# Eckmann-Hilton argument / Grothendieck

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 ?

• Where's this stated in SGA I? – user40276 Jul 12 '18 at 2:49
• Exposé XI (Exemples et compléments), section 2 (Variétés abéliennes), bottom of p. 287 (Springer edition) : « le foncteur $X \mapsto \pi_1(X)$ transforme un groupe dans la première catégorie dans un groupe dans la catégorie des groupes, i.e. en un groupe commutatif. » – ACL Jul 12 '18 at 9:22
• I might note in view of the connection between the Eckmann-Hilton argument and the Godement relation that Roger Godement and Grothendieck were close friends. – Carlo Beenakker Jul 12 '18 at 18:23
• Carlo, what do you mean with “the Godement relation” ? – ACL Jul 12 '18 at 18:25
• There is a somewhat obscure article by Dan Kan from 1958 ("On monoids and their dual", Bull Soc Math Mexcana) which appears to prove more or less the same result: that monoid objects in groups are abelian groups. (At least according to MathSciNet, as the article isn't easily available to me.) – Charles Rezk Jul 13 '18 at 14:37