Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006, as well as Joyal's own account in The Theory of Quasi-Categories and its Applications from 2008.

I have seen unsourced claims that this model structure may originate in 1980s, possibly in the correspondence between Joyal and Grothendieck. However, an examination of the only available letter from Joyal to Grothendieck does not reveal any material on quasicategories. I was unable to find any other correspondence between Joyal and Grothendieck.

In his 2008 notes cited above, Joyal says “The results presented here are the fruits of a long term research project which began around thirty years ago.”, which appears to indirectly corroborate the above unsourced claims.

When did the Joyal model structure on simplicial sets originate? Is there any written source that confirms its existence in 1980s?

  • 6
    $\begingroup$ Is there some reason for not asking Joyal about this? $\endgroup$ Jun 20, 2022 at 7:44
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    $\begingroup$ @AndrejBauer 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 $\endgroup$
    – David Roberts
    Jun 20, 2022 at 9:02
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    $\begingroup$ Sending an email to Joyal is even obviouser. $\endgroup$ Jun 20, 2022 at 9:05
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    $\begingroup$ @AndrejBauer: One reason is to see whether I missed something, before I start bothering him with questions. $\endgroup$ Jun 20, 2022 at 14:02
  • $\begingroup$ That's a fair point. $\endgroup$ Jun 20, 2022 at 14:38

2 Answers 2


Here is what André Joyal wrote in an email to me:

No, I have not discovered the model structure for quasi-categories in the 1980's. I became interested in quasi-categories (without the name) around 1980 after attending a talk by Jon Beck on the work of Boardman and Vogt. I wondered if category theory could be extended to quasi-categories. In my mind, a crucial test was to show that a quasi-category is a Kan complex if its homotopy category is a groupoid. All my attempts at showing this have failed for about 15 years, until I stopped trying hard! I found a proof after extending to quasi-categories a few basic notions of category theory. This was around 1995. The model structure for quasi-categories was discovered soon after. I did not publish it immediately because I wanted to show that it could be used for proving something new in homotopy theory. I am a bit of a perfectionist (and overly ambitious?). I was hoping to develop a synthesis between category theory and homotopy theory (hence the name quasi-categories). I met Lurie at a conference organised by Carlos Simpson in Nice (in 2001?). I gave a talk on the model structure and Lurie asked for a copy of my notes afterward. I intuitively understood that he could develop the theory of quasi-categories more and better than I could. He was young and a better mathematician than I was. I do not regret it.


My suspicion is now that it was some time between 2004 and 2006. I have a lot more citations in this blog post, but I note three points, in reverse chronological order:

  1. Multiple experts are referring to the Joyal model structure in mid-2006, in print (Joyal–Tierney, Verity, Lurie,...).

  2. At the June 2004 IMA conference, the joint Joyal–May–Porter presentation on models for weak higher categories, including quasi-categories, made no mention on the slides of the model structure. Further, Tim Porter's own 2004 notes linked to that talk call the question of the relation between quasi-categories and Segal categories a "vague" one. This question was answered by Joyal and Tierney in 2006 giving the Quillen equivalence of model structures.

  3. There is no mention of a model structure in Joyal's 2002 paper on quasi-categories and Kan complexes.


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