# Segal spaces, spans, and semicategories

## TL;DR

This paper establishes a connection between Segal spaces, category objects in $
$-categories, and associative algebras in spans, demonstrating that identities are a property rather than additional structure in non-unital higher categories.

## Contribution

It reveals that Segal spaces and category objects in $
$-categories correspond to associative algebras in spans, and shows identities are a property in non-unital $(
)$-categories.

## Key findings

- Segal spaces are equivalent to associative algebras in spans.
- Having identities is a property, not extra structure, in non-unital $(
)$-categories.
- Category objects in an $
$-category can be characterized via algebraic structures in spans.

## Abstract

We show that Segal spaces, and more generally category objects in an $\infty$-category $\mathcal{C}$, can be identified with associative algebras in the double $\infty$-category of spans in $\mathcal{C}$. We use this observation to prove that "having identities" is a property of a non-unital $(\infty,n)$-category.