Triangulated structures induced by simplicial descent categories

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.

💡 Research Summary

The paper develops a comprehensive homotopical and triangulated framework for the notion of a simplicial descent category ((\mathcal D,s,\mathcal E)). A simplicial descent category consists of a base category (\mathcal D), a “simple” functor (s) that turns simplicial objects into objects of (\mathcal D), and a distinguished class (\mathcal E) of morphisms called equivalences. These data are required to satisfy a list of axioms (simplicial identities, exactness, compatibility with homotopy, etc.) that mimic the behavior of the total complex functor in homological algebra.

The first major result is that (\mathcal E) admits a calculus of left fractions after passing to the quotient of (\mathcal D) by the homotopy relation (\sim). Concretely, one forms the homotopy category (\mathcal D/!\sim) and then formally inverts the images of (\mathcal E). The resulting category (\operatorname{Ho}(\mathcal D) = (\mathcal D/!\sim)