nLab
omegacomplete poset
Contents
Context
$(0,1)$Category theory
Limits and colimits
limits and colimits
1Categorical

limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit, wide pullback

preserved limit, reflected limit, created limit

product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum

finite limit

Kan extension

weighted limit

end and coend

fibered limit
2Categorical
(∞,1)Categorical
Modelcategorical
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 $\bot \leq a$;

a function
$\Vee_{n \colon \mathbb{N}} ()(n) \;\colon\;
(\mathbb{N} \to P) \to L$
exhibiting the existence of denumerable/countable joins in the poset, namely such that

for every natural number $n \in \mathbb{N}$ and every sequence $s \colon \mathbb{N} \to P$ we have
$s(n)
\leq
\Vee_{n \colon \mathbb{N}} s(n)
\,;$

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
$\Vee_{n \colon \mathbb{N}}
s(n) \leq a
\,.$
See also
References
 Partiality, Revisited: The Partiality Monad as a Quotient InductiveInductive Type (arXiv:1610.09254)
Last revised on June 10, 2022 at 06:10:20.
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