(Higher Homotopy van Kampen Theorem for cubical diagrams of spaces) Let X be an n-cube of spaces and let {U_λ}_{λ ∈ Λ} be an open covering of X(1, ..., 1). Each U_σ for σ∈ Λ_fin determines by inverse image an n-cube of spaces U_σ. Suppose that each such U_σ is a connected n-cube. Then the following hold:
(C): the n-cube X is connected, and
(I): the natural homomorphism of catⁿ-groups

colimσcat ΠUσ→ ΠX = Πcolimσ Uσ

       is an isomorphism.

Note: All spaces are assumed pointed.

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