CatDat

locally essentially small

A category is locally essentially small when for every pair of objects $A,B$ the collection of morphisms $A \to B$ is isomorphic to a set. (Here, we work with a set-theoretic foundation in which there are sets and collections. Categories are based on collections of objects and morphisms.) Equivalently, the category is equivalent to a locally small category. In contrast to being locally small, this condition is invariant under equivalences of categories. This is why we have added it to the database. For instance, every algebraic category is locally essentially small, but not necessarily locally small. This indicates that this is the "right" notion to work with.

• Dual property: locally essentially small (self-dual)
• Related properties: locally small

Relevant implications

• elementary topos andlocally essentially small implies well-copowered
• essentially discrete implies connected colimits andlocally essentially small
• essentially discrete implies connected limits andlocally essentially small
• essentially small implies locally essentially small andwell-copowered andwell-powered
• Grothendieck topos is equivalent to coproducts andelementary topos andgenerating set andlocally essentially small
• locally essentially small andsubobject classifier implies well-powered
• locally presentable implies cocomplete andcomplete andgenerating set andlocally essentially small andwell-copowered andwell-powered
• locally small implies locally essentially small
• preadditive implies locally essentially small andzero morphisms

Examples

There are 63 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 fields
• category of finite abelian groups
• category of finite orders
• category of finite 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 groups
• category of Hausdorff spaces
• 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 posets
• category of prosets
• category of rings
• category of rngs
• category of schemes
• 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 vector spaces
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• discrete category on two objects
• dual of the category of sets
• empty category
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal 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 of morphisms
• walking span

Counterexamples

There are 2 categories without this property.

• category of Z-functors
• delooping of the additive monoid of ordinal numbers

Unknown

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

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