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, à 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?
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3$\begingroup$ Boardman and Vogt were the first to discover quasicategories, which they called Weak Kan complexes. However, I think that the original observation that you reference is due either to Boardman and Vogt, Dan Kan, or Eilenberg and Mac Lane. However, I'm inclined to believe that it is due to Dan Kan. $\endgroup$– Harry GindiCommented Feb 25, 2011 at 0:25
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$\begingroup$ I think the observation that $\Delta$ plays a special role among categories is much older than Weber. I remember reading it in an old copy of Baez's This Week's Finds. It was a 'walking something', meaning it carried some structure or property and that was all - the minimal such category with that structure/property (initial in some way). $\endgroup$– David Roberts ♦Commented Feb 25, 2011 at 0:49
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$\begingroup$ Thanks Harry. Can anyone confirm, was it Kan? Anyone have a reference? I am just curious. Thank you. $\endgroup$– David CarchediCommented Feb 25, 2011 at 0:49
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1$\begingroup$ Ah, I've found it. $\Delta$ is the free monoidal category on a monoid. It's here: math.ucr.edu/home/baez/week117.html which is from 1998. It may appear earlier in a 1995 paper by Street, "Higher categories, strings, cubes and simplex equations", but I'm sure it's much older. I wouldn't be surprised if it was in CWM. $\endgroup$– David Roberts ♦Commented Feb 25, 2011 at 0:56
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1$\begingroup$ @David: I think you're talking about is when $\Delta$ includes the empty set. This is slightly different. $\endgroup$– David CarchediCommented Feb 25, 2011 at 1:11
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2 Answers
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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.
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$\begingroup$ Graeme Segal also attributes this observation to Grothendieck, in the start of Section 2 of "Classifying spaces and spectral sequences". The citation given there is a bit garbled, but he intends to reference "Techniques de construction et théorèmes d’existence en géométrie algébrique III : préschémas quotients", Proposition 4.1. $\endgroup$ Commented May 19, 2023 at 11:09
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According to Mac Lane (see p19 of Topology and logic as a source of algebra, Bull. Amer. Math. Soc. 82 (1976) 1-40) they were introduced by Eilenberg-Zilber in 1950 under the name complete semisimplicial complexes.
