From Categories to Homotopy Theory
"Category theory has at least two important features. The first one is that it allows us to structure our mathematical world. Many constructions that you encounter in your daily life look structurally very similar, like products of sets, products of topological spaces, products of modules and then you might be delighted to learn that there is a notion of a product of objects in a category and all the above examples are actually just instances of such products, here in the category of sets, topological spaces and modules, so you don't have to reprove all the properties products have, because they hold for every such construction. So category theory helps you to recognize things as what they are. It also allows you to express objects in a category by something that looks apparently way larger. For instance the Yoneda Lemma describes a set of the form F(C) (where C is an object of some category and F is a functor from that category to the category of sets) as the set of natural transformations between another nice functor and F. This might look like a bad deal, but in this set of natural transformations you can manipulate things and this reinterpretation for instance gives you cohomology operations as morphisms between the representing objects. Another feature is that you can actually use category theory in order to build topological spaces and to do homotopy theory. A central example is the nerve of a (small) category: You view the objects of your category as points, every morphism gives a 1-simplex, a pair of composable morphisms gives a 2-simplex and so on. Then you build a topological space out of this by associating a topological n-simplex to an n-simplex in the nerve, but you do some non-trivial gluing, for instance identity morphisms don't really give you any information so you shrink the associated edges. In the end you get a CW complex BC for every small category C. Properties of categories and functors translate into properties of this space and continuous maps between such spaces. For instance a natural transformation between two functors gives rise to a homotopy between the induced maps and an equivalence of categories gives a homotopy equivalence of the corresponding classifying spaces. Classifying spaces of categories give rise to classifying spaces of groups but you can also use them and related constructions to build the spaces of the algebraic K-theory spectrum of a ring, you can give models for iterated based loop spaces and you can construct explicit models of homotopy colimits and much more"--