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 cohomol