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Stream: theory: category theory

Topic: Terminology for "double hom sets"?

Chad Nester (Feb 02 2022 at 10:41):

Let $\mathbb{D}$ be a double category. What are things of the form $\mathbb{D}({\scriptstyle f} {g \atop k} {\scriptstyle h})$ -- being the collection of all cells of $\mathbb{D}$ with left boundary $f$, right boundary $h$, top boundary $g$, and bottom boundary $k$ -- called?

For example if $\mathbb{C}$ is a category then we call things of the form $\mathbb{C}(A,B)$ hom-sets, and I'm wondering if there's a similar term for the collections of cells. If not, what's a good name? cell-sets?

I know that we could write some version of $\mathbb{D}(A,B)(X,Y)$ and be done with it, but I'd rather not in this instance.

Chad Nester (Feb 02 2022 at 10:46):

... tilesets?

Mike Shulman (Feb 02 2022 at 15:41):

I'd probably go with something like cell-sets.

Chad Nester (Feb 03 2022 at 08:26):

I guess that's what I'll do then :)