André-Quillen Cohomology and $k$-invariants of simplicial categories

TL;DR

This work reframes the Dwyer–Kan–Smith cohomology of a simplicial category as spectral André-Quillen cohomology and uses cube-based decompositions to make the $k$-invariants explicit. It proves that any fibrant simplicial category $\mathscr{X}$ is the homotopy colimit of its non-degenerate cubes, yielding a concrete cochain complex for $H^*_Q(\mathscr{X},\pi_n\mathscr{X})$ and a tangible obstruction $\beta$ to lifting along Postnikov stages. By examining the boundary of cubes, it derives explicit higher-order invariants such as differentials in spectral sequences and Toda/Whitehead-type operations, showing how $k$-invariants encode rich higher structure. The paper also develops obstruction theory for upgrading ${\mathbb E}_1$- to ${\mathbb E}_2$-algebras via cube-based colimit decompositions, connecting operadic realizations with cubical models.Overall, the cube-centric approach provides a hands-on toolkit for computing and interpreting higher homotopical data in simplicial categories and related algebraic structures.

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 André-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$ take value. Given a map of simplicial categories $φ:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of $\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to study the obstruction to lifting $φ$ 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 $\mathcal{X}$.

Theorems & Definitions (90)

• Theorem 1.1
• proof
• Definition 1.2
• Remark 1.5
• Definition 1.6
• Example 1.7
• Definition 1.8
• Remark 1.10
• Proposition 1.11
• proof : Sketch of Proof