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Stream: learning: questions

Topic: Collection level interactions from individual interactions

Adittya Chaudhuri (Mar 15 2025 at 17:50):

I apologise priorly, if my question sounds a little vague. But I am searching for the right mathematical language to describe it.

Say, a game(like Soccer) is being played among $n$ teams $T_1,T_2, \ldots T_n$ (unlike $n$=2 as the usual case) in a huge play ground. A collection of audiances is watching and trying to analyse the game. Now, the audiance is seeing interaction among players (both intra and inter teams). Now, in a way after watching the game for a "sufficient period of time", audiances started making statements about "how each team interacts with other teams" (zooming out from the level of players). I am thinking of this analysis as an "emergence of collection level interactions" from a "family of individual level interactions".

Now, if we represent the "player level interactions"as a labeled directed graph , then the induced collection level interaction would be labeled directed hypergraph. I was thinking of a functorial (functorial upto certain factors) relationships between labeled directed graphs and labeled directed hypergraphs to describe "this sort of transition from individual interactions to collection level interactions".

Has there been any existing study in this direction? What could be a right mathematicial language to study such interactions?

I was also vaguely thinking about sheaves/stacks (if we want to relax the locally determined condition)-but not sure!!

Thanks in advance.

Morgan Rogers (he/him) (Mar 17 2025 at 08:25):

Concepts from "coarse geometry", which I learned about in a course on geometric group theory, come to mind. Roughly (unapologetic pun) the idea is to capture when things 'look the same from far away'

Adittya Chaudhuri (Mar 17 2025 at 09:57):

Interesting! Thank you @Morgan Rogers (he/him) . I am not familiar with this area of geometry. I will look into it.

Morgan Rogers (he/him) (Mar 17 2025 at 15:31):

It's not exactly what you asked for, in that coarse geometry is typically not expressed in a categorical language, and for good reason: it can he overly constraining to demand that the associations that you describe are functorial. See also (the earlier parts of) the discussion in #theory: philosophy about categorifying analogies, where spans of maps were considered: that might be the closest relevant categorical formalism for your situation.

Adittya Chaudhuri (Mar 17 2025 at 17:51):

Thank you so much @Morgan Rogers (he/him) for referrring to this super interesting discussion on categorifying analogies.

Little out of topic, but if we think of a directed graph $G:=(V, E, s,t \colon E \rightarrow V)$ as a way to describe "analogies/interrelations" between "set $V$ of vertices (here individuals)", then the span $P(V) \xleftarrow{s'} E' \xrightarrow{t'} P(V)$ , where $P$ is the covariant power set functor, representing a directed hypergraph, can be thought of as an analogy/interaction between a collection of individuals in $V$.

I am interested in the construction of some examples of producing $\lbrace P(V) \xleftarrow{s'_{T}} E_{T}' \xrightarrow{t'_{T}} P(V) \rbrace_{T \in [0,K]}$ from the data $\lbrace G_{T}=(E_T, V, s_T, t_{T} \colon E_{T} \rightarrow V) \rbrace_{T \in [0,K]}$, where $T$ is a time in the time interval $[0,K], K \in \mathbb{R}^{+}$. Here, I am excluding the trivial case: $E'_{T}=E_{T}$, $s'_{T}=s_{T}$ and $t'_{T}=t_{T}$ for all $T$.

Adittya Chaudhuri (Mar 18 2025 at 01:12):

If we define a directed hypergraph as a span $P(V) \xleftarrow{s} E \xrightarrow{t} P(V)$ in Set, where $P(V)$ is the power set of $V$, then, I think the covariant power-set functor $P \colon Set \to Set$ (that takes a function to its image function) induces a functor $F \colon Gph \to HypGph$, where $Gph$ is the category of directed multigraphs and $HypGph$ is the category of directed multi-hypergraphs, whose morphisms are the morphisms of spans. Hence, may be this is an example (but not so interesting from the perspective of my initial question),

$\lbrace G_{T}=(E_T, V, s_T, t_{T} \colon E_{T} \rightarrow V) \rbrace_{T \in [0,K]} \mapsto \lbrace F(G_{T})\rbrace_{T \in [0,K]}= \lbrace P(V) \xleftarrow{P(s_{T})} P(E_{T}) \xrightarrow{P(t_{T})} P(V) \rbrace_{T \in [0,K]}$