In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are commutative — more generally, that homotopy groups of H-spaces are commutative, in particular, higher homotopy groups are commutative.

This argument takes its name from its appearance in a 1962 Eckmann-Hilton paper (Group-like structures in general categories, I. Multiplications and comultiplications, *Mathematische Annalen*, June 1962, Volume 145, Issue 3, pp 227–255) but stems, it seems, from previous work from the same authors (*Homotopy and duality*, 1959).

It also appears explicitly in algebraic geometry, in Grothendieck's 1961 SGA 1 seminar *Revêtements étales et groupe fondamental,* devoted to the theory of the fundamental group of schemes. Namely, in Exposé XI (Examples and complements), Section 2 (Abelian varieties), Grothendieck writes (bottom of page 287 of the Springer-Verlag edition) : “On the other hand, since the functor $X\mapsto \pi_1(X)$ from pointed schemes $X$ to groups commutes with product, it transforms a group in the first category to a group in the category of groups, i.e., a commutative group.”

Does anybody know whether the two groups of people were aware of one another? Was this categorical argument already clear at that time ?