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dg-nerve

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Idea

Any pretriangulated dg-category π’ž\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 π’ž\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 π’ž\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 Ξ” k\Delta^k with coefficients in π’ž\mathcal{C} (flat ∞-connections with coefficients in π’ž\mathcal{C}). The formula is just as for Lie integration of L-infinity algebroids.

References

Definition 2.2 in

Construction 1.3.1.6 in

In the following paper, the definition of the dg-nerve is extended to A-infinity categories, and it is proved that the dg-nerve maps pretriangulated dg-categories to stable (∞,1)-categories.

Last revised on January 22, 2016 at 09:35:47. See the history of this page for a list of all contributions to it.