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  "body": "\n## Summary\n\nA forgetful functor sends a mathematical object to a less structured object by dropping some of its structure.\nIt sends each morphism to the same map, regarded in the poorer setting.\nThe basic example sends an algebra to its underlying set and a homomorphism to its underlying function.\nForgetful functors often have left adjoints, the [[free functors|N353JF8RBK]], and the pair produces a monad that encodes the algebraic theory.\nThey also make a category of algebras into a [[concrete category|ZD0GTPSJS1]].\n\n## Explanations\n\n### L1 human\n\nA group is a set together with a multiplication.\nForgetting the multiplication leaves just the set, and a group homomorphism is still a function between the sets.\nThe forgetful functor from groups to sets is this act of forgetting, done for all groups and homomorphisms at once.\nOne can forget less: a ring can be sent to its additive group, forgetting only the multiplication.\n",
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