Segal condition (Rev #3)


A Segal condition is a (condition defining a) relation on a functor. In motivating cases these relations describe how a value F(A)F(A) of the functor FF may be constructed (up to equivalence) by values of subobjects- or truncated versions of AA.



A groupoid object in 𝒞\mathcal{C} is a simplicial object in an (∞,1)-category

X:Δ op𝒞 X : \Delta^{op} \to \mathcal{C}

that satisfies the groupoidal Segal conditions, meaning that for all nn \in \mathbb{N} and all partitions [n]SS[n] \simeq S \cup S' that share a single element SS={s}S \cap S' = \{s\}, the (∞,1)-functor XX exhibits an (∞,1)-pullback

X([n])X(S)× X(SS)X(S). X([n]) \simeq X(S) \times_{X(S \cap S')} X(S') \,.

Write Grpd(𝒞)Grpd(\mathcal{C}) for the (∞,1)-category of groupoid objects in 𝒞\mathcal{C}, the full sub-(∞,1)-category of simplicial objects on the groupoid objects.


An internal precategory XX in an (,1)(\infty,1)-topos 𝒞\mathcal{C} is a simplicial object in an (∞,1)-category

X:Δ op𝒞 X : \Delta^{op} \to \mathcal{C}

such that it satifies the Segal condition, hence such that for all nn \in \mathbb{N} XX exhibits X([n])X([n]) as the (∞,1)-limit / iterated (∞,1)-pullback

X([n])X({0,1})× X([0])× X[0]X({n1,n}). X([n]) \simeq X(\{0,1\}) \times_{X([0])} \cdots \times_{X[0]} X(\{n-1,n\}) \,.

Write PreCat(𝒞)PreCat(\mathcal{C}) for the (,1)(\infty,1)-category of internal pre-categories in 𝒞\mathcal{C}, the full sub-(∞,1)-category of the simplicial objects on the internal precategories.


An internal category in an (,1)(\infty,1)-topos 𝒞\mathcal{C} is an internal pre-category XX, def. \ref{Pre Category Object?} such that its core Core(X)Core(X) is in the image of the inclusion 𝒞Grpd(𝒞)\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:{1d 1d 00} opSetD:\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}\to Set

Revision on November 2, 2012 at 01:48:42 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.