Bridgeland-Stability Condition: Why is the Harder-Narasimhan filtration unique? I'm trying to understand Bridgeland's notion of stability condition on a triangulated category as defined in Definition 1.1 of the paper Stability conditions on triangulated categories.

See also Definition 3.3 in the same paper.
The decompositions of a non-zero object as in Definition 3.3(c) are the Harder-Narasimhan filtrations.


  Question: Why is the Harder-Narasimhan filtration of any non-zero object unique (up to isomorphism of the semistable factors)?


Thank you for your help!
The short answer is that the filtrations are unique because they arise as factorization of an orthogonal, multiple factorization system: see here §4.1.

The even shorter answer is that you can prove it for triangulated categories, but it's unsatisfying.
The short answer is that the filtrations are unique because they arise as factorization of an orthogonal, multiple factorization system: see here §4.1.

The even shorter answer is that you can prove it for triangulated categories, but it's unsatisfying.