Some remarks on groupoids and small categories

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This unpublished note contains some materials taken from my old study note on groupoids and small categories. It contains a proof for the fact that any groupoid is a group bundle over an equivalence relation. Moreover, the action of a category G on a category H as well as the resulting semi-direct product category H×αG will be defined (when either G is a groupoid or H⁽⁰⁾ = G⁽⁰⁾). If both G and H are groupoids, then H×αG is also a groupoid. The reason of producing this note is for people who want to check some details in a recent work of Li.