Spahn category object

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 $colim\dashv const$.

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

(…)

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 codomain fibration of $\mathcal{C}$ is an (∞,2)-sheaf when restricted to $\mathcal{X}$: its classifying functor $\chi : \mathcal{C}^{op} \to$ (∞,1)Cat preserves (∞,1)-limits when restricted along $\mathcal{X} \hookrightarrow \mathcal{C}$.