Let M be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space H * (LM ). We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively.We also define a generalized cup product on the Hochschild cohomology HH * (A, A ∨ ) of a commutative Gorenstein algebra A and show that over Q, HH * (A P L (M ), A P L (M ) ∨ ) is isomorphic as algebras to H * (LM ). Thus, when M is a Poincaré duality space, we recover the isomorphism of algebras H * (LM ; Q) ∼ = HH * (A P L (M ), A P L (M )) of Félix and Thomas.
The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. Based on this notion, we introduce a new homotopy invariant of spaces over a fixed space, called the level of a map. Moreover we provide a method to compute the invariant for spaces over a K-formal space. This enables us to determine the level of the total space of a bundle over the 4dimensional sphere with the aid of Auslander-Reiten theory for spaces due to Jørgensen. We also discuss the problem of realizing an indecomposable object in the derived category of the sphere by the singular cochain complex of a space. The Hopf invariant provides a criterion for the realization.
We propose the notion of association schemoids generalizing that of association schemes from small categorical points of view. In particular, a generalization of the Bose-Mesner algebra of an association scheme appears as a subalgebra in the category algebra of the underlying category of a schemoid. In this paper, the equivalence between the categories of groupoids and that of thin association schemoids is established. Moreover linear extensions of schemoids are considered. A general theory of the Baues-Wirsching cohomology deduces a classification theorem for such extensions of a schemoid. We also introduce two relevant categories of schemoids into which the categories of schemes due to Hanaki and due to French are embedded, respectively.2010 Mathematics Subject Classification: 18D35, 05E30
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