Spahn category object (Rev #1)

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 $colim\dashv const$.