Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Something which isn't a globe

Brendan Murphy (Oct 21 2023 at 19:15):

For a directed graph $G$ we can form the "derived graph" or "line graph" $G'$ whose vertices are the edges of $G$ and where there is an edge $e \to f$ whenever the target of $e$ is the source of $f$. If we represent $G$ as a pair of (jointly injective) maps $s, t : E \to V$ then $E' = E \times_{t, s} E$ and the pullback projections $s', t' : E' \to E$ give the directed graph $G'$. We can of course iterate this process, and get a diagram that looks like

$$\dots \rightrightarrows E'' \underset{t''}{\overset{s''}{\rightrightarrows}} E' \underset{t'}{\overset{s'}{\rightrightarrows}} E \underset{t}{\overset{s}{\rightrightarrows}} V.$$

In fact requiring injectivity is a little artifical, we could start with any endo-span $s, t : Y \to X$ and get a diagram like the above. The maps here will satisfy the relation $s' \circ t = t' \circ s$, or more generally $s^{(n+1)} \circ t^{(n)} = t^{(n+1)} \circ s^{(n)}$. I don't believe there are any other relations. Now to me this construction feels a lot like the cech nerve, but it isn't that since we're not taking pullbacks of a map along itself. The cech nerve produces a simplicial object: what does this process produce? Ie, what is the indexing category of this diagram, or rather it's opposite? Is there a nice way of seeing it as "shapes" like simplices, globes, cubes, etc?

We can also view this construction as taking powers of the object $Y \xrightarrow{(s, t)} X\times X$ in the monoidal category $\mathsf{Set}/(X\times X)$, under span composition, but it's not clear to me how we see the higher maps $s^{(n)}, t^{(n)}$ from this pov.

David Egolf (Oct 21 2023 at 20:33):

This is interesting! I'd never heard of a "derived graph" or a "line graph" before. Super cool!

This reminds me of the exposition in The algebra of entanglement and the geometry of composition around page 4 (or page 18 in the PDF), where the author is explaining the concept of a globular set. However, I think that the relations that are satisfied in the case you describe are different than the "globularity conditions".

Regarding the relations satisfied, you indicate that $s' \circ t = t' \circ s$. But $s': E \to E$ and $t: E \to V$. So I don't see how we can compose $s'$ after $t$. When trying to understand which relations are satisfied, I think I need to know which pullback projection $s'$ is and which pullback projection $t'$ is. If our pullback diagram is as follows

pullback diagram

then we have $t \circ s' = s \circ t'$. Is this the relation that you were referring to above?

Brendan Murphy (Oct 21 2023 at 20:35):

Yes, my bad! I just got my order of composition swapped

Brendan Murphy (Oct 21 2023 at 20:35):

(and I agree they're not the globularity conditions, see the post title :P)

Brendan Murphy (Oct 21 2023 at 22:20):

I think we can view the indexing category of the (backwards) diagram as a subcategory of the simplex category, the one with all objects [0], [1], [2], ... and just the first and last coface maps [n] -> [n+1]

Brendan Murphy (Oct 21 2023 at 22:21):

Er not quite, that's not closed under composition. But maybe the subcategory generated by those maps

Brendan Murphy (Oct 21 2023 at 22:23):

Ah, so the subcategory of the simplex category whose maps are the injections with image an interval

David Egolf (Oct 21 2023 at 22:45):

I wonder if there is some way to get the category you just described by viewing each object in it as a category, and the morphisms as all functors between these. In the current description (as a subcategory of the simplex category), we have to restrict the morphisms "manually" in the sense that there could be more morphisms that respect the structure of the objects - which I am thinking as totally ordered sets (but we declare we only want injective morphisms that have as their image an interval). It might be nice if the "shape" of the objects naturally restricted what morphisms (functors) are available.

Brendan Murphy (Oct 21 2023 at 22:48):

hm, my mind goes to the undirected graph 0 — 1 — ... — n

Brendan Murphy (Oct 21 2023 at 22:49):

Or maybe not undirected uhh

Brendan Murphy (Oct 21 2023 at 22:50):

Basically, there are too many maps in Δ because we're considering [n] = { 0 -> 1 -> ... -> n } as a poset instead of just a DAG (the transitive reduction or hasse diagram of the poset)

Brendan Murphy (Oct 21 2023 at 22:50):

A morphism of directed graphs [n] -> [m] is necessarily an interval-embedding, right?

Brendan Murphy (Oct 21 2023 at 22:52):

We need to get rid of the reflexive edges in the underlying directed graph of the category [n] to guarantee morphisms are injective, and we need to get rid of the composite edges to ensure we adjacent vertices remain adjacent (ie the image is an interval)

Brendan Murphy (Oct 21 2023 at 23:14):

Well I think I'm satisfied with that answer but I'd be happy to hear any thoughts other people have

Eric Forgy (Oct 22 2023 at 02:16):

Brendan Murphy said:

For a directed graph $G$ we can form the "derived graph" or "line graph" $G'$ whose vertices are the edges of $G$ and where there is an edge $e \to f$ whenever the target of $e$ is the source of $f$.

In computational electromagnetics, this sounds like the barycentric subdivision.

Brendan Murphy (Oct 22 2023 at 06:20):

I think the nerve construction I was outlining above can be understood as right kan extension along the inclusion of the walking quiver into the opposite of my category of chains. But I haven't carefully checked the details