Categorical properties of the complex numbers Export

@misc{citeulike:3025211, abstract = {Given the success of categorical approaches to quantum theory, and given that quantum theory is underpinned by the complex numbers, it is interesting to study the categorical properties of the complex numbers directly. We describe natural categorical conditions under which the scalars of a monoidal dagger-category gain many of the features of the complex numbers. Central to our approach is the requirement that the dagger-functor be compatible with the construction of particular limits in the category; we show that this implies nondegeneracy of the dagger-functor, as well as cancellable hom-set addition. Our main theorem is that in a nontrivial monoidal dagger-category with finite dagger-biproducts and finite dagger-equalisers, for which the monoidal unit has no proper dagger-subobjects, the scalars have an involution-preserving embedding into an involutive field with characteristic 0 and orderable fixed field, and therefore embed into the complex numbers if they are at most of continuum cardinality.}, archivePrefix = {arXiv}, author = {Vicary, Jamie}, citeulike-article-id = {3025211}, citeulike-linkout-0 = {http://arxiv.org/abs/0807.2927}, citeulike-linkout-1 = {http://arxiv.org/pdf/0807.2927}, day = {18}, eprint = {0807.2927}, keywords = {category-theory, mathematics, printed}, month = {Jul}, posted-at = {2008-07-21 19:45:34}, priority = {3}, title = {Categorical properties of the complex numbers}, url = {http://arxiv.org/abs/0807.2927}, year = {2008} }

TY - ELEC ID - citeulike:3025211 L3 - citeulike-article-id:3025211 N2 - Given the success of categorical approaches to quantum theory, and given that quantum theory is underpinned by the complex numbers, it is interesting to study the categorical properties of the complex numbers directly. We describe natural categorical conditions under which the scalars of a monoidal dagger-category gain many of the features of the complex numbers. Central to our approach is the requirement that the dagger-functor be compatible with the construction of particular limits in the category; we show that this implies nondegeneracy of the dagger-functor, as well as cancellable hom-set addition. Our main theorem is that in a nontrivial monoidal dagger-category with finite dagger-biproducts and finite dagger-equalisers, for which the monoidal unit has no proper dagger-subobjects, the scalars have an involution-preserving embedding into an involutive field with characteristic 0 and orderable fixed field, and therefore embed into the complex numbers if they are at most of continuum cardinality. TI - Categorical properties of the complex numbers KW - category-theory KW - mathematics KW - printed AU - Vicary, Jamie PY - 2008/Jul/18/ UR - http://arxiv.org/abs/0807.2927 ER -