Andr\'{e}-Quillen Cohomology and $k$-invariants of simplicial categories

TL;DR

This paper develops a cochain complex framework for Andre9-Quillen cohomology of simplicial categories, linking it to $k$-invariants and providing tools to analyze higher homotopy invariants.

Contribution

It introduces a new cochain complex for Andre9-Quillen cohomology of simplicial categories and relates it to $k$-invariants and obstruction theory.

Findings

Explicit description of obstructions for lifting maps between Postnikov sections.

Recovery of higher homotopy invariants from boundary cube obstructions.

Connection between cohomology, $k$-invariants, and homotopy invariants.

Abstract

Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the Andr\'{e}-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$ take value. Given a map of simplicial categories $\phi:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of $\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to study the obstruction to lifting $\phi$ to $P^{(n)}\mathcal{X}$. In particular, an explicit description of this obstruction for the boundary of a cube can be used to recover various higher homotopy invariants of $\mathcal{X}$.