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

• essentially small implies   well-powered and  well-copowered and  locally essentially small

  trivial
• locally small implies   locally essentially small

  trivial
• essentially discrete implies   locally essentially small and  connected limits

  trivial
• locally presentable implies   locally essentially small and  well-powered and  well-copowered and  complete and  cocomplete and  generator

  For the non-trivial conclusions see Adamek-Rosicky, Thm. 1.20, Cor. 1.28, Rem. 1.56, Thm. 1.58.
• subobject classifier and  locally essentially small implies   well-powered

  Mac Lane & Moerdijk, Prop. I.3.1
• elementary topos and  locally essentially small implies   well-copowered

  This follows from Mac Lane & Moerdijk, Theorem IV.7.8 (and Prop. I.3.1).
• Grothendieck topos is equivalent to   elementary topos and  coproducts and  generator and  locally essentially small

  Mac Lane & Moerdijk, Appendix, Prop. 4.4
• preadditive implies   locally essentially small and  zero morphisms

  trivial
• essentially discrete implies   locally essentially small and  connected colimits

  [dualized] trivial

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• 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 left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• 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
• empty category
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism
• walking parallel pair of morphisms

Counterexamples

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

Unknown

For these categories the database has no info if they satisfy this property or not.

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