Functors

Functor

A functor is a generalized homomorphism.

A functor preserves category laws while providing functions which take objects to objects and morphisms to morphisms between category C and some other category C'.

Forgetful functors

You can use a smartphone as a paperweight.

One can "downgrade" categories to their closely related cousins:

One can downgrade the "context of interaction."

• Grp to Mon. Forget every element has an inverse.
• Mon to Set. Forget multiply function around underlying set of a monoid.
• Grp to Set. Forget inverse property and forget multiply function.