nLab dg-nerve

Idea

This section is no good…

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.

Properties

Compatibility with the simplicial nerve

Recall:

Remark

From a relative category $(\mathcal{C}, \mathrm{W})$ (a category with weak equivalences $\mathrm{W} \subset Mor(\mathcal{C})$) one obtains:

1. the sSet-enriched category $\mathcal{C}[\mathrm{W}^{-1}]$ which is its simplicial localization,

2. 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:

1. the simplicial nerve $N \mathcal{C}$,

2. its localization as a quasi-category $N\mathcal{C}[\mathrm{W}^{-1}]$.

Now:

Proposition

For $\mathcal{A}$ an additive category, write

1. $Ch_\bullet(\mathcal{A})$ for the plain category of chain complexes in $\mathcal{A}$,

2. $\mathbf{Ch}_\bullet(\mathcal{A})$ 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),

3. $\mathrm{W}_{che} \,\subset\, Mor\big(Ch_\bullet(\mathcal{A})\big)$ 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. :

$$N_{dg} \big( \mathbf{Ch}_\bullet(\mathcal{A}) \big) \;\; \simeq \;\; N Ch_\bullet(\mathcal{A})[\mathrm{W}^{-1}_{che}] \,.$$

[Lurie (2017), Prop. 1.3.4.5]

Similarly:

Proposition

Given $\mathbf{C}$ a $Ch_\bullet(R Mod)$-enriched model category (over some ring $R$) all whose objects are cofibrant, then

$$N_{dg}(\mathbf{C}) \;\; \simeq \;\; N C[\mathrm{W}_{che}^{-1}]$$

where $C$ denotes the underlying ordinary category and $\mathrm{W}_{che} \subset Mor(C)$ again the class of chain homotopy equivalences.

[Gwilliam & Pavlov (2018), Prop. 5.17]

References

The definition originates with:

• Vladimir Hinich, Vadim Schechtman, On homotopy limit of homotopy algebras, in: K-Theory, Arithmetic and Geometry – Seminar, Moscow University, 1984–1986, Lecture Notes in Mathematics 1289, Spinger (2006) 240–264 [doi:10.1007/BFb0078363]

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.

• Giovanni Faonte, Simplicial nerve of an $\mathcal{A}_\infty$-category, Theory and Applications of Categories 32 2 (2017) 31-52 [arXiv:1312.2127, tac:32-02]

Further discussion:

• Owen Gwilliam, Dmitri Pavlov, Enhancing the filtered derived category, Journal of Pure and Applied Algebra 222 11 (2018) 3621-3674 [arXiv:1602.01515, doi:10.1016/j.jpaa.2018.01.004]