Homotopy push-out squares and exact triangles are colimits I read somewhere that homotopy push-out squares and exact triangles in a triangulated category can both be interpreted as special cases of higher categorical colimits. Why is this true?
Please note that I am indeed a beginner in category theory, so a rough intuition of the analogy would be more than enough for me. I am fully aware this would sacrifice precise formalism, but I am looking for the general idea here.
 A: Oh, boy, this is a very long story to tell!
I will indeed sacrifice formalism to convey intuition: roughly speaking, a triangulated category arises as a certain kind of homotopy category, i.e. a category $C$ where you have some notion of weak equivalence, i.e. a class of maps you want to treat as if they were invertible.
Now, you can think of homotopy categories as arising as "1-dimensional shadows" of higher categorical concepts: intuitively, a 1-category is objects+morphisms; a higher category has higher-dimensional "cells" "filling" "commutative" "diagrams" (each of these words is better left undefined).
$$
\begin{array}{c}
\text{higher categories} \ni \mathcal K\\ 
\downarrow \\
\text{1-categories} \ni K
\end{array}
$$ where the correspondence "flattens" all the higher-dimensional information, forgetting all the stuff that lies beyond dimension 1.
If you're familiar with simplicial complexes, the intuition behind truncating a simplicial complex carries over here, again with some magic spell muttered here and there.
Now, a fundamental tenet of Platonism, as well as of higher category theory is that very little can be said about an object, judging the shadow it casts: so, it turns out that construction in triangulated categories, like distinguished triangles and homotopy colimits, lack the universal properties they had "upstairs" when regarded as shadows.
In short: the object $\mathcal K$ has a true and very clear universal property, but it is extremely difficult to define and handle the place where it belongs to; otoh, its shadow $K$ is extremely unnatural to define (it's not a real colimit, just a shadow thereof), but the place where it lives is way more ergonomic.
