CatDat

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

• discrete implies   essentially discrete and  locally small and  skeletal

  trivial
• essentially discrete is equivalent to   thin and  groupoid

  trivial
• essentially discrete implies   locally essentially small and  connected limits

  trivial
• essentially discrete and  connected implies   trivial

  trivial
• trivial implies   finitary algebraic and  Grothendieck topos and  split abelian and  self-dual and  essentially discrete and  essentially finite

  trivial
• essentially discrete implies   locally essentially small and  connected colimits

  [dualized] trivial

Examples

• discrete category on two objects
• empty category
• trivial category
• walking isomorphism

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 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

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

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