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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, for example, atlases with isotropy, boundary and corners, or lack of differentiable structure.

The topology of Kuranishi atlases