Interchange law

Template:More citations needed In mathematics, specifically category theory, the interchange law regards the relationship between vertical and horizontal compositions of natural transformations.

Let $𝐅,𝐆,𝐇:ℂ⟶𝔻$ and $\overline{𝐅},\overline{𝐆},\overline{𝐇}:𝔻⟶𝔼$ where $𝐅,𝐆,𝐇,\overline{𝐅},\overline{𝐆},\overline{𝐇}$ are functors and $ℂ,𝔻,𝔼$ are categories. Also, let $\mathit{\alpha }:𝐅\mathbf{⟶}𝐆$ and $\mathit{\beta }:𝐆\mathbf{⟶}𝐇$ while $\overline{\mathit{\alpha }}:\overline{𝐅}\mathbf{⟶}\overline{𝐆}$ and $\overline{\mathit{\beta }}:\overline{𝐆}\mathbf{⟶}\overline{𝐇}$ where $\mathit{\alpha },\mathit{\beta },\overline{\mathit{\alpha }},\overline{\mathit{\beta }}$ are natural transformations. For simplicity's and this article's sake, let $\overline{\mathit{\alpha }}$ and $\overline{\mathit{\beta }}$ be the "secondary" natural transformations and $\mathit{\alpha }$ and $\mathit{\beta }$ the "primary" natural transformations. Given the previously mentioned, we have the interchange law, which says that the horizontal composition ($\circ$) of the primary vertical composition ($\bullet$) and the secondary vertical composition ($\bullet$) is equal to the vertical composition ($\bullet$) of each secondary-after-primary horizontal composition ($\circ$); in short, $(\mathit{\beta }\bullet \mathit{\alpha })\circ (\overline{\mathit{\beta }}\bullet \overline{\mathit{\alpha }})=(\overline{\mathit{\beta }}\circ \mathit{\beta })\bullet (\overline{\mathit{\alpha }}\circ \mathit{\alpha })$.

The word "interchange" stems from the observation that the compositions and natural transformations on one side are switched or "interchanged" in comparison to the other side. The entire relationship can be shown in the following diagram.

The interchange law in category theory.

If we apply this context to functor categories, and observe natural transformations $\mathit{\alpha }:𝐅\mathbf{⟶}𝐆$ and $\mathit{\beta }:𝐆\mathbf{⟶}𝐇$ within a category $V$ and $\overline{\mathit{\alpha }}:\overline{𝐅}\mathbf{⟶}\overline{𝐆}$ and $\overline{\mathit{\beta }}:\overline{𝐆}\mathbf{⟶}\overline{𝐇}$ within a category $W$, we can imagine a functor $\mathrm{\Gamma }:V⟶W$, such that

the natural transformations are mapped like such:

• $\mathrm{\Gamma }(\mathit{\alpha })⟶\overline{\mathit{\alpha }},$
• $\mathrm{\Gamma }(\mathit{\beta })⟶\overline{\mathit{\beta }},$
• and $\mathrm{\Gamma }(\mathit{\beta }\bullet \mathit{\alpha })⟶(\overline{\mathit{\beta }}\bullet \overline{\mathit{\alpha }})$.

The functors are also mapped accordingly as such:

• $\mathrm{\Gamma }(𝐅)⟶(\overline{𝐅}),$
• $\mathrm{\Gamma }(𝐆)⟶(\overline{𝐆}),$
• and $\mathrm{\Gamma }(𝐇)⟶(\overline{𝐇})$.