forgetful functor

A forgetful functor is the formalization of the everyday act of ignoring structure. A group is a set with extra data — a multiplication, an identity, inverses — and if you simply forget that extra data you are left with a plain set and a plain function. The forgetful functor performs precisely this discarding, systematically and on morphisms too, sending each homomorphism to its underlying function. Despite the modest name, forgetful functors organize a huge amount of algebra.

There is no formal definition isolating “forgetful” from other functors; the term is descriptive. Typical examples are U : Grp -> Set, U : Ring -> Ab (forget multiplication, keep the additive group), U : Top -> Set, and U : Ab -> Grp (forget commutativity, an inclusion that forgets a property rather than data). Forgetful functors are generally faithful — they do not lose track of which morphism is which — but rarely full.

Their importance is that they so often have a left adjoint, the corresponding free functor, which builds the universal object carrying the forgotten structure with no relations imposed. Free group, free module, free monoid, tensor algebra, and group ring all arise as left adjoints to forgetful functors. Because forgetful functors are right adjoints in these situations, they preserve all limits — which is why the underlying set of a product of groups is the product of the underlying sets.