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

    of the codiagonal morphism

    pt⨿pt[Id,Id]ptpt \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

    :III\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]:pt⨿ptI[0, 1]\colon pt \amalg pt \to I

is a cofibration and ϵ:Ipt 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):

(HH)H H(HH) H H HH H HH\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 }
IH 0H HH H0 HI 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&& }
IH 1H HH H1 HI Iϵ ϵI II I 1 H 1 I II\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}
HH ϵϵ II I 0 H 1 id ϵ H ϵ I H ϵ 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}
Created on November 8, 2012 17:57:07 by Stephan Alexander Spahn (