Segal condition (Rev #3)

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 (∞,1)-category

$X : \Delta^{op} \to \mathcal{C}$

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

$X([n]) \simeq X(S) \times_{X(S \cap S')} X(S')
\,.$

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.

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

$X : \Delta^{op} \to \mathcal{C}$

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

$X([n]) \simeq X(\{0,1\}) \times_{X([0])} \cdots \times_{X[0]} X(\{n-1,n\})
\,.$

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.

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?}.

This is called a *complete Segal space object* in (Lurie, def. 1.2.10).

A directed graph is a presheaf

$D:\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}\to Set$