Spahn Segal condition (Rev #4, changes)

Showing changes from revision #3 to #4: Added | Removed | Changed

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  PreCat(\mathcal{C}) 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 def. that \ref{ itsPre 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).

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

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).