Toen and Vezzosi: HAG1 and HAG2. Files in Toen web publ. Lots of content.
Ref: Toen: Essen talk. Main idea: Further generalizing Relative algebraic geometry, one can replace rings by homotopy ring-like object, and get a corresponding notion of algebraic geom. More precisely, replace ring by commutative monoid in a symmetric monoidal category endowed with equivalences. Sometimes monoid here should be understood as up-to-homotopy monoid, e.g. E-infty rings. Examples include cdgas, E-inft algebras, E-infty ring spectra, and symmetric monoidal categories. See also section 5.1 for more examples including brave new algebraic geometry.
Toen and Vezzosi: Algebraic geometry over model categories. Early paper, looks very nice! Applications to interpreting DG-schemes, and to defining etale K-theory of E-infty algebras. Expectation to extend the classical work for E-infty algebras to the more general setting of AG over a model cat, for the following concepts: tangent Lie algebra, cotangent complex, Hochschild cohom, K-theory, A-Q cohomology. Would like to do AG over a symmetric monoidal infty-cat, need strictification results. An E-infty alg should be a monoid in a SM infty-cat. Pp 34: Short nice review of operads and E-infty stuff.
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.
nLab page on Homotopical algebraic geometry