Riv. Mat. Univ. Parma, Vol. 11, No. 1, 2020

Simone Borghesi [a]

Homotopy theory and complex geometry

Pages: 89-98
Received: 12 January 2019
Accepted in revised form: 7 May 2019
Mathematics Subject Classification (2010): 14D22, 14D23, 18G55, 18G30, 32Q45.
Keywords: Hyperbolicity, Presheaves, Stacks, Holotopy.
Authors address:
[a]: Università degli Studi di Milano-Bicocca, Edificio U5, via R. Cozzi 55, 20126 Milano, Italia

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Abstract: Abstract homotopy theory provides a framework in which any site \(\mathcal{S}\) can be embedded and its objects studied. In this context, objects of the site appear distinguished among more general ones. The purpose of this framework is to create an environment where the site \(\mathcal{S}\) and supplementary ''combinatorial/homotopical'' data can blend together. Such a blending is achieved by a well-established mathematical procedure, the localization. The challenge in using this framework in practice is to find close ties between these general objects and the ones in \(\mathcal{S}\). I will describe the results that G. Tomassini and I have obtained on these topics.

S. Borghesi and G. Tomassini, Extended Hyperbolicity, Ann. Mat. Pura Appl. (4) 191 (2012), 261-284. MR2909798
S. Borghesi and G. Tomassini, The coarse moduli space of a flat analytic groupoid, Ann. Mat. Pura Appl. (4) 194 (2015), 247-257. MR3303014
S. Borghesi and G. Tomassini, Analytic stacks and hyperbolicity, Ann. Mat. Pura Appl. (4) 196 (2017), 1273-1306. MR3673667
D. Dugger, S. Hollander and D. C. Isaksen Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004), 9-51. MR2034012
S. Hollander, A homotopy theory for stacks, Israel J. Math. 163 (2008), 93-124. MR2391126
F. Morel and V. Voevodsky, \(\mathbb{A}^1\)-Homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45-143. MR1813224

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