# 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 $𝒞=$ sSet, equipped with the standard model structure on simplicial sets.

###### Definition

A semi-Segal space is a semi-simplicial object in $𝒞$ such that

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

2. it satisfies the Segal conditions be weak equivalences.

###### Remark

A semi-simplicial object ${X}_{•}$ being Reedy fibrant means that for each $n\in ℕ$ the morphisms

$\begin{array}{c}{X}_{n}\\ {↓}^{\left({\partial }_{0},\cdots ,{\partial }_{n}\right)}\\ {X}^{\partial {\Delta }^{n}}\end{array}$\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 ${X}_{•}$ such that the two morphisms

${X}_{1}^{\mathrm{inv}}↪{X}_{1}\stackrel{{\partial }_{1},{\partial }_{0}}{\to }{X}_{0}$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. 2 is quasi-unital, def. 3.

###### Remark

A morphism of complete semi-Segal spaces ${f}_{•}:{X}_{•}\to {Y}_{•}$ is quasi-unital if it preserves the weak equivalences, hence if

$\begin{array}{ccc}{X}_{1}^{\mathrm{inv}}& ↪& {X}_{1}\\ {↓}^{{f}_{1}{\mid }_{\mathrm{inv}}}& & {↓}^{{f}_{1}}\\ {Y}_{1}^{\mathrm{inv}}& ↪& {Y}_{1}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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)