Triangulated structures induced by simplicial descent categories

TL;DR

This paper explores the homotopy category of simplicial descent categories, establishing triangulated structures and fiber/cofiber sequences, with applications to various algebraic and topological contexts.

Contribution

It proves that the class of equivalences admits a calculus of fractions, and extends triangulated structures to homotopy categories of diagram categories, broadening their applicability.

Findings

Homotopy category of a stable simplicial descent category is triangulated.

Triangulated structures extend to derived categories and stable categories.

Examples include derived categories of abelian categories and fibrant spectra.

Abstract

The present paper is devoted to study the homotopy category associated with a simplicial descent category (D,s,E) (arXiv:0808.3684v2). We prove that the class E of equivalences has a calculus of left fractions over a quotient category of D modulo homotopy. We study the fiber/cofiber sequences induced by a (co)simplicial descent structure. Examples of such fiber/cofiber sequences are deduced for (commutative) differential graded algebras, simplicial sets or topological spaces. We prove that the homotopy category of a stable simplicial descent category is triangulated. In addition, these triangulated structures may be extended to the homotopy categories of diagram categories of D. As a corollary, we obtain the triangulated structures on: (filtered) derived categories of abelian categories, the derived category of DG-modules over a DG-category, the stable derived category of fibrant spectra and the localized category of mixed Hodge complexes.