CatDat

ℵ₁-filtered

A category is $\aleph_1$-filtered if every countable diagram admits a cocone. This is the special case of the notion of a $\kappa$-filtered category for $\kappa = \aleph_1$, the first uncountable cardinal. An equivalent characterization is given as Theorem 2.2 at the nLab.

• Dual property: ℵ₁-cofiltered
• Related properties: cocomplete, filtered, ℵ₁-filtered colimits
• nLab Link

Relevant implications

• coequalizers andcountable coproducts implies ℵ₁-filtered
• essentially countable andthin andℵ₁-filtered implies terminal object
• self-dual andℵ₁-cofiltered implies ℵ₁-filtered
• self-dual andℵ₁-filtered implies ℵ₁-cofiltered
• terminal object implies ℵ₁-filtered
• ℵ₁-filtered implies filtered

Examples

There are 66 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of left modules over a division ring
• category of left modules over a ring
• category of locally ringed spaces
• 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 metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of pointed sets
• category of pointed topological spaces
• category of posets
• category of prosets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sets
• category of sets and relations
• 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 vector spaces
• category of Z-functors
• dual of the category of sets
• dual of the category of topological spaces
• poset [0,1]
• poset of extended natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• simplex category
• trivial category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking idempotent
• walking isomorphism
• walking morphism
• walking splitting

Counterexamples

There are 15 categories without this property.

• category of fields
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of sets with finite-to-one maps
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• empty category
• poset of natural numbers
• walking fork
• walking parallel pair
• walking span

Unknown

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

—