I have a pretty easy historical question about simplicial sets. Unless I am mistaken, simplicial sets first came out of topology, explicitly from combinatorial topology and the study of simplicial complexes. However, simplicial sets, as we all know, are intimately linked with the study of categories. On one hand, a la Joyal we know that they provide a model for quasicategories, or in Lurie's terminology, $\infty$-categories. And on the other hand, some machinery of Mark Weber takes the free category monad and spits out the category $\Delta$ on which simplicial sets are based, showing that they belong just as much to category theory as they do to topology. My question is, when was the connection of simplicial sets to the study of categories first noticed, and by whom?
I once heard a talk by Ezra Getzler in which he attributed to Grothendieck (in the 1950s) the observation that a small category is the same thing as a simplicial set with unique fillings for inner horns. I believe I've seen a citation since then, but I can't locate one now: google searches involving the words "Grothendieck" and "Category" aren't particularly effective.
According to Mac Lane (see p19 of this paper) they were introduced by Eilenberg-Zilber in 1950 under the name complete semisimplicial complexes.