# nLab omega-complete poset

Contents

(0,1)-category

(0,1)-topos

## Theorems

#### Limits and colimits

limits and colimits

# Contents

## Idea

An $\omega$-complete poset, in the following sense, is a poset with countable joins, hence a countably cocomplete poset.

## Definition

A $\omega$-complete poset is a poset $(P, \leq)$ with

• an initial element $\bot \in P$ (bottom), hence such that for every element $a \in P$ we have $\bot \leq a$;

• $\Vee_{n \colon \mathbb{N}} (-)(n) \;\colon\; (\mathbb{N} \to P) \to L$

exhibiting the existence of denumerable/countable joins in the poset, namely such that

1. for every natural number $n \in \mathbb{N}$ and every sequence $s \colon \mathbb{N} \to P$ we have

$s(n) \leq \Vee_{n \colon \mathbb{N}} s(n) \,;$
2. for every element $a \in P$ and sequence $s \colon \mathbb{N} \to P$ of elements $\leq a$,

$\prod_{n \colon \mathbb{N}} (s(n) \leq a) \,,$

we have

$\Vee_{n \colon \mathbb{N}} s(n) \leq a \,.$