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A Segal condition is a (condition defining a) relation on a functor. In motivating cases these relations describe how a value $F(A)$ of the functor $F$ may be constructed (up to equivalence) by values of subobjects- or truncated versions of $A$.
A groupoid object in $\mathcal{C}$ is a simplicial object in an (β,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 (β,1)-functor $X$ exhibits an (β,1)-pullback
Write $Grpd(\mathcal{C})$ for the (β,1)-category of groupoid objects in $\mathcal{C}$, the full sub-(β,1)-category of simplicial objects on the groupoid objects.
An internal precategory $X$ in an $(\infty,1)$-topos $\mathcal{C}$ is a simplicial object in an (β,1)-category
such that it satifies the Segal condition, hence such that for all $n \in \mathbb{N}$ $X$ exhibits $X([n])$ as the (β,1)-limit / iterated (β,1)-pullback
Write $\mathrm{PreCatPre}\mathrm{Cat}(\mathrm{\pi \x9d\x92\x9e})$ PreCat(\mathcal{C}) Pre Cat(\mathcal{C}) for the $(\infty,1)$-category of internal pre-categories in $\mathcal{C}$, the full sub-(β,1)-category of the simplicial objects on the internal precategories.
An internal category in an $(\infty,1)$-topos $\mathcal{C}$ is an internal pre-category $X$ , such def. that \ref{ itsPre 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?}.
This is called a complete Segal space object in (Lurie, def. 1.2.10).
A directed graph is a presheaf
Complete Segal spaces were originally defined in
The relation to quasi-categories is discussed in
A survey of the definition and its relation to equivalent definitions is in section 4 of
See also pages 29 to 31 of
Jacob Lurie, On the Classification of Topological Field Theories
ncafΓ©, univalence is a Segal condition