Definition Midpoint algebra [efr-6W7X]

Let \mathcal {C} be a Cartesian category. Then a midpoint algebra is an object A equipped with m: A \times A \to A, so that: m(a,a) = a m(a,b) = m(b,a) m(m(a,b),m(c,d)) = m(m(a,c), m(b,d))

We will call a homomorphism of midpoint algebras (a morphism satisfying m(f(a),f(a')) = f(m(a,a'))) a midpoint homomorphism