The higher Segal degree of a partial groupoid equals the Helly number of the closure space of a characteristic action, and the method gives explicit degrees for punctured Weyl groups.
The decomposition space perspective
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abstract
This paper provides an introduction to decomposition spaces and 2-Segal spaces, unifying the two perspectives. We begin by defining decomposition spaces using the active-inert factorization system on the simplicial category, and show their equivalence to 2-Segal spaces. Key results include the path space criterion, which characterizes decomposition spaces in terms of their upper and lower d\'ecalages, and the edgewise subdivision criterion. We also introduce free decomposition spaces arising from outer face complexes, providing a rich source of examples. Formal prerequisites are minimal -- readers should have a working knowledge of simplicial methods and basic category theory.
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Higher Segal spaces and partial groups
The higher Segal degree of a partial groupoid equals the Helly number of the closure space of a characteristic action, and the method gives explicit degrees for punctured Weyl groups.