  Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the
purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic
curves that cannot be regularized by geometric methods. Starting from the same
core idea (patching local finite dimensional reductions) we develop a theory of
topological Kuranishi atlases and cobordisms that transparently resolves
algebraic and topological challenges in this virtual regularization approach.
It applies to any Kuranishi-type setting, e.g. atlases with isotropy, boundary
and corners, or lack of differentiable structure.
