multfolds

Multfolds are a generalization of smooth manifolds designed to model spaces that locally look like several overlapping sheets of smooth manifolds rather than a single chart. They arise in contexts where moduli spaces or geometric structures exhibit branching, multiplicity, or nontrivial sheet structure that cannot be captured by ordinary manifolds or even orbifolds.

From a formal standpoint, a multfold can be described as a Lie groupoid presentation G = (Obj, Mor) where Obj is a disjoint union of smooth manifolds called branches, and Mor consists of smooth maps encoding identifications between branches. The orbit space |G|, obtained by identifying points related by Mor, is the underlying topological space of the multfold. This formulation captures both the local branching (several sheets over a base point) and the global identifications between sheets.

Examples include ordinary smooth manifolds (trivial branching with a single branch and identity arrows) and simple branched spaces where a base manifold is equipped with multiple local sheets that may coincide along sets of positive codimension. More elaborate multfolds appear in polyfold theory and the study of moduli spaces of pseudoholomorphic curves, where they provide a framework for achieving a transversality in a setting with branching and multiple covers.

Relation to existing concepts: multfolds extend manifolds and orbifolds by incorporating branching and sheet multiplicities beyond quotient constructions by finite group actions. They are related to branched manifolds, stacks, and groupoid approaches to geometry.

See also: manifolds, orbifolds, branched manifolds, polyfolds, groupoids.