The Language of Category Theory (Part II)

We sort of paused after our first category theory post. We left off with the definition of a functor. What should a functor be? Like group homomorphisms, ring homomorphisms, linear transformations, continuous maps, etc., it should be something that “preserves the structure of the category”. This is reflected in the following formal definition:

Definition: Let \mathscr{C} and \mathscr{D} be categories. Then a functor is a map that takes \mathrm{Ob}(\mathscr{C})\to\mathrm{Ob}(\mathscr{D}) and \mathrm{Mor}(\mathscr{C})\to\mathrm{Mor}(\mathscr{D}), that preserves the identity map and composition laws.

What’s an example of this? Let \mathbf{Top} and \mathbf{Ab} denote the categories of topological spaces and abelian groups. Then homology is simply a collection of functors \mathbf{Top}\to\mathbf{Ab}. Ok, what about cohomology? Observe that cohomology is like the “opposite” of homology, in the sense that a map X\to Y induces a map H^n(Y)\to H^n(X).

This brings us to the notion of the opposite of a category. If \mathscr{C} is a category, then \mathscr{C}^{op} is the category whose objects are the same as that of \mathscr{C}, but whose morphisms are reversed.

Then cohomology is a collection of functors \mathbf{Top}^{op}\to\mathbf{Ab}. Functors \mathscr{C}^{op}\to\mathscr{D} have a special name – contravariant functors. (The ordinary functors are called covariant functors, but no one really says that. They just say functor.) Let’s give another example: for i>0, we see that homotopy is simply a functor \mathbf{Top}_\ast\to\mathbf{Grp}, where \mathbf{Top}_\ast  is the category of pointed spaces.

There are special kinds of functors, for which we need to introduce some more concepts (like hom-sets, etc.) This gives rise to a notion of equivalence between categories, which is much weaker (and more useful) than the (obvious) notion of an isomorphism of categories. This’ll be done in the next post.




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