retract-trivial
We call a category retract-trivial if it satisfies one of the following equivalent conditions:
- If , are morphisms with , then .
- Every split monomorphism is an isomorphism.
- Every split epimorphism is an isomorphism.
- Every split idempotent is the identity.
Dual retract-trivial (self-dual) Related one-way, regular-quotient-trivial, regular-subobject-trivial External nLab link
Relevant implications
Examples
There are 32 categories with this property.
- category of fields
- category of fields of characteristic zero
- category of finite sets and bijections
- category of finite sets and injections
- category of finite sets and surjections
- category of large families of sets and bijections
- category of sets and bijections
- delooping of a group
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category of sets
- discrete category on two objects
- empty category
- forked commutative square
- indiscrete category of sets
- 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]
- trivial category
- walking commutative square
- walking composable pair
- walking fork
- walking idempotent
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
Counterexamples
There are 107 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 connected sequences of sets
- category of coproducts of Euclidean spaces
- category of countable groups
- category of countable sets
- category of countable-dimensional vector spaces
- category of countably infinite sets
- category of empty-or-finite pairs of sets
- category of extended metric spaces
- category of F(I)-sets
- 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 of cardinality a power of 3
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite Z-sets
- 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 groups
- 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 families of abelian groups
- category of large families of sets
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- category of large families of vector spaces which are mostly zero
- 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 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 quivers
- category of quivers with finite components
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of sequences of abelian groups
- category of sequences of sets
- category of set functions and commutative squares
- category of sets
- category of sets and relations
- category of sets equipped with a binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of sets equipped with an irreflexive binary relation
- 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
- category of Z-sets
- category of ω-sets
- cocompletion of a discrete–pair join
- delooping of the bicyclic monoid
- delooping of the double bicyclic monoid
- dual of the category of abelian groups
- dual of the category of sets
- dual of the category of topological spaces
- simplex category
- walking coreflexive pair
- walking splitting
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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