or would that be a different category

Yes, You are allowed to upload full game runs / single difficulty runs that are Multiplayer.

-TechNation Ex. SPEEDRUN Flood Escape Mod

Thanks Tech

@TechNation follow up question, if i did a single difficulty no items run and someone else was using items, would that be allowed?

Does all players need to beat the difficulty in order for it to count?

If you're doing, as an example, single difficulty easy, then only you need to beat it. This is specifically for public servers though. For private server runs, they should be solo. This is specified since not everyone would be able to afford a private server to play on.