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.
Relevant implications
Examples
There are 12 categories with this property.
- category of finite sets and bijections
- category of finite sets and surjections
- delooping of a group
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- discrete category of sets
- discrete category on two objects
- empty category
- trivial category
- walking idempotent
- walking isomorphism
Counterexamples
There are 100 categories without this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- category of cochain complexes of abelian groups
- category of combinatorial species
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of compact Hausdorff spaces
- category of coproducts of Euclidean spaces
- category of countable groups
- category of countable sets
- category of countable-dimensional vector spaces
- category of fields
- category of fields of characteristic zero
- 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 sets of cardinality a power of 3
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite-dimensional vector spaces
- 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 finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of finitely generated projective modules over the ring of dual numbers
- category of free abelian groups
- category of graded abelian groups
- category of graded modules over a graded ring
- category of groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of large vector spaces over a large field with a small basis
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of left modules over a ring
- category of locally ringed spaces
- category of long transfinite sequences of abelian groups
- 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 partially ordered sets
- category of partially ordered sets without isolated points
- category of pointed sets
- category of pointed topological spaces
- category of preordered sets
- category of pseudo-metric spaces with non-expansive maps
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of sequences of abelian groups
- category of set functions and commutative squares
- category of set-indexed families of abelian groups
- category of sets
- category of sets and relations
- category of sets with a distinguished subset
- 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 uniform spaces
- category of vector spaces
- category of Z-functors
- cocompletion of a discrete–pair join
- 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
- forked commutative square
- partially ordered collection of ordinal numbers
- partially ordered set of extended natural numbers
- partially ordered set of natural numbers
- preordered set of integers w.r.t. divisibility
- real interval [0,1]
- 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.
—