essentially discrete
A category is essentially discrete if it is equivalent to a discrete category. Equivalently, it is a thin groupoid. Notice that the nLab calls this property simply "discrete". In contrast to being discrete, clearly this property is invariant under equivalences of categories. An essentially discrete category is the same as a setoid (a set equipped with an equivalence relation).
- Dual property: essentially discrete (self-dual)
- Related properties: discrete
- nLab Link
Relevant implications
- connected and essentially discrete implies trivial
- discrete implies essentially discrete and locally small and skeletal
- essentially discrete implies connected colimits and locally essentially small
- essentially discrete implies connected limits and locally essentially small
- essentially discrete is equivalent to groupoid and thin
- trivial implies essentially discrete and essentially finite and finitary algebraic and Grothendieck topos and self-dual and split abelian
Examples
There are 4 categories with this property.
Counterexamples
There are 47 categories without this property.
- 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 a non-trivial finite group
- 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
- walking morphism
- walking parallel pair of morphisms
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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