Semi-coarse Spaces: Fundamental Groupoid and the van Kampen Theorem

Abstract: In algebraic topology, the fundamental groupoid is a classical homotopy invariant which is defined using continuous maps from the closed interval to a topological space. In this paper, we construct a semi-coarse version of this invariant, using as paths a finite sequences of maps from $\mathbb{Z}_1$ to a semi-coarse space, connecting their tails through semi-coarse homotopy. In contrast to semi-coarse homotopy groups, this groupoid is not necessarily trivial for coarse spaces, and, unlike coarse homotopy, it is well-defined for general semi-coarse spaces. In addition, we show that the semi-coarse fundamental groupoid which we introduce admits a version of the Van Kampen Theorem.