalences are the Morita equivalences and the cofibrations are the *-functors which are injective on objects. As an application, we obtain an elegant description of Brown–Green–Rieffelʼs Picard group in the associated homotopy category Ho(M[subscript Mor]). We then prove that Ho(M[subscript Mor]) is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category Ho(M[subscript Mor])[superscript −1] and a composite functor C1*cat→Ho(M[subscript Mor][superscript −1] which is characterized by two simple properties: inversion of Morita equivalences and preservation of all finite products. Finally, we prove that the classical Grothendieck group functor becomes co-represented in Ho(M[subscript Mor])[superscript −1] by the tensor unit object. NEC Corporation (NEC Award 2742738) Fundação para a Ciência e a Tecnologia (Portugal) (PEst-OE/MAT/UI0297/2011) 2016-08-24T18:01:26Z 2016-08-24T18:01:26Z 2013-10 2012-12 Article http://purl.org/eprint/type/JournalArticle 00218693 http://hdl.handle.net/1721.1/103964 DellʼAmbrogio, Ivo, and Gonçalo Tabuada. "Morita homotopy theory of C*-categories." Journal of Algebra 398 (January 2014): 162–199. https://orcid.org/0000-0001-5558-9236 en_US http://dx.doi.org/10.1016/j.jalgebra.2013.09.022 Journal of Algebra Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier arXiv |
| spellingShingle | DellʼAmbrogio, Ivo Trigo Neri Tabuada, Goncalo Jo Morita homotopy theory of C*-categories |
| title | Morita homotopy theory of C*-categories |
| title_full | Morita homotopy theory of C*-categories |
| title_fullStr | Morita homotopy theory of C*-categories |
| title_full_unstemmed | Morita homotopy theory of C*-categories |
| title_short | Morita homotopy theory of C*-categories |
| title_sort | morita homotopy theory of c categories |
| url | http://hdl.handle.net/1721.1/103964 https://orcid.org/0000-0001-5558-9236 |
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