5.2 Vertical and horizontal composition of natural transformations

Natural transformations are like bridges between functors. Vertical composition stacks these bridges, creating a direct path from one functor to another. It's a way to simplify multiple transformations into one.

Horizontal composition, on the other hand, chains functors and transformations together. It's like connecting different bridges to create a longer path. This allows us to combine transformations across different categories.

Vertical Composition of Natural Transformations

Vertical composition of transformations

• Combines two natural transformations $\alpha: F \Rightarrow G$ and $\beta: G \Rightarrow H$ between functors $F, G, H: \mathcal{C} \rightarrow \mathcal{D}$ into a single natural transformation $\beta \circ \alpha: F \Rightarrow H$
• Defined componentwise for each object $X$ in $\mathcal{C}$, where $(\beta \circ \alpha)_X = \beta_X \circ \alpha_X$
  • $\alpha_X: F(X) \rightarrow G(X)$ and $\beta_X: G(X) \rightarrow H(X)$ are the component morphisms of $\alpha$ and $\beta$ at $X$ respectively
• Vertical composition essentially "stacks" the natural transformations on top of each other, connecting the component morphisms in a chain

Associativity in vertical composition

• Vertical composition of natural transformations is associative
• Given natural transformations $\alpha: F \Rightarrow G$, $\beta: G \Rightarrow H$, and $\gamma: H \Rightarrow K$, we have $(\gamma \circ \beta) \circ \alpha = \gamma \circ (\beta \circ \alpha)$
• Associativity follows from the associativity of composition of morphisms in the codomain category $\mathcal{D}$
  • The order in which we compose the component morphisms does not matter, as long as the overall sequence is maintained

Examples of transformation compositions

• In the category of sets $\mathbf{Set}$, natural transformations between functors $F, G, H: \mathbf{Set} \rightarrow \mathbf{Set}$ can be vertically composed
  • The component functions of the resulting natural transformation are the compositions of the component functions of the original natural transformations (e.g., $(\beta \circ \alpha)_X(x) = \beta_X(\alpha_X(x))$ for each set $X$ and element $x \in F(X)$)
• In the category of groups $\mathbf{Grp}$, natural transformations between functors $F, G, H: \mathbf{Grp} \rightarrow \mathbf{Grp}$ can be vertically composed
  • The component group homomorphisms of the resulting natural transformation are the compositions of the component group homomorphisms of the original natural transformations (e.g., $(\beta \circ \alpha)_G(g) = \beta_G(\alpha_G(g))$ for each group $G$ and element $g \in F(G)$)

Horizontal Composition of Natural Transformations

Horizontal composition of transformations

• Combines natural transformations $\alpha: F \Rightarrow G$ and $\beta: F' \Rightarrow G'$ between functors $F, G: \mathcal{C} \rightarrow \mathcal{D}$ and $F', G': \mathcal{D} \rightarrow \mathcal{E}$ into a single natural transformation $\beta * \alpha: F' \circ F \Rightarrow G' \circ G$
• Defined componentwise for each object $X$ in $\mathcal{C}$, where $(\beta * \alpha)_X = \beta_{G(X)} \circ F'(\alpha_X) = G'(\alpha_X) \circ \beta_{F(X)}$
  • $\alpha_X: F(X) \rightarrow G(X)$ is the component morphism of $\alpha$ at $X$
  • $\beta_{F(X)}: F'(F(X)) \rightarrow G'(F(X))$ and $\beta_{G(X)}: F'(G(X)) \rightarrow G'(G(X))$ are the component morphisms of $\beta$ at $F(X)$ and $G(X)$ respectively
• Horizontal composition "chains" the functors and natural transformations together, applying the second natural transformation to the result of the first

Interchange law for compositions

• The interchange law relates vertical and horizontal composition of natural transformations
• Given natural transformations $\alpha: F \Rightarrow G$, $\beta: G \Rightarrow H$, $\gamma: F' \Rightarrow G'$, and $\delta: G' \Rightarrow H'$, the interchange law states that $(\delta * \beta) \circ (\gamma * \alpha) = (\delta \circ \gamma) * (\beta \circ \alpha)$
• The order in which we apply vertical and horizontal composition does not matter, as long as the overall structure is maintained
• Proof involves showing that for each object $X$ in $\mathcal{C}$, the component morphisms of both sides of the equation are equal by using the associativity of composition in $\mathcal{E}$ and the functoriality of $G'$

Examples of transformation compositions

• In the category of sets $\mathbf{Set}$, given functors $F, G: \mathbf{Set} \rightarrow \mathbf{Set}$ and $F', G': \mathbf{Set} \rightarrow \mathbf{Set}$, and natural transformations $\alpha: F \Rightarrow G$ and $\beta: F' \Rightarrow G'$
  • The horizontal composition $\beta * \alpha: F' \circ F \Rightarrow G' \circ G$ has component functions that are the compositions of the component functions of $\beta$ and $F'$ applied to the component functions of $\alpha$ (e.g., $(\beta * \alpha)_X(x) = \beta_{G(X)}(F'(\alpha_X)(x))$ for each set $X$ and element $x \in F'(F(X))$)
• In the category of groups $\mathbf{Grp}$, given functors $F, G: \mathbf{Grp} \rightarrow \mathbf{Grp}$ and $F', G': \mathbf{Grp} \rightarrow \mathbf{Grp}$, and natural transformations $\alpha: F \Rightarrow G$ and $\beta: F' \Rightarrow G'$
  • The horizontal composition $\beta * \alpha: F' \circ F \Rightarrow G' \circ G$ has component group homomorphisms that are the compositions of the component group homomorphisms of $\beta$ and $F'$ applied to the component group homomorphisms of $\alpha$ (e.g., $(\beta * \alpha)_G(g) = \beta_{G(G)}(F'(\alpha_G)(g))$ for each group $G$ and element $g \in F'(F(G))$)