Algebraic Geometry over model categories (a general approach to derived algebraic geometry) Export

@article{citeulike:4814666, abstract = {For a (semi-)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory of E\_{\infty}-ring spectra. This first version is very preliminary and might be considered as a detailed research announcement. Some proofs, more details and more examples will be added in a forthcoming version.}, archivePrefix = {arXiv}, author = {Toen, Bertrand and Vezzosi, Gabriele}, citeulike-article-id = {4814666}, citeulike-linkout-0 = {http://arxiv.org/abs/math/0110109}, citeulike-linkout-1 = {http://arxiv.org/pdf/math/0110109}, day = {10}, eprint = {math/0110109}, keywords = {algebraic-geometry, category-theory, math}, month = {Oct}, posted-at = {2009-06-11 17:53:41}, priority = {2}, title = {Algebraic Geometry over model categories (a general approach to derived algebraic geometry)}, url = {http://arxiv.org/abs/math/0110109}, year = {2001} }

TY - JOUR ID - citeulike:4814666 L3 - citeulike-article-id:4814666 N2 - For a (semi-)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory of E_{\infty}-ring spectra. This first version is very preliminary and might be considered as a detailed research announcement. Some proofs, more details and more examples will be added in a forthcoming version. TI - Algebraic Geometry over model categories (a general approach to derived algebraic geometry) KW - algebraic-geometry KW - category-theory KW - math AU - Toen, Bertrand AU - Vezzosi, Gabriele PY - 2001/Oct/10/ UR - http://arxiv.org/abs/math/0110109 ER -