# Two $\infty$-categories of chain complexes

In the literature, I've mostly seen two quasicategories coming from $$\text{Ch}_R$$:

1. By considering $$\text{Ch}_R$$ with weak equivalences $$\mathcal W = \text{quasi-isomorphisms}$$, we can consider its Dwyer-Kan localization $$L^H(\text{Ch}_R)$$, a simplicial category. Then, from the Quillen equivalence $$|-|:\text{sSet}_{\text{Joyal}}\leftrightarrows\text{sSet-Cat}_{\text{Bergner}}:N_\Delta,$$ we can define, taking a fibrant replacement if necessary, $$N_\Delta(L^H(\text{Ch}_R))\in\text{quasicategories}$$

2. In the Dold-Kan equivalence $$N:\text{sAb}\leftrightarrows\text{Ch}_R:\Gamma.$$, $$\Gamma$$ is a monoidal functor (between cartesian categories), thus it induces a functor $$\Gamma_*:\text{Ch}_R\text{-Cat}\to \text{sAb-Cat}.$$ The underlying set of a simplicial abelian group is a Kan complex thus $$N_\Delta(\Gamma_*(\text{Ch}_R))\in\text{quasicategories}$$

Are the quasicategories $$N_\Delta(L^H(\text{Ch}_R)$$ and $$N_\Delta(\Gamma_*(\text{Ch}_R)$$ equivalent? Is this trivial?

• Are you restricting to complexes of projectives? If not, then the latter doesn't turn quasi-isomorphisms into equivalences - its homotopy category is the "chain homotopy category" $K(R)$ instead of the derived category $D(R)$. Jan 25 at 23:04
• @TylerLawson: This is an answer, In my opinion Jan 26 at 0:01
• @TylerLawson Thank you for the answer, the question was wrong and I updated Example 2. What about now Jan 26 at 14:46
• @DanielPlácido Your edit does not address Tyler Lawson's point, which is still correct: in your example 2 you're only inverting homotopy equivalences of complexes, not quasi-isomorphisms. Jan 26 at 15:34

The two categories you describe are not equivalent in the fashion that you hope.

No matter what kind of simplicial category $$C$$ is, the quasicategory $$N_\Delta(C)$$ has an explicit description of its homotopy category: namely, it has the same objects as $$C$$, and $$Hom_{hN_\Delta C}(X,Y) = \pi_0 Hom_C(X,Y).$$ (This is true even if we need to take a fibrant replacement of $$C$$ first.)

Consider first case (1). Whenever $$C$$ is a category with a subcategory $$W$$ of weak equivalences, the Dwyer-Kan localization $$L^H C$$ has morphism spaces that are explicitly described: the 0-simplices are composites of morphisms in $$C$$ with formal inverses of morphisms in $$C$$, and the 1-simplices are natural transformations ("hammocks") built out of commutative diagrams involving these morphisms in $$C$$ or $$W$$. Taking $$\pi_0$$, we find that the homotopy category is formed by taking the morphisms in $$C$$ and formally inverting the morphisms in $$W$$: it is the localization. $$hN_\Delta L^H C \cong C[W^{-1}]$$

Now consider case 2. If $$C$$ is a dg category, then we can also be explicit about the simplices in $$Hom_{\Gamma_* C}(X,Y)$$. Namely, a 0-simplex is a 0-cycle $$f$$ in the Hom-complex $$\underline{Hom}_C(X,Y)$$, and an edge from $$f$$ to $$g$$ is a 1-chain $$h$$ in $$\underline{Hom}_C(X,Y)$$ with $$\partial h = g-f$$. Taking $$\pi_0$$, we find that the homotopy category is formed by taking the morphisms in $$C$$ and modding out by the relation of chain homotopy equivalence $$Hom_{hN_\Delta \Gamma_* C}(X,Y) \cong H_0 \underline{Hom}_C(X,Y).$$ This makes it the "chain homotopy category".

Applied to $$Ch(R)$$, the first construction is the derived category $$D(R)$$, and the second construction is the chain homotopy category $$K(R)$$. There is a natural functor $$K(R) \to D(R)$$, but it is not an equivalence: for example, there is a map of complexes from $$(\dots \to 0 \to \Bbb Z \to \Bbb Z \to 0)$$ to $$(\dots \to 0 \to 0 \to \Bbb Z/2 \to 0)$$ that is a quasi-isomorphism; there are no nonzero maps of chain complexes in the opposite direction in $$Ch(\Bbb Z)$$ and hence no nonzero maps in $$K(\Bbb Z)$$ either.

I believe that if you restrict to the subcategory of "cofibrant" complexes (e.g. bounded-below complexes of projectives) then you get an equivalence. Most of the versions of this result that I know, however, are for simplicial model categories (which $$Ch_R$$ is not), and so I don't have a reference handy.

• One can use a trick to answer your last question in the affirmative: an exact functor between stable ∞-category is an equivalence iff it induces an equivalence on the homotopy categories, and it's quite easy to see that both the dg-nerve of cofibrant complexes and the DK localization are stable, and that the functor preserves cofibers and direct sums (the latter because it is the DK-localization of a stable model category). Jan 29 at 18:42
• @DenisNardin Good point! Jan 30 at 1:26