I don't seem to have access to the forge playlist in the EDZ. When talking to Ada-1 the weekly bounties say "Require forge access". I've seen screenshots of where the Forge playlist is supposed to be in the bottom left of the EDZ, but I have nothing there. It's my understanding that this is supposed to be unlocked automatically with Shadowkeep.
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Did you start playing the game with New Light or had you played the game prior to the release of Shadowkeep? If you are a New Light player you should have a New Light quest-line. During that quest you will be tasked with visiting Ada-1 at which point she will give you a machine gun frame. You need to forge that frame before the playlist icon becomes available to freely run Forges.