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Derived Smooth Manifolds
"... ... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable ..."
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... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a PontrjaginThom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A ∩ B of submanifolds A, B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ⌣ [B] = [A ∩ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.
Homotopy theory of C ∗algebras
, 2008
"... In this work we construct from ground up a homotopy theory of C ∗algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy t ..."
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In this work we construct from ground up a homotopy theory of C ∗algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C ∗algebras. The spaces in C ∗homotopy theory are certain hybrids of functors represented by C ∗algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C ∗algebra circle of complexvalued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C ∗homotopy theory. The stable homotopy category of C ∗algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We
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"... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C ∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable nor ..."
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and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C ∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a PontrjaginThom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A ∩ B of submanifolds A, B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ⌣ [B] = [A ∩ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.
Homotopy theory of presheaves of Γspaces
, 2008
"... ... in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on simplicial presheaves. If a local model structure on simplicial presheaves is monoidal, the corresponding stable model structure on presheaves of Γ ..."
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... in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on simplicial presheaves. If a local model structure on simplicial presheaves is monoidal, the corresponding stable model structure on presheaves of Γspaces is monoidal and satisfies the monoid axiom. This allows us to lift the stable model structures to categories of algebras and modules over commutative algebras.