Interchange law

 ( Category Theory ) 

In mathematics, specifically category theory, the interchange law (or exchange law) regards the relationship between vertical and horizontal compositions of natural transformations.

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

It also appears in monoidal categories wherein classical composition ($\backslash circ$) and the tensor product ($\backslash otimes$) take their places in lieu of the horizontal composition and vertical composition partnership and is denoted $(g\; \backslash circ\; f)\; \backslash otimes\; (j\; \backslash circ\; h)\; =\; (g\; \backslash otimes\; j)\; \backslash circ\; (f\; \backslash otimes\; h)$.

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 within the following diagram.

If we apply this context to functor categories, and observe natural transformations $\backslash boldsymbol\{\backslash alpha\}\; :\; \backslash mathbf\{F\; \backslash longrightarrow\; G\}$ and $\backslash boldsymbol\{\backslash beta\}\; :\; \backslash mathbf\{G\; \backslash longrightarrow\; H\}$ within a category $V$ and $\backslash boldsymbol\backslash bar\backslash alpha\; :\; \backslash mathbf\{\backslash bar\{F\}\; \backslash longrightarrow\; \backslash bar\{G\}\}$ and $\backslash boldsymbol\backslash bar\backslash beta\; :\; \backslash mathbf\{\backslash bar\{G\}\; \backslash longrightarrow\; \backslash bar\{H\}\}$ within a category $W$, we can imagine a functor $\backslash Gamma\; :\; V\; \backslash longrightarrow\; W$, such that

the natural transformations are mapped like such:

• $\backslash Gamma(\backslash boldsymbol\{\backslash alpha\})\; \backslash longrightarrow\; \backslash boldsymbol\{\backslash bar\{\backslash alpha\}\},\backslash ,$

• $\backslash Gamma(\backslash boldsymbol\{\backslash beta\})\; \backslash longrightarrow\; \backslash boldsymbol\{\backslash bar\{\backslash beta\}\},\backslash ,$
• and $\backslash Gamma(\backslash boldsymbol\backslash beta\backslash \; \backslash bullet\backslash \; \backslash boldsymbol\backslash alpha)\; \backslash longrightarrow\; (\backslash boldsymbol\backslash bar\backslash beta\backslash \; \backslash bullet\backslash \; \backslash boldsymbol\backslash bar\backslash alpha)$;

functors are also mapped accordingly:

• $\backslash Gamma(\backslash boldsymbol\{\backslash mathbf\{F\}\})\; \backslash longrightarrow\; (\backslash boldsymbol\{\backslash mathbf\{\backslash bar\{F\}\}\}),\backslash ,$

• $\backslash Gamma(\backslash boldsymbol\{\backslash mathbf\{G\}\})\; \backslash longrightarrow\; (\backslash boldsymbol\{\backslash mathbf\{\backslash bar\{G\}\}\}),\backslash ,$

• and $\backslash Gamma(\backslash boldsymbol\{\backslash mathbf\{H\}\})\; \backslash longrightarrow\; (\backslash boldsymbol\{\backslash mathbf\{\backslash bar\{H\}\}\})$.

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