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Stream: learning: questions

Topic: The free double category on a virtual double category

Chad Nester (Dec 31 2025 at 11:40):

I'm looking at section 6.6 of Leinster's "Higher Operads, Higher Categories", and I'm trying to figure out how the adjunction between $T$-multicategories and $T$-structured categories works concretely in the case of the free category monad $\mathbf{fc}$. In particular, I'm confused about the action of the left adjoint $F$ on objects.

Given a virtual double category $\mathbb{X}$ (i.e., an $\mathbf{fc}$-mulicategory), I expect to obtain a strict double category $F(\mathbb{X})$. It seems that cells of $F(\mathbb{X})$ must be sequences $(t_1,\ldots,t_n)$ of cells of $\mathbb{X}$ in which the right boundary of $t_i$ matches the left boundary of $t_{i+1}$ for all $i$. Here I have in mind the usual pasting-diagram-like way of picturing the cells of $\mathbb{X}$. These sequences are really paths in a graph with vertices the tight morphisms of $\mathbb{X}$ and edges between $f$ and $g$ given by cells with left boundary $f$ and right boundary $g$. In particular this means that the empty path from $f$ to $f$ gives a cell $()_f$ of $F(\mathbb{X})$, which I imagine is the horizontal identity on $f$.

What I do not understand is how to define vertical composition in $F(\mathbb{X})$ in the presence of cells like $()_f$. In particular, if $t$ is a cell of $\mathbb{X}$ whose top boundary is the empty sequence, then should be able to vertically compose $()_f$ and $(t)$ in $F(\mathbb{X})$ (with $()_f$ "on top"). I don't see what sequence/path of cells of $\mathbb{X}$ this could possibly be.

Where am I going wrong here?

James Deikun (Dec 31 2025 at 14:35):

The result is $(t \circ f)$. In an $\mathbf{fc}$-multicategory there is a "whiskering" composition of triangles with vertical arrows, which is actually just the normal composition of cells, but the way it works out to be this is not obvious to everybody and descriptions of $\mathbf{fc}$-multicategories seldom mention it specifically.

Chad Nester (Jan 01 2026 at 14:36):

Thank you!

This was indeed not obvious to me, although I suppose it should have been.

Mike Shulman (Jan 02 2026 at 07:35):

It's "obvious" once you realize that "a list of horizontally-composable cells of length zero" means a single vertical arrow. Which is "obvious" from the fact that a list of composable arrows in a quiver means a single object. But it's one of those things that often only seems obvious in hindsight.

fosco (Jan 02 2026 at 09:53):

It's the same as "the tuple $(x_1,\dots,x_n)$", and if $n=0$ this means the empty tuple

Mike Shulman (Jan 02 2026 at 16:56):

But the thing that's sometimes surprising is that in the cases I mentioned, even an "empty tuple" contains nontrivial information: there's more than one "empty tuple".