2008/04/14 by Beatriz Rodríguez González, Beatriz Rodriguez Gonzalez, Gonzalez, Beatriz Rodriguez
Computer Science · Mathematics · #14F35 (Primary) 18G30 #18D99 (Secondary) #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #math.AG #math.CT #msc:14F35 #msc:18D99 #msc:18G30
paper · pdf · doi:10.48550/arxiv.0804.2154
237 pages, author's PhD thesis (translated and revised). Suggestions and comments are welcome
arxiv created 2008/04/14 · openalex publication_date 2008/04/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 = E2-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.