ℵ₁-cofiltered

morphism behavior

A category is $\aleph_1$ cofiltered if every countable diagram admits a cone. This is the special case of the notion of a

\kappa

cofiltered category for

\kappa = \aleph_1

the first uncountable cardinal. An equivalent characterization is given as the dual of Theorem 2.2 at the nLab.

Dual ℵ₁-filtered Related cofiltered, complete External nLab link

Relevant implications

countable products andequalizers implies ℵ₁-cofiltered
essentially countable andthin andℵ₁-cofiltered implies initial object
initial object implies ℵ₁-cofiltered
self-dual andℵ₁-cofiltered implies ℵ₁-filtered
self-dual andℵ₁-filtered implies ℵ₁-cofiltered
ℵ₁-cofiltered implies cofiltered

Examples

There are 115 categories with this property.

category of abelian groups
category of abelian sheaves
category of algebras
category of Banach spaces with linear contractions
category of cochain complexes of abelian groups
category of combinatorial species
category of commutative algebras
category of commutative monoids
category of commutative rings
category of compact Hausdorff spaces
category of coproducts of Euclidean spaces
category of countable groups
category of countable sets
category of countable-dimensional vector spaces
category of empty-or-finite pairs of sets
category of extended metric spaces
category of F(I)-sets
category of fields of characteristic zero
category of filtered vector spaces
category of finite abelian groups
category of finite groups
category of finite ordered sets
category of finite sets
category of finite sets and injections
category of finite sets of even cardinality
category of finite Z-sets
category of finite-dimensional vector spaces
category of finite-dimensional vector spaces [countable field]
category of finite-dimensional vector spaces [finite field]
category of finite-dimensional vector spaces [uncountable field]
category of finitely generated abelian groups
category of finitely generated free abelian groups
category of finitely generated free modules over Z x Z
category of finitely generated groups
category of finitely generated projective modules over the ring of dual numbers
category of free abelian groups
category of graded abelian groups
category of graded modules over a graded ring
category of groups
category of Hausdorff spaces
category of Jónsson-Tarski algebras
category of large families of abelian groups
category of large families of sets
category of large families of sets which are mostly empty
category of large families of sets which are mostly singletons
category of large families of vector spaces which are mostly zero
category of large vector spaces over a large field with a small basis
category of left modules over a division ring
category of left modules over a non-semisimple ring
category of left modules over a ring
category of locally ringed spaces
category of long transfinite sequences of abelian groups
category of M-sets
category of measurable spaces
category of metric spaces with continuous maps
category of metric spaces with non-expansive maps
category of monoids
category of pairs of sets
category of partially ordered sets
category of partially ordered sets without isolated points
category of pointed sets
category of pointed topological spaces
category of preordered sets
category of pseudo-metric spaces with non-expansive maps
category of quivers
category of quivers with finite components
category of rings
category of rngs
category of schemes
category of semigroups
category of sequences of abelian groups
category of sequences of sets
category of set functions and commutative squares
category of sets
category of sets and relations
category of sets equipped with a binary relation
category of sets equipped with a reflexive binary relation
category of sets equipped with a symmetric binary relation
category of sets equipped with a symmetric irreflexive binary relation
category of sets equipped with a symmetric reflexive binary relation
category of sets equipped with an irreflexive binary relation
category of sets with a distinguished subset
category of sets with finite-to-one maps
category of sheaves
category of simplicial sets
category of small categories
category of smooth manifolds
category of topological spaces
category of torsion abelian groups
category of torsion-free abelian groups
category of uniform spaces
category of vector spaces
category of Z-functors
category of Z-sets
category of ω-sets
cocompletion of a discrete–pair join
delooping of the additive monoid of ordinal numbers
dual of the category of sets
dual of the category of topological spaces
forked commutative square
indiscrete category of sets
partially ordered collection of ordinal numbers
partially ordered set of extended natural numbers
partially ordered set of natural numbers
preordered set of integers w.r.t. divisibility
real interval [0,1]
trivial category
walking commutative square
walking composable pair
walking fork
walking idempotent
walking isomorphism
walking morphism
walking span
walking splitting

Counterexamples

There are 20 categories without this property.

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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