Let $\mathcal{C}$ be an $(\infty,1)$-category. A *groupoid object in $\mathcal{C}$* is defined to be a simplicial object satifying the *groupoidal [[Segal condition]]* (we could call this condition also ''horn partition condition''. We will see, that a *precategory object* reps. a *category object* will be defined to satisfy a weaker ''pair horn condition'' resp. a ''inner horn partition condition''.) In the left half of the below diagram $const$ is a full and faithful functor. If $\mathcal{C}$ has small colimits $const$ has a left adjoint $colim\dashv const$. $$ \mathcal{C} \stackrel{\overset{colim}{\leftarrow}}{\underset{const}{\hookrightarrow}} Grpd(\mathcal{C}) \stackrel{\hookrightarrow}{\underset{Core}{\leftarrow}} PreCat(\mathcal{C}) $$ (...) +-- {: .num_defn} ###### Definition (choice of internal groupoids) Let $\mathcal{C}$ be a presentable $(\infty,1)$-category. A *choice of internal groupoids* is a choice of a presentable full sub $(\infty,1)$-category $Disc\mathcal{X}\hookrightarrow \mathcal{C}$ satisfying * $Disc$ has a left- and a right adjoint $(\Pi\dashv Disc \dashv \Gamma)$ * For all $f:Y\to X$ with $Y\in \mathcal{C}$, $X\in\mathcal{X}$ base change $f^*: \mathcal{X}/X\to \mathcal{C}/Y$ preserves colimits. * The [[nLab:codomain fibration]] of $\mathcal{C}$ is an [[nLab:(∞,2)-sheaf]] when restricted to $\mathcal{X}$: its [[nLab:(∞,1)-Grothendieck construction|classifying functor]] $\chi : \mathcal{C}^{op} \to $ [[nLab:(∞,1)Cat]] preserves [[nLab:(∞,1)-limits]] when restricted along $\mathcal{X} \hookrightarrow \mathcal{C}$. =--