We discussed Communism recently, which is one type of left wing belief.
I would suggest that the left’s core belief is:
Everyone should have a good life and society should work to make that happen.
What different left ideologies are arguing about his the means more than the end. A Communist believes the only way for this to be achieved is for the proletariat to control the means of production. (This makes them effectively the left oppositional image of capitalists.)