Smooth branched $n$-manifolds are a generalization of smooth manifolds which may have ill-defined tangent spaces at certain “branch loci”, where however they are required to have a well defined tangent $n$-plane.
Branched $n$-manifolds arise for instance as quotients of foliations.
1) a collection $\{U_i\}$ of closed subsets of $X$
2) for each $U_i$ a finite collection $\{D_{i j}\}$ of closed subsets of $U_i$
3). for each $i$ a map $\p_i:U_i\to D_i^n$ to a closed $n$-disk of class $C^k$ in $\mathbb{R}^n$
a) $\cup_i D_{i j}=U_i$ and $\cup_i U^\circ_i =X$
b) $p_i |_{D_{i j}}$ is a homeomorphism onto its image which is a closed $C^k$ n-disk relative to $\partial D_i^n$.
c) There is a “cocycle” of diffeomorphisms $\{\alpha_{i^' i}\}$ of class $C^k$ such that $p_{i^'}=\alpha_{i^',i}\circ p_i$ when defined. The domain of $\alpha_{i^' i}$ is $p_i(U_i\cap U_{i^'})$.
Then $X$ is called branched n-manifold of class $C^k$.
This appears as Williams, def. 1.0 ns.
If $X$ satisfies this set of axioms with b) replaced by
$b_{ns})$ $p_i |_{D_{i j}}$ is a homeomorphisms onto $D_i^n$
$X$ is called nonsingular branched $n$-manifold of class $C^k$.
Discussion relating to orbifolds is in