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Abstract: Much of the homotopical and homological structure of the categories of chaincomplexes and topological spaces can be deduced from the existence andproperties of the -simple- functors Tot : {double chain complexes} -> {chaincomplexes} 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 weintroduce the notion of -cosimplicial descent category-. It is inspired byGuillen-Navarros-s -cubical descent categories-. The key ingredients in acosimplicial descent category D are a class E of morphisms in D, calledequivalences, and a -simple- functor s : {cosimplicial 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 = filteredquasi-isomorphisms or E = E 2-isomorphisms, commutative differential gradedalgebras with s = Navarro-s Thom-Whitney simple, DG-modules over aDG-category and mixed Hodge complexes, where s = Deligne-s simple. From thesimplicial descent structure we obtain homotopical structure on D, as cone andcylinder objects. We use them to i explicitly describe the morphisms ofHoD=DE^{-1} similarly to the case of calculus of fractions; ii endow HoDwith a non-additive pre-triangulated structure, that becomes triangulated inthe stable additive case. These results use the properties of a -totalfunctor-, which associates to any biaugmented bisimplicial object a simplicialobject. It is the simplicial analogue of the total chain complex of a doublecomplex, and it is left adjoint to Illusie-s -decalage- functor.

Author: Beatriz Rodriguez Gonzalez


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