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I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble I've had is computing things like the join, product, coproduct, pullback, pushout, and so forth. I understand them as far as their universal properties, and maybe have a little intuition because the category of simplicial sets is a presheaf category over the simplex category, but Lurie uses geometrical language, so I can't even compute like when working with presheaves. So, could you lot recommend some books or lecture notes, preferably with suggested sections, that won't go too deep into simplicial homotopy theory, but deep enough for me to learn how to compute?

Note: Most of the time I waste is sitting around with that book is trying to make sense of the computations, while I understand the arguments just fine. So please, don't suggest an entire book detailing all of simplicial homotopy theory from start to finish. I have a goal in mind here, and I'm only trying to learn as much as is necessary for me to continue reading HTT.

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While Charles's answer is good, the whole point of making this community wiki was to get more answers. – Harry Gindi Dec 13 '09 at 15:21
up vote 14 down vote accepted

I'm not really sure what you are asking for. Colimits and limits are easy to compute in simplicial sets, because it's a presheaf category (as you say). But if you want "geometrical" intuition about simplicial sets (including "pictures" of joins, etc.), you want to know about the geometric realization functor from simplicial sets to spaces. In particular, the fact that geometric realization preserves colimits (it's a left adjoint), and that it preserves finite limits (essentially a theorem of Milnor).

The limit result is proved in the book by Gabriel-Zisman. (Goerss-Jardine mention the result in chapter 1, but don't seem to prove it.)

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This doesn't seem to be widely known (even though the nLab tells anyone who googles "quasi-category" or similar) but André Joyal's very nice and accessible pre-book/lecture notes (reference 44 in HTT) is available online

The pdf is here:

André Joyal, The theory of quasi-categories and its applications (dead link) (mirror)

This is bundled with lecture notes by Ieke Moerdijk on $(\infty,1)$-operads and by Betrand Toën on derived stacks over simplicial rings in one remarkable collection of lecture notes, that was supposed to have been officially published already, but keeps being delayed:

Simplicial methods in higher categories (dead link) (Barcelona, 2008): Course Notes (dead link). (Update:) The book was published in 2010, MathSciNet, Springer Link.

Have a look. This is a very nice resource. But notice that these lecture notes were distributed before that course started and meanwhile improved and polished versions exist, likely available on request from the authors.

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In fact, I just asked Prof. Joyal about that. I assume you were posting w/r/t that request. – Harry Gindi Jan 24 '10 at 12:54
Yes, I saw your question on the nForum, but I thought it would be more appropriate for me to reply here and let Prof. Joyal reply to you himself there. Also, here more people will see this. But you should all be using the nLab more: if you want literaturee on quasi-categories, look uo the references section on quasi-categories! :-) – Urs Schreiber Jan 24 '10 at 13:08
Lurie is a really hard read for me. I don't know if it's like that for everyone, or if most people just skip the proofs for a lot of things, but he rarely defines the terms he uses, which is really frustrating. I'm not sure if it's a mathematical maturity issue or just the fact that he assumes you've at least heard of these things before. – Harry Gindi Jan 24 '10 at 13:31
I think the book is not meant really meant to be introductory in the way that you may be looking for. It is meant to develop some technology. But have a look at Joyal's lectures. And have you spent some time with Goerss-Jardine "Simplicial homotopy theory"? You should probably start with that. It's online available, too. See the nLab page on simplicial sets for links. – Urs Schreiber Jan 24 '10 at 13:45
I've read through some of Goerss-Jardine, but they don't really talk about the "handed" fibrations (i.e. left, right, mid). A lot of that book seems devoted to showing that the Kan model structure on SSet is Quillen equivalent to CGHaus with Serre fibrations. That's something I can take on faith without reading it. If I'm wrong about the book focusing on that, let me know. – Harry Gindi Jan 24 '10 at 15:04

With Heiner Kamps, I had a go at bridging some of the gap which you say you are suffering from. Our book is Abstract Homotopy and Simple Homotopy Theory. (Look at my web page and the publication list.) There are some relevant things also in the Menagerie notes. An early version of these is available via my n-Lab page. If you have any other difficulties, I will try to give pointers.

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Echoing what Urs said, "But you should all be using the nLab more: if you want literature on quasi-categories, look up the references section on quasi-categories! :-) " and in regard to your question in particular, "See the nLab page on simplicial sets for links."

By looking at this nLab page on simplicial sets, in the references section you'll have found (as of this date, July 21, 2011 10:19 pm, EST):

Greg Friedman: An elementary illustrated introduction to simplicial sets

which is chock full of pictures illustrating the geometric ideas underlying the combinatorics of simplicial sets. As Friedman discusses in the introduction on the second and third pages of this paper,

"Here, for the most part, you won't find many complete proofs of theorems, and so these notes will not be completely self-contained. Rather, I try primarily to show by example how the very basic combinatorics, including the definitions, arise out of geometric ideas and to show the geometric ideas underlying the most elementary proofs and properties."

Suffice it to say, those notes may be the end of your search in finding a concise introduction to simplicial sets that also helps develop your geometric intuition and the computation of products. You also might find the references I gave in answer to the question here helpful.

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One thing that might help is to develop some intuition about triangulable spaces and the analogies with simplicial sets. Taking the geometric realization and drawing some pictures might hel you get comfortable with these ideas.

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Harry,have you looked at Gelfland and Manin's Methods of Homological Algebra (2nd ed)? There's a pretty impressive final chapter introduction to homotopical algebra and a good part of the book is concerned with homological algebra via simplicial sets. It probably contains just what you need to know.I'm carefully reading it for the first time now-just skimmed it before now. Another book also worth checking out that has a chapter on the basics of simplical sets and thier relations to topological spaces is Wiebel's An Introduction To Homological Algebra. This book needs to be in every topologist's library.

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