essentially small
A category is essentially small when it is equivalent to a small category. In particular, there is a set of objects such that every object is isomorphic to an object in this set. In contrast to the property of being small, being essentially small is invariant under equivalences of categories.
- Dual property: essentially small (self-dual)
- Related properties: locally essentially small, small
- nLab Link
Relevant implications
- cocomplete andessentially small andthin implies complete
- complete andessentially small andinfinitary distributive andthin implies cartesian closed
- complete andessentially small andthin implies cocomplete
- coproducts andessentially small implies thin
- essentially finite implies essentially small
- essentially small andproducts implies thin
- essentially small implies cogenerating set
- essentially small implies generating set
- essentially small implies locally essentially small andwell-copowered andwell-powered
- small implies essentially small andlocally small
Examples
There are 25 categories with this property.
- category of combinatorial species
- 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
- 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
- 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 of morphisms
- walking span
Counterexamples
There are 40 categories without this property.
- category of abelian groups
- category of abelian sheaves
- 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 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
- category of Z-functors
- delooping of the additive monoid of ordinal numbers
- dual of the category of sets
- poset 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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