A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor\_{C}(d,e)=\{0\}\ \forall d\in D$; which "operations" respect $E$? Note that $E$ is "extension-stable" (i.e., for any $C$-triangle $e\_1\to e\_2\to e\_3$ the object $e\_2$ belongs to $E$ if $e\_1$ and $e\_3$ do). Besides, $E$ is Karoubi-closed in $C$ (i.e., if $e$ is an element of $E$ then any $C$-retract of $e$ belongs to $E$ also).


I have an example of a more explicit (and general) "operation" of this sort. Let us say that $M$ is a "pseuo-extension" of an object $N$ by $A$ if there exists a distinguished triangle $A\stackrel{g}{\to} M\stackrel{f}{\to} M'$ such that $f$ factors through $N$ (this is actually equivalent to assuming the existence of a triangle $B\stackrel{g}{\to} M\stackrel{f}{\to} M'$ such that $f$ factors through $N$ and $g$ factors through $A$; so, the notion is self-dual). Then $E$ is closed with respect to "pseudo-extensions". Have anybody previously studied constructions of this sort? If I am the first to consider this notion, does "pseudo-extension" sounds fine to you?


P.S. Surprisingly, being closed under direct summands and extensions characterizes precisely those classes of objects of $C$ that are zero sets of a cohomological functor $H$ from $C$ into abelian groups (see the paper <https://arxiv.org/abs/1508.04427>). This probably implies that all "finite operations" of the sort described above reduce to retractions and extensions. 



On various "extension closures" and "orthogonals" in triangulated categories
The argument described in the proof of Proposition 1.3 of
 Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317,
yields that any pseudo-extension (of $N$ by $A$, in the sense described in my question) is a retract of an "honest extension" (of $N$ by $A$). So, pseudo-extensions do not yield a new "operation". This statement seems to be quite important (at least, historically): whereas Verdier essentially defined his epaisse subcategories of triangulated categories in terms of pseudo-extensions (certainly, he did not use this term), Rickards' result gave a much more convenient criterion for a (triangulated) subcategory to be epaisse.


Yet I am still interested in the "general" form of my question.


