Given a pushout diagram of nice topological spaces (such as CW complexes), does the infinity groupoid functor $\Pi(-)$ commute with the pushout? More precisely, does the pushout diagram get sent to a homotopy pushout?
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3$\begingroup$ No: squares of top spaces that are sent to pushouts by Π are called homotopy pushouts. Not all pushouts are homotopy pushouts, no matter how nice the spaces are... $\endgroup$– Marc HoyoisCommented Dec 23, 2016 at 23:54
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$\begingroup$ A sufficient condition is that both maps are open embeddings, or that one of them is a relative CW complex. As far as I know there is no simple characterization of homotopy pushouts. $\endgroup$– Marc HoyoisCommented Dec 24, 2016 at 0:00
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$\begingroup$ Okay cool, this works for the example I was thinking about: attaching a 2-cell to a torus by the inclusion of a circle homotopic whose attaching map is homotopic to a point. This would give the fundamental groupoid $\Pi(X) \cong \Pi(S^2 \vee (S^1 \times S^1))$ $\endgroup$– 54321userCommented Dec 24, 2016 at 0:03
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3$\begingroup$ You can even cook up plenty of counterexamples using just discrete spaces with finitely many points. Probably the simplest example is the pushout of the map from a two-point space to a one-point space along itself. The pushout is again just a one-point space, while the homotopy pushout is $S^1$: you get it by taking two copies of the one-point space and identifying them in two ways, resulting in two arcs. $\endgroup$– Tobias FritzCommented Dec 24, 2016 at 0:03
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