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Any pretriangulated dg-category $\mathcal{C}$ presents a stable (infinity,1)-category. A plain dg-category only presents a spectrally enriched (infinity,1)-category. One way to construct this is to apply the Dold-Kan correspondence on each hom-object to produce a fibrant sSet-enriched category and then, if desired, form the homotopy coherent nerve of that to obtain a quasi-category.
On the other hand, the dg-nerve of $\mathcal{C}$ is a more direct construction that directly sends the dg-category to a simplicial set which is the quasi-category incarnation of the corresponding stable (∞,1)-category.
To the extent that one may think of $\mathcal{C}$ as analogous to a category of chain complexes, the dg-nerve may be thought of producing the simplicial set whose $k$-simplices are the local systems on $\Delta^k$ with coefficients in $\mathcal{C}$ (flat ∞-connections with coefficients in $\mathcal{C}$). The formula is just as for Lie integration of L-infinity algebroids.
Recall:
From a relative category $(\mathcal{C}, \mathrm{W})$ (a category with weak equivalences $\mathrm{W} \subset Mor(\mathcal{C})$) one obtains:
the sSet-enriched category $\mathcal{C}[\mathrm{W}^{-1}]$ which is its simplicial localization,
the quasi-category which is the homotopy coherent nerve $N \mathcal{C}[\mathrm{W}^{-1}]$ of that.
or alternatively, up to equivalence of quasi-categories:
the simplicial nerve $N \mathcal{C}$,
its localization as a quasi-category $N\mathcal{C}[\mathrm{W}^{-1}]$.
Now:
For $\mathcal{A}$ an additive category, write
$Ch_\bullet(\mathcal{A})$ for the plain category of chain complexes in $\mathcal{A}$,
$\mathbf{Ch}_\bullet(\mathcal{A})$ for the dg-category obtained form the self-enrichment of $Ch_\bullet(\mathcal{A})$
(via its symmetric monoidal structure given by the tensor product of chain complexes),
$\mathrm{W}_{che} \,\subset\, Mor\big(Ch_\bullet(\mathcal{A})\big)$ for the class of chain homotopy equivalences
(which for $\mathcal{A} = k$ Vect coincide with the quasi-isomorphisms, but not otherwise).
Then the dg-nerve of $\mathbf{Ch}_\bullet(\mathcal{A})$ is equivalent as a quasi-category to the (homotopy coherent) nerve of the (simplicial) localization of $Ch_\bullet(\mathcal{A})$ at $\mathrm{W}_{che}$ in the sense of Rem. :
Similarly:
Given $\mathbf{C}$ a $Ch_\bullet(R Mod)$-enriched model category (over some ring $R$) all whose objects are cofibrant, then
where $C$ denotes the underlying ordinary category and $\mathrm{W}_{che} \subset Mor(C)$ again the class of chain homotopy equivalences.
The definition originates with:
Further discussion:
Jonathan Block, Aaron M. Smith, Def. 2.3 (2.4 in the preprint) of: The higher Riemann–Hilbert correspondence, Advances in Mathematics 252 (2014) 382-405 [arXiv:0908.2843, doi:10.1016/j.aim.2013.11.001]
Jacob Lurie, Construction 1.3.1.6 in: Higher Algebra (2017)
Extension to $A_\infty$-categories, proof that the dg-nerve maps pretriangulated dg-categories to stable (∞,1)-categories.
Further discussion:
Last revised on April 18, 2024 at 04:06:54. See the history of this page for a list of all contributions to it.