4.2 Functor composition and identity functors

Functor composition and identity functors are key concepts in category theory. They allow us to combine and simplify complex relationships between categories, creating new functors that preserve structure and behavior.

These ideas build on the basics of functors, showing how they can be combined and manipulated. Understanding composition and identity functors is crucial for grasping more advanced category theory concepts and their applications in mathematics and computer science.

Functor Composition and Identity Functors

Composition of functors

• Functor composition combines two functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{E}$ to create a new functor $G \circ F: \mathcal{C} \to \mathcal{E}$
  • $(G \circ F)(A) = G(F(A))$ applies the functors sequentially to objects $A$ in $\mathcal{C}$ ($F$ first, then $G$)
  • $(G \circ F)(f) = G(F(f))$ applies the functors sequentially to morphisms $f$ in $\mathcal{C}$ ($F$ first, then $G$)
• Functor composition is associative, meaning $(H \circ G) \circ F = H \circ (G \circ F)$ for functors $F: \mathcal{C} \to \mathcal{D}$, $G: \mathcal{D} \to \mathcal{E}$, and $H: \mathcal{E} \to \mathcal{F}$
  • $((H \circ G) \circ F)(A) = (H \circ G)(F(A)) = H(G(F(A))) = H((G \circ F)(A)) = (H \circ (G \circ F))(A)$ for objects $A$ in $\mathcal{C}$
  • $((H \circ G) \circ F)(f) = (H \circ G)(F(f)) = H(G(F(f))) = H((G \circ F)(f)) = (H \circ (G \circ F))(f)$ for morphisms $f$ in $\mathcal{C}$

Identity functors and composition

• The identity functor $1_{\mathcal{C}}: \mathcal{C} \to \mathcal{C}$ maps objects and morphisms in a category $\mathcal{C}$ to themselves
  • $1_{\mathcal{C}}(A) = A$ for objects $A$ in $\mathcal{C}$
  • $1_{\mathcal{C}}(f) = f$ for morphisms $f$ in $\mathcal{C}$
• Identity functors serve as identity elements under functor composition
  • For any functor $F: \mathcal{C} \to \mathcal{D}$, $F \circ 1_{\mathcal{C}} = F$ and $1_{\mathcal{D}} \circ F = F$
    • $(F \circ 1_{\mathcal{C}})(A) = F(1_{\mathcal{C}}(A)) = F(A)$ and $(1_{\mathcal{D}} \circ F)(A) = 1_{\mathcal{D}}(F(A)) = F(A)$ for objects $A$ in $\mathcal{C}$
    • $(F \circ 1_{\mathcal{C}})(f) = F(1_{\mathcal{C}}(f)) = F(f)$ and $(1_{\mathcal{D}} \circ F)(f) = 1_{\mathcal{D}}(F(f)) = F(f)$ for morphisms $f$ in $\mathcal{C}$

Functor composition and identities

• Functor composition respects identity functors
  • For functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{E}$, $G \circ 1_{\mathcal{D}} \circ F = G \circ F$
    • $(G \circ 1_{\mathcal{D}} \circ F)(A) = G(1_{\mathcal{D}}(F(A))) = G(F(A)) = (G \circ F)(A)$ for objects $A$ in $\mathcal{C}$
    • $(G \circ 1_{\mathcal{D}} \circ F)(f) = G(1_{\mathcal{D}}(F(f))) = G(F(f)) = (G \circ F)(f)$ for morphisms $f$ in $\mathcal{C}$
• This property ensures that composing with an identity functor does not change the overall behavior of the composition

Examples of functor composition

• Composing forgetful and free functors ($\mathbf{Grp}$ and $\mathbf{Set}$)
  • The forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$ maps a group to its underlying set and a group homomorphism to its underlying function
  • The free functor $F: \mathbf{Set} \to \mathbf{Grp}$ maps a set to the free group generated by that set and a function to the induced group homomorphism
  • $U \circ F: \mathbf{Set} \to \mathbf{Set}$ maps a set to the underlying set of its free group
  • $F \circ U: \mathbf{Grp} \to \mathbf{Grp}$ maps a group to the free group generated by its underlying set
• Composing power set and distribution functors ($\mathbf{Set}$)
  • The power set functor $\mathcal{P}: \mathbf{Set} \to \mathbf{Set}$ maps a set to its power set and a function to the induced function between power sets
  • The distribution functor $D: \mathbf{Set} \to \mathbf{Set}$ maps a set to the set of distributions (finite additive measures) on that set and a function to the induced function between distributions
  • $D \circ \mathcal{P}: \mathbf{Set} \to \mathbf{Set}$ maps a set to the set of distributions on its power set
  • $\mathcal{P} \circ D: \mathbf{Set} \to \mathbf{Set}$ maps a set to the power set of its set of distributions