Quiver describing perverse sheaves on $\mathbb C$ I have two sources, which claim that the category of perverse sheaves on $\mathbb C$ constructible with respect to the stratification $0$ and $\mathbb C^*$ is equivalent to the category of certain representations of a quiver. In both cases the quiver consists of two dots with one arrow $u,v$ in each direction between them. The first source considers such representation such that both 1+uv and 1+vu are invertible. The second source only wants 1+uv to be invertible. Now I have to admit that I understand neither of the proofs completely so my question is which description is correct?

Suppose $1+uv$ is invertible, inverse $w$. Let your imagination run wild:

$1/(1+vu)= 1-vu+vuvu - vuvuvu + ... = 1 - v(1-uv + uvuv - ...)u = 1-vwu$

Now you can check $1-vwu$ really is inverse to $1+vu$.

There's a name for this trick; I have forgotten it but someone here will know.