It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a chain $G_0\le G_1\le \dots G_n\to\dots$ of subgroups $(G_n)_{n\in\mathbb N}$ then the cohomological dimension $\mathrm{cd}(G)$ of $G$ is at most $1$ greater than $\sup_n\mathrm{cd}(G_n)$. This result can be proved by using the chain of subgroups to define a tree with an action of $G$: there results a Mayer-Vietoris sequence and various homological corollaries. On the other hand there is a theorem in Gruenberg's "Notes on group cohomology" that the same conclusion about $G$ can be drawn if the sequence of groups $G_0\to G_1\to G_2\to\dots$ is connected by maps that are not necessarily injective. The Mayer–Vietoris sequence is not available, and there is no useful action on a tree. For example, consider the Baumslag–Solitar group $\mathrm{BS}(2,3)=\langle x,y;\ y^{-1}x^2y=x^3\rangle$, and the chain in which each map is the map determined by $x\mapsto x^6$ and $y\mapsto y$. Gruenberg's theorem then correctly predicts that the cohomological dimension of the colimit $G$ is $\le 3$ (in fact it is $3$) but without an action on a tree there is no easy way of proving that the dimension is $3$ (which is a result of Gildenhuys and Strebel) without using additional methods.
Is this a territory where $\infty$-categories can bring a way of connecting Gruenberg's point of view and the Mayer–Vietoris sequence point of view? Is there a connection with this mo question: Mayer-Vietoris sequence in group cohomology for arbitrary pushout squares of groups?
