Frames in Pretriangulated Dg-Categories
Lukas Heidemann

TL;DR
This paper establishes an explicit equivalence between two methods of converting pretriangulated dg-categories into quasicategories, enhancing understanding of their homotopical and higher-categorical structures.
Contribution
It constructs an explicit, natural equivalence between the dg-nerve and the quasicategory of frames for pretriangulated dg-categories, providing computational tools and conceptual insights.
Findings
Constructs Reedy-cofibrant resolutions for simplices in the dg-nerve.
Proves the equivalence of the two quasicategory constructions.
Provides an explicit, calculable framework for understanding higher structures.
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 -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…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Fuzzy and Soft Set Theory
