how do I delete a shared photo stream?

Hello nick baccante,

Welcome to the Apple Support Communities!

I understand that you have questions about shared albums using Photo Stream. If you would like to delete specific photos or videos from the shared album, please review the first attached article.

iCloud: Delete photos or videos from a shared album

If you would like to remove a shared album completely, please refer to the next attached article.

iCloud: Remove a shared album

Have a great day,

Joe

Right-Click or Control_Click and delete album.

If the one that's empty is created by enabling Family Sharing in iCloud prefs, then you can't delete it. You would have to disable Family Sharing.

If Family Sharing is what the album is for and you want to keep it, move the photos from the other album to it, and delete the other you created.

I've never enable it myself, so I don't know how good it works. If you didn't want to enable Family Sharing, you can disable it:

Leave Family Sharing - Apple Support

You don't need to enable family sharing to share albums, as I'm sure you know.