nLab dyadic interval coalgebra

Definition

A dyadic interval coalgebra is a set $I$ with a linear order $\lt$, elements $0 \in I$ and $1 \in I$ and functions $z_0:I \to I$ and $z_1:I \to I$, such that $z_0(0) = 0$, $z_1(0) = 0$, $z_0(1) = 1$, $z_1(1) = 1$, * for all elements $a \in I$, $0 \lt a$ or $a \lt 1$, and for all elements $a \in I$, it is false that both $0 \lt z_0(a)$ and $z_1(a) \lt 1$.

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, such as the decimal interval coalgebra.

Examples

• The initial dyadic interval coalgebra is the unit interval on the dyadic rational numbers

• The terminal dyadic interval coalgebra is the unit interval on the Dedekind real numbers

See also

• dyadic rational number
• rational interval coalgebra
• decimal interval coalgebra
• unit interval
• coalgebra

References

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)