8 1. Categories and Functors led to new insights. Another algebraic example is given by group actions, in which the category of groups is studied using the category of sets play- ing these two categories off of one another is how the Sylow theorems are generally proved. A functor is a formalism for comparing categories you can think of a functor intuitively as a morphism from one category to another one. Just as a homomorphism respects the algebraic structure of a group, and a continuous map respects the topological structure of a space, a functor must respect the key features of categories. Functors take objects to objects and morphisms to morphisms they respect composition and preserve identity morphisms. There are actually two kinds of functors—those which reverse the direc- tion of morphisms and those which don’t. A covariant functor F : C D consists of a function F : ob(C) −→ ob(D) and, for each X, Y ob(C), a function F : morC(X, Y ) −→ morD(F (X),F (Y )). These must satisfy the following conditions: (1) F (g f) = F (g) F (f). (2) F (idX) = idF (X) for any X ob(C). Notice that if F : C D is a covariant functor and f : X Y is a morphism in C, then F (f) : F (X) F (Y ) thus, F carries the domain of f to the domain of F (f), and similarly for the targets. In other words, F (f) points in ‘the same direction’ as f. This is the meaning of the word ‘covariant’. Exercise 1.10. (a) Let D be a diagram in a category A, and let D be the category obtained from it as in Exercise 1.3(d). Show that D is commutative if and only if for any two objects X, Y D, mor D (X, Y ) has at most one element. (b) Show that if you apply a covariant functor F : A B to a commutative diagram in A, the result is a commutative diagram in B. Problem 1.11. Let F : C D be a functor. Show that if f : X Y is an equivalence in C, then F (f) is an equivalence in D. Is it possible for F (f) to be an equivalence without f being an equivalence? Exercise 1.12. Let G be a group, and think of it as a category with one object, as in Exercise 1.3. Interpret functors F : G Sets in terms of familiar concepts in algebra.
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