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 $\mathrm{pt}$ for the tensor unit in $H$ (not necessarily the terminal object).

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

• a factorization

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

  of the codiagonal morphism

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

  from the coproduct of $\mathrm{pt}$ with itself that sends each component identically to $\mathrm{pt}$.

• together with an associative morphsim

  $$\vee :I\otimes I\to I$$\vee : I \otimes I \to I

  which has 0 as its neutral and 1 as its absorbing element, and for which $ϵ$ is a counit.

If $H$ is equipped with the structure of a model category then a segment object is an interval in $H$ if

$$[0,1]:\mathrm{pt}⨿\mathrm{pt}\to I$$[0, 1]\colon pt \amalg pt \to I

is a cofibration and $ϵ:I\to \mathrm{pt}$ 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):

$$\begin{array}{ccc}(H\otimes H)\otimes H& {\to }^{\sim }& H\otimes (H\otimes H)\\ {\downarrow }^{\vee \otimes H}& & {\downarrow }_{H\otimes \vee }\\ H\otimes H& \stackrel{\vee }{\leftarrow }H\stackrel{\vee }{⟵}& H\otimes H\end{array}$$\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 }

$$\begin{array}{ccccc}I\otimes H& {\to }^{0\otimes H}& H\otimes H& {\leftarrow }^{H\otimes 0}& H\otimes I\\ & {\searrow }_{\sim }& {\downarrow }_{\vee }& {\swarrow }_{\sim }& \\ & & H& & \end{array}$$\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&& }

$$\begin{array}{ccccccccc}& & I\otimes H& {\to }^{1\otimes H}& H\otimes H& {\leftarrow }^{H\otimes 1}& H\otimes I& & \\ & {\swarrow }^{I\otimes ϵ}& \downarrow & & {\downarrow }_{\vee }& & \downarrow & {\searrow }^{ϵ\otimes I}& \\ I\otimes I& {\to }^{\sim }& I& {\to }^1& H& {\leftarrow }^1& I& {\leftarrow }^{\sim }& I\otimes I\end{array}$$\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}

$$\begin{array}{ccccccc}H\otimes H& {\to }^{ϵ\otimes ϵ}& I\otimes I& & I& {\to }^0& H\\ {\downarrow }^{\vee }& & {\downarrow }_{\sim }& & {\downarrow }_1& {\searrow }^{\mathrm{id}}& {\downarrow }_{ϵ}\\ H& {\to }^{ϵ}& I& & H& {\to }^{ϵ}& I\end{array}$$\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}