Effective generalized Seifert-Van Kampen: how to calculate ΩX

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Suppose X is a 1-connected simplicial set with finitely many nondegenerate simplices. We give an effective algorithm to calculate a simplicial set with the n-type of the loop space ΩX. Iterating gives an algorithm to calculate the πᵢ(X), different from the algorithms already known due to E. Brown and Kan-Curtis. The method is an effective version of the generalized Seifert-Van Kampen theorem of alg-geom/9704006. This can be viewed as a Van Kampen statement concerning the loop space ΩX with its delooping structure. We use Segal's delooping machinery but at the end we speculate on extensions to other delooping machinery.