Does quest progress in co-op carry over to single player?

For example, if my friend hosts a game and I join it and we go through the entirety of Act I, will I have to beat Act I all over again once I leave the co-op session?

Solution 1:

While in CO-OP.

• You gain achievements for co-op and for single.
• You quest log count for co-op and for single.

So you gain more achievements playing as a group than solo. You don't lose nothing. But remember monsters get 75% more health for each player in the party.

About the quest progress.

• You are starting ACT III quest 1
• Your friend is starting ACT III quest 4
• You jump in your friend game and finish ACT III with him

You'll be able to start all quest beyond 4 and the first quest in ACT III. But You'll need to complete the quests 1, 2 and 3 from ACT III.

Solution 2:

You won't have to redo the act. Every quest your character completes is completed both when playing with, and without, friends.