from the coproduct of $pt$ with itself that sends each component identically to $pt$.
together with an associative morphsim
$\vee : I \otimes I \to I$
which has 0 as its neutral and 1 as its absorbing element, and for which $\epsilon$ is a counit.
If $H$ is equipped with the structure of a model category then a segment object is an interval in $H$ if
$[0, 1]\colon pt \amalg pt \to I$
is a cofibration and $\epsilon : I \to pt$ a weak equivalence.
The axioms of a segment are expressed by the commutativity of the following five diagrams (all isomorphisms being induced by the symmetric monoidal structure):
$\array{
(H\otimes H)\otimes H&\to^\sim&H\otimes(H\otimes H)\\\downarrow^{\vee\otimes H}&&\downarrow_{H\otimes\vee}\\H\otimes H&\overset{\vee}{\leftarrow} H\overset{\vee}{\longleftarrow}&H\otimes H
}$