I have to like and suggest ideas connected with the Seifert-van Kampen theorem for the fundamental group and its extensions to groupoids and higher groupoids. An anomaly in traditional approaches, centred on the fundamental group, was that this theorem did not compute the fundamental group of the circle, which is THE basic example in algebraic topology. The reason is the the circle cannot be represented as the union of two path connected sets with path connected intersection. The solution was to use many base points rather than just one, and so to work in the context of groupoids; this dates from 1967 and a fairly recent account is in the book Topology and Groupoids.

This extension led to ideas of using higher groupoids in homotopy theory, and so to define higher homotopy groupoids, with higher order Seifert-van Kampen theorems. By 1984 this led to the idea of a nonabelian tensor product of groups which act on each other, see the bibliography. As an example, if $M,N$ are normal subgroups of the group $P$, then the commutator map $[\;,\;]: M \times N \to P$ is a biderivation and so factors through a universal biderivation, a morphism $\kappa: M \otimes N \to P$. For example if $M=N=P$ then Ker $\kappa$ is isomorphic to $\pi_3SK(P,1)$. Thus taking groupoids seriously in algebraic topology has led to new algebraic ideas.