When was the Joyal model structure on sSet born?

Back in 2012, I was under the impression that the Joyal model structure was described in a letter to Grothendieck in the early/mid 1980s. There is a letter from Joyal to Grothendieck describing a model structure, but it was the model structure on simplicial sheaves. Based on my MO answer, the nLab was edited to include this claim. But Dmitri Pavlov started asking questions on the nForum and under my answer, and now I have to retract my statement! Now he has asked an MO question of his own, looking for a definitive answer. Here is my attempt to track things down, written before Pavlov’s MO question landed.

Joyal seems to be citing his own work “in preparation” in the ’00s for the source of the model structure whose fibrant objects are the quasicategories, at latest in 2006, based on the citation of Theory of quasi-categories I in the (arXived) 2006 Quasi-categories vs Segal spaces. In the (published in) 2002 paper Quasi-categories and Kan complexes Joyal cites Theory of quasi-categories (not Theory of quasi-categories I), but doesn’t say anything about the model structure yet. However, in the CRM notes from 2008 (partly based on a manuscript—the IMA lectures—from 2004, plus the then draft of the book Theory of quasi-categories; there is also a 2007 version of the notes with the model structure) Joyal says “The results presented here are the fruits of a long term research project which began around thirty years ago.”

Verity, in his (arXived 2006) paper Weak Complicial Sets A Simplicial Weak ω-Category Theory Part I: Basic Homotopy Theory writes

we round out our presentation by localising our model structure and transporting it to the category of simplicial sets itself, in order to provide an independent construction of a model category structure on that latter category whose fibrant objects are Joyal’s quasi-categories [10].

where [10] is Joyal’s 2002 paper, so that the model structure was known to experts at least by 2006, even if not announced in 2002.

So I’m tempted to guess that the whole 1980s origin of the Joyal model structure (i.e. in a letter to Grothendieck, as stated on Wikipedia at the time of writing) for quasi-categories might be an urban myth.

I couldn’t find a mention of a model structure for quasicategories in the 2004 slides from IMA for the talk of Joyal/May/Porter, except for the closing sentence:

Baby camparison should give that the hammock localizations of all models for weak categories have equivalent hammock localizations. Model category theory shows how.

Tim Porter’s 2004 IMA notes likewise don’t seem to mention the model structure. So perhaps the date for the model structure can be pinned down to between (June) 2004 and (July) 2006, at least as far as going by Joyal’s public statements. One point in favour of this is that Tim’s notes include the open question

In general, what is the precise relationship between quasi categories (a weakening of categories) and Segal categories (also a weakening of categories)? (This question is vague, of course, and would lead to many interpretations)

which is what Joyal and Tierney’s 2006 paper pins down, in terms of a Quillen equivalence of model categories. If the question in 2004 had been merely one of trying to match up existing model structures (the Segal category one existed in 1998), I doubt Tim would have called it a vague question!

PS: I don’t know how long Lurie took to write the 589-page version 1 of Higher Topos Theory, put on the arXiv at the start of August 2006, but he refers to Joyal’s model structure there, citing Theory of quasi-categories I.

2 thoughts on “When was the Joyal model structure on sSet born?

    1. As I responded to the same suggestion on MathOverflow, the categories mailing list is the obvious place to ask, but I found it quicker to try to reconstruct upper and lower bounds from the available literature.


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