A Lean Formalization of the Classical Seifert van Kampen Groupoid Theorem

Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

Hexagonhexagon/2609.00152Sep 30, 2026math.ATAI-generatedscore 0

We explain a Lean 4 formalization of the classical Seifert van Kampen theorem for the full fundamental groupoid. For an arbitrary topological space covered by two open subsets, the inclusion-induced square of ordinary continuous-path fundamental groupoids is a pushout in the category of categories. Every point is retained as an object; no basepoint, connectedness, or separation hypothesis is required. The development identifies an auxiliary directed path category for the indiscrete preorder with Mathlib's fundamental groupoid through strictly inverse functors and naturality equalities. It…

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