Contents

# 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 $\mathcal{C} =$ 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

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

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

###### Definition

A semi-Segal space is quasiunital if (…)

###### Proposition

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

###### 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

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

The notion is mentioned in

More details are spelled out in