Mr Fish

can you not have auth left and lib left viewpoints?

Yes, but actually no. The distinction is fundamentally unstable. If the left is constantly questioning power structures, it will inevitably turn to whatever structure the auth left comes up with.

what’s wrong with the definition of left and right on the political compass?

It’s specially economic left/right, which is almost always defined by taxation, government spending, and social welfare. While leftists usually say social welfare is a good thing, it’s not changing the fundamentals of how capitalism works, which is the current dominant power structure that leftists are against.

do auth left not also take issue with the power capitalist businesses have?

Yes, but they usually put something just as bad in its place. You might have heard people saying that the USSR was “state capitalist rather than communist”. This means that the workers and customers had just as little say in how things are run than they would under capitalists, only is was directly with the state rather than individual business owners.

Only on the political compass, which uses a definition of left vs right that a lot of leftists disagree with. Really, the entire history of “left wing” politics has been about questioning and dismantling authority. The terms “left wing” and “right wing” come from the French revolution, when the people in favour of simply reforming the monarchy sat on the right side of the room, while the people who wanted to fully dismantling the monarchy sat on the left. A lot of more modern leftist thought is about questioning the power that capitalist businesses have.