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Eilenberg-MacLane spaces, that classify the singular cohomology groups of topological spaces, admit natural constructions in the framework of simplicial sets. The existence of similar spaces for the intersection cohomology groups of a stratified space is a long-standing open problem asked by M. Goresky and R. MacPherson. One feature of this work is a construction of such simplicial sets. From works of R. MacPherson, J. Lurie and others, it is now commonly accepted that the simplicial set of singular simplices associated to a topological space has to be replaced by the simplicial set of singular simplices that respect the stratification. This is encoded in the category of simplicial sets over the nerve of the poset of strata. For each perversity, we define a functor from it, with values in the category of cochain complexes over a commutative ring. This construction is based upon a simplicial blow up and the associated cohomology is the intersection cohomology as it was defined by M. Goresky and R. MacPherson in terms of hypercohomology of Deligness sheaves. This functor admits an adjoint and we use it to get classifying spaces for intersection cohomology. Natural intersection cohomology operations are understood in terms of intersection cohomology of these classifying spaces. As in the classical case, they form infinite loop spaces. In the last section, we examine the depth one case of stratified spaces with only one singular stratum. We observe that the classifying spaces are Joyals projective cones over classical Eilenberg-MacLane spaces. We establish some of their properties and conjecture that, for Goresky and MacPherson perversities, all intersection cohomology operations are induced by classical ones.
We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the Dwyer-Kan-Smith cohomological obstructions to rectifying homotopy-commutative diagrams.
There are two main approaches to the problem of realizing a $Pi$-algebra (a graded group $Lambda$ equipped with an action of the primary homotopy operations) as the homotopy groups of a space $X$. Both involve trying to realize an algebraic free simp
This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given oper
Let $X^{2n}subseteq mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}pi{_*}mathbb{Q}$, where $pi$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(mat
Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.