CatDat

ℵ₁-accessible

This is the special case of the notion of $\kappa$-accessible categories, where $\kappa = \aleph_1$ is the first uncountable cardinal.

• Related properties: accessible, finitely accessible, locally ℵ₁-presentable
• nLab Link

Relevant implications

• essentially countable andthin implies ℵ₁-accessible
• finitely accessible implies ℵ₁-accessible
• generalized variety implies ℵ₁-accessible
• locally ℵ₁-presentable is equivalent to cocomplete andℵ₁-accessible
• ℵ₁-accessible implies accessible

Examples

There are 42 categories with this property.

• category of abelian groups
• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• category of finite abelian groups
• category of finite groups
• category of finite sets
• category of groups
• category of left modules over a division ring
• category of left modules over a ring
• category of M-sets
• 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 posets
• category of prosets
• category of rings
• category of rngs
• category of sets
• category of simplicial sets
• category of small categories
• category of vector spaces
• delooping of a non-trivial finite group
• discrete category on two objects
• empty category
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking fork
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span

Counterexamples

There are 9 categories without this property.

• category of pointed topological spaces
• category of schemes
• category of sets and relations
• category of topological spaces
• category of Z-functors
• delooping of the additive monoid of ordinal numbers
• dual of the category of sets
• poset of ordinal numbers
• walking idempotent

Unknown

There are 19 categories for which the database has no information on whether they satisfy this property. Please help us fill in the gaps by contributing to this project.

• category of abelian sheaves
• category of combinatorial species
• category of finite ordered sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of Hausdorff spaces
• category of locally ringed spaces
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of pseudo-metric spaces with non-expansive maps
• category of sheaves
• category of smooth manifolds
• delooping of an infinite countable group
• delooping of the additive monoid of natural numbers
• simplex category