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

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

2012-04-13T15:57:10Z

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.

Answer by David Carchedi for how to make the category of chain complexes into an $\infty$-category

2012-04-13T16:32:03Z

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.

Answer by Yosemite Sam for how to make the category of chain complexes into an $\infty$-category

2012-04-13T18:15:19Z

[ 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!

Answer by Hiro Lee Tanaka for how to make the category of chain complexes into an $\infty$-category

2012-06-20T15:56:46Z

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.