Definition 4.37.2. Let $\mathcal{C}$ be a category. Suppose that $F : \mathcal{C}^{opp} \to \textit{Groupoids}$ is a functor to the $2$-category of groupoids. We will write $p_ F : \mathcal{S}_ F \to \mathcal{C}$ for the category fibred in groupoids constructed in Example 4.37.1. A *split category fibred in groupoids* is a category fibred in groupoids isomorphic (!) over $\mathcal{C}$ to one of these categories *$\mathcal{S}_ F$*.

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