The realization space of an unstable coalgebra

TL;DR

This paper investigates the realization problem for unstable coalgebras over the Steenrod algebra, describing the moduli space of realizations through cohomological invariants by comparing homotopy theories of cosimplicial structures.

Contribution

It provides a detailed description of the realization space of unstable coalgebras using cohomological invariants and compares homotopy theories of cosimplicial objects.

Findings

Characterization of the realization space via cohomological invariants

Comparison of homotopy theories of cosimplicial unstable coalgebras and spaces

Description of the moduli space of realizations

Abstract

Unstable coalgebras over the Steenrod algebra form a natural target category for singular homology with prime field coefficients. The realization problem asks whether an unstable coalgebra is isomorphic to the homology of a topological space. We study the moduli space of such realizations and give a description of this in terms of cohomological invariants of the unstable coalgebra. This is accomplished by a thorough comparative study of the homotopy theories of cosimplicial unstable coalgebras and of cosimplicial spaces.