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A bicategory of reduced orbifolds from the point of view of differential geometry Tommasini, Matteo in Journal of Geometry & Physics (2016), 108 Detailed reference viewed: 65 (0 UL)Some insights on bicategories of fractions: representations and compositions of 2-morphisms Tommasini, Matteo in Theory and Applications of Categories (2016), 31(10), 257-329 Detailed reference viewed: 28 (0 UL)Some insights on bicategories of fractions – I Tommasini, Matteo E-print/Working paper (2014) In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory C together with a suitable class of morphisms W, one can construct a ... [more ▼] In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory C together with a suitable class of morphisms W, one can construct a bicategory C[W^{-1}], where all the morphisms of W are turned into internal equivalences, and that is universal with respect to this property. Most of the descriptions leading to such a construction were long and heavily based on the axiom of choice. In this paper we simplify considerably the constructions of associators, vertical and horizontal compositions in a bicategory of fractions, thus proving that the axiom of choice is not needed under certain conditions. The simplified description of associators and 2-compositions will also play a crucial role in the next papers of this series. [less ▲] Detailed reference viewed: 44 (1 UL)Some insights on bicategories of fractions - II Tommasini, Matteo E-print/Working paper (2014) We fix any bicategory A together with a class of morphisms W_A, such that there is a bicategory of fractions A[W_A^{-1}] (as described by D. Pronk). Given another such pair (B,W_B) and any pseudofunctor F ... [more ▼] We fix any bicategory A together with a class of morphisms W_A, such that there is a bicategory of fractions A[W_A^{-1}] (as described by D. Pronk). Given another such pair (B,W_B) and any pseudofunctor F : A → B, we find necessary and sufficient conditions in order to have an induced pseudofunctor G : A[W_A^{-1}]→ B[W_B^{-1}]. Moreover, we give a simple description of G in the case when the class W_B is “right saturated”. [less ▲] Detailed reference viewed: 22 (0 UL)Weak fiber products in bicategories of fractions Tommasini, Matteo E-print/Working paper (2014) We fix any pair (C,W) consisting of a bicategory and a class of morphisms in it, admitting a bicalculus of fractions, i.e. a “localization” of C with respect to the class W. In the resulting bicategory of ... [more ▼] We fix any pair (C,W) consisting of a bicategory and a class of morphisms in it, admitting a bicalculus of fractions, i.e. a “localization” of C with respect to the class W. In the resulting bicategory of fractions, we identify necessary and sufficient conditions for the existence of weak fiber products. [less ▲] Detailed reference viewed: 28 (1 UL)Some insights on bicategories of fractions - III Tommasini, Matteo E-print/Working paper (2014) We fix any bicategory A together with a class of morphisms W_A, such that there is a bicategory of fractions A[W_A^{-1}] (as described by D. Pronk). Given another such pair (B,W_B) and any pseudofunctor F ... [more ▼] We fix any bicategory A together with a class of morphisms W_A, such that there is a bicategory of fractions A[W_A^{-1}] (as described by D. Pronk). Given another such pair (B,W_B) and any pseudofunctor F : A → B, we find necessary and sufficient conditions in order to have an induced equivalence of bicategories from A[W_A^{-1}] to B[W_B^{-1}]. In particular, this gives necessary and sufficient conditions in order to have an equivalence from any bicategory of fractions A[W_A^{-1}] to any given bicategory B. [less ▲] Detailed reference viewed: 28 (0 UL)The Hodge-Deligne polynomials of some moduli spaces of coherent systems Tommasini, Matteo Scientific Conference (2013) After proving the existence of (non-split, non-degenerate) universal families of extensions of coherent systems (in the spirit of Lange). we are able to describe wallcrossing for some moduli spaces of ... [more ▼] After proving the existence of (non-split, non-degenerate) universal families of extensions of coherent systems (in the spirit of Lange). we are able to describe wallcrossing for some moduli spaces of coherent systems. This allows us to compute the Hodge-Deligne polynomials of the moduli spaces of coherent systems of type (1,d,n) for n=2,3 and of type (2,d,2) for d odd big enough [less ▲] Detailed reference viewed: 31 (1 UL)Universal families of extensions of coherent systems Tommasini, Matteo E-print/Working paper (2013) We prove a result of cohomology and base change for families of coherent systems over a curve. We use that in order to prove the existence of (non-split, non-degenerate) universal families of extensions ... [more ▼] We prove a result of cohomology and base change for families of coherent systems over a curve. We use that in order to prove the existence of (non-split, non-degenerate) universal families of extensions for families of coherent systems (in the spirit of the paper "Universal families of extensions" by H. Lange). Such results will be applied in subsequent papers in order to describe the wallcrossing for some moduli spaces of coherent systems. [less ▲] Detailed reference viewed: 34 (1 UL)A bicategory of reduced orbifolds from the point of view of differential geometry - I Tommasini, Matteo E-print/Working paper (2013) We describe a bicategory (Red Orb) of reduced orbifolds in the framework of differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using ... [more ▼] We describe a bicategory (Red Orb) of reduced orbifolds in the framework of differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we first define a 2-category (Red Atl) whose objects are reduced orbifold atlases (on paracompact, second countable, Hausdorff topological spaces). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl, while the definition of 2-morphisms and compositions of them is new in this setup. Using the bicalculus of fractions described by D. Pronk, we are able to construct from such a 2-category the bicategory (Red Orb). We prove that it is equivalent to the bicategory of reduced orbifolds described in terms of proper, effective, étale Lie groupoids by D. Pronk and I. Moerdijk and to the 2-category of reduced orbifolds described by several authors in the past in terms of a suitable class of differentiable Deligne-Mumford stacks. [less ▲] Detailed reference viewed: 37 (2 UL) |
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