I read Bridgelands *Stability conditions on K3 surfaces*, in the **Lemma 4.4** a complete quasi-abelian subcategory appears $ mathscr {A} subset mathscr {D} $ a triangulated category $ mathscr {D} = mathscr {D} (X) $ of a smooth sort $ X $, Then he considers a "strict" short exact order

$$ 0 to A to B to C to 0 tag {$ * $} $$

With $ A, B, C in mathscr {A} $,

**1 question:** How exactly do you define (short) exact consequences in a quasi-abelian category?

The definition of abelian categories $ operatorname {Ker} (f_i) = operatorname {im} (f_ {i + 1}) $ seems problematic because I do not know how to define the picture. In general it seems like that $ operatorname {ker coker} f neq operatorname {ker coker} f $see Wikipedia.

**2nd question:** What does "strict" mean in this context? Does it have anything to do with the first question, or does it just mean that? $ A neq 0, B neq 0 $these are non-trivial sub-objects / quotients?

Then he argues further $ f (B) = f (A) + f (C) $, from where $ f: K ( mathscr {D}) to mathbb {R} $ is an additive function.

**3rd question:** Why does a short exact sequence $ (*) $ in the $ mathscr {A} $ induce a triangle in $ mathscr {D} $?

I know that this is true, though $ mathscr {A} $ is Abelian and $ mathscr {D} = mathscr {D} ( mathscr {A}) $ is the derived category of $ mathscr {A} $but I see no reason why this is true.