Some of the earliest writings on the [Joyal model structure on simplicial sets](https://ncatlab.org/nlab/show/model+structure+for+quasi-categories) include Jacob Lurie's account in [Higher Topos Theory](https://arxiv.org/abs/math/0608040v1) from 2006,
as well as Joyal's own account in [The Theory of Quasi-Categories and its Applications](https://mat.uab.cat/%7Ekock/crm/hocat/advanced-course/Quadern45-2.pdf) 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](http://webusers.imj-prg.fr/%7Egeorges.maltsiniotis/ps/lettreJoyal.pdf) 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?**



When did the Joyal model structure on simplicial sets originate?
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.
> 
> 
> 


