Categorical properties of the complex numbers
<p>Given the success of categorical approaches to quantum theory, it is interesting to consider why the complex numbers are special from a categorical perspective. We describe natural categorical conditions under which the scalars of a monoidal †-category gain many of the features of the compl...
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| Format: | Journal article |
| Language: | English |
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2011
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| _version_ | 1797072953165217792 |
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| author | Vicary, J |
| author_facet | Vicary, J |
| author_sort | Vicary, J |
| collection | OXFORD |
| description | <p>Given the success of categorical approaches to quantum theory, it is interesting to consider why the complex numbers are special from a categorical perspective. We describe natural categorical conditions under which the scalars of a monoidal †-category gain many of the features of the complex numbers. Central to our approach are †-<em>limits</em>, certain types of limits which are compatible with the †-functor; we explore their properties and prove an existence theorem for them. Our main theorem is that in a nontrivial monoidal †-category with finite †-limits and simple tensor unit, and in which the self-adjoint scalars satisfy a completeness condition, the scalars are valued in the complex numbers, and scalar involution is exactly complex conjugation.</p> |
| first_indexed | 2024-03-06T23:15:06Z |
| format | Journal article |
| id | oxford-uuid:66da9dc5-afda-41b8-8f0f-168b07925767 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:15:06Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:66da9dc5-afda-41b8-8f0f-168b079257672022-03-26T18:34:28ZCategorical properties of the complex numbersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:66da9dc5-afda-41b8-8f0f-168b07925767Quantum theory (mathematics)EnglishOxford University Research Archive - Valet2011Vicary, J<p>Given the success of categorical approaches to quantum theory, it is interesting to consider why the complex numbers are special from a categorical perspective. We describe natural categorical conditions under which the scalars of a monoidal †-category gain many of the features of the complex numbers. Central to our approach are †-<em>limits</em>, certain types of limits which are compatible with the †-functor; we explore their properties and prove an existence theorem for them. Our main theorem is that in a nontrivial monoidal †-category with finite †-limits and simple tensor unit, and in which the self-adjoint scalars satisfy a completeness condition, the scalars are valued in the complex numbers, and scalar involution is exactly complex conjugation.</p> |
| spellingShingle | Quantum theory (mathematics) Vicary, J Categorical properties of the complex numbers |
| title | Categorical properties of the complex numbers |
| title_full | Categorical properties of the complex numbers |
| title_fullStr | Categorical properties of the complex numbers |
| title_full_unstemmed | Categorical properties of the complex numbers |
| title_short | Categorical properties of the complex numbers |
| title_sort | categorical properties of the complex numbers |
| topic | Quantum theory (mathematics) |
| work_keys_str_mv | AT vicaryj categoricalpropertiesofthecomplexnumbers |