Examples of (covariant) functors

Last time we had a slightly awkward definition of a (covariant) functor, so I thought I’d clear some things up with some definitions and motivation. Don’t worry if you don’t understand why I keep writing “covariant” in parentheses. You’ll understand soon enough.

The idea of a functor is that it translates the diagrams we can draw in one category into diagrams in another category. Often times this lets us convert ideas and theorems in one category into another.

  • Given any category \mathcal C, we have the identity functor. It takes an object C\in Ob(\mathcal C) to itself, and the morphism f:X\to Y to itself. Pretty lame.
  • Let \textsc{Grp} be the category of groups with group homomorphisms, and let \textsc{Set} be the category of sets with set maps. Then there is a functor F:\textsc{Grp}\to\textsc{Set} which takes each group G to the set of elements in G, and each group homomorphism \phi to it’s underlying set map. In some sense, this functor shows us something obvious. It’s saying that every group is really just a set with more structure, and every group homomorphism is a set map but with more structure. Then F says, take these things, and forget about the group structure; instead just think of them as sets. For this reason, we call it a forgetful functor.
  • Similarly, there are forgetful functors from rings to sets, fields to sets, fields to rings, rings to groups, etc.
  • One of my favorite examples of a forgetful functor is the functor from \textsc{Cat} (the category of small categories) to \textsc{Dig} (the category of directed graphs). It takes the “diagram of the category,” and forgets about the identity/associativity rules, and just keeps the underlying directed graph.
  • If you know any algebraic topology, then you probably already know this fact, but any homology group H_n(-) is a functor from \textsc{Top} (the category of topological spaces with continuous maps) to \textsc{Ab} (the category of abelian groups with group homomorphisms).
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