Two $\infty$-categories of chain complexes

Question

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?

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

Answer 1

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".

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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.

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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.