essentially finite
A category is essentially finite if it is equivalent to a finite category. Equivalently, there are only finitely many objects up to isomorphism, and the collection of morphisms between any two objects is isomorphic to a finite set. In contrast to being finite, this property is invariant under equivalences of categories.
- Dual property: essentially finite (self-dual)
- Related properties: finite
Relevant implications
essentially finite and finite products implies thin
Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case.finite implies small and essentially finite
trivialessentially finite implies essentially small
trivialtrivial implies finitary algebraic and Grothendieck topos and split abelian and self-dual and essentially discrete and essentially finite
trivialessentially finite and finite coproducts implies thin
[dualized] Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case.
Examples
- delooping of a non-trivial finite group
- discrete category on two objects
- empty category
- trivial category
- walking isomorphism
- walking morphism
- walking parallel pair of morphisms
Counterexamples
- 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
- category of Z-functors
- delooping of an infinite group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- 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
Unknown
For these categories the database has no info if they satisfy this property or not.
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