A Categorical Construction of the Real Unit Interval
Abstract
The real unit interval is the fundamental building block for many branches of mathematics like probability theory, measure theory, convex sets and homotopy theory. However, a priori the unit interval could be considered an arbitrary choice and one can wonder if there is some more canonical way in which the unit interval can be constructed. In this paper we find such a construction by using the theory of effect algebras. We show that the real unit interval is the unique noninitial, nonfinal irreducible algebra of a particular monad on the category of bounded posets. The algebras of this monad carry an order, multiplication, addition and complement, and as such model much of the operations we need to do on probabilities. On a technical level, we show that both the categories of omegacomplete effect algebras as well as that of omegacomplete effect monoids are monadic over the category of bounded posets using Beck's monadicity theorem. The characterisation of the real unit interval then follows easily using a recent representation theorem for omegacomplete effect monoids.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.10094
 Bibcode:
 2021arXiv210610094V
 Keywords:

 Mathematics  Category Theory
 EPrint:
 13 pages + 2 page appendix