## Definition ## A **dyadic interval coalgebra** is a type $T$ with a [[strict order]] $\lt$, terms $0$ and $1$, functions $z_0:T \to T$ and $z_1:T \to T$, identities $z_0(0) = 0$, $z_1(0) = 0$, $z_0(1) = 1$, $z_1(1) = 1$, inequality $0 \lt 1$, and terms $$\alpha:\prod_{a:T} \left[(0 \lt z_0(a)) + (z_1(a) \lt 1)\right] \to \emptyset$$ This is called simply an interval coalgebra by Peter Freyd, however there exist similarly defined interval coalgebras with $n+1$ terms and $n$ zooming operations. ## Examples ## * The initial dyadic interval coalgebra is the [[unit interval]] on the [[dyadic rational numbers]] * The terminal dyadic interval coalgebra is the [[real unit interval]] ## See also ## * [[rational interval coalgebra]] * [[decimal interval coalgebra]] * [[real unit interval]] ## References ## * Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))