Spahn Segal condition (Rev #3)

Example

A groupoid object in $\mathcal{C}$ is a simplicial object in an (∞,1)-category

that satisfies the groupoidal Segal conditions, meaning that for all $n \in \mathbb{N}$ and all partitions $[n] \simeq S \cup S'$ that share a single element $S \cap S' = \{s\}$, the (∞,1)-functor $X$ exhibits an (∞,1)-pullback

Write $Grpd(\mathcal{C})$ for the (∞,1)-category of groupoid objects in $\mathcal{C}$, the full sub-(∞,1)-category of simplicial objects on the groupoid objects.

Example

An internal precategory $X$ in an $(\infty,1)$-topos $\mathcal{C}$ is a simplicial object in an (∞,1)-category

such that it satifies the Segal condition, hence such that for all $n \in \mathbb{N}$ $X$ exhibits $X([n])$ as the (∞,1)-limit / iterated (∞,1)-pullback

Write $PreCat(\mathcal{C})$ for the $(\infty,1)$-category of internal pre-categories in $\mathcal{C}$, the full sub-(∞,1)-category of the simplicial objects on the internal precategories.

Definition

An internal category in an $(\infty,1)$-topos $\mathcal{C}$ is an internal pre-category $X$, def. \ref{Pre Category Object?} such that its core $Core(X)$ is in the image of the inclusion $\mathcal{C} \hookrightarrow Grpd(\mathcal{C})$, prop. \ref{Embedding Of Constant Groupoid Objects?}.