Frames in Pretriangulated Dg-Categories

Abstract

Triangulated categories arising in algebra can often be described as the homotopy category of a pretriangulated dg-category, a category enriched in chain complexes with a natural notion of shifts and cones that is accessible with all the machinery of homological algebra. Dg-categories are algebraic models of $\infty$-categories and thus fit into a wide ecosystem of higher-categorical models and translations between them. In this paper we describe an equivalence between two methods to turn a pretriangulated dg-category into a quasicategory. The dg-nerve of a dg-category is a quasicategory whose simplices are coherent families of maps in the mapping complexes. In contrast, the cycle category of a pretriangulated category forgets all higher-degree elements of the mapping complexes but becomes a cofibration category that encodes the homotopical structure indirectly. This cofibration category then has an associated quasicategory of frames in which simplices are Reedy-cofibrant resolutions. For every simplex in the dg-nerve of a pretriangulated dg-category we construct such a Reedy-cofibrant resolution and then prove that this construction defines an equivalence of quasicategories which is natural up to simplicial homotopy. Our construction is explicit enough for calculations and provides an intuitive explanation of the resolutions in the quasicategory of frames as a generalisation of the mapping cylinder.