Today I Learned

Some of the things I've learned every day since Oct 10, 2016

86: Forgetful Functors

Forgetful functors are, as the name implies, functors between categories that forget something about the structure or properties of the objects and arrows in the source category of the functor.

Examples:

  • The functor U: \mathbf{Grp} \rightarrow \mathbf{Set}, which maps groups to their underlying sets and group homomorphisms to themselves. U forgets the group structure and that group homomorphisms are anything other than functions between sets.
  • The functor V: \mathbf{Ab} \rightarrow \mathbf{Grp} from the category of abelian groups to the category of groups. V simply maps groups and group homomorphisms to themselves, essentially forgetting that the source groups are abelian.
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