CatDat

essentially countable

A category is essentially countable if it is equivalent to a countable category. This means that up to isomorphism there is only a countable collection of objects, and for every pair of objects $A,B$ the collection $\Hom(A,B)$ is countable.

• Dual property: essentially countable (self-dual)
• Related properties: countable, essentially finite

Relevant implications

• Cauchy complete andessentially countable andlocally finite implies ℵ₁-accessible
• countable copowers andessentially countable implies thin
• countable coproducts andessentially countable andthin implies coproducts
• countable powers andessentially countable implies thin
• countable products andessentially countable andthin implies products
• countable implies essentially countable
• essentially countable andthin andℵ₁-cofiltered implies initial object
• essentially countable andthin andℵ₁-filtered implies terminal object
• essentially countable implies essentially small
• essentially finite implies essentially countable

Examples

There are 31 categories with this property.

• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finitely generated abelian groups
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of the additive monoid of natural numbers
• discrete category on two objects
• empty category
• forked commutative square
• partially ordered set of extended natural numbers
• partially ordered set of natural numbers
• preordered set of integers w.r.t. divisibility
• simplex category
• trivial category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking idempotent
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 64 categories without 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 countable groups
• category of countable sets
• category of fields
• category of filtered vector spaces
• category of finite-dimensional vector spaces [uncountable field]
• 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 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 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 partially ordered sets
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of sets and relations
• 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
• delooping of an infinite uncountable group
• delooping of the additive monoid of ordinal numbers
• dual of the category of sets
• dual of the category of topological spaces
• partially ordered collection of ordinal numbers
• real interval [0,1]

Undecidable categories

There are 2 categories for which it cannot be decided if this property is satisfied or not.

• category of finite-dimensional vector spaces
• delooping of a group

Unknown

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

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