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: small
- nLab Link
Relevant implications
- cocomplete and essentially small and thin implies complete
- complete and essentially small and infinitary distributive and thin implies cartesian closed
- complete and essentially small and thin implies cocomplete
- coproducts and essentially small implies thin
- essentially finite implies essentially small
- essentially small and products implies thin
- essentially small implies locally essentially small and well-copowered and well-powered
- small implies essentially small and locally small
Examples
There are 21 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
- partial order [0,1]
- partial order of extended natural numbers
- partial order of natural numbers
- preorder of integers w.r.t. divisiblity
- trivial category
- walking isomorphism
- walking morphism
- walking parallel pair of morphisms
Counterexamples
There are 30 categories without this property.
- category of abelian groups
- category of Banach spaces with linear contractions
- category of commutative rings
- category of fields
- 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
- category of Z-functors
- delooping of the additive monoid of ordinal numbers
- partial order 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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