Ricci flow as functor

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.