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 $colim\dashv const$. $$ \mathcal{C} \stackrel{\overset{colim}{\leftarrow}}{\underset{const}{\hookrightarrow}} Grpd(\mathcal{C}) \stackrel{\hookrightarrow}{\underset{Core}{\leftarrow}} PreCat(\mathcal{C}) $$