Functors are the backbone of category theory, bridging different mathematical structures. They map objects and morphisms between categories, preserving essential properties. Understanding functors is crucial for grasping how mathematical concepts relate across diverse fields.
Covariant and contravariant functors are two key types that differ in how they handle morphism direction. Covariant functors maintain direction, while contravariant functors reverse it. This distinction is vital for understanding how information flows between categories.
Functors and Categories
Defining Functors and Their Components
- Functor maps objects and morphisms from one category to another category while preserving the structure and composition of morphisms
- Category consists of a collection of objects and morphisms between those objects, along with a composition operation that satisfies associativity and identity laws
- Morphism is an arrow between two objects in a category representing a structure-preserving map or transformation (functions between sets)
- Object is a basic element or entity within a category upon which morphisms act (sets in the category of sets)
- Identity functor maps each object and morphism in a category to itself, serving as the identity element under functor composition
Functor Properties and Identity Functor
- Functors preserve the identity morphism of each object, mapping it to the identity morphism of the corresponding object in the target category
- Functors respect the composition of morphisms, ensuring that the composition of mapped morphisms equals the mapping of the composed morphisms: $F(g \circ f) = F(g) \circ F(f)$
- Identity functor, often denoted as $1_C$ or $Id_C$, maps every object and morphism in a category $C$ to itself
- The identity functor serves as the identity element under functor composition, meaning for any functor $F: C \to D$, we have $F \circ 1_C = F$ and $1_D \circ F = F$
Types of Functors
Covariant and Contravariant Functors
- Covariant functor preserves the direction of morphisms, mapping morphisms from $A \to B$ in the source category to morphisms $F(A) \to F(B)$ in the target category
- Covariant functors are the most common type of functor and are often referred to simply as "functors" when the context is clear
- Contravariant functor reverses the direction of morphisms, mapping morphisms from $A \to B$ in the source category to morphisms $F(B) \to F(A)$ in the target category
- Contravariant functors are sometimes called "cofunctors" and can be thought of as functors that "go against the grain" of the original category
Functor Composition and Examples
- Functor composition allows functors to be composed together, creating a new functor that combines their actions
- Given functors $F: C \to D$ and $G: D \to E$, their composition $G \circ F: C \to E$ is defined by $(G \circ F)(A) = G(F(A))$ for objects and $(G \circ F)(f) = G(F(f))$ for morphisms
- Functor composition is associative, meaning $(H \circ G) \circ F = H \circ (G \circ F)$ for functors $F: C \to D$, $G: D \to E$, and $H: E \to F$
- Examples of covariant functors include the power set functor $\mathcal{P}: \mathbf{Set} \to \mathbf{Set}$ and the fundamental group functor $\pi_1: \mathbf{Top}_ \to \mathbf{Grp}$
- An example of a contravariant functor is the dual vector space functor $(-)^: \mathbf{Vect}_K^{op} \to \mathbf{Vect}_K$, which maps a vector space to its dual space and reverses the direction of linear transformations