In section 4 of
- Clemens Berger, Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories (arXiv)
the following definition is given:
Let be a monoidal model category and write for the tensor unit in (not necessarily the terminal object).
A segment (object) in a monoidal model category is
of the codiagonal morphism
from the coproduct of with itself that sends each component identically to .
together with an associative morphsim
which has 0 as its neutral and 1 as its absorbing element, and for which is a counit.
If is equipped with the structure of a model category then a segment object is an interval in if
is a cofibration and 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):