quotient-trivial
A category is quotient-trivial if every epimorphism is an isomorphism. Equivalently, the poset of quotients of any object is trivial. This is no standard terminology. We have added it to the database since it clarifies the relationship between several related properties.
- Dual property: subobject-trivial
- Related properties: epi-regular, groupoid
Relevant implications
- co-Malcev andquotient object classifier implies quotient-trivial
- coequalizers andquotient-trivial implies thin
- cofiltered limits andquotient-trivial implies cofiltered-limit-stable epimorphisms
- groupoid implies quotient-trivial
- left cancellative andmono-regular implies quotient-trivial
- quotient-trivial andself-dual implies subobject-trivial
- quotient-trivial implies coreflexive equalizers andreflexive coequalizers
- quotient-trivial implies epi-regular
- self-dual andsubobject-trivial implies quotient-trivial
Examples
There are 9 categories with this property.
- category of finite sets and bijections
- category of finite sets and injections
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- discrete category on two objects
- empty category
- trivial category
- walking idempotent
- walking isomorphism
Counterexamples
There are 69 categories without this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of countable sets
- category of fields
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of finite sets and surjections
- category of finitely generated abelian groups
- category of free abelian groups
- category of groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- 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 pointed topological spaces
- category of posets
- category of prosets
- category of pseudo-metric spaces with non-expansive maps
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of sets
- category of sets and relations
- category of sets with finite-to-one maps
- category of sheaves
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of vector spaces
- category of Z-functors
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- dual of the category of sets
- poset [0,1]
- poset of extended natural numbers
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- simplex category
- walking commutative square
- walking composable pair
- walking coreflexive pair
- walking fork
- walking morphism
- walking parallel pair
- walking span
- walking splitting
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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