The center of the Yoshida algebra of a finite groupoid is isomorphic to the center of its groupoid algebra, and the crossed Burnside ring of the groupoid surjects onto that center.
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The crossed Burnside ring of a finite groupoid decomposes as a product of the crossed Burnside rings of its isotropy groups.
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Yoshida algebra for groupoids
The center of the Yoshida algebra of a finite groupoid is isomorphic to the center of its groupoid algebra, and the crossed Burnside ring of the groupoid surjects onto that center.
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Crossed Burnside rings for groupoids
The crossed Burnside ring of a finite groupoid decomposes as a product of the crossed Burnside rings of its isotropy groups.