Abstract:
We explore the concept of conjugation between subgroupoids, providing several characterizations of the conjugacy relation (Theorem A in §1.2). We show that two finite groupoid-sets, over a locally strongly finite groupoid, are isomorphic, if and only if, they have the same number of fixed points with respect to any subgroupoid with a single object (Theorem B in §1.2).
Lastly, we examine the ghost map of a finite groupoid and the idempotents elements of its Burnside algebra. The exposition includes an Appendix where we gather
the main general technical notions that are needed along the paper.
Key Words: The monoidal category of groupoid-bisets,
conjugation between subgroupoids, Burnside Theorem, Burnside ring of
finite groupoids, table of marks, the ghost map and the
idempotents, Laplaza categories.
2010 Mathematics Subject Classification: Primary
18B40, 20L05, 19A22; Secondary 20C15, 20D05.
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