Simplicial Descent Categories

TL;DR

This paper introduces the concept of (co)simplicial descent categories to abstract the homotopical and homological structures of various mathematical categories, providing a unified framework for their analysis.

Contribution

It formalizes (co)simplicial descent categories with axioms and demonstrates their applicability to diverse mathematical contexts, extending the theory of homotopical and homological structures.

Findings

Defines (co)simplicial descent categories with key axioms.

Establishes homotopical structures like cones and cylinders within these categories.

Constructs a non-additive pre-triangulated structure that becomes triangulated in stable cases.

Abstract

Much of the homotopical and homological structure of the categories of chain complexes and topological spaces can be deduced from the existence and properties of the 'simple' functors Tot : {double chain complexes} -> {chain complexes} and geometric realization : {sSets} -> {Top}, or similarly, Tot : {simplicial chain complexes} -> {chain complexes} and | | : {sTop} -> {Top}. The purpose of this thesis is to abstract this situation, and to this end we introduce the notion of '(co)simplicial descent category'. It is inspired by Guillen-Navarros's '(cubical) descent categories'. The key ingredients in a (co)simplicial descent category D are a class E of morphisms in D, called equivalences, and a 'simple' functor s : {(co)simplicial objects in D} -> D. They must satisfy axioms like 'Eilenberg-Zilber', 'exactness' and 'acyclicity'. This notion covers a wide class of examples, as chain complexes, sSets, topological spaces, filtered cochain complexes (where E = filtered quasi-isomorphisms or E = E_2-isomorphisms), commutative differential graded algebras (with s = Navarro's Thom-Whitney simple), DG-modules over a DG-category and mixed Hodge complexes, where s = Deligne's simple. From the simplicial descent structure we obtain homotopical structure on D, as cone and cylinder objects. We use them to i) explicitly describe the morphisms of HoD=D[E^{-1}] similarly to the case of calculus of fractions; ii) endow HoD with a non-additive pre-triangulated structure, that becomes triangulated in the stable additive case. These results use the properties of a 'total functor', which associates to any biaugmented bisimplicial object a simplicial object. It is the simplicial analogue of the total chain complex of a double complex, and it is left adjoint to Illusie's 'decalage' functor.