20 - Functors

Summary

In this chapter we look at the concept of morphisms between categories, using the principle of preserving structure. We give the definition in two ways, one for each of our two approaches to defining categories (by homsets or by graphs). We look at functors between small examples of categories, including functors between posets, monoids, and groups, expressed as categories. We consider functors from small drawable categories and show that they produce a diagram of that shape in the target category. We think about the category consisting of a single non-trivial isomorphism, and see that a functor out of it picks out an isomorphism in the target category. We describe free and forgetful functors, including the free monoid functor. We define the concept of functors preserving and reflecting structure, and show that not all functors preserve epics, but they all preserve split epics. We consider whether the above forgetful functors preserve terminal and initial objects. Further topics include the fundamental group functor, and Van Kampen’s theorem reframed as preservation of pushouts under certain circumstances. We introduce contravariant functors.