## Idea 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 $A$. ## Examples +-- {: .num_defn #GroupoidObject} ###### Example A _[[nLab:groupoid object in an (∞,1)-category|groupoid object]]_ in $\mathcal{C}$ is a [[nLab:simplicial object in an (∞,1)-category]] $$ X : \Delta^{op} \to \mathcal{C} $$ that satisfies the groupoidal _[[nLab: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 [[nLab:(∞,1)-functor]] $X$ exhibits an [[nLab:(∞,1)-pullback]] $$ X([n]) \simeq X(S) \times_{X(S \cap S')} X(S') \,. $$ Write $Grpd(\mathcal{C})$ for the [[nLab:(∞,1)-category]] of groupoid objects in $\mathcal{C}$, the [[nLab:full sub-(∞,1)-category]] of [[nLab:simplicial object in an (∞,1)-category|simplicial objects]] on the groupoid objects. =-- +-- {: .num_defn #PreCategoryObject} ###### Example An **internal precategory** $X$ in an $(\infty,1)$-topos $\mathcal{C}$ is a [[nLab:simplicial object in an (∞,1)-category]] $$ X : \Delta^{op} \to \mathcal{C} $$ such that it satifies the _[[nLab:Segal condition]]_, hence such that for all $n \in \mathbb{N}$ $X$ exhibits $X([n])$ as the [[nLab:(∞,1)-limit]] / iterated [[nLab:(∞,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 [[nLab:full sub-(∞,1)-category]] of the [[nLab:simplicial objects]] on the internal precategories. =-- +-- {: .num_defn #CategoryObject} ###### Definition An **internal category** in an $(\infty,1)$-topos $\mathcal{C}$ is an internal pre-category $X$, def. \ref{PreCategoryObject} such that its [[nLab:core]] $Core(X)$ is in the image of the inclusion $\mathcal{C} \hookrightarrow Grpd(\mathcal{C})$, prop. \ref{EmbeddingOfConstantGroupoidObjects}. =-- This is called a _[[nLab:complete Segal space]] object_ in ([Lurie, def. 1.2.10](#Lurie)). ## General theory A directed graph is a presheaf $$D:\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}\to Set$$