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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.
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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 simplicial resolution $G_bullet$ of $Lambda$ by a simplicial space $W_bullet$ and proceed by induction on the simplicial dimension. The first provides a sequence of Andr{e}-Quillen cohomology classes in $H_{AQ}^{n+2}(Lambda;Omega^{n}Lambda)$ for $n geq 1$ as obstructions to the existence of successive Postnikov sections for $W_bullet$ by work of Dwyer, Kan and Stover. The second gives a sequence of geometrically defined higher homotopy operations as the obstructions by earlier work of Blanc; these were identified with the obstruction theory of Dwyer, Kan and Smith in earlier work of the current authors. There are also (algebraic and geometric) obstructions for distinguishing between different realizations of $Lambda$. In this paper we 1) provide an explicit construction of the cocycles representing the cohomology obstructions; 2) provide a similar explicit construction of certain minimal values of the higher homotopy operations (which reduce to long Toda brackets), and 3) show that these two constructions correspond under an evident map.
In this paper we provide an explicit general construction of higher homotopy operations in model categories, which include classical examples such as (long) Toda brackets and (iterated) Massey products, but also cover unpointed operations not usually considered in this context. We show how such operations, thought of as obstructions to rectifying a homotopy-commutative diagram, can be defined in terms of a double induction, yielding intermediate obstructions as well.
We develop foundations for the category theory of $infty$-categories parametrized by a base $infty$-category. Our main contribution is a theory of indexed homotopy limits and colimits, which specializes to a theory of $G$-colimits for $G$ a finite group when the base is chosen to be the orbit category of $G$. We apply this theory to show that the $G$-$infty$-category of $G$-spaces is freely generated under $G$-colimits by the contractible $G$-space, thereby affirming a conjecture of Mike Hill.
We discuss an approach to the emph{covering} and emph{vanishing} theorems for the comparison map from bounded cohomology to singular cohomology, based on the observation that the comparison map is the coassembly map for bounded cohomology.
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.
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