symmetric monoidal (∞,1)-category of spectra
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.
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
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