Franklin's Notes

Comma category

Given categories $\mathsf{C},\mathsf{D},\mathsf{E}$ with functors $F:\mathsf{D}\to\mathsf{C}$ and $G:\mathsf{E}\to\mathsf{C}$, the comma category $F\downarrow G$ is defined as the category

The functors $F$ and $G$ can be thought of as "translating" elements of $\mathsf{D}$ and $\mathsf{E}$ into the "common language" of $\mathsf{C}$ so that morphisms can be found between them. Here's another way of visualizing how morphisms $(h,k)$ interact with objects $(d,e,f)$ of the comma category:

Notice that the slice categories can be constructed as special cases of the comma category. In particular, if $\mathbf{1}$ denotes the discrete category with one object, we have that:

The above graphic can be modified to show the slice categories as a special case of the comma category:



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