Homotopy coherent structures

TL;DR

This paper introduces homotopy coherent category theory, explaining how it captures higher homotopical information to restore functoriality in diagrams that are only commutative up to homotopy, with applications to algebraic topology.

Contribution

It provides a comprehensive introduction to homotopy coherent category theory, including the homotopy coherent nerve and its applications to algebraic structures.

Findings

Homotopy coherent diagrams can be characterized via the homotopy coherent nerve.

Diagrams in homotopy coherent nerves are automatically homotopy coherent.

Homotopy coherent adjunctions relate to classical cobar and bar resolutions.

Abstract

Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such "up to homotopy" settings. These notes provide a three-part introduction to homotopy coherent category theory. The first part surveys the classical theory of homotopy coherent diagrams of topological spaces. The second part introduces the homotopy coherent nerve and connects it to the free resolutions used to define homotopy coherent diagrams. This connection explains why diagrams valued in homotopy coherent nerves or more general $\infty$-categories are automatically homotopy coherent. The final part ventures into homotopy coherent algebra, connecting the newly discovered notion of homotopy coherent adjunction to the classical cobar and bar resolutions for homotopy coherent algebras.