## Functors

Each category describes, in some sense, the paradigm for one area of mathematics. The bridges between the various areas are realized by functors, cf. the definition and examples below.

Definition 45.1. A functor Φ : C→ C˜ between two categories C = (X(0), X(1), σ, τ, µ, ), and C˜ = (X˜(0), X˜(1), σ˜, τ˜ , µ˜, ˜)

is a pair of maps (Φ(0),Φ(1)),

Φ(0) : X(0) → X˜(0), Φ(1) : X(1) → X˜(1) such that the following diagrams are commutative:

X(1) Φ(1)−→ X˜(1)

σ ↓ ↓ σ˜ X(0)

Φ(0)−→ X˜(0) ,

X(1) Φ(1)−→ X˜(1)

τ ↓ ↓ τ˜ X(0)

Φ(0)−→ X˜(0) ,

X(1) Φ(1)−→ X˜(1)

↑ ↑ ˜ X(0)

Φ(0)−→ X˜(0)

X(2) Φ(2)−→ X˜(2)

µ ↓ ↓ µ˜ X(1)

Φ(1)−→ X˜(1) where Φ(2) : X(2) → X˜(2) is the naturally induced map. One usually denotes both Φ(0) and Φ(1) by Φ. So compatibility with µ and µ˜ reads

Φ(a ? b) = Φ(a) ? Φ(b),

for all (a, b) ∈ X(2). Here are a few examples of functors. We start with some “forgetful” functors

whose effect is to “forget” part of the structure:

Example 45.2. Consider the “forgetful” functor

Φ : {commutative unital rings} → {Abelian groups} defined as follows. For (R,+,×,−, 0, 1) a commutative unital ring we let

Φ(R,+,×,−, 0, 1) = (R,+,−, 0), which is an Abelian group. For any ring homomorphism F we set Φ(F ) = F , viewed as a group homomorphism.