Ricci flow as functor

TL;DR

This paper proposes a categorical framework for Ricci flow, modeling it as a sequence of functors connecting Riemannian metrics to geometric decompositions, and introduces a stratification of the differentiable stack of metrics.

Contribution

It introduces a novel categorical perspective on Ricci flow, utilizing functors and differentiable stacks to formalize the flow's structure.

Findings

Ricci flow can be modeled as a sequence of functors.

The framework stratifies the stack of Riemannian metrics.

A rigorous foundation using differentiable stacks on Banach manifolds.

Abstract

In this note we attempt to propose a categorical framework for the Ricci flow, treating it as a sequence of functors connecting the stack of Riemannian metrics to the category of geometric decompositions via singular flow spacetimes. To rigorize the domain of the flow, we adapt the definition of differentiable stacks to the site of Banach manifolds. We demonstrate that the Ricci flow defines a stratification of this stack.