nLab semi-Segal space

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

Definition

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

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

2. it satisfies the Segal conditions be weak equivalences.

Remark

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

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

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 $X_\bullet$ such that the two morphisms

$$X_1^{inv} \hookrightarrow X_1 \stackrel{\partial_1, \partial_0}{\to} X_0$$

are weak equivalences.

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

Definition

A semi-Segal space is quasiunital if (…)

(Harpaz 12, p. 38)

Proposition

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

(Harpaz 12, Cor 4.1.11).

Remark

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

$$\array{ X_1^{inv} &\hookrightarrow& X_1 \\ \downarrow^{\mathrlap{f_1|_{inv}}} && \downarrow^{\mathrlap{f_1}} \\ Y_1^{inv} &\hookrightarrow & Y_1 } \,.$$

Related concepts

• semi-category

• semi-simplicial object

References

The notion is mentioned in

• Jacob Lurie, On the Classification of Topological Field Theories

More details are spelled out in

• Yonatan Harpaz, Quasi-unital $\infty$-Categories (arXiv:1210.0212)

See also

• Hiro Lee Tanaka, Functors (between $\infty$-categories) that aren’t strictly unital (arXiv:1606.05669)

• Wolfgang Steimle, Degeneracies in quasi-categories, arxiv:1702.08696