The Psycore plays a major role in the Protomon world. It was first introduced in Protomon Gen 1; Beta and Theta, but can be found in all the Protomon games. It refers to more than one place, but just saying Psycore means that you're talking about the waiting room of the Psycore Lab. Go to this link to see all the Psycore areas.
"The Psycore started when Blight Diro made the Protanium Magnet. Blight tested it out on the driveway to space he made, but it malfunctioned and pulled a local planet to the driveway, which is now called Planet Psycore. Professors settled there and built what is locally known as the Psycore Lab. The Psycore Lab became one of the biggest industries in Protomon history, causing tourism, and the tourists brought Protomon with them. The Protomon bred at Planet Psycore, resulting in Planet Psycore being their natural habitat."