Recently, the French president Macron has offered to mediate between the Syrian Kurdish force YPG and the Turkish government.
How will Macron or France benefit from mediating between the YPG Kurds and the Turkish government?
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The Kurdish YPG is a core part of the SDF, the Syrian Democratic Force; and the French and the USA are supporting this militia with weapons and training in the fight against IS - it's this that's the benefit to the two western powers.
For example, Macrons office "paid tribute to the sacrifices and determining role of the SDF" in the fight against IS.
Macrons negotiations on behalf of the SDF with Turkey is further support; what troubles Turkey is alleged links of the SDF with the PKK, the Kurdistans Workers Party even though the USA is backing the YPG insistence that they have nothing to do with the PKK. Turkey calls the PKK a terrorist organisation (as is common today) and say that both the PKK & YPG are attempting to legitimise themselves under the umbrella of the SDF.
Source: the BBC