Spahn Segal condition (Rev #4)

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([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 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

References

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

André Joyal, Myles Tierney, Quasi-categories vs Segal spaces (arXiv:0607820)

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).