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PSL is only one that has access in several states. SEP and SWP are both Trotskyists, and I haven't heard of the other parties.
This question probably deserves its own thread, but why do the Trotskyists have a bad rap? Whenever I hear them mentioned on a podcast (these might include The Antifada, Delete Your Account, Well There's Your Problem, or Chapo Trap House) it's always accompanied with an almost audible eye roll. Are there major critiques against the Trotskyists that they just can't address or is it just me and maybe I need to broaden my perspective a little bit?
Trotskyism in the imperial core is broadly the politics of rhetorically supporting the theoretical tenets of Marxism while materially siding with liberal imperialism against every actually existing communist movement, party, or state in history. It's a way to adopt countercultural aesthetics from a position of privilege while fighting to preserve the material conditions that allow your aesthetics to remain countercultural.
Is there any relationship to the real guy or do they just use his name?
Trotskyists generally believe in permanent revolution and dictatorship of the proletariat alone instead of a dictatorship of the proletariat and peasantry, which both come from Trotsky. Trotsky called the USSR a "degenerated workers' state" which meant it still had a socialist economy but the state did not represent the workers. Some modern Trotskyists still follow this but others consider the USSR and modern socialist countries like Cuba and the DPRK to be state capitalist and then often abandon Marxism entirely (the Trotskyist to neocon pipeline followed by Tony Blair and others). Trotsky also wanted rapid collectivization and to continue war communism after the civil war but Trotskyists then criticize Stalin for collectivizing agriculture. There are also Marcyists who originated out of Trotskyism but are now fully pro-AES and essentially align with MLs on most issues.
In the West, Trotskyism is also seen as an acceptable alternative to “Stalinism” and the Soviet Union.
It’s the reason mindset behind why liberals idealise failed revolutions but despise those that succeed.