Categorical Properties of The Complex Numbers

https://doi.org/10.1016/j.entcs.2011.01.030Get rights and content
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Abstract

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.

Keywords

Quantum theory
category theory
complex numbers

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Financial support from the EPSRC and the ONR is gratefully acknowledged.