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Invariants additifs de dg-categories  [PDF]
Goncalo Tabuada
Mathematics , 2005,
Abstract: With the help of the tools of Quillen's homotopical algebra, we construct `the universal additive invariant', namely a functor from the category of small dg categories to an additive category, that inverts the Morita dg functors, transforms the semi-orthogonal decompositions in the sense of Bondal-Orlov into direct sums and which is universal for these properties. We compare our construction with that of Bondal-Larsen-Lunts. ----- A l'aide des outils de l'algebre homotopique de Quillen, on construit `l'invariant additif universel', c'est-a-dire un foncteur defini sur la categorie des petites dg-categories et a valeurs dans une categorie additive qui rend inversibles les dg-foncteurs de Morita, transforme les decompositions semi-orthogonales au sens de Bondal-Orlov en sommes directes et qui est universel pour ces proprietees. Nous comparons notre construction a celle de Bondal-Larsen-Lunts.
A new Quillen model for the Morita homotopy theory of DG categories  [PDF]
Goncalo Tabuada
Mathematics , 2007,
Abstract: We construct a new Quillen model, based on the notions of Drinfeld's DG quotient and localization pair, for the Morita homotopy theory of DG categories. This new Quillen model carries a natural closed symmetric monoidal structure and allows us to re-interpret Toen's construction of the internal Hom-functor for the homotopy category of DG categories as a total right derived internal Hom-functor.
An explicit construction of the Quillen homotopical category of dg Lie algebras  [PDF]
Boris Shoikhet
Mathematics , 2007,
Abstract: Let $\g_1$ and $\g_2$ be two dg Lie algebras, then it is well-known that the $L_\infty$ morphisms from $\g_1$ to $\g_2$ are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra $\Bbbk(\g_1,\g_2)$. Then the gauge action by exponents of the zero degree component $\Bbbk(\g_1,\g_2)^0$ on $MC\subset\Bbbk(\g_1,\g_2)^1$ gives an explicit "homotopy relation" between two $L_\infty$ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are $L_\infty$ algebras and morphisms are $L_\infty$ morphisms modulo the gauge relation) is well-defined, and is a localization of the category of dg Lie algebras and dg Lie maps by quasi-isomorphisms. As localization is unique up to an equivalence, it is equivalent to the Quillen-Hinich homotopical category of dg Lie algebras [Q1,2], [H1,2]. Moreover, we prove that the Quillen's concept of a homotopy coincides with ours. The last result was conjectured by V.Dolgushev [D].
Modeles De Croissance économique Par Les Investissements
Marin Andreica,Sorin Briciu,Romulus Andreica,Liliana Todor
Communications of the IBIMA , 2009,
Abstract: L’ investissement, dans une large acception, est un concept primordial, synonyme de dotation, allocation, faire des placements ou des dépanses longue durée. L’efficience est la propriété d’un système d’obtenir des résultats optimaux avec des dépanses ayant un niveau suffisament petit. Les investissements répresentent le suport materiélede la croissance économique. Les dimensions de l’invesstisement, leur rythme, la manière d’allocations dans les secteurs d’activité économique et leur efficience determinant la croissance économique.
DG-modules over de Rham DG-algebra  [PDF]
Sergey Rybakov
Mathematics , 2013,
Abstract: For a morphism of smooth schemes over a regular affine base we define functors of derived direct image and extraordinary inverse image on coderived categories of DG-modules over de Rham DG-algebras. Positselski proved that for a smooth algebraic variety $X$ over a field $k$ of characteristic zero the coderived category of DG-modules over $\Omega^\bullet_{X/k}$ is equivalent to the unbounded derived category of quasi-coherent right ${\mathscr D}_X$-modules. We prove that our functors correspond to the functors of the same name for ${\mathscr D}_X$-modules under Positselski equivalence.
Theories homotopiques de Quillen combinatoires et derivateurs de Grothendieck  [PDF]
Olivier Renaudin
Mathematics , 2006,
Abstract: We construct a pseudo-localization of the 2-category of combinatorial Quillen model categories with respect to Quillen equivalences, and then verify that it embeds in a 2-category of Grothendieck derivators.
The bar derived category of a curved dg algebra  [PDF]
Pedro Nicolas
Mathematics , 2007,
Abstract: Curved A-infinity algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A-infinity algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.
Seicento e Novecento. Le categorie moderne della politica  [cached]
Pierangelo Schiera
Scienza & Politica : per una Storia delle Dottrine , 2002, DOI: 10.6092/issn.1825-9618/2887
Abstract: Seicento e Novecento. Le categorie moderne della politica
On the Quillen determinant  [PDF]
Kenro Furutani
Mathematics , 2003, DOI: 10.1016/j.geomphys.2003.07.001
Abstract: We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line bundles and trivial on some subspaces. Also we remark a relation of the {\it Quillen determinant line bundle} and the {\it Maslov line bundle}.
DG coalgebras as formal stacks  [PDF]
V. Hinich
Mathematics , 1998,
Abstract: The category of unital (unbounded) dg cocommutative coalgebras over a field of characteristic zero is provided with a structure of simplicial closed model category. This generalizes the model structure defined by Quillen in 1969 for 2-reduced coalgebras. In our case, the notion of weak equivalence is structly stronger than that of quasi-isomorphism. A pair of adjoint functors connecting the category of coalgebras with the category of dg Lie algebras, induces an equivalence of the corresponding homotopy categories. The model category structure allows one to consider dg coalgebras as very general formal stacks. The corresponding Lie algebra is then interpreted as a tangent Lie algebra which defines the formal stack uniquely up to a weak equivalence. An example of the coalgebra of formal deformaions of a principal $G$-bundle on a scheme $X$ is calculated.
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