nLab segment object

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 $H$ be a monoidal model category and write $pt$ for the tensor unit in $H$ (not necessarily the terminal object).

A segment (object) $I$ in a monoidal model category $H$ is

• a factorization

  $$pt \amalg pt \stackrel{[0 , 1]}{\to} I \stackrel{\epsilon}{\to} pt$$

  of the codiagonal morphism

  $$pt \amalg pt \stackrel{[Id , Id]}{\to} pt$$

  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):

$$\array{ (H\otimes H)\otimes H&\to^\sim&H\otimes(H\otimes H)\\\downarrow^{\vee\otimes H}&&\downarrow_{H\otimes\vee}\\H\otimes H&\overset{\vee}{\leftarrow} H\overset{\vee}{\longleftarrow}&H\otimes H }$$

$$\array{I\otimes H&\rightarrow^{0\otimes H}& H\otimes H&\leftarrow^{H\otimes 0}&H\otimes I\\&\searrow_\sim&\downarrow_\vee&\swarrow_\sim&\\&&H&& }$$

$$\array{&&I\otimes H&\rightarrow^{1\otimes H}&H\otimes H&\leftarrow^{H\otimes 1}&H\otimes I&&\\&\swarrow^{I\otimes\epsilon}&\downarrow&&\downarrow_\vee&&\downarrow&\searrow^{\epsilon\otimes I}&\\I\otimes I&\rightarrow^\sim&I&\rightarrow^1&H&\leftarrow^1&I&\leftarrow^\sim&I\otimes I}$$

$$\array{H\otimes H&\rightarrow^{\epsilon\otimes\epsilon}&I\otimes I&\quad&I&\rightarrow^0&H\\\downarrow^\vee&&\downarrow_\sim&\quad&\downarrow_1&\searrow^{id}&\downarrow_\epsilon\\H&\rightarrow^\epsilon&I&\quad&H&\rightarrow^\epsilon&I}$$