Forgetful functor

Summary

A forgetful functor sends a mathematical object to a less structured object by dropping some of its structure. It sends each morphism to the same map, regarded in the poorer setting. The basic example sends an algebra to its underlying set and a homomorphism to its underlying function. Forgetful functors often have left adjoints, the free functors, and the pair produces a monad that encodes the algebraic theory. They also make a category of algebras into a concrete category.