Are the genres "co-op" and "multiplayer" different? [duplicate]

I had an argument with a Steam friend, and the conversation was about the topic of so-called "multiplayer" games. I brought up games like TF2, Dota2, and SR3 as games that define what multiplayer is. My friend, however disagreed, and insisted that co-op and multiplayer are different categories; co-op, where you play along players, and multiplayer, meaning playing against someone else.

Am I wrong to think these are two seperate genres of games? Or are co-op and multiplayer virtually the same?

Co-op is a subcategory or game mode of Multiplayer.

Multiplayer covers all games where you are more than one player.

Co-op is short for cooperative and refers a multiplayer game where you play together with other players to complete a common goal against a non-player enemy.

So Co-op games are multiplayer games, but multiplayer games don't have to be co-op games.

Co-op mode is commonly found in shooter games as a way of playing the story mode in multiplayer. Examples of this is the multiplayer/co-op mode in Quake, Saints Row: The Third and Gears of Wars series.

You also find Co-op in many Hack and slash, rpg and music games, some examples of these are the Diablo series, Secret of Mana and Guitar Hero.

I think co-op is a subcategory of multiplayer. More precisely:

• Multiplayer Basically every video game with more than one player, be it through a network connection or at the same PC.
• Co-op Several players cooperate in solving a task and do not compete with each other. If there are opponents or enemies, they are controlled by an AI.
• PVP Players compete with each other. Note that this does not mean that there can be no teams.

So games like DOTA would be multiplayer PVP, while for example Portal 2 would qualify as co-op.