Categorical properties of the complex numbers

Journal article

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 †-limits, 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.

Authors: Vicary, J

• Publication status: Published
• Peer review status: Peer reviewed
• Journal: Electronic notes in theoretical computer science
• Volume: 270
• Issue: 2
• Pages: Pages 163–189
• Publication date: 2011-02-01
• Edition: Publisher's version
• DOI: 10.1016/j.entcs.2011.01.030
• ISSN: 1571-0661
• Language: English
• Keywords: complex numbers; category theory; Quantum theory
• Subjects: Quantum theory (mathematics)