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A groupoid is a category where all morphisms are invertible. This notion can also be seen as an extension of the notion of group. A motivating example is the fundamental groupoid of a topological space with respect to several base points, compared to the usual fundamental group.

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Does the $\infty$-groupoid functor $\Pi$ commute with pushouts of nice spaces?

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 …
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