2,455 research outputs found
Eilenberg–MacLane mapping algebras and higher distributivity up to homotopy
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 -distributivity, we provide a new construction of a
derivation of degree of the mod Steenrod algebra.Comment: v3: Minor changes. Final versio
Moduli of -algebras
We describe a homotopy-theoretic approach to the moduli of -algebras of
Blanc-Dwyer-Goerss using the -category of
product-preserving presheaves on finite-wedges of positive-dimensional spheres,
reproving all of their results in this new setting
Cellular Cohomology in Homotopy Type Theory
We present a development of cellular cohomology in homotopy type theory.
Cohomology associates to each space a sequence of abelian groups capturing part
of its structure, and has the advantage over homotopy groups in that these
abelian groups of many common spaces are easier to compute. Cellular cohomology
is a special kind of cohomology designed for cell complexes: these are built in
stages by attaching spheres of progressively higher dimension, and cellular
cohomology defines the groups out of the combinatorial description of how
spheres are attached. Our main result is that for finite cell complexes, a wide
class of cohomology theories (including the ones defined through
Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This
result was formalized in the Agda proof assistant
Can one classify finite Postnikov pieces?
We compare the classical approach of constructing finite Postnikov systems by
k-invariants and the global approach of Dwyer, Kan, and Smith. We concentrate
on the case of 3-stage Postnikov pieces and provide examples where a
classification is feasible. In general though the computational difficulty of
the global approach is equivalent to that of the classical one.Comment: 13 page
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