See Toen AIM talk on Homotopy types of algebraic varieties.
Toen, Katzarkov, Pantev: Schematic homotopy types and nonabelian Hodge theory. File Toen web publ nht.pdf. Constructs a “Hodge decomposition” (a certain action) on the schematic homotopy type of a smooth projective complex variety. This recovers many other Hodge invariants, short review of these.
http://www.ncatlab.org/nlab/show/schematic+homotopy+type
TKP: Algebraic and topological aspects etc. File Toen web publ small.pdf. Discusses various things related to schematic htpy types, including two types of algebraic models.
Toen: Homotopical and higher categorical structures in algebraic geometry. File Toen web unpubl hab.pdf. Treats general philosophical background, various forms of homotopy theories, Segal categories, Waldhausen Kth briefly, Hochshild cohomology of Segal categories and of model cats, S-cats, Segal topoi, Tannakian duality for Segal cats, and schematic homotopy types. Also letter to May about n-cats.
Toen lecture at MSRI, see file Toen web unpubl msri2002-2.pdf.
Toen: Champs affines. File Toen web publ chaff.pdf. Among other things, contains constructions of schematic homotopy types for the classical cohomologies (Betti, de Rham, Hodge, crystalline, l-adic), extending the classical notions of fundamental group. Also stuff on rational homotopy theory and p-adic homotopy theory, nonabelian Abel-Jacobi map and nonabelian period, and much more on schematic homotopy types and stacks.
Toen: Higher and derived stacks, a global overview. File web publ seatt.pdf. Also contains an intro to Segal cats, and some material on schematic homotopy types.
Question for Toen in Seville: Do schematic homotopy types only make sense for smooth projective vars? Answer: No, this should not be essential. Reason for assuming proj was that Higgs bundle correspondence was not worked out in the general case, but this has now been done by Mochizuki. Pridham has done stuff on the singular case. See also Olsson: “towards p-adic homotopy”, and “F-isocristal”.
arXiv:0912.4844 The de Rham homotopy theory and differential graded category from arXiv Front: math.AT by Syunji Moriya 1 person liked this This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan’s minimal model. We also provide some examples. We prove an equivalence between fiberwise rationalizations and closed dg-categories with subsidiary data.
nLab page on Schematic homotopy type