nLab dg-nerve

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Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Enriched category theory

Stable Homotopy theory

Contents

Idea

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Any pretriangulated dg-category 𝒞\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 𝒞\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 𝒞\mathcal{C} as analogous to a category of chain complexes, the dg-nerve may be thought of producing the simplicial set whose kk-simplices are the local systems on Δ k\Delta^k with coefficients in 𝒞\mathcal{C} (flat ∞-connections with coefficients in 𝒞\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 (𝒞,W)(\mathcal{C}, \mathrm{W}) (a category with weak equivalences WMor(𝒞)\mathrm{W} \subset Mor(\mathcal{C})) one obtains:

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

  2. the quasi-category which is the homotopy coherent nerve N𝒞[W 1]N \mathcal{C}[\mathrm{W}^{-1}] of that.

or alternatively, up to equivalence of quasi-categories:

  1. the simplicial nerve N𝒞N \mathcal{C},

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

Now:

Proposition

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

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

  2. Ch (𝒜)\mathbf{Ch}_\bullet(\mathcal{A}) for the dg-category obtained form the self-enrichment of Ch (𝒜)Ch_\bullet(\mathcal{A})

    (via its symmetric monoidal structure given by the tensor product of chain complexes),

  3. W cheMor(Ch (𝒜))\mathrm{W}_{che} \,\subset\, Mor\big(Ch_\bullet(\mathcal{A})\big) for the class of chain homotopy equivalences

    (which for 𝒜=k\mathcal{A} = k Vect coincide with the quasi-isomorphisms, but not otherwise).

Then the dg-nerve of Ch (𝒜)\mathbf{Ch}_\bullet(\mathcal{A}) is equivalent as a quasi-category to the (homotopy coherent) nerve of the (simplicial) localization of Ch (𝒜)Ch_\bullet(\mathcal{A}) at W che\mathrm{W}_{che} in the sense of Rem. :

N dg(Ch (𝒜))NCh (𝒜)[W che 1]. 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 C\mathbf{C} a Ch ( R Mod ) Ch_\bullet(R Mod) -enriched model category (over some ring RR) all whose objects are cofibrant, then

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

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

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

References

The definition originates with:

Further discussion:

Extension to A 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.