Abstract
In this note we present a work in progresswhosemain purpose is to establish a categorified version of sheaf theory.We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of O-modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.
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References
Ando M, Morava J (1999) A renormalized Riemann–Roch formula and the Thom isomorphism for the free loop space. Topology, geometry, and algebra: interactions and new directions, Stanford, CA, pp 11–36, Contemp. Math., 279, Amer. Math. Soc., Providence, RI, 2001.
Ausoni C, Rognes J (2008) The chromatic red-shift in algebraic K-theory. Enseign Math 54(2):9–11
Tillmann U (ed) (2004) Two-vector bundles and forms of elliptic cohomology. Topology, Geometry and quantum field theory, London Mathematical Society Lecture Note Series, 308, Cambridge University Press, 18–45
Ben-Zvi D, Nadler D Loops spaces and Langlands parameters, preprint arXiv:0706. 0322
Bergner J A survey of (1, ∞)-categories. To appear in the proceedings of the IMA workshop on n-categories, arXiv preprint math.AT/0610239
Getzler E (1993) Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology. In: Quantum deformations of algebras and their representations, Israel Mathematical Conference Proceedings vol. 7, Bar-Ilan University, Ramat-Gan, pp 65–78
Kontsevich M, Soibelmann Y Notes on A-infinity algebras, A-infinity categories and non-commutative geometry, I. arXiv Preprint math.RA/0606241
Jones JDS (1987) Cyclic homology and equivariant homology. Invent Math 87: 403–423
Laumon G, Moret-Bailly L (2000) Champs algébriques. Series of modern surveys in mathematics vol. 39, Springer, Berlin, p 208
Loday J-L (1992) Cyclic homology. Springer, Berlin
Lurie J A survey of elliptic cohomology. preprint available at http://www-math.mit.edu/∼lurie/papers/survey.pdf
Lurie J Derived algebraic geometry. Thesis
Lurie J (2007) Talk at the Abel symposium, Oslo. Expository article on topological field theories. Available at http://math.mit.edu/∼lurie/cobordism.pdf
Neeman A (2001) Triangulated categories. Annals of mathematical studies vol. 148, Princeton University Press, Princeton, NJ, pp viii, 449
Stolz S, Teichner P (2004) What is an elliptic object? In: Tillmann U (ed) Topology, geometry and quantum field theory. In: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Segal G, London Mathematical Society Lecture Note Series, 308, pp 247–344
Tabuada G (2005) Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories. Comptes Rendus de l'Acadmie de Sciences de Paris 340:15–19
Toën B Higher and derived stacks: a global overview. To appear in Proceedings of the Seattle conference on Algebraic Geometry, arXiv preprint math.AG/0604504
Toën B (2007) The homotopy theory of dg-categories and derived Morita theory. Invent Math 167(3):615–667
Toën B, Vaquié M Moduli of objects in dg-categories. To appear in Ann Scient Ecole Norm Sup, arXiv preprint math.AG/0503269
Toën B, Vezzosi G (2005) Homotopical algebraic geometry I: topos theory. Adv Math 193:257–372
Toën B, Vezzosi G Homotopical algebraic geometry II: derived stacks and applications. To appear in Mem. AMS, arXiv preprint math.AG/0404373
Walker M (2002) Semi-topological K-homology and Thomason's theorem. K-theory 26:207–286
Witten E (1988) The index of the Dirac operator on the loop space. In Elliptic curves and modular forms in algebraic topology. Lecture Notes in Mathematics 1326, Springer, pp 161–181
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Toën, B., Vezzosi, G. (2009). Chern Character, Loop Spaces and Derived Algebraic Geometry. In: Baas, N., Friedlander, E., Jahren, B., Østvær, P. (eds) Algebraic Topology. Abel Symposia, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01200-6_11
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DOI: https://doi.org/10.1007/978-3-642-01200-6_11
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