locally essentially small

A category is locally essentially small when for every pair of objects $A,B$ the collection $\Hom(A,B)$ of morphisms $A \to B$ is essentially small, i.e. isomorphic to a set; see here for the set-theoretic foundation of category theory we are working with. A category is locally essentially small if and only if it 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, from a structural perspective, this is the right notion to work with.

Relevant implications

• accessible implies locally essentially small
• coaccessible implies locally essentially small
• cocomplete andgenerating collection andlocally essentially small andwell-copowered implies total
• cogenerating collection andcomplete andlocally essentially small andwell-powered implies cototal
• cogenerating collection andlocally essentially small implies concretizable
• concretizable implies locally essentially small
• cototal implies locally essentially small
• elementary topos andlocally essentially small implies well-copowered
• essentially discrete implies locally essentially small
• essentially discrete implies locally essentially small
• essentially small implies locally essentially small
• essentially small implies locally essentially small
• generating collection andlocally essentially small implies concretizable
• Grothendieck abelian andlocally essentially small implies cogenerator
• Grothendieck abelian andlocally essentially small implies locally presentable
• Grothendieck topos is equivalent to coproducts andelementary topos andgenerating collection andlocally essentially small
• locally essentially small andquotient object classifier implies well-copowered
• locally essentially small andsubobject classifier implies well-powered
• locally finite implies locally essentially small
• locally small implies locally essentially small
• total implies locally essentially small

Examples

There are 126 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 connected sequences of sets
• 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 F(I)-sets
• category of fields
• 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 bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finite sets of cardinality a power of 3
• category of finite sets of even cardinality
• category of finite sets of odd 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 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 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 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 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 bijections
• 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-sets
• cocompletion of a discrete–pair join
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• delooping of the additive monoid of natural numbers
• discrete category of sets
• discrete category on two objects
• dual of the category of sets
• dual of the category of topological spaces
• empty category
• 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]
• 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 7 categories without this property.

• category of large families of abelian groups
• category of large families of sets
• category of large families of sets and bijections
• category of large vector spaces over a large field with a small basis
• category of long transfinite sequences of abelian groups
• 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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