Simplicial Descent Categories
Beatriz Rodriguez Gonzalez

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…
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Taxonomy
TopicsAdvanced Algebra and Logic
