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A FORGETFUL FUNCTOR IS A TYPE OF FUNCTOR IN MATHEMATICS. THE NOMENCLATURE IS SUGGESTIVE OF SUCH A FUNCTOR'S BEHAVIOUR: GIVEN SOME ALGEBRAIC OBJECT AS INPUT, SOME OR ALL OF THE OBJECT'S STRUCTURE IS 'FORGOTTEN' IN THE OUTPUT. FOR AN ALGEBRAIC STRUCTURE OF A GIVEN SIGNATURE, THIS MAY BE EXPRESSED BY CURTAILING THE SIGNATURE IN SOME WAY: THE NEW SIGNATURE IS AN EDITED FORM OF THE OLD ONE. IF THE SIGNATURE IS LEFT AS AN EMPTY LIST, THE FUNCTOR IS SIMPLY TO TAKE THE UNDERLYING SET OF A STRUCTURE; THIS IS IN FACT THE MOST COMMON CASE. FOR EXAMPLE, THE FORGETFUL FUNCTOR FROM THE CATEGORY OF RINGS TO THE CATEGORY OF ABELIAN GROUPS ASSIGNS TO EACH RING R THE UNDERLYING ADDITIVE ABELIAN GROUP OF R. TO EACH MORPHISM OF RINGS IS ASSIGNED THE SAME FUNCTION CONSIDERED MERELY AS A MORPHISM OF ADDITION BETWEEN THE UNDERLYING GROUPS. A COMMON SUBCLASS OF FORGETFUL FUNCTORS IS AS FOLLOWS. LET BE ANY CATEGORY BASED ON SETS, E.G. GROUPS - SETS OF ELEMENTS - OR TOPOLOGICAL SPACES - SETS OF 'POINTS'. AS USUAL, WRITE FOR THE OBJECTS OF AND WRITE FOR THE MORPHISMS OF THE SAME. CONSIDER THE RULE: A IN THE UNDERLYING SET OF A, U IN THE MORPHISM, U, AS A MAP OF SETS. THE FUNCTOR IS THEN THE FORGETFUL FUNCTOR FROM TO , THE CATEGORY OF SETS. FORGETFUL FUNCTORS ARE ALWAYS FAITHFUL. CONCRETE CATEGORIES HAVE FORGETFUL FUNCTORS TO THE CATEGORY OF SETS -- INDEED THEY MAY BE DEFINED AS THOSE CATEGORIES WHICH ADMIT A FAITHFUL FUNCTOR TO THAT CATEGORY. FORGETFUL FUNCTORS TEND TO HAVE LEFT ADJOINTS WHICH ARE 'FREE' CONSTRUCTIONS. FOR EXAMPLE, THE FORGETFUL FUNCTOR FROM (THE CATEGORY OF R-MODULE) TO HAS LEFT ADJOINT F, WITH , THE FREE R-MODULE WITH BASIS X. FOR A MORE EXTENSIVE LIST, SEE [MAC LANE]. [EDIT] REFERENCES [MAC LANE] CATEGORIES FOR THE WORKING MATHEMATICIAN, SAUNDERS MAC LANE, SPRINGER GRADUATE TEXTS IN MATHEMATICS 5, 1997.

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