## 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 $Pre Cat(\mathcal{C})$ 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 $X$such that its [[nLab:core]] $Core(X)$ is in the image of the inclusion $\mathcal{C} \hookrightarrow Grpd(\mathcal{C})$. =-- This is called a _[[nLab:complete Segal space]] object_ in ([Lurie, def. 1.2.10](#Lurie)). ## General theory A directed graph is a presheaf $$D:\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}\to Set$$ ## References Complete Segal spaces were originally defined in * [[nLab:Charles Rezk]], _A model for the homotopy theory of homotopy theory_ , Trans. Amer. Math. Soc., 353(3), 973-1007 ([pdf](http://www.math.uiuc.edu/~rezk/rezk-ho-models-final-changes.pdf)) {#Rezk} The relation to [[nLab:quasi-categories]] is discussed in * [[nLab:André Joyal]], [[nLab:Myles Tierney]], _Quasi-categories vs Segal spaces_ ([arXiv:0607820](http://arxiv.org/abs/math/0607820)) A survey of the definition and its relation to equivalent definitions is in section 4 of * [[nLab:Julia Bergner]], _A survey of $(\infty, 1)$-categories_ ([arXiv](http://arxiv.org/abs/math.AT/0610239)). See also pages 29 to 31 of * [[nLab:Jacob Lurie]], _[[nLab:On the Classification of Topological Field Theories]]_ * [nforum, Segal condition](http://nforum.mathforge.org/discussion/3740/segal-condition/) * ncafé, univalence is a Segal condition