I am following Lurie's Higher Algebra and having trouble understanding Definition 1.1.2.11 (I wanted to write it down here but I don't know how to draw a diagram), which defines distinguished triangles in a homotopy category of a pointed $\infty$-category with cofibers.
My (possibly trivial) question is, why can't we define them using the same pushout diagram with $W$ is replaced by $X[1]$ (i.e. the diagram obtained by composing the equivalence $W\to X[1]$) and just say it represents $X\to Y\to Z\to X[1]$? Isn't it equivalent, or am I just confused?
Ok, I was half-correct and half-confused.
The following is an equivalent definition: There exists a fiber sequence $X\xrightarrow{f} Y\xrightarrow{g} Z\xrightarrow{h} X[1]$ such that the composition of the two pushout diagrams is the prescribed one, i.e. the induced self-equivalence $X[1]\to X[1]$ is homotopic to the identity.
This is not really simpler than the definition given in HA.
The (obvious but subtle) point is that, if we just take two random pushout diagrams, it does not produce a right definition, since if the induced self-equivalence of $X[1]$ is not the identity in the homotopy category it will disturb $h$.