Nerve construction, A-infinity functors and homotopy theory of differential graded categories [electronic resource].

9781369091137

1 online resource (156 p.)

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In this dissertation we prove that the derived enrichment of the homotopy category of differential graded categories (dg-categories) with respect to a certain model category structure defined by G. Tabuada, is computed by the dg-category of A-infinity functors. This result was originally formulated by M. Kontsevich but rio formal proof was available in the literature. We further put this construction into the framework of higher categories. We introduce a functor, called the simplicial nerve of an A-infinity category, defined on the category of (small) A-infinity categories and taking values in simplicial sets. This construction extends the nerve construction for dg-categories of J. Lurie. We prove that the nerve of any A-infinity category is an (infinity,1)-category and that the nerve of a pretriangulated dg-category in the sense of A. Bondal and M. Kapranov is a stable (infinity, 1)-category in the sense of J. Lurie. We then provide a series of applications of these results. We introduce an enhancement of the categories dgCat and Ainfinity Cat, of dg and A-infinity categories, to (infinity, 2)-categories. We show that the (infinity, 1)-category associated to the (infinity, 2)-category of dg-categories is a model for the simplicial localization at the class of weak-equivalences of the Tabuada's model structure. Finally, we prove that the homotopy groups of the self-mapping space of the identity functor in the (infinity, 2)-category of A-infinity categories compute the Hochschild cohomology for A-infinity categories.

Books / Online / Dissertations & Theses

English

January 19, 2017

Thesis (Ph.D.)--Yale University, 2016.

Yale University.