Interchange law

In category theory, the interchange law is a law regarding the relationship between vertical and horizontal compositions of natural transformations.

Let ${\textstyle \mathbf {F,\,G,\,H}$ :\mathbb {C} \longrightarrow \mathbb {D} } and ${\textstyle \mathbf {{\bar {F}},\,{\bar {G}},\,{\bar {H}}}$ :\mathbb {D} \longrightarrow \mathbb {E} } where ${\textstyle \mathbf {F,\,G,\,H,\,{\bar {F}},\,{\bar {G}},\,{\bar {H}}} }$ are functors and ${\textstyle \mathbb {C} ,\,\mathbb {D} ,\,\mathbb {E} }$ are categories. Also, let ${\displaystyle {\boldsymbol {\alpha }}:\mathbf {F\longrightarrow G} }$ and ${\displaystyle {\boldsymbol {\beta }}:\mathbf {G\longrightarrow H} }$ while ${\displaystyle {\boldsymbol {\bar {\alpha }}}:\mathbf {{\bar {F}}\longrightarrow {\bar {G}}} }$ and ${\displaystyle {\boldsymbol {\bar {\beta }}}:\mathbf {{\bar {G}}\longrightarrow {\bar {H}}} }$ where ${\displaystyle {\boldsymbol {\alpha }},\,{\boldsymbol {\beta }},\,{\boldsymbol {\bar {\alpha }}},\,{\boldsymbol {\bar {\beta }}}}$ are natural transformations. For simplicity's and this article's sake, let us call ${\displaystyle {\boldsymbol {\bar {\alpha }}}}$ and ${\displaystyle {\boldsymbol {\bar {\beta }}}}$ the "secondary" natural transformations of the "primary" natural transformations ${\displaystyle {\boldsymbol {\alpha }}}$ and ${\displaystyle {\boldsymbol {\beta }}}$. Given the previously mentioned, we have the interchange law, which says that the horizontal composition (${\displaystyle \circ }$) of the primary vertical composition (${\displaystyle \bullet }$) and the secondary vertical composition (${\displaystyle \bullet }$) is equal to the vertical composition (${\displaystyle \bullet }$) of each secondary-after-primary horizontal composition (${\displaystyle \circ }$). ^[1]

In short, ${\textstyle ({\boldsymbol {\beta }}\ \bullet \ {\boldsymbol {\alpha }})\ \circ \ ({\bar {\boldsymbol {\beta }}}\ \bullet \ {\bar {\boldsymbol {\alpha }}})=({\bar {\boldsymbol {\beta }}}\ \circ \ {\boldsymbol {\beta }})\ \bullet \ ({\bar {\boldsymbol {\alpha }}}\ \circ \ {\boldsymbol {\alpha }})}$. ^[1] 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 whole, entire relationship can be shown in the following diagram:

The interchange law in category theory.

If we consider the previously mentioned context within the category of functors, and specifically observe natural transformations ${\displaystyle {\boldsymbol {\alpha }}:\mathbf {F\longrightarrow G} }$ and ${\displaystyle {\boldsymbol {\beta }}:\mathbf {G\longrightarrow H} }$ within a category ${\displaystyle V}$ and ${\displaystyle {\boldsymbol {\bar {\alpha }}}:\mathbf {{\bar {F}}\longrightarrow {\bar {G}}} }$ and ${\displaystyle {\boldsymbol {\bar {\beta }}}:\mathbf {{\bar {G}}\longrightarrow {\bar {H}}} }$ within a category ${\displaystyle W}$, we can imagine a functor ${\displaystyle \Gamma :V\longrightarrow W}$, such that:

the natural transformations are mapped like such:

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

the functors are also mapped accordingly like:

• ${\displaystyle \Gamma ({\boldsymbol {\mathbf {F} }})\longrightarrow ({\boldsymbol {\mathbf {\bar {F}} }}),\,}$
• ${\displaystyle \Gamma ({\boldsymbol {\mathbf {G} }})\longrightarrow ({\boldsymbol {\mathbf {\bar {G}} }}),\,}$
• and ${\displaystyle \Gamma ({\boldsymbol {\mathbf {H} }})\longrightarrow ({\boldsymbol {\mathbf {\bar {H}} }})}$.

References

1. Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Springer Science+Business Media New York. p. 43. ISBN   978-1-4419-3123-8.