Building on the theory of parity sheaves due to Juteau–Mautner– Williamson, we develop a formalism of “mixed modular perverse sheaves” for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a “Koszul-type” derived equivalence relating a given flag variety to the Langlands dual flag variety, and (2) a formality theorem for the modular derived category o...

The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified, Treumann, based on unpublished ideas of MacPherson, constructed an ‘exit category’ (in the terminology of this paper, the ‘fundamental category’) which classifies constructible sheaves, equivalently stratified etale covers. This pap...

We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit co...

Given a sheaf of unital commutative and associative algebras A, first we construct the k-th Grassmann sheaf GA(k, n) of An, whose sections induce vector subsheaves of An of rank k. Next we show that every vector sheaf over a paracompact space is a subsheaf of A. Finally, applying the preceding results to the universal Grassmann sheaf GA(n), we prove that vector sheaves of rank n over a paracomp...

In the first part of this Thesis we develop the theory of locally boolean domains and bistable maps (as introduced in [Lai05b]) and show that the category of locally boolean domains and bistable maps is equivalent to the category of Curien-Lamarche games and observably sequential functions (cf. [CCF94]). Further we show that the category of locally boolean domains has inverse limits of ω-chains...

We identify Morita cohomology, which is a categorification of the cohmology of a topological space X, with the category of homotopy locally constant sheaves of perfect complexes on X.

In this article, we introduce a new class of ideal convergent sequence spaces using an infinite matrix, Musielak-Orlicz function and a new generalized difference matrix in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We also give some relations related to these sequence spaces.

In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number ζ an abelian artinian category FS. We call its objects finite factorizable sheaves. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility (”factorization”) and finiteness conditions. In Chapter 2 the tensor structure on FS is defined using funct...