Moduli stack of stable curves from a stratified homotopy viewpoint

TL;DR

This paper demonstrates that a category of stable curves accurately captures the stratified homotopy type of the moduli stack of stable curves, providing a new perspective on its topological and sheaf-theoretic structure.

Contribution

It extends previous rational homology results to show the category encodes the full stratified homotopy type of the moduli stack.

Findings

The category classifies constructible sheaves via exodromy.

It strengthens the connection between stable curves and the moduli stack.

Provides a stratified homotopy equivalence for the moduli stack.

Abstract

In 1984, Charney and Lee defined a category of stable curves and exhibited a rational homology equivalence from its geometric realisation to (the analytification of) the moduli stack of stable curves, also known as the Deligne-Mumford-Knudsen compactification. We strengthen this result by showing that, in fact, this category captures the stratified homotopy type of the moduli stack. In particular, it classifies constructible sheaves via an exodromy equivalence.