nLab
semi-Segal space

Contents

Idea

A semi-Segal space is like a Segal space but without specified identities/degeneracies. It is to semicategories as Segal spaces are to categories.

Definition

Let 𝒞=\mathcal{C} = sSet, equipped with the standard model structure on simplicial sets.

Definition

A semi-Segal space is a semi-simplicial object in 𝒞\mathcal{C} such that

  1. it is a fibrant object in the Reedy model structure on 𝒞 Δ + op\mathcal{C}^{\Delta^{op}_+};

  2. it satisfies the Segal conditions be weak equivalences.

Remark

A semi-simplicial object X X_\bullet being Reedy fibrant means that for each nn \in \mathbb{N} the morphisms

X n ( 0,, n) X Δ n \array{ X_{n} \\ \downarrow^{\mathrlap{(\partial_0, \cdots, \partial_{n})}} \\ X^{\partial \Delta^n} }

are fibrations.

Remark

Equivalently this says that it is a semi-simplicial object which satisfies the Segal conditions by homotopy pullbacks. This is just as for Segal spaces, see there for details.

Definition

A complete semi-Segal space is a semi-Segal space X X_\bullet such that the two morphisms

X 1 invX 1 1, 0X 0 X_1^{inv} \hookrightarrow X_1 \stackrel{\partial_1, \partial_0}{\to} X_0

are weak equivalences.

This is the completeness/univalence condition just as for complete Segal spaces.

Definition

A semi-Segal space is quasiunital if (…)

(Harpaz, p. 38)

Proposition

A complete semi-Segal space, def. 2 is quasi-unital, def. 3.

(Harpaz, Cor 4.1.11).

Remark

A morphism of complete semi-Segal spaces f :X Y f_\bullet \colon X_\bullet \to Y_\bullet is quasi-unital if it preserves the weak equivalences, hence if

X 1 inv X 1 f 1 inv f 1 Y 1 inv Y 1. \array{ X_1^{inv} &\hookrightarrow& X_1 \\ \downarrow^{\mathrlap{f_1|_{inv}}} && \downarrow^{\mathrlap{f_1}} \\ Y_1^{inv} &\hookrightarrow & Y_1 } \,.

References

The notion is mentioned in

More details are spelled out in

Revised on November 23, 2012 13:02:01 by Urs Schreiber (82.169.65.155)