Simplicial Descent Categories
| dc.creator | Gonzalez, Beatriz Rodriguez | |
| dc.date | 2008-04-14 | |
| dc.date.accessioned | 2026-07-07T09:32:14Z | |
| dc.date.available | 2026-07-07T09:32:14Z | |
| dc.description | 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. | |
| dc.description | 237 pages, author's PhD thesis (translated and revised). Suggestions and comments are welcome | |
| dc.identifier | https://arxiv.org/abs/0804.2154 | |
| dc.identifier | http://arxiv.org/abs/0804.2154 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158739 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Category Theory | |
| dc.subject | 14F35 (Primary) 18G30, 18D99 (Secondary) | |
| dc.title | Simplicial Descent Categories | |
| dc.type | text |