Segal condition (changes)

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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 π’ž\mathcal{C} is a simplicial object in an (∞,1)-category

X:Ξ” opβ†’π’ž X : \Delta^{op} \to \mathcal{C}

that satisfies the groupoidal Segal conditions, meaning that for all nβˆˆβ„•n \in \mathbb{N} and all partitions [n]≃SβˆͺSβ€²[n] \simeq S \cup S' that share a single element S∩Sβ€²={s}S \cap S' = \{s\}, the (∞,1)-functor XX exhibits an (∞,1)-pullback

X([n])≃X(S)Γ— X(S∩Sβ€²)X(Sβ€²). X([n]) \simeq X(S) \times_{X(S \cap S')} X(S') \,.

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


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

X:Ξ” opβ†’π’ž X : \Delta^{op} \to \mathcal{C}

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

X([n])≃X({0,1})Γ— X([0])β‹―Γ— X[0]X({nβˆ’1,n}). X([n]) \simeq X(\{0,1\}) \times_{X([0])} \cdots \times_{X[0]} X(\{n-1,n\}) \,.

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


An internal category in an (∞,1)(\infty,1)-topos π’ž\mathcal{C} is an internal pre-category XXsuch that its core Core(X)Core(X) is in the image of the inclusion π’žβ†ͺGrpd(π’ž)\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←d 1←d 00} opβ†’SetD:\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}\to Set


(description of this

Cat β†ͺNβŠ₯←τ PSh(Ξ”)≃sSet Uβ†“βŠ’β†‘ F j *β†“βŠ’β†‘ j ! Graph≃Sh(Ξ” 0) β†ͺi *βŠ₯←i * PSh(Ξ” 0) \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

See also pages 29 to 31 of

Last revised on December 9, 2012 at 22:23:49. See the history of this page for a list of all contributions to it.