CatDat

subobject-trivial

A category is subobject-trivial if every monomorphism is an isomorphism. Equivalently, the poset of subobjects 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: quotient-trivial
• Related properties: groupoid, mono-regular

Relevant implications

• epi-regular andright cancellative implies subobject-trivial
• equalizers andsubobject-trivial implies thin
• filtered colimits andsubobject-trivial implies filtered-colimit-stable monomorphisms
• groupoid implies subobject-trivial
• Malcev andsubobject classifier implies subobject-trivial
• quotient-trivial andself-dual implies subobject-trivial
• self-dual andsubobject-trivial implies quotient-trivial
• subobject-trivial implies coreflexive equalizers andreflexive coequalizers
• subobject-trivial implies mono-regular

Examples

There are 10 categories with this property.

• category of finite sets and bijections
• category of finite sets and surjections
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• discrete category on two objects
• empty category
• trivial category
• walking idempotent
• walking isomorphism

Counterexamples

There are 76 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 compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of fields
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and injections
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• 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
• dual of the category of topological spaces
• 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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