right cancellative
A category is right cancellative if for every morphism and every parallel pair of morphisms with we have . Equivalently, every morphism is an epimorphism.
- Dual property: left cancellative
- Related properties: groupoid
- nLab Link
Relevant implications
right cancellative and initial object implies strict initial object
Let be a morphism. Let be the unique morphism. It is an epimorphism by assumption. Also, since is initial. But then is a split monomorphism and an epimorphism, hence an isomorphism.left cancellative and right cancellative and balanced implies groupoid
trivialthin implies coequalizers and right cancellative
[dualized] Any two parallel morphisms are equal, so their equalizer is the identity, and every morphism is a monomorphism as well.right cancellative and equalizers implies thin
[dualized] If are two parallel morphisms, then their coequalizer is a regular epimorphism, but also a monomorphism by assumption, so it must be an isomorphism. But this means that .right cancellative and terminal object implies strict terminal object
[dualized] It suffices to prove that in general any monomorphism into an initial object is an isomorphism. If is the unique morphism, then since is initial. But then is a split epimorphism and a monomorphism, hence an isomorphism.right cancellative implies Cauchy complete
[dualized] Any idempotent monomorphism must be the identity and therefore splits.groupoid implies self-dual and epi-regular and pushouts and filtered colimits and right cancellative and well-copowered
[dualized] easyself-dual and left cancellative implies right cancellative
trivial by self-dualityself-dual and right cancellative implies left cancellative
trivial by self-duality
Examples
- category of finite sets and bijections
- category of finite sets and surjections
- delooping of a non-trivial finite group
- delooping of an infinite group
- delooping of the additive monoid of natural numbers
- discrete category on two objects
- empty category
- 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
- trivial category
- walking isomorphism
- walking morphism
- walking parallel pair of morphisms
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 injections
- 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 the additive monoid of ordinal numbers
Unknown
For these categories the database has no info if they satisfy this property or not.
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