I mean… I wouldn’t really say Doom or Duke Nukem or whatever were arena shooters, though? Something like Quake 3 certainly is, but I feel like anything that vaguely has a map that you progress through instead of having a round of a fixed length / kill count in a smaller space doesn’t scream “arena shooter” to me.
Doom had multiplayer over lan that I would say fits in that definition. For me personally arena shooters are more about movement speed and how weapons/health are managed as power ups to be found and managed as a recourses. I don’t consider multiplayer to be a requirement for the genre basically.