To incorporate ideas from homotopy theory into areas such as topological data analysis and topological lattice field theory, it is crucial to have effective constructions of concepts defined only transcendentally. During this talk,
we will present a concrete construction of a structure on cochains that, after
Mandell, encodes the entire homotopy type of a space in its quasi-isomorphism
type. Additionally, we will examine applications of this ideas to the aforementioned areas as well as to knot theory, higher category theory, and toric geometry.
