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Register a new userA constructive approach to higher homotopy operations
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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.
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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.
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.
Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sense. As an application of $2$-distributivity, we provide a new construction of a derivation of degree $-2$ of the mod $2$ Steenrod algebra.
This is an introduction to Homotopy Type Theory and Univalent Foundations for philosophers, written as a chapter for the book Categories for the Working Philosopher (ed. Elaine Landry)
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a higher homotopy structure for $X$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $X$ up to weak equivalence. It can also be used to distinguish between different maps $f$ from $X$ to $Y$ which induce the same morphism on homotopy groups $f_*$ from $pi_* X$ to $pi_* Y$.
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