I'm reading Bridgeland's *Stability conditions on K3 surfaces*. In **Lemma 4.4** there appears a full quasi-abelian subcategory $\mathscr{A} \subset \mathscr{D}$ of a triangulated category $\mathscr{D} = \mathscr{D}(X)$ of a smooth variety $X$. Then he considers a "strict" short exact sequence
$$0 \to A \to B \to C \to 0, \tag{$\*$}$$
with $A, B, C \in \mathscr{A}$.


**1. Question:** How exactly does one define (short) exact sequences in a quasi-abelian category?


The definition of abelian categories $\operatorname{Ker}(f\_i) = \operatorname{Im}(f\_{i+1})$ seems problematic, because I don't know how to define the image. In general it seems that $\operatorname{coker ker} f \neq\operatorname{ker coker} f$, see [Wikipedia](https://en.wikipedia.org/wiki/Quasi-abelian_category#Properties).


**2. Question:** What does "strict" mean in that context? Does it have anything to do with the first question, or does it just mean that $A \neq 0 \neq B$, i.e. those are non-trivial subobjects / quotients?


After that Bridgeland goes on to argue that $f(B) = f(A) + f(C)$, where $f: K(\mathscr{D}) \to \mathbb{R}$ is an additive function.


**3. Question:** Why does a short exact sequence $(\*)$ in $\mathscr{A}$ induce a triangle in $\mathscr{D}$?


I know that this is true if $\mathscr{A}$ 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 in our case.



About "strict" short exact sequences in quasi-abelian subcategory of a derived category
I found the answer two 1. and 2. in Bridgeland's previous work *Stability Conditions on Triangulated Categories*:


Let $\mathscr{A}$ be an additive category with kernels and cokernels. A morphism $f: A \to B$ is called *strict*, if the canonical map $\operatorname{coker ker} f \to \operatorname{ker coker} f$ is an isomorphism.


$\mathscr{A}$ is called *quasi-abelian* if the pullback of every strict epi is a strict epi, and the pushout of every strict mono is a strict mono. Then a strict short exact sequence is a diagram
$$ 0 \to A \xrightarrow{i} B \xrightarrow{j} C \to 0$$
in which $i$ is the kernel of $j$ and $j$ is the cokernel of $i$. In particular $i$ is mono and $j$ is epi, so $\operatorname{ker} i = 0$ and $\operatorname{coker} j = 0$. Hence
\begin{align}
\operatorname{ker coker} i = \operatorname{ker}j & = i = \operatorname{coker ker} i \\
\operatorname{coker ker} j = \operatorname{coker} i & = j = \operatorname{ker coker} j,
\end{align}
so both $i$ and $j$ are strict.
In Lemma 4.3 Bridgeland then proceeds to prove that in exactly the situation which appears in the K3 surface paper, strict short exact sequences in $\mathscr{A}$ are in one-to-one correspondence to exact triangles in $\mathscr{D}$.


