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The Heywood reading talks about Culture Wars having to do with third wave feminism and how they distinguish themselves from second wave feminists. Culture wars also can refer to a conflict or struggle for dominance between groups within a society or between societies, arising from their differing beliefs, practices, etc. This of course ties into feminism since feminists are a group within society trying to fight for equal rights to their male counterparts.
I agree with your view on culture wars. I think that feminism on its own has a culture. Culture wars definitely play a role in third wave feminism and create drawbacks for those of color and those that are unprivileged.
I hold a similar definition on what culture wars are and agree with your definition. These wars do play a role within society and the way the privileged and underprivileged are treated.