how to make the category of chain complexes into an $\infty$-category

I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.

Has anyone ever written down (more or less explicitly) what the simplicial set corresponding to the quasi-category associated with the category of (say unbounded) chain complexes on an abelian category looks like?

I am not looking for an enhancement of the derived category or anything like this, I'm thinking of the much simpler infinity category where higher morphisms correspond to homotopies between complexes. My understanding is that the derived category should then be constructed as a localization of this $\infty$-category.

I am guessing my problem lies with the coherent nerve for simplicial categories.

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"I am not looking for an enhancement of the derived category or anything like this, I'm thinking of the much simpler infinity category where higher morphisms correspond to homotopies between complexes." What??? Your understanding of enhancements of triangulated categories and/or quasi-categories is not very accurate if you write sentences like this one. What you describe as "much simpler" is exactly an enhancement of the derived category. If I were you, I'd go for more basic stuff before getting into these topics. – Fernando Muro Apr 13 '12 at 19:54
Fernando: seems pretty clear to me -- Yosemite Sam's lookin' to enhance the chain homotopy category, not the derived category. Yosemite Sam: I think Section 1.3.1 of Lurie's book math.harvard.edu/~lurie/papers/HigherAlgebra.pdf has what you want. – Dustin Clausen Apr 13 '12 at 20:06
@fernando: chill out. I have already gone through the basics (although it doesn't mean I master them) and Dustin is correct, notice I never wrote quasi-isomorphism. @dustin: yep, thanks for that, I was reading DAG I, but I guess now we are all supposed to read HA now. :) – Yosemite Sam Apr 13 '12 at 20:10
Any enhancement of the homotopy category contains an enhancement of the derived category, since the derived category is a full triangulated subcategory of the homotopy category. Your "much simpler" indicates that I'm right in my appreciation. Believe me, you (Yosemite and Dustin) will profit from going to simpler stuff instead of the advanced references you're looking at. – Fernando Muro Apr 13 '12 at 21:01
Fernando: Which out of the two is simpler is a matter of taste and perspective. Same as with presheaves and sheaves, I guess. – Dustin Clausen Apr 13 '12 at 22:08

As everybody's said, there's an obvious thing to do. As Yosemite Sam cites, it's done in Section 13 of the ArXiv version of DAG I -- you think of chain complexes as enriched over simplicial sets via Dold-Kan, and then apply the nerve construction.

But there's an explicit thing you can do for any dg category, and I find it useful because it's given in terms of formulas. Moreover there's an obvious (if tedious) way to generalize this formula for any $A_\infty$-category so it's a cool thing to know. It's in the latest (February 2012) version of Higher Algebra. Since chain complexes obviously form a dg-category, this explicit method might be what you're looking for in case you want to produce some simplices in your quasi-category.

Specifically, Construction 1.3.1.6 tells you how to get a quasi-category from any dg category. Then Construction 1.3.13 and Remark 1.3.1.12 should convince you that it's equivalent to the "Dold-Kan + Simplicial nerve" construction cited by everybody else. (Lurie summarizes this equivalence in Proposition 1.3.1.17.) I would write out the formulas here but I don't want to re-TeX the long discussions. So here's at least a link to the latest Higher Algebra.

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For any simplicial model category $C$, let $C^0$ denote the fullsubcategory on its fibrtant and cofibrant objects. It may be considered as a simplicial category via its simplicial-enrichment. Via the corner axioms for this enrichment, it follows that the Hom-complexes of $C^0$ are Kan complexes, so that $C^0$ is a fibrant simplicial category in the Bergner model structure on simplicial categories. Then apply the homotopy coherent nerve to $C^0$ as you suggest. Since it is the right-Quillen pair of the Quillen equivalence between the Bergner model structure on simplicial categories and the Joyal model structure on simplicial sets, the result, $N_{hc}\left(C^0\right)$ will be a fibrant simplicial set in the Joyal structure, i.e. a quasi-category.

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yeah, this was exactly the kind of thing I wanted to avoid. I mean, it's really annoying having to take resolutions (but perhaps I'm just still confused). I am wishing for a world where all the concepts (limits, colimits, derived functors etc) are elegantly defined using infinity categories (instead of model categories). Only later I want to take resolutions to actually compute things. I just feel uncomfortable being forced to do so right from the get-go. – Yosemite Sam Apr 13 '12 at 16:52
On the other hand it is true that taking an injective resolution is entirely analogous to sheafifying a presheaf, so maybe I just need to get over it. – Yosemite Sam Apr 13 '12 at 16:53

[ Edit: Section 13 of DAG I had everything I was looking for http://arxiv.org/pdf/math/0608228v5.pdf ]

I think I now partially understand why I'm confused (I'd like to thank David's answer for providing a more high-brow perspective, I'm sure it will be useful to me as soon as I try and understand stable/derived categories)

The $\infty$-category one should obtain has for vertices complexes $A,B, \ldots$, the 1-simplices are given by chain maps $A \to B$, the 2-simplices are given by maps $A \to B \to C, A \to C$ (not necessarily commuting) together with a homotopy, the 3-simplices are given by maps $A \to B \to C \to D, A \to C, B \to D, A \to D$ together with a homotopies and homotopies among homotopies and so on (perhaps I got something wrong but you get the idea).

I'm pretty sure this is the coherent nerve construction for simplicial categories but I need to understand it better first.

This should give the right thing, an $\infty$-enhancement of the category of complexes such that $\pi_0$ of it is the homotopy category of complexes (so no resolutions and no model categories were harmed in the process).

If someone corrects and/or has a better way of writing this please do so!

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I think what you're looking for is done in Higher Algebra (Lurie), via the Dold-Kan correspondence: View Ch(A) as simplicially enriched via the Dold-Kan correspondence, and then take the coherent nerve. That's how you make your idea of "homotopies between homotopies" etc., precise. I wrote something about this here which you may or may not find helpful: docs.google.com/a/uw.edu/… – Dylan Wilson Apr 13 '12 at 18:51
I had actually stumbled upon it earlier, will check it out again, thanks. – Yosemite Sam Apr 13 '12 at 19:19