# Spahn Segal condition (changes)

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## 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

###### Example

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.

###### Example

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

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

Write $Pre Cat(\mathcal{C})$ 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 $X$such that its core $Core(X)$ is in the image of the inclusion $\mathcal{C} \hookrightarrow Grpd(\mathcal{C})$.

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

## General theory

A directed graph is a presheaf

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

(…)

(description of this

$\array{ Cat &\stackrel{\overset{\tau}{\leftarrow}}{\underoverset{N}{\bottom}{\hookrightarrow}}& PSh(\Delta) \simeq sSet \\ {}^{\mathllap{U}}\downarrow \vdash \uparrow^{\mathrlap{F}} && {}^{\mathllap{j^*}}\downarrow \vdash \uparrow^{\mathrlap{j_!}} \\ Graph \simeq Sh(\Delta_0) &\stackrel{\overset{i^*}{\leftarrow}}{\underoverset{i_*}{\bottom}{\hookrightarrow}}& PSh(\Delta_0) }$

diagram from

(…)

Complete Segal spaces were originally defined in

• Charles Rezk, A model for the homotopy theory of homotopy theory , Trans. Amer. Math. Soc., 353(3), 973-1007 (pdf)

The relation to quasi-categories is discussed in

A survey of the definition and its relation to equivalent definitions is in section 4 of

• Julia Bergner, A survey of $(\infty, 1)$-categories (arXiv).