The diagonal of cellular spaces and effective algebro-homotopical constructions

TL;DR

This survey explores homotopy coherent enhancements of the coalgebra structure on cellular chains, emphasizing effective constructions that encode geometric and combinatorial data relevant to multiple mathematical and physical fields.

Contribution

It provides a comprehensive overview of effective methods to construct homotopy coherent coalgebra structures on cellular chains, linking algebraic models to geometric and combinatorial insights.

Findings

Effective constructions encode geometric and combinatorial information.

Homotopy coherent coalgebra structures control rational and integral homotopy theories.

Applications span deformation theory, higher categories, and physics.

Abstract

In this survey article we discuss certain homotopy coherent enhancements of the coalgebra structure on cellular chains defined by an approximation to the diagonal. Over the rational numbers, $C_\infty$-coalgebra structures control the $\mathbb Q$-complete homotopy theory of spaces, and over the integers, $E_\infty$-coalgebras provide an appropriate setting to model the full homotopy category. Effective constructions of these structures, the focus of this work, carry geometric and combinatorial information which has found applications in various fields including deformation theory, higher category theory, and condensed matter physics.