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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.

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  • $\begingroup$ As I am sure you know, $E$ can be described as the $D$-acyclic objects. If $D$ can be replaced by just a set worth of objects which it suffices to test Mor(d,-) against then you can use the machinery of Verdier localization and Bousfield localization to get a lot more traction on $E$. See for example Neeman's book on triangulated categories. In that case $E$ is a colocalizing subcategory, so it's also closed under arbitrary products. I don't know about these pseudo-extensions. It reminds me of projective modules, so maybe already covered by retract closure? $\endgroup$ – David White May 21 '15 at 19:05
  • $\begingroup$ Certainly, in some cases retract-closures or extension-closures are sufficient to "generate" $E$ starting from a smaller $E'$ that is orthogonal to $D$ (and such that $D$ is the maximal class orthogonal to it). Yet my pseudo-extensions seem to give "more" objects in general. $\endgroup$ – Mikhail Bondarko May 21 '15 at 19:31
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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.

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    $\begingroup$ Actually, Verdier knew the alternative criterion for a subcategory to be épaisse (although I didn't know that when I rediscovered it): it's in his thesis, that wasn't published in full until 1996. $\endgroup$ – Jeremy Rickard Jun 3 '15 at 15:00

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